Atlas of Finite Groups: Maximal Subgroups and Ordinary Characters for Simple Groups

~ I

OF

FINITE GROUPS Maximal Subgroups and Ordinary Characters for Simple Groups BY

1. H. CONWAY

R. T. CURTIS S. P. NORTON R. A. PARKER R. A. WILSON with computational assistance from

J. G. THACKRAY

CLARENDON PRESS· OXFORD

1985

Oxford University Press, Walton Street, Oxford OX26DP Oxford New York Toronto Delhi Bombay Calcutta Madras Karachi Kuala Lumpur Singapore Hong Kong Tokyo Nairobi Dar es Salaam Cape Town Melbourne Auckland and associated companies in Beirut Berlin Ibadan Nicosia Oxford is a trade mark of Oxford University Press Published in the United States by Oxford University Press, New York

© J. H. Conway, R. T. Curtis, S. P. Norton, R A. Parker, and R

A. Wilson 19&5

All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any fonn or by any means, electronic, mechanical, photocopying, recording, or otherwise, without the prior pennission of Oxford University Press This book is sold subject to the condition that it shall not, by way

of trade or otherwise, be lent, re-sold, hired out or otherwise circulated without the publisher's prior consent in any fonn of binding or cover other than that in which it is published and without a similar condition induding this condition being imposed on the subsequent purchaser British Library Cataloguing in Publication Data Atlas of finite groups: maximal subgroups and ordinary characters for simple groups. 1. Finite groups I. Conway, John H. 512~2 ()Jl171

ISBN 0-19-853199-0 Library of Congress Cataloging in Publication Data Main entry under title: Atlas of finite groups. Bibliography: p. Indudes index. 1. Finite groups. I. Conway, John Horton. ()JlI71.Jl86 1985 512'.2 ISBN 0-19-853199-0

85-11559

Preliminary pages typeset by The Universities Press, (Belfast) Ltd Printed in Great Britain by Thomson Litho (East Kilbride)

Contents

INTRODUCTION

The reader who wishes merely to understand how to use an A lr D...A§ character table should turn directly to Chapter 7.

1.

2.

A survey of the finite simple groups

vii

Note to Chapters 6 and 7

1. 2. 3. 4.

vii vii viii ix

6. How to read this A lr IL A§: the map of an

Preliminaries The finite simple groups The sporadic groups Finite fields

The classical groups 1. The groups OLn(q), SL n(q), POLn (q), aud PS4 (q) = L n(q) 2. The groups OUn(q), SUn(q), POUn(q), and PSUn(q) = Un(q) 3. The groups SPn(q) and PSPn(q)=Sn(q) 4. The groups OOn(q), SOn(q), POOn(q), PSOn(q), and On(q) 5. Classification of points and hyperplanes in orthogonal spaces

6. The Clifford algebra and spin group 7. Structure tables for the classical groups 8. Other notations for the simple groups

3.

4.

x x x x xi

xiii

1. 2. 3. 4. 5. 6. 7.

xiv xiv

1. Order, multiplier, and outer automorphism group 2. Constructions 3. Notation for types of point and hyperplane for orthogonal groups 4. Projective dimensions and projective counting 5. Coordinates for vectors 6. Quaternionic spaces 7. Presentations

xxi

The case 0 The case p. 0 The case O.p The case M. 0 The case O.A The geueral case M. O. A Isoclinism

xxi xxi xxi xxi xxii xxii

xxiii

Guide to the sections of Chapter 7

xxiv

7. How to read this A lr ILA§: character tables

xxv

1. 2. 3. 4. 5. 6. 7.

xiii

xiv

How to read this AlrILA§: the 'constructions' sections

A lr ILA§ character table

xii xiii

The Chevalley and twisted Chevalley groups The untwisted groups The twisted groups Multipliers and automorphisms of Chevalley groups Orders of the Chevalley groups The simple groups enumerated Parabolic subgroups The fundamental representations

xv xv xv xv xv

xvii xvii xvii xvii

xvii xviii xviii xviii

1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18.

Column width and type markers Centralizer orders Power maps p'-parts Class names Character values, for characters of 0 Lifting order rows for a group m. 0 Character values for a group m. 0 Follower cohorts The A 'If D...A§ notation for algebraic numbers A note on algebraic conjugacy The character number column(s) The indicator column for 0, and for m. 0 Columns for classes of 0 The detachment of columns for a group 0.2 The detachment of columns for a group O. a, a;" 3 Follower cosets The portion of an A'If D.. A§ table for a bicyclic extension m. O. a 19. The precise rules for proxy classes and characters 20. Abbreviated character tables

8.. Concluding remarks 5.

How to read this A lr ILA§: information about subgroups and their structure 1. The maximal subgroups sections 2. Notations for group structures

xx

xxv xxv xxv xxv xxv xxv xxvi xxvi xxvi xxvii

xxvii xxvii xxviii

xxviii xxviii

xxviii xxix

xxix xxix xxx

xxxi

xix

1. Format 2. Ordering of classes and characters

xxxi xxxi

3. Some other 'virtuous' properties

xix xx

4. The reliability of A 'If D...A§ information 5. Acknowledgements

xxxi xxxi xxxii

THE GROUPS A number in parentheses denotes the start of the text for a group,

when this is not the first page of the entire entry for that group. As"" L 2 (4) "" L 2 (5) L 3 (2) "" LP) A 6 "" L,(9) "" S4(2)' L,(8) "" R(3)' 1-,(11) L 2 (13) L,(17)

A7 1-,(19) 1-,(16) L,(3) U 3 (3) "" G 2 (2)' L 2 (23)

1-,(25)

Mll L,(27)

1-,(29) L 2 (31) As ""L4 (2) L 3 (4) U 4(2) "" S4(3) Sz(8) L 2 (32) U 3 (4) M '2

U 3 (5) J,

A9 L 3 (5) M 22

J 2 =HJ=Ps_ S.(4) S6(2)

AlO L 3 (7) U 4 (3)

G2 (3) S4(5)

U 3 (8) UP)

L 4 (3) L s(2)

M23 U s(2)

L 3 (8) 2P4 (2)' All

2 3

4 6 7 8 9 10 11 12 13 14 15 16 18 18 20 21

22 23 26 28 29 30 31 34 36 37 38 39 42

44 46 48 50 52 60 61 64 (66) 66 68 70 71 72 74 74 75

Sz(32) L 3 (9)

77 78 79 80 82 84 85 88 89 91 91 94 97 100 102 104 106 110 114 115 122 123 126 128 132 134 136 141 142 147 154 156 164 167 174 176 177 180 188 191 191 200 208 219 220 235

U3 (9) HS J3 U 3 (11) 0~(2)

0;;(2)

3D4 (2) L 3 (11)

A12 M24 GM) MCL

A13 He

0 7 (3) S6(3) G,(5) U 6 (2) R(27) Ss(2)

Ru Suz O'N

C0 3 0~(3)

0;;(3)

Oro(2) 0 10(2)

CO 2 Pi 22 HN=Ps+ P 4 (2)

Ly Th=P313 Pi23 Co , = P 2 J4 2E6 (2)

E6 (2) Fi 2A =F3 + B=P2 + E,(2)

M=P, E s (2)

Partitions and the classes and characters of So Involvement of sporadic groups in one another List of orders of simple groups Bibliography Index of groups

236 238 239 243 252

(89)

(104) (108) (112)

(123)

(131)

(140) (146)

(162) (166) (170) (177)

(190)

(206) (216) (228)

INTRODUCTION

1 A survey of the finite simple groups .,n 1. Prelimaries J.. The theory of groups is the theory of the different possible kinds of symmetry, and as such finds applications throughout mathematics and the sciences whenever symmetrical objects or theories are being discussed. We originally conceived this A lr ILA§ as a work whose aim would be to convey every interesting fact about every interesting finite group. This, of course, was a tall order, only partly fulfilled by the present volume! The most interesting groups are the simple groups, and perhaps the most impressive result of finite group theory is the recent complete classification of finite simple groups. This volume tabulates, for various finite simple groups G, the following information: (i) the order of G, and the orders of its Schur multiplier and outer automorphism group; (ii) various 'constructions' for G, or for concepts closely associated with G; (iii) information about the subgroups of G and of its automorphism groups (often the complete list of maximal subgroups); (iv) a 'compound character table' from which it is easy to read off the ordinary character table of G and those of various closely related groups. Some of the terminology used here will be defined in later sections of this introduction. In general, we expect the reader to be familiar with the basic concepts of group theory and representation theory. The main aim of our definitions is to introduce the notation used later in the A lr ILA§, which is sometimes idiosyncratic. According to the classification theorem, the finite simple groups fall into certain infinite families, with 26 extremely interesting exceptions called the sporadic groups. Many of the early terms in the infinite families display intriguing special behaviour. Within each family, we have continued until the groups became either too big or too boring, and have tried hard to include any group that had a property or properties not explained by its membership within that family. In doubtful cases, our rule was to think how far the reasonable person would go, and then go a step further. For the smaller families, this has had the effect that the last group included gives a very good idea of the structure of the typical element of the family. We have included all of the sporadic groups. We have chosen to print the particular information described above partly for its obvious utility, and partly because we could manage to obtain a reasonably uniform coverage. Our first priority was to print the ordinary character table, which is beyond doubt the most compendious way of conveying information about a group to the skilled reader. Moreover, the computation of character tables is almost routine for groups of moderate size, while for several notorious groups of inunoderate size it has been achieved after heroic efforts. The complete list of maximal subgroups is very useful in internal investigations of the group, mainly because it reduces the difficult problem of deciding what group certain elements generate to the mere enumeration of cases. It is considerably harder to find all maximal subgroups than to compute the character table, but the answers are now known for a surpris-

ingly large number of groups, and useful partial information is often available for other cases. The Schur multipliers and outer automorphism groups of all finite simple groups are now known. (It should be noted that the computation of multipliers is delicate, and there have been several amendments to previously published lists.) For most of the groups treated in the A lr IL A§ we also give the character tables for the corresponding covering groups and extensions by automorphisms. The individual tables are easily read off from the handy 'compound character table' into which we have combined them. The typical expert in this subject knows a vast number of isolated facts about particular groups, but finds it very hard to convey this rather formless information to the earnest student. We hope that such students will find that our 'constructions' answer some of their questions. Usually the object constructed is the simple group itself or a close relative, but we allow it to be any concept connected with the group. Most of the interesting facts about our groups are fairly naturally described in these constructions, and we have not hesitated to stretch the 'constructions' format so as to swallow the odd counter-example. We plead indulgence for the many defects in these rather hasty descriptions. The 'constructions' format arose almost by oversight at a fairly late stage in the preparation of our material, and we have not really had time to organize the information properly. However, we feel that they are already useful enough to be presented to the reader now, rather than postponed to a later volume. In the rest of this introduction, we shall describe the finite simple groups and their classification, and then explain how to read the tables and text in the main body of the A lr IL A§.

2. The finite simple groups Since finite simple groups are the main topic of this A lr ILA§, we had better describe them! Combining the results of a large number of authors, we have:

The classification theorem for finite simple groups The finite simple groups are to be found among: the cyclic groups of prime order the alternating groups of degree at least 5 the Chevalley and twisted Chevalley groups, and the Tits group the 26 sporadic simple groups. We briefly describe these classes. More detailed descriptions appear in the subsequent sections of this introduction, and in the A lr ILA§ entries for the individual groups. The cyclic groups of prime order are the only abelian (commutative) simple groups. There are so many ways in which these groups behave differently to the non-abelian simple groups that the term 'simple group' is often tacitly understood to mean 'non-abelian simple group'. The fact that the alternating groups of degree at least 5 are non-abelian simple groups has been known since Galois, and implies that generic algebraic equations of degree at least 5 are insoluble by radicals. In fact the true origin of finite group theory is Galois theory.

A SURVEY OF THE FINITE SIMPLE GROUPS

viii

The Chevalley and twisted Chevalley groups generalize the familiar 'classical' groups (the linear, unitary, symplectic, and orthogonal groups), which are particular cases. There is a sense in which the vast majority of finite simple groups belong to this class. The Tits group, which might have been considered as a sporadic group, is a simple subgroup of index 2 in the twisted group 2F4 (2). We shall describe all the Chevalley groups in later sections.

3. The sporadic gronps The sporadic simple groups may be roughly sorted as the Mathieu groups, the Leech lattice groups, Fischer's 3-transposition groups, the further Monster centralizers, and the halfdozen oddments. For about a hundred years, the only sporadic simple groups were the Mathieu groups Mu, M 12 , M 22, M 23 , M 24 • These were described by Emil Mathieu in 1861 and 1873 as highly transitive permutation groups, the subscripts indicating the numbers of letters permuted. (It is interesting to note that Mathieu, in the early language of Galois theory, spoke of highly transitive functions of several letters, rather than highly transitive groups.) One of the first great exercises in representation theory was the computation by Frobenius of the character table of M 24 • Frobenius also showed that all the Mathieu groups were subgroups of M 24 (the containment of M 12 in M 24 was not known to Mathieu). The group M 24 is one of the most remarkable of all finite groups. Many properties of the larger sporadic groups reduce on examination to properties of M 24 • This cen~enarian group can still startle us with its youthful acrobatics. The automorphism group of the Leech lattice, modulo a centre of order 2, is the Conway group Co" and by stabilizing sublattices of dimensions 1 and 2 we obtain the other Conway groups Co2 , Co 3 , the McLaughlin group MCL, and the HigmanSims group HS. The sporadic Suzuki group Suz, and the HanJanko group HJ = J 2 , can also be obtained from the Leech

lattice by enlarging the ring of definition. The Leech lattice is a 24-dimensional Euclidean lattice which is easily defined in terms of the Mathieu group M 24 • The Fischer groups Fi~, Fi", Fi 22 , were discovered by B. Fischer in the course of his enumeration of 3-transposition groups (groups generated by a conjugacy class of involutions whose pairwise products have orders at most 3). Fi 22 and Fi" are 3-transposition groups, and Fi~4 has index 2 in the 3transposition group Fi 24 • There is a set S of 24 transpositions in n24 such that the elements of Fi 24 that fix S as a whole realize exactly the permutations of M 24 on S. The Monster group, or Friendly Giant, which was independently discovered by B. Fischer and R. Griess, is the largest of the sporadic groups, with order 808017424794512875886459904961710757005754368000000000 ~

246.320.59.76.112.13' . 17 . 19.23.29.31.41.47.59.71.

It involves all the sporadic groups so far mentioned. Many of

them can be described as the non-abelian composition factors in the centralizers of various elements of the Monster. The remaining sporadic groups that can be so obtained are M= B = Th = HN = He =

F,: the Monster itself

F2+: Fischer's 'Baby Monster' F 313: the Thompson group F 5 +: the Harada-Norton group, and F 7 +: the Held group.

The subscript on our F symbol specifies the relevant Monster element. In the same notation, we have Co, = F2-, Fi~4 = F3 +. It hardly needs to be said that the Mathieu group M 24 plays a vital role in the structure of the Monster. The groups we referred to as the oddments are the remaining groups J" J 3 , J 4 discovered by Janko, and those groups Ru, O'N, Ly discovered by Rudvalis, O'Nan, and Lyons. These were originally found in a variety of ways, but are probably now best constructed via matrix groups over finite fields. We

Table 1. The sporadic groups Order

Group Mu M'2 M 22 M 23 M 24 J2

Suz HS McL C0 3 CO 2 Co, He Fi 22 Fi" Fi~

RN Th B

M J, O'N J3

Ly Ru J4

24 .32 .5.11 2 6 .33 .5.11 27 • 32 . 5 .7 . 11 2 7 • 32 . 5 .7 . 11 .23 2'0.33 .5.7.11 .23 27 .33 .5 2 .7 2 13 • 37 • 52 . 7 . 11 . 13 2 9 . 32 . 53 . 7 . 11 27 • 36 . 53 . 7 . 11 2'0.37 • 53 . 7 . 11 . 23 2'8.36 . 53 . 7 . 11 .23 2 2 ' • 39 . 54 . 7 2 . 11 . 13 . 23 2'0 . 33 . 52 . 7 3 . 17 2 17 • 39 . 52 . 7 . 11 . 13 2'8 . 3 13 • 52 . 7 . 11 . 13 . 17 . 23 22' . 3'6 . 52 . 7 3 . 11 . 13 . 17 . 23 . 29 2 14 • 36 . 56 .7.11 .19

Investigators

M

A

Mathieu Mathieu Mathieu Mathieu Mathieu

1 2 12 1 1

1 2 2 1 1

Hall, Janko Suzuki Higman, Sims McLaughlin Conway Conway Conway, Leech

2 6 2 3 1 1 2

2 2 2 2 1 1 1

Held/Higman, McKay Fischer Fischer Fischer

1 6 1 3

2 2 1 2

1 1 2 1

2 1 1 1

1 3 3 1 2 1

1 2 2 1 1 1

Harada, Norton/Smith 2 15 • 3'0 . 53 . 7 2 .13. 19.31 Thompson/Smith 13 4 6 2 Fischer/Sims, Leon 2 '.3 .5 .7 .11.13 .17 .19.23.31.47 246.320.59.76.112.133.17 .19.23 .29.31.41.47 .59.71 Fischer, Griess 2 3 . 3 . 5 .7 . 11 . 19 Janko 9 4 3 2 .3 .7 .5.11 . 19.31 O'Nan/Sims Janko/Higman, McKay 27 • 35 • 5 . 17 . 19 28 • 37 • 56 . 7 . 11 . 31 . 37 . 67 Lyons/Sims 2'4. 33 . 53 . 7 . 13 . 29 Rudvalis/Conway, Wales 22'.3 3 .5.7.113 .23.29.31.37.43 Janko/Norton, Parker, Benson, Conway, Thackray

A SURVEY OF THE FINITE SIMPLE GROUPS

give a brief table of the appropriate dimensions and fields: group: J , dimension: 7

J3 9

field: 0=11

0=4

J 4 Ru 112 28 0=2

0=2

O'N Ly 45 111 0=7

0=5

The 112-dimensional matrices for J 4 are rather complicated,

bu~ explicit matrices for the other five cases can be found in the appropriate 'constructions' sections of this AT"- A§. The constr\Iction for J 4 makes heavy use of the Mathieu group M 24 , while those for the O'Nan and Lyons groups involve Mu. The indicated representation of the Rudvalis group lifts to a complex representation of its double cover. It has often happened that a sporadic group was predicted to exist some time before its construction, in sufficient detail to give its order, various local subgroups, and sometimes the character table. In such cases we have separated names by a slash in Table 1. Those who were mainly concerned with the prediction appear before the slash; those mainly concerned with the construction after it. Elsewhere in the AT "- A§, we usually name the group after its predictor(s) only, in order to avoid a multiplicity of names, and we recommend this practice. The columns headed 'M' and 'A' give the orders of the Schur multiplier and outer automorphism group, which are both cyclic for each sporadic group. For the simple alternating groups (n ~ 5) we ha-:e M = 2 (see page 236) except that A 6 and A 7 have M = 6, and A = 2 except that A 6 has A = 2 2 •

4. Finite fields The description of the classical groups, and of the Chevalley and twisted Chevalley groups that generalize them, involves heavy use of the properties of finite fields. We summarize the main facts here.

IX

The order of any finite field is a prime power, and for each prime power q there is up to isomorphism just one such field, which we call 0=qo The existence of these fields was established by Galois, and their uniqueness for each possible order by Moore. These fields are often referred to as Galois fields, and Dickson's notation GF(q) is often used for 0=qo If p is prime, the field O=p is just lL/plL, the integers taken modulo p, while for q = pt, the field 0=q may be constructed from 0= p by adjoining a root of any irreducible eq~ation of degree f over 0= p. It may also be defined non-constructIvely as the set of solutions of x q = x in the algebraic closure of 0= p. Since 0=q is a vector space of dimension f over 0=p, its additive group is the direct sum of f cyclic groups of order p. The multiplicative group is cyclic of order q - 1, and a generator for this group is called a primitive root for 0=qo There are exactly (q -1) primitive roots, where (n) is Euler's totient function. Although primitive roots are easily found by inspection in any particular case, there is no simple formula which gives a primitive root in 0=q for an arbitrary given q. The automorphism group of 0=q is cyclic of order f, and consists of the maps x ---7 x' (r = 1, p, p2, ... , pt-I), which are called the Frobenius maps. The subfields of 0=q are those 0=, for which q is a power of r, and each subfield is the fixed field of. t~e corresponding Frobenius map. If q = r', then the characterIstIc polynomial over 0=, of an element x of IFq is

(t - x)(t- x') . .. (t - X,H), The coefficients of this polynomial lie in IF,. In particular, the trace and norm functions from IFq to IF, are defined by

+x + .. . +x ,.'-1 r,2 ,.'-1 = X(r'-l)!(r-l) N IFq_D=,() X -x. x . x . .... X •

Tr.,~dx)=x+x

r,2

Any linear function from IFq to IF, can be written as Tr.,~• .(kx) for a unique k in IFqo

2 The classical groups The Linear, Unitary, Symplectic, and Orthogonal groups have been collectively known as 'The classical groups' since the publication of Hermann Weyl's famous book of that name, which discussed them over the real and complex fields. Most of their theory has been generalized to the other Chevalley and twisted Chevalley groups. However, the classical definitions require little technical knowledge, lead readily to invariant treatments of the groups, and provide many techniques for easy calculations inside them. In this Pi. If IL Pi.§ we take a severely classical viewpoint, for the most part. Later in this introduction, however, we shall quickly describe the larger class of groups, and the present section contains some forward references.

1. The groups GL,,(q), SL,,(q), PGL,,(q), and PSL,,(q) = L,,(q) The general linear group GL,,(q) consists of all the n x n matrices with entries in IFq that have non-zero determinant. Equivalently . it is the group of all linear automorphisms of an n-dimensional vector space over IFq- The special linear group SL" (q) is the subgroup of all matrices of determinant 1. The projective general linear group PGL,,(q) and projective special linear group PSL,,(q) are the groups obtained from GL,,(q) and SL,,(q) on factoring by the scalar matrices contained in those groups. For n;;" 2 the group PSL" (q) is simple except for PSL,(2) = S3 and PSL,(3) = A 4 , and we therefore also call it L,,(q), in conformity with Artin's convention in which single-letter names are used for groups that are 'generally' simple. The orders of the above groups are given by the formulae !GL,,(q)! = (q -1)N,

!SL,,(q)! = !PGL,,(q)1 = N,

!PSL,,(q)1 = 1L,,(q)1 =

N

d'

It is called singular if there is some Xo f 0 such that f(xo, y) = 0 for all y. The kernel is the set of all such xo. The nullity and rank are the dimension and codimension of the kernel. A Hermitian form F(x) is any function of the shape f(x, x), where f(x, y) is a conjugate-symmetric sesquilinear form. Since

either of the forms F and f determines the other uniquely, it is customary to transfer the application of adjectives freely from one to the other. Thus F(x) = f(x, x) is termed non-singular if and only if f(x, y) is non-singular. Coordinates can always be chosen so that a given non-singular Hermitian form becomes x,x, + X,X2 + ... + XnXn.

The general unitary group GUn (q) is the subgroup of all elements of GL(q2) that fix a given non-singular Hermitian form, or, equivalently, that fix the corresponding non-singular conjugate-symmetric sesquilinear form. If the forms are chosen to be the canonical one above, then a matrix U belongs to GUn (q) (is unitary) just if U-' = fr, the matrix obtained by replacing the entries of U' by their qth powers. The determinant of a unitary matrix is necessarily a (q + l)st root of unity. The special unitary group SUn(q) is the subgroup of unitary matrices of determinant 1. The projective general unitary group PG Un (q) and projective special unitary group PSUn(q) are the groups obtained from GUn(q) and SUn(q) on factoring these groups by the scalar matrices they contain. For n ;;,,2, the group PSUn(q) is simple with the exceptions PSU,(2) = S3,

[GUn(q)1 =

q,n(n-l)(qn -1)(qn-l_l) ... (q2-1),

and d = (q-l, n). L,,+1(q) is the adjoint Chevalley group An (q), with Dynkin diagram 0-0---0 ... Q----{) (n nodes). 1 2 3 n-l n

The maximal parabolic subgroup correlated with the node labelled k in the diagram corresponds to the stabilizer of a kdimensional vector subspace.

2.

ne groups GUn(q), SUn(q), PGUn(q), and PSUn(q) = Un(q)

Let V be a vector space over IFq'. Then a function f(x, y) which is defined for all x, y in V and takes values in If'q' is called a conjugate-symmetric sesquilinear form if it satisfies f(A,X, + A2X2, y) = Ad(x" y) + A2f(X2, y)

(linearity in x), and f(y, x) = f(x, y)

PSU3(2) = 32 : Q8,

and so we also give it the simpler name Un (q). We have U,(q) = L,(q). The orders of the above groups are given by

where N

PSU2(3) = A 4,

=

(q + I)N,

\SUn(q)1 = [PGUn(q)\ = N,

N IPSUn(q)1 = \Un(q)[ =d'

where N

=

q,n(n-'l(qn _ (_I)")(qn-l_ (-I)n-') ... (q3 + 1)(q2-1),

and d=(q+l,n). U n+1(q) is the twisted Chevalley group 2 An(q), with the Dynkin diagram and twisting automorphism indicated:

~~~

0--0-0 ... 0-0---0 (n nodes). 1 2 3 3 2 1

The maximal parabolic subgroup correlated with the orbit of nodes labelled k in the diagram corresponds to the stabilizer of a k-dimensional totally isotropic subspace (Le. a space on which F(x) or equivalently f(x, y) is identically zero).

3. The groups Sp.(q) and PSp.(q) = 8,,(q)

(conjugate-symmetry), where x -'> x = x q is the automorphism of IFq' whose fixed field is IFq- Such a form is necessarily semilinear in y, that is

An alternating bilinear form (or symplectic form) on a vector space V over IFq is a function f(x, y) defined for all x, y in V and taking values in IFq' which satisfies

f(x, A,y, + A2Y2) = Ad(x, y,) + A2f(x, y,).

f(A,X, + A2X2, y) = Ad(x" y) + A2f(X2, y)

THE CLASSICAL GROUPS

(linearity in x), and also

fey, x) = -f(x, y)

and

f(x, x) = 0

(skew-symmetry and alternation). It is automatically linear in y also (and so bilinear). The kernel of such a form is the subspace of x such that f(x, y) = 0 for all y, and the nullity and rank of f are the diIDension and codimension of its kernel. A form is called noil-singular if its nullity is zero. The rank of a symplectic form is necessarily an even number, say 2m, and coordinates can be chosen so that the form has the shape XIYm+l

+ XZYm+2 + ... + XmYZm -

Xm+1Yl - Xm+2YZ - ••• - X2mYm o

For an even number n = 2m, the symplectic group Sp.(q) is defined as the group of all elements of GL" (q) that preserve a given non-singular symplectic form f(x, y). Any such matrix necessarily has determinant 1, so that the 'general' and 'special' symplectic groups coincide. The projective symplectic group PSp.(q) is obtained from Sp.(q) on factoring it by the subgroup of scalar matrices it contains (which has order at most 2). For 2m ;;;02, PSPZm(q) is simple with the exceptions

PSPz(2) = S3,

PSPz(3) = A 4,

PSp4(2) = S6

and so we also call it SZm(q). We have Sz(q)=L,(q). If A, B, C, D are m

X

m matrices, then

(~ ~) belongs to

the symplectic group for the canonical symplectic form above just if A'C-C'A=O,

A'D-C'B=I,

B'D-D'B=O,

where M denotes a transposed matrix. The orders of the above groups are given by

!SPZm(q)! = N,

IPSPzm (q)\ = !SZm (q)1 =

N

d'

where N = qm\qZm _1)(qZm-Z_l) ... (qZ-l) and d = (q-I, 2). SZm (q) is the adjoint Chevalley group Cm.{q) with Dynkin diagram 0----0---0 ... ~ (m nodes). 1 2 3 m-2 m-I m

The maximal parabolic group correlated with the node labelled k corresponds to the stabilizer of a k -dimensional totally isotropic subspace (that is, a space on which f(x, y) is identically zero). 4. The groups GO.(q), SO.(q), PGO.(q), PSO.(q), and O.(q) A symmetric bilinear form on a space V over IFq is a function f(x, y) defined for all x, y in V and taking values in IFq which satisfies

f(A,x, + Azxz, y) = Ad(x" y) + Azf(xz, y) (linearity in x), and also

f(y, x) = f(x, y) (symmetry). It is then automatically linear in y. A quadratic form on V is a function F(x) defined for x in V and taking values in IFq, for which we have

F1:JI.x + fLY) =

AZF(x) + AfLf(x, y) + fLZF(y)

for some symmetric bilinear form f(x, y). The kernel of f is the subspace of all x such that f(x, y) = 0 for all y, and the kernel of F is the set of all x in the kernel of f for which also F(x) = O. When the characteristic is not 2, F and f uniquely determine each other, so that the two kernels coincide. The literature contains a bewildering variety of terminology adapted to describe the more complicated situations that can hold in characteristic 2. This can be greatly simplified by using only a few

Xl

standard terms (rank, nullity, non-singular, isotropic), but always being careful to state to which of f and F they apply. Thus we define the nullity and rank of either f or F to be the dimension and codimension of its kernel, and say that f or F is non-singular just when its nnllity is zero. A subspace is said to be (totally) isotropic for f if f(x, y) vanishes for all x, y in that subspace, and (totally) isotropic for F if F(x) vanishes for all x in the subspace. When the characteristic is not 2 our adjectives can be freely transferred between f and F. The Witt index of a quadratic form F is the greatest dimension of any totally isotropic subspace for F. It turns out that if two non-singular quadratic forms on the same space over IFq have the same Witt index, then they are equivalent to scalar multiples of each other. The Witt defect is obtained by subtracting the Witt index from its largest possible value, Hn]. For a nonsingular form over a finite field the Witt defect is 0 or 1. The general orthogonal group GO.(q, F) is the subgroup of all elements ofGL,,(q) that fix the particular non-singular quadratic form F. The determinant of such an element is necessarily ±I, and the special orthogonal group SO. (q, F) is the subgroup of all elements with determinant 1. The projective general orthogonal group PGO. (q, F) and projective special orthogonal group PSO.(q, F) are the groups obtained from GO.(q, F) and SO.(q, F) on factoring them by the groups of scalar matrices they contain. In general PSO.(q, F) is not simple. However, it has a certain subgroup, specified precisely later, that is simple with finitely many exceptions when n;;;O 5. This subgroup, which is always of index at most 2 in PSO. (q, F), we call 0. (q, F). When n = 2m + 1 is odd, all non-singular quadratic forms on a space of dimension n over IFq have Witt index m and are equivalent up to scalar factors. When n = 2m is even, there are up to equivalence just two types of quadratic form, the plus type, with Witt index m, and the minus type, with Witt index m-I. (These statements make use of the finiteness of IFw) Accordingly, we obtain only the following distinct families of groups: When n is odd GO.(q), SO. (q), PGO.(q), PSO.(q), O.(q), being the values of GO. (q, F) (etc.) for any non-singular F. When n is even GO~(q), SO~(q), PGO~(q), PSO~(q), O~(q) for either sign e = + or -, being the values of GO.(q, F) (etc.) for a form F of plus type or minus type respectively. We now turn to the problem of determining the generally simple group O.(q) or O~(q). This can be defined in terms of the invariant called the spinor norm, when q is odd, or in terms of the quasideterminant, when q is even. We define these below, supposing n;;;o 3 (the groups are boring for n <;;2). A vector r in V for which F(r) of 0 gives rise to certain elements of GO. (q, F) called reflections, defined by the formula

x

---,>

f(x, r) x - F(r) . r.

We shall nOw define a group a~(q) of index I or 2 in SO~(q). The image P~(q) of this group in PSO~(q) is the group we call O~(q), which is usually simple. The a, po. notation was introduced by Dieudonne, who defined a~~o be the commutator subgroup of SO~ (q), but we have changed the definition so as to obtain the 'correct' groups (in the Chevalley sense) for small n. For n;;;o 5 our groups agree with Dieudonne's. When q is odd, ~(q) is defined to be the set of all those g in SO~ (q) for which, when g is expressed in any way as the product of reflections in vectors rh rz, ... , r" we have

F(r,) . F(rz) ..... F(r,)

a square in IFw

The function just defined is called the spinor norm of g, and takes values in 1F~/(1F~j2, where IF~ is the multiplicative group of IFw Then a~(q) has index 2 in SO~(q). It contains the scalar matrix -1 just when n = 2m is even and (4, qm - e) = 4. When q is even, then if n =2m + I is odd, SO.(q) = GO.(q) is isomorphic to the (usually simple) symplectic group SPZm(q), and we define a.(q) = SO.(q). To obtain the isomorphism, observe that the associated symmetric bilinear form f has a onedimensional kernel, and yields a non-singular symplectic form on VI ker(f).

xii

THE CLASSICAL GROUPS

led k,,;; m - 2 corresponds to the stabilizer of an isotropic kspace for F. Those correlated with the nodes labelled ml and m2 correspond to stabilizers of members of the two families of isotropic m-spaces for F. OZm(q) is the twisted Chevalley group 2Dm (q), with the Dynkin diagram and twisting automorphism

For arbitrary q, and n = 2m even, we define the quasideterminant of an element to be (-1)", where k is the dimension of its fixed space. Then for q odd this homomorphism agrees with the determinant, and for q even we define n~ to be its kernel. The quasideterminant can also be written as (-l)D, where D is a polynomial invariant called the Dickson invariant, taking values in 1F2 • An alternative definition of the quasideterminant is available. When e = + there are two families of mayJmal isotropic subspaces for F, two spaces being in the same family just if the codimension of their intersection in either of them is even. Then the quasideterminant of an element is 1 or -1 according as it preserves each family or interchanges the two families. When e = -, the maximal isotropic spaces defined over IFq have dimension m-I, but if we extend the field to IFq', we obtain two families of m-dimensional isotropic spaces and can use the same definition. When n = 2m + 1 is odd, the groups have orders

0-0-0 '"

I

O;i(q) = L 2(q) x L,(q), O.(q) = L,(q2), O,(q) = S.(q), O~(q) = L 4(q), O;;(q) = U 4(q).

O,(q) = L,(q),

The group On(q) is simple for n;;' 5, with the single exception that 0,(2) = S.(2) is isomorphic to the symmetric group S6.

Inn (q)1 = IPOn(q)\ = 10n(q)1 = Nld, where N

=

qm'(q2m _1)(q2m-2_1) ... (q2_1)

5. Oassification of points and hyperplanes in orthogonal spaces

and d = (2, q -1). 02m+1(q) is the adjoint Chevalley group Bm(q), with Dynkin diagram 0-0-0 ...

1 2

3

09=0

Let V be a space equipped with a non-singular quadratic form F. Then for fields of odd characteristic many authors classify the vectors of V into three classes according as

(m nodes).

m-I m

F(v) =0,

The maximal parabolic subgroup correlated with the node labelled k corresponds to the stabilizer of an isotropic k-space for F. When n = 2m is even, the groups have orders IGO~(q)1 = 2N,

F(v) a non-zero square, F(v) a non-square,

ISO~(q)1 = IpGO~(q)1 = 2Nle,

2

IPSO~(q)1 = 2Nle , Ipn~(q)1 =

since these correspond exactly to the three orbits of projective points under the orthogonal group of F. In this A"If IL A§ we prefer a different way of making these distinctions, which is independent of the choice of any particular scalar multiple of the quadratic form F, and which does the correct thing in characteristic 2. We say that a subspace H of even dimension on which F is non-singular is of plus type or minus type according to the type of F when restricted to H, and if H is the hyperplane perpendicular to a vector v, we apply the same adjectives to v. Thus our version of the above classification is v isotropic (or null),

\n~(q)\ = Nle,

10n(q)1 = Nld,

where N = qm(m-l)(q'" _ e)(q--2_1)(q2m-4_1) ... (q2_1)

and d=(4,qm- e ), e=(2,qm- e ). O'i:m(q) is the adjoint Chevalley group Dm(q), with Dynkin diagram ml

0-0-0

123

(m nodes).

( m-2

Am -1 (m nodes).

m-2'o)m~1

3

The maximal parabolic subgroup correlated with the orbit of nodes labelled k,,;; m-I corresponds to the stabilizer of an isotropic k-space for F. For n";; 6, the orthogonal groups are isomorphic to other classical groups, as follows:

ISOn(q)1 = IPGOn(q)1 = IPSOn(q)\ = N,

IGOn(q)\ = dN,

2

v of plus type,

m2 The maximal parabolic subgroup correlated with the node label-

v of minus type.

Table 2. Structures of classical groups GL,.(q)

d (n,q-1)

PGL,.(q) SL,.(q)

d. (q-1 T X G). d

PSL,. (q) = L,. (q)

d.G

G.d

G

PGUn(q) SUn(q) PSUn(q) = Un(q)

d

GUn(q)

(n, q+ 1)

d. (q+1 -d-XG) .d

G.d

d.G

G

d

Spn(q)

PSpn(q) = Sn(q)

(2, q-1) =2 (2, q-1) = 1

2.G G

G G

d

GO~(q)

PGO~(q)

SO~(q)

PSO~(q)

(4,qm- e )=4 (4,qm- e )=2 (4,qm- e )=1 (2, q-1) =2 (2, q-1) =1

2.G.22 2x G.2 G.2 2xG.2 G

G.22 G.2 G.2 G.2 G

2.G.2 2xG G.2 G.2 G

G.2 G G.2 G.2 G

n~(q) pn~(q)

2.G G G G G

=

O~(q)

g}n=2m g}n=2m+1

Xlll

THE CLASSICAL GROUPS

6. The Cliftord algebra and the spin group The Clifford algebra of F is the associative algebra generated by the vectors of V with the relations x 2 = F(x), which imply xY-l-yx=f(x,y). If V has basis e" ... ,e", then the Clifford algebra is 2· -dimensional, with basis consisting of the formal products The vectors r with F(r) 'I 0 generate a subgroup of the Clifford algebra which is a central extension of the orthogonal group, the 1vector r in the Clifford algebra mapping to the negative of the 'reflection in r. When the ground field has characteristic '12, this, remark can be used to construct a proper double cover of theorthogonal group, called the spin group.

7. Structure tables for the classical groups Table 2 describes the structure of all the groups mentioned, in terms of the usually simple group G, which is the appropriate one of L,,(q), U.(q), S.(q), O.(q).

8. Other uotations for the simple groups There are many minor variations such as L" (IFq) or L(n, q) for L" (q) which should give little trouble. However, the reader should be aware that although the 'smallest field' convention

which we employ in this AT D...A§ is rapidly gammg ground amongst group theorists, there are still many people who write U.(q2) or U(n, q2) for what we call U.(q). Artin's 'single letter for simple group' convention is not universally adopted, so that many authors would use U.(q) and O.(q) for what we call GU.(q) and GO.(q). The notations E 2(q) and E 4 (q) have sometimes been used for G 2(q) and F.(q). Dickson's work has had a profound influence on group theory, and his notations still have some currency, but are rapidly becoming obsolete. Here is a brief dictionary: (Linear fractional) LF(n, q) = L,,(q) (Hyperorthogonal) HO(n, q2) = U.(q) (Abelian linear) A(2m, q) = S2m(q) · (FIrst orthogona1){FO(2m + 1, q) = 02m+1(q) • FO(2m, q) = 02m(q)} where e = ± 1,

(Second orthogonal) SO(2m, q) = O,::,(q) qm == e modulo 4. (First hypoabelian) FH(2m, q) =

O~m(q)

(Second hypoabelian) SH(2m, q) = O'm(q) Dickson's first orthogonal group is that associated with the quadratic form xi+ x~+ . .. + x~. His orthogonal groups are defined only for odd q, and his hypoabelian groups only for even q. He uses the notation GLH(n, q) (General Linear Homogeneous group) for GL,,(q).

3 The Chevalley and twisted Chevalley groups 1. The untwisted groups The Chevalley and twisted Chevalley groups include and neatly generalize the classical families of linear, unitary, symplectic, and orthogonal groups. In the entries for individual groups in the A lr IL A§, we have preferred to avoid the Chevalley notation, since it requires considerable technical knowledge, and since most of the groups we discuss also have classical definitions. The classical descriptions, when available, permit easy calculation, and lead readily to the desired facts about subgroups, etc. However, for a full understanding of the entire set of finite simple groups, the Chevalley theory is unsurpassed. In particular the isomorphisms such as L.(q) = ot(q) between different classical groups of the same characteristic become evident. The full Chevalley theory is beyond the scope of this A lr ILA§, but in the next few pages we give a brief description for those already acquainted with some of the terminology of Lie groups and Lie algebras. We reject any reproach for the incompleteness of this treatment. It is intended merely to get us to the point where we can list all the groups, and the isomorphisms among them, and also te specify their Schur multipliers and outer automorphism groups. In 1955 Chevalley discovered a uniform way to define bases for the complex simple Lie algebras in which all their structure constants were rational integers. It follows that analogues of these Lie algebras and the corresponding Lie groups can be defined over arbitrary fields. The resulting groups are now known as the adjoint Chevalley groups. Over finite fields, these groups are finite groups which are simple in almost all cases. The definition also yields certain covering groups, which are termed the universal Chevalley groups. If a given finite simple group can be expressed as an adjoint Chevalley group, then in all but

finitely many cases its abstract universal cover is the corresponding universal Chevalley group. In the standard notation, the complex Lie algebras are An

Bn

c" D n Oz

F4

E6

E,

E8

where to avoid repetitions we may demand that n ;;;>1, 2, 3, 4 for An> Bn> c", D n respectively. The corresponding adjoint Chevalley groups are denoted by An(q), Bn(q), c,,(q), Dn(q), OzCq), F.(q), E 6 (q), E,(q), E 8 (q).

The corresponding Dynkin diagrams, which specify the structure of the fundamental roots, appear in Table 3. In the cases when a p-fold branch appears (p = 2 or 3), the arrowhead points from long roots to short ones, the ratio of lengths being -Jp. We have included the non-simple case Dz=A,EBA" and the repetitions A, = B , = Cl, B z = C z, A 3 = D 3 , since these help in the understanding of the relations between various classical groups.

2. The twisted groups Steinberg showed that a modification of Chevalley's procedure could be made to yield still more finite groups, and in particular, the unitary groups. Any symmetry of the Dynkin diagram (preserving the direction of the arrowhead, if any) yields an automorphism of the Lie group or its Chevalley analogues, called an ordinary graph automorphism. Let us suppose that ex is such an automorphism, of order t, and call it the twisting automorphism. We now define the twisted Chevalley group 'Xn (q, q') to be set of elements of X n (q') that are fixed by the quotient of the twisting automorphism and the field automorphism induced by the Frobenius map x .....;.x q of F q ,. The particular cases are ZAn(q, qZ) = Un+l(q),

Table 3. Dynkin diagrams

ZDn(q, qZ) = 02n(q), 0

A, 0

B, 0

C,

0-0 Az

0-0--0

~

~

~

~

~

~

~

~

~

o-o-c(

A3

Bz

B3

Cz

C3

<

0 0

Dz

D3

~

~

Oz

F4

‫ס‬--‫ס‬-o---o

A4

B4

C4

D4

o---o--o--o--D As

Bs

Cs

Ds

~~

~

3D4 (q, q3), ZE6 (q, qZ),

the last two families being discovered by Steinberg. We usually abbreviate 'Xn(q, q') to 'Xn(q). A further modification yields the infinite families of simple groups discovered by Suzuki and Ree. If the Dynkin diagram of X n has a p-fold edge (p = 2,3), then over fields of characteristic p the Chevalley group is independent of the direction of the arrowhead on that edge. In other words, there is an isomorphism between Xn(pf) and Yn(pf), where Y n is the diagram obtained from X n by reversing the direction of the arrowhead. Thus for example, B n(2 f ) = c,,(2f ), or in classical notation OZn+I(2f ) = SZn(2f ), as we have already seen. In the three cases B z = C 2 , O 2 , F 4 , the diagram has an automorphism reversing the direction of the p-fold edge, and so over fields of the appropriate characteristic p, the resulting . Chevalley groups have a new type of graph automorphism, which we call an extraordinary graph automorphism, whose square is the field automorphism induced by the Frobenius map x ---?o x p •

xv

THE CHEVALLEY AND TWISTED CHEVALLEY GROUPS

Now when q = pZm+' is an odd power of this p, the field automorphism induced by x --i> x p - · ' has the same square, and the elements of Xn(pZm+') fixed by the quotient of these two al!tomorphisms form a new type of twisted Chevalley group, cal1ed ZXn(*, pZm+'), and usually abbreviated to ZXn(pZm+1). The p~rticular cases are ZB z( *, 2 Zm +1) = ZCz( *, 2 Zm +'), a Suzuki group, zGz( *, 3Zm +1), a Ree group of characteristic 3, ZF.( *, 2 Zm +1), a Ree group of characteristic 2. These groups are often written Sz(q), R,(q), Rz(q),

ZBz(2) = 5 :4, the Frobenius group of order 20, zGz(3) = Ld8) : 3, the extension of the simple group Lz(8) of order 504 by its field automorphism; ZF 4 (2) = T. 2, where T is a simple group not appearing elsewhere in the classification of simple groups, called the Tits group. Table 4 gives the Dynkin diagrams and twisting automorphisms for all the twisted groups. We remark that what we have called simply the twisted Chevalley groups are more fully called Table 4. Diagrams and twisting automorphisms for twisted groups

°

zA,

~ zA 3

2A z

0) °

zDz

4

<) zD3

e)

~

~) zD4

~ 2A s

~) zDs

~~

~

3D4

zBz=zCz

~ zA

Eo

&iD zG z

4. Orders of the Chevalley groups In Table 6, the parameters have been chosen so as to avoid the 'generic' isomorphisms. N is the order of the universal Chevalley group. N/d is the order of the adjoint Chevalley group.

5. The simple groups euumerated

where q = pZm+', but the subscripts 1 and 2 for· the two types of Ree group can be omitted without risk of confusion. It turns out that the first group in each of these families is not simple, but all the later ones are simple. The cases are

o=D

the (extraordinary) graph automorphism squares to the generating field automorphism. The groups of orders d, e, i, g are cyclic except that 3! indicates the symmetric group of degree 3, and orders written as powers indicate the corresponding direct powers of cyclic groups.

~ zF 4

the adjoint twisted Chevalley groups, and that, like the untwisted groups, they have certain multiple covers called the universal twisted Chevalley groups. We have included the untwisted group zA, =A, in the table, and also the case zDz obtained by twisting a disconnected diagram, and the repetition ZA 3 = ZD 3 • The reason is again that these special cases illuminate relations between some classical groups.

3. Multipliers and automorpbisms of Chevalley groups The Schur multiplier has order de, and the outer automorphism group has order dig, where the order of the base field is q = pf (p prime), and the numbers d, i, g are tabulated in Table 5. (An entry '2 if ... ' means 1 if not.) The Schur multiplier is the direct product of groups of orders d (the diagonal multiplier) and e (the exceptional multiplier). The diagonal multiplier extends the adjoint group to the corresponding universal Chevalley group. The exceptional multiplier is always a p-group (for the above p), and is trivial except in finitely many cases. The outer automorphism group is a semidirect product (in this order) of groups of orders d (diagonal automorphisms), i (field automorphisms), and g (graph automorphisms modulo field automorphisms), except that for B,(2f ), Gz(3f), F 4 (2f )

The exact list of finite simple groups is obtained from the union of the list of Chevalley groups (Table 5) the alternating groups An' for n '" 5 the cyclic groups of prime order the 26 sporadic groups, and finally the Tits simple group T = ZF 4 (2)' by taking into account· the exceptional isomorphisms below. Each of these isomorphisms is either between two of the above groups, or between one such group and a non-simple group. For the reader's convenience, we give both the Chevalley and classical notations. The exceptional isomorphisms: A,(2) = L z(2) "'S3 A,(3) = 1.2 (3) "'A 4

B 2 (2) = S4(2) '" S6

A,(4) = L z(4) "'As

G 2(2) ",2 Az(3) .2= U 3 (3) . 2 2Az(2) = U (2) '" 32 . Q

A,(5) = 1.2 (5) "'As

2A 3 (2) = U.(2) "'Bz(3) = S.(3)

3

8

A,(7) = Lz(7) "'Az(2) = 1.3 (2) ZB2(2) = Sz(2) "'5:4 A,(9) = Lz(9) "'A 6 2Gz(3) = R(3) ",A,(8) . 3 = Lz(8) .3 A 3 (2)

=

1.4 (2) '" As

Z

F.(2) = R (2) '" T. 2.

6. Parabolic subgroups With every proper subset of the nodes of the Dynkin diagram there is associated a parabolic subgroup, which is in structure a p"group extended by the Chevalley group determined by the subdiagram on those nodes. (Here p, as always, denotes the characteristic of the ground field IF..) The maximal parabolic subgroups are those associated to the sets containing all but one of the nodes-we shall say that such a subgroup is correlated to the remaining node. According to the Bore/-Tits theorem the maximal p-local subgroups of a Chevalley group are to be found among its maximal parabolic subgroups. These statements hold true for the twisted groups, provided we replace 'node' by 'orbit of nodes under the twisting automorphism'.

7. The fundameutal representations The ordinary representation theory of Chevalley groups, as recently developed by Deligne and Lusztig, is very complex, and the irreducible modular representations are not yet completely described. But certain important representations (not always irreducible) can be obtained from the representation theory of Lie groups. The irreducible representations of a Lie group are completely classified, and can be written as 'polynomials' in certain iundamental representations, one for each node of the Dynkin diagram. The degrees of the representations for the various Lie groups X n are given in Table 7. By change of field, we obtain the so-called fundamental representations of the universal Chevalley group X n (q), which are representations over the field IF q having the given degrees. In the classical cases these are fairly easily described geometrically-for example in An(q) = 4+1(q) the fundamental representation corresponding to the rth node is that of S4+1(q) on the rth exterior power of the original vector space V.

xvi

THE CHEVALLEY AND TWISTED CHEVALLEY GROUPS

Table 5. Automorphisms and multipliers of the Chevalley groups

, d

Condition Group

Definitions . of

A 1 (q) (2, q-1) n~2 A,,(q) (n+ 1, q-1) 2A,,(q) (n+1,q+1) n~2 B,(q) (2, q-1) 2B (q) 1 fodd 2 n~3 B,,(q) (2, q-1) n~3 c,,(q) (2, q-1) Diq) (2, q-1)' 3D4 (q) 1 D,,(q) (2,q-1)' n>4,even n>4,odd D,,(q) (4, q" -1) 2D,,(q) (4,q"+1) n;;::4 G 2(q) 1 2G2(q) 1 f odd Fiq) 1 2Fiq) 1 fodd B 6 (q) (3, q-1) 2B6 (q) (3, q+ 1) E,(q) (2, q-1) Bo(q) 1

f

q=pl 1 q=pl 2 q2= pi 1 q=pl 2ifp=2 q=21 1 q=pl 1 q=pl 1 q=pl 3! q3= pi 1 q=pl 2 q=pl 2 q2= pt 1 q=pl 2ifp=3 q=31 1 q=pl 2ifp=2 q=21 1 q=pl 2 q2=p 1 q=pl 1 q=pl 1

Table 6. Orders of Chevalley groups G

N

d

fI (qHl_l) q"' fI (q" -1) q"' fI (q2< -1) "-, q"'"-"(q"-1) n (q'i_1) q,,
A,,(q), n '" 1

Cases when e f 1

g

AM)--> 2, A 1 (9) --> 3 A,(2) -->2, A 2(4) --> 42 , A 3(2) -->~ 2A 3(2)-> 2, 2 A 3(3) -> 32 , 2A 5 (2) -> 22 B 2(2)--> 2 2 2 B 2(8) --> 2 B 3(2) -->2, B 3(3) --> 3 C 3 (2) -->2 D 4 (2)--> 22 none none none none G 2 (3)-->3, G,(4)--> 2 none F 4 (2)--> 2 none none 2B6 (2) --> 22 none none

Table 7. Degrees of the fundamental representations of Lie and Chevalley groups. (The degrees are the numbers below or to the right of the nodes.)

(n+l,q-l)

1

;=1

(2,q-1)

l;:,l

A" B"

i=1

q6(q6_1)(q'_1)

Fiq)

q24(q>2_1)(q8 -1)(q6_1)(q2_1)

1

E 6 (q)

q36(q>2_1)(q9 _1)(q8_1)(q6 -1)(q5_1)(q2_1)

(3,q-1)

E,(q)

q ''''(q'''-1)(q24 -1)(q'" -1)(q" -1) (q>4_1)(q>2_1)(q8_1)(q2_1) q,,(n+l)12

ft (qi+l_( _1)1+1)

n

n+1

(n;l}·t;l).. t~~)

n+1

6-----6-:::~:::~

(4,q"-1)

G,(q)

q63(q" -1)(q>4-1)(q>2_1)(q>O -1)(q8_1)(q6_1)(q2_1)

n-1

(2,q-1)

;=1

E,(q)

r

2

0 - - - - 0 - ::: - 0 - : : ; - 0 - - - 0

2n+1

(2n2+1) ...(2nr+1) ..

1

{2:~~)

2"

(2,q-1)

1

(n+1,q+1)

·1,=1

q2(q2+1)(q_1)

2Giq), q =

3 2m + 1

2Piq), q = 22m + 1

'E6(q)

n

1

(q2< -1) i=l q>2(q8+ q 4+ 1)(q6_1)(q2_1)

(4,q" +1)

q3(q3+1)(q_1)

1

q>2(q6+ 1)(q4_1)(q3+ 1)(q-1)

1

q"'"-"(q" + 1)

3Diq)

"-,

q36(q>2_1)(q9+ 1)(q8-1)(q6-1)(q'+ 1)(q2-1)

248

1

G2

(3,q+1)

1r4

!'

273 78

F4

27 351 2925

351 1274 B. 27 52

912 E,

56 1539 27664 365750 8645 133

30380 2450240 146325270 6899079264

147250 6696000 Bo 3875

4 How to read this AillAS: the 'constructions' sections The typical {SI, If IL{SI,§ entry has four parts, giving: 1. the order of G, and its Schur multiplier and outer automorphism group; 2. various constructions for G, or related groups or concepts; 3. the maximal subgroups of G and its automorphic extensions; 4. the compound character table, from which the ·ordinary character tables for various extensions of G can be read off. The parts are usually in this order, but the exigencies of pagination have occasionally forced uS to put the character table first. For some large groups, we can only give partial information. Below, we discuss the type of information contained.

which these groups are the automorphism groups. The style is telegraphic, and the reader must expect to fill in many details for himself. However, we have tried wherever possible to make the definitions so explicit that a sufficiently determined reader can completely recover the group or object from our description of it. For the classical definitions, say of the unitary groups, we have been content to give merely the definition itself, and the structure of the various groups in terms of the simple group G, for example

1. Order, multiplier, and outer automorpbism group

The most interesting constructions, however, are those that relate the group to some special combinatorial object whose existence is not immediate from the classical definition; for example in the case of U 3 (5) the Hoffman-Singleton graph of 50 vertices, each joined to 7 others, whose automorphism group is G. 2 = P L U 3 (5). There is no reasonable analogue of this for the other unitary groups. In such cases, the thing we construct may be the combinatorial object, its automorphism group, or preferably both. The combinatorial object is usually a set of some kind, and often a set of vectors in some space. In the simplest cases we have been able to give explicit names for all the members of this set, and also for all their automorphisrns, but more usually we have only been able to specify a generating set of automorphisms. We have usually tried to make this generating set as large as we can, rather than as small as we can, since computations are then easier. Readers who want minimal systems of generators must look elsewhere! In some large cases we have ouly been able to give a description of the relevant object rather than an explicit construction. We are pleased to report that in several cases where our description was originally of this kind we have managed, sometimes at the proof stage, to insert a more explicit construction. (The publishers of the {SI, If IL{SI,§ were not quite so pleased!) In very complicated cases the work of the construction has been spread over several sections, each individually called a 'construction' because it constructs some portion of the final object. In this way we have even managed to fit a complete and explicit construction for the Monster onto a few {SI, If IL{SI,§ pages! In the remainder of this chapter, we briefly describe some notations and ideas that appear repeatedly in the 'constructions' sections.

Every reader of this {SI, If IL {SI,§ should be familiar with the notion of the order of a finite group! The set of all automorphisms (isomorphisms from G onto itself) of a given group G forms a group Aut(G) called the automorphism group of G. Any element g of G yields a particular automorphism x -> g-'xg called transformation or conjugation by g, or the inner automorphism determined by g, and the set of all such automorphisms is a normal subgroup Inn(G) of Aut(G), the inner automorphism group. It is abstractly isomorphic to the quotient of G by its centre. The quotient group Aut(G)/Inn(G) = Out(G) is the outer automorphism group of G. A central extension, or covering group, of G is a group H with a central subgroup C whose quotient is G. It is called a proper covering group if C is contained in the commutator subgroup of H. The groups C that arise for proper covers of G are each of them quotients of a largest one, called the Schur multiplier (or multiplicator), Mult(G), of G. The corresponding central extension is only determined up to isoclinism, see Chapter 6, Section 7. However, for a perfect group G (and in particular for a simple G), we do have a unique universal covering group, of which any proper cover of G is a quotient, and for which C is the Schur multiplier of G. The multiplier may be computed in the following way: Let G be defined by the relations

R,(a" a2, ...) = 1, ... , R/a" a2, ...) = 1, ... in generators a" ... , a" ••.. Then the modified relations

R/a" ...) =

Zj,

define a group H which is a central extension of G by the subgroup C = (z" Z2, •.•) of H. Then the Schur multiplier of G is the intersection of C with the commutator subgroup of H. Moreover, if we factor H by a subgroup of C that is maximal subject to its not intersecting the commutator subgroup, we obtain one of the proper covers of G by its multiplier. The multiplier plays an important role in representation theory-see later sections.

GU3 (5)=2 x 3. G. 3, PGU3 (5) = G. 3, SU3 (5) = 3 . G, PSU3 (5) = U 3 (5) = G.

3. Notation for types of point and hyperplane for orthogonaI groups We attach the terms 'plus' and 'minus' to various notions connected with orthogonal groups. This usage is explained in Chapter 2, Section 5.

4. Projective dimensions and projective counting 2. Constructions Under this heading, we give brief descriptions of various ways of constructing G, various closely related groups, or objects of

Since the group stabilizing a vector also stabilizes its multiples, it is often sensible to use projective terminology, in which the (k + I)-dimensional subspaces of an (n + I)-dimensional vector

xviii

CONSTRUCTIONS

space are called the k -dimensional subspaces of an ndimensional projective space. To avoid confusion, we usually use the terms point, line, plane, ... to indicate a space of projective dimension 0, 1, 2, .... In projective counting, a vector is identified with its non-zero scalar multiples. Thus the 8 vectors from the centre of a cube to its vertices arise in 4 pairs {v, -v}, and so projectively speaking they yield just 4 distinct points. In such a situation we might write'... 2 x 4 vectors .. .' or '... , projectively, 4 vectors .. .' or, when we wish to indicate just which scalars are to be neglected ' ... 4 (±)-vectors .. .'. Formally, a (±)-vector is a pair {v, -v}, and an w-vector (where w 3 = 1) is a triple {v, wv, w 2v}. 5. Coordinates for vectors We adopt various compact notations for vectors. For example (28 0 '6) indicates a vector, or set of vectors, having some eight coordinates 2 and the remaining sixteen coordinates O. On occasion we indicate that we intend to apply all permutations of the appropriate Symmetric, Alternating, Cyclic, or Dihedral group by the corresponding superscript letter. Thus (x, y, z, t)A represents the twelve vectors obtained by all even permutations of the coordinates. 6. Quaternionic spaces A fair number of our constructions involve quaternionic spaces, and as some readers will not be familiar with the necessary distinctions between the roles of the two kinds of multiplication, we state our conventions here. (Care is needed in comparisons with other sources.) We use left multiplication for scalar products, and right multiplication for group actions. Thus if x = (x" ... , x,,) is a vector, A a scalar, and S = diag(S" ... , 5,,) a diagonal group element, then the scalar product of x by A is

7. Presentations We write presentations in the format (generators Irelations) thus for example (a, b I a 2 = b 3 = (ab)S = 1)

is a presentation for As. Relations placed in square brackets are redundant. Coxeter-type presentations We represent certain relations by diagrams, in which each node refers to a generator g, and indicates the relation g2 = 1. For any two such generators g and h, we have the further relation gh = hg, or (gh)2 = 1, if the corresponding nodes are not joined, (gh)3 = 1, if they are joined by an unmarked edge, (gh)" = 1, if they are joined by an edge marked n.

This notation is often combined with the '(... 1 . . .)' format, and additional relations may be explicitly adjoined. On occasions we specify certain relations, and then write '+ more relations(?)'this means that the group certainly satisfies the specified relations, but we do not know whether they suffice for a presentation. Many interesting presentations for particular groups can be deduced from the information on pages 232-3. Notations for particular presentations (after Coxeter and Moser) (I, m, n) (I, m, n) (I, m

I n, k)

(I, m, n; q)

R' = srn = (RS)" = (R- 1 S)k = 1 R' = srn = T n =RST= (TSR)q = 1, or

R' = srn = (Rs)n = (R- 1 S-lRS)q = 1

AX = (AX" ... , Ax,,),

AP =Bq = C' = (AB)2=(BC)2= (CA?

while the image of x under S is

= (ABC? = 1, or 2 X = y 2 =(xy?= 1, and

xS = (x , S" ... , x"Sn)'

The reader is cautioned that the map x --? AX is not a linear operation. Again, in a lattice L over some quaternionic ring R, we might say that x == y (modulo r) to mean that x - y E rL. This does not usually imply that AX==Ay (modulo r), but if S is a group operation of some group acting on L, it will imply that xS==yS (modulo r).

R' =srn = T n = RST= 1,0r R' = srn = (RS)" = 1 R' =srn = T n =RST

GI1,q,r,s,t,u,V

1 = T 2 = (XT)P = (YT)q = (XYTY X 2 = y2 = Z2 = (xy)2 = (YZ)2 = (ZX)2

1, and 1 = T 2 = (XT)P = (YT)q = (XYTY = =

(ZT)'

= (XZT)' = (YZT)" = (XYZT)"

5 How to read this AillAS: information about subgroups and their structure 1. The maximal subgroups sections A maximal subgroup H of G is a proper subgroup of G that is contained in no strictly larger proper subgroup of G. Whenever we have been able to do so, under the heading 'maximal subgroups' we have listed all the maximal subgroups of G up to conjugacy. On other occasions, we have modified this heading appropriately, or made some explicit qualification. For each of the listed subgroups H of G, we usually give, in successive columns: its order its index in G its structure in the notation described at the end of this section for each automorphic extension G. A of G, the fusion behaviour of H under A, and the associated maximal subgroups of G.A the permutation character of G on the cosets of H, when this is sufficiently simple, and finally various ways to specify a subgroup of type H in G. In the column for an automorphic extension G. A, a splitting symbol (:) indicates that some conjugate of H is invariant under automorphisms of G that suffice to generate G. A modulo G. (It then gives rise to an 'ordinary' maximal subgroup H. A of G. A.) a fusion join (I) is used to show that several isomorphic groups H, that are not conjugate in G become conjugate in G . A. These symbols may be followed by structure-symbols for the 'ordinary' and 'novel' maximal subgroups of G. A. The maximal subgroups K of G. A are of three types: Trivialities: these are the groups G. B, where B is a maximal subgroup of A. They are ignored in our tables. Ordinaries: if H is invariant under some automorphisms of G that generate G. A modulo G, it extends to a maximal subgroup K = H. A of G. A. In this case, we write the structure of K immediately after the corresponding splitting symbol. The letter H, when used, refers to the corresponding maximal subgroup of G. Novelties. It can happen that the intersection of K with G is a non-maximal proper subgroup of G. In such cases K is called a novel subgroup of G. A, and we have written its structure in the row corresponding to one of the maximal subgroups H of G that contains K n G. Any maximal subgroup of G. A that occurs after a fusion marker is a novelty; but novelties can also occur after a splitting symbol, in which case they are written after the corresponding 'ordinary' maximal subgroup H. A. The permutation character of G associated with H is the character of the permutation representation of G acting by right multiplication on the right cosets of H in G. The irreducible constituents of this representation are indicated by their degrees followed by lower case letters a, b, c, ... , which indicate the successive irreducible representations of G of that degree, in the order in which they appear in the IS>.. 1r ILIS>..§ character table. A sequence of small letters (not necessarily distinct) after a single number indicates a sum of irreducible constituents all of the same degree. Thus la + 45ab + 216a + 342aac

would be used to abbreviate la +45a +45b+216a + 342a + 342a + 342c. The specifications of subgroups H give information which might help to locate a copy of H inside G. An abstract specification expresses H as the normalizer in G of some element or subgroup, or alternatively as the centralizer in G of an outer automorphism. We indicate these (respectively) by writing N(. ..) or C(.. .), where the parentheses contain information about the conjugacy classes of the groups being normalized or centralized. The form C(...) is only used for the centralizers of outer automorphisms. The class of a cyclic subgroup is. indicated by its order, followed by the tag letters of its generators (see 'Class names', Chapter 7, Section 5): subscripts are used to count cyclic subgroups in a larger group, and superscripts indicate direct powers of groups. Thus the symbols below would be used for the normalizers in G of elements or groups of the indicated forms: N(2A): C(2B): C(2): N(3A): N(5AB): N(2A 2 ): N(3') = N(3AB 4 C,D6 ):

N(2A, 3B, 5CD):

an involution in G, of class 2A. an outer automorphism of G, of class 2B. an outer automorphism of order 2 and unspecified class a group of order 3, with each generator in class 3A. a group of order 5, containing both classes 5A and 5B. a fourgroup, whose involutions are in class 2A. an elementary abelian group of order 27, whose 13 cyclic subgroups number 4 containing both classes 3A and 3B, 3 containing 3C only, 6 containing 3D only (neglecting the identity element). an elementary abelian group of order 26 , unspecified classes. an As, containing elements of classes 2A, 3B, 5C, 5D.

N(2A, 3B, 5CD)':

the direct product of two such As groups. N(2A, 2C, 3A, 3B, ...): a group containing elements of the indicated classes, among others. The type of the group being normalized, when it is not obvious, can usually be deduced from the structure information about H. In the case of a non-abelian simple group S that is treated elsewhere in this IS>.. 1r ILIS>..§, the conjugacy classes in G of the non-trivial cyclic subgroups of S are listed in the order in which the classes appear in the IS>.. 1r IL IS>..§ character table for S. The remaining specifications describe SUbgroups, where possible, as the stabilizers of objects that arise naturally in the various constructions for G and related groups. The appropriate column is headed by a keyword which hints at the construction(s) referred to. The entry against a group H hints at an object whose stabilizer is a copy of H. These stabilizers should always be understood in the widest possible sense, so that for instance the stabilizer of a 'vector' would be the set of all group elements that fix that vector up to scalar multiplication (its projective

xx

SUBGROUPS AND THEIR STRUCTURE

stabilizer). For this reason, we have a slight bias towards projective terminology in these columns. There are many generic involvements of small Chevalley groups S in greater ones G. Thus X n(q2) involves Xn(q), Amn-l(q) involves An-l(q) wr Sm, and B,(q) involves G,(q), which in turn involves both A 2(q) and 2 A,(q). We indicate these by writing just the name of S in the appropriate column of the maximal subgroup table for G, even though the actual maximal subgroup of G is often not just the group S, but a more complicated group involving S in its composition series. The structure column should supply the extra information.

Central and diagonal products may have more than two factors. The notation !(A x B) denotes a diagonal product in which D has order 2.

2. Notations for group structures There are many different ways in which structures of groups are indicated in the literature. We recommend the convenient ones below, and have used many of them in our tables of maximal subgroups and elsewhere in this AT ILA§. The notations c;,. (cyclic), D m (dihedral), Om (quatemionic, or dicyclic), Em (elementary abelian), and X m or X;;' or X;;, (extraspecial), are often used for groups of order m with the indicated structures. (The reader should note that some authors use D m and Om for the dihedral and quaternionic groups of order 2m rather than m.) The cyclic, dihedral, and quaternionic groups are defined by the presentations

c;,.: Dm:

a m =l aim = 1,

Om:

= 1,

a!m

b 2 = 1, b 2 = aim,

into the centres of A and B, and A 0 B is the quotient of A x B by the set of ordered pairs of the form (a(c), (3 (c» (c E C). When it is omitted from the notation, C is usually understood to take the largest possible value. ALB or ALaB indicates a diagonal product of A and B over their common homomorphic image D. Formally, we have homomorphisms y: A ~ D and 8: B ~ D from A and B onto D, and A L B is the subgroup ofAx B consisting of the (a, b) for which y(a) = 8(b). When D is omitted from the notation, it is usually understood to take the largest possible value.

b-lab = a- l

(m even)

b-lab = a-

(4 divides m),

l

and the elementary abelian and extraspecial groups are described later. The particular group O. is the quatemion group, presented by ij = k, jk = i, ki = j, or by i 2 = j2 = k 2 = ijk, in which the common value is -1, a central element of order 2. There are various ways to combine groups: A x B is the direct product, or Cartesian product, of A and B. It may be defined as the set of ordered pairs (a, b) (a E A, bE B), with (a, b)(a', b') = (aa', bb'). A . B or AB denotes any group having a normal subgroup of structure A, for which the corresponding quotient group has structure B. This is called an upward extension of A by B, or a downward extension of B by A. A : B or A:"B indicates a case of A. B which is a split extension, or semi-direct product. The structure can be completely described by giving the homomorphism cl> : B ~ Aut(A) which shows how B acts by conjugation on A. It may be defined to consist of the ordered pairs (b, a) (b E B, a E A), with (b, a)(b', a') = (bb', a"(b1 a '). A . B indicates any case of A . B which is not a split extension. AoB or AocB indicates the central product of A and B over their common central subgroup C. Formally, we have injections a: C ~ A and (3: C ~ B of the abstract group C

A &.B or A Oc La B can be used for a central-diagonal product. For suitable groups, and with notations as above, this consists of the ordered pairs (a, b) for which y(a) = 8(b) in D, taken modulo the pairs of form (a (c), (3(c» (c E C). A 1B, or A wr B, the wreath product of A and B, is defined when B is a permutation group on n letters. It is the split extension An: B, where A n is the direct product of n copies of A, on which B acts by permuting the factors. In our structure symbols for complicated groups, we use the abbreviations: [m] for m an integer, denoting an arbitrary group of order m m denoting a cyclic group of order m A n for the direct product of n groups of structure A. In particular pn where p is prime, indicates the elementary abelian group of that order.

Thus the two types of group of order four become 4 (the cyclic group),

and

2 2 (the fourgroup).

pm+n indicates a case of pm. pn, and so on. Thus 2 3 + 2 + 6 would indicate a 2-group having an elementary abelian normal subgroup of order 8 whose quotient contains a normal fourgroup modulo which becomes elementary abelian of order 2 6 • p'+2n or p~+2n or p:'+2n is used for the particular case of an

extraspecial group. For each prime number p and positive n, there are just two types of extraspecial group, which are central products of n non-abelian groups of order p3. For odd p, the subscript is + or - according as the group has exponent p or p2. For p = 2, it is + or - according as the central product has an even or odd number of quaternionic factors. Products of three or more groups are left-associated, and should only be used when the indicated decomposition is invariant. Thus A . B : C means (A· B): C, and implies the existence of a normal subgroup A.

Note to Chapters 6 and 7 The more complicated parts of these chapters are intended to be read in conjunction with one another. In these chapters, G usually denotes a simple group, with Schur multiplier M and outer automorphism group A. If '" is a root of unity, we shall sometimes say that a representation represents a certain group element by '" when we mean that it represents that element by '"

times an identity matrix. The same phrase is applied to the character of that representation. Two characters of M. G are said to belong to the same cohort if they represent the elements of M by the same roots of unity. The unqualified word 'class' usually refers to a conjugacy class in a group G or G. a, while 'character' usually refers to a character of a group G or m. G.

6 How to read this ATLAS: The map of an ATLAS character table With each character table printed in this A If"- A§ we give a small map that explains to which groups the various portions of the table refer. The shape of this map depends on the Schur multiplier M and outer automorphism group A. We describe some of the simplest and most common cases first.

4. The case M. G (arbitrary multiplier, trivial automorphism group) We illustrate the cases 4. G, 22 • G, and 6 . G:

1. The case G (that is, trivial M and A) The map is a square

(0),

c:J' r

where the right-hand number r is the number of characters of G, and the lower one the number of classes. Of course these two numbers are equal, and usually called the class number of G.

r

2. The case p. G (that is, M cyclic of prime order, A trivial) The map is a 'tower' of two squares

r

[:::1 El

(p. G):

r

The upper square gives the same information as before. The lower one refers to one cohort of characters faithful on the central element, and tells us that there are s characters in that cohort. Since there are p - 1 such cohorts, the class number of p. G is r + (p -1)s. The orders of the elements in these classes are visible in the lifting order rows for p. G.

3. The case G. P (M trivial, A cyclic of prime order) The map consists of two squares side by side:

r

In the first case (4 . G), there is a cohort of r representations in which the central element is mapped to + 1, and which we regard as representations of G, another cohort of s representations mapping that element to -1, which we regard as representations of 2 . G, and two cohorts of t representations each, mapping the central element to i or - i, which are faithful representations of 4 . G. The class numbers of G, 2. G, and 4 . G are r, r + s, and r+s+2t. In the second case (22 • G), no irreducible representation can be faithful, since the central fourgroup must be represented by a group of roots of unity, which is necessarily cyclic. The map is a tower of four squares, the topmost one corresponding to the representations that are trivial on the central fourgroup, and so can be regarded as representations of G, and the other three corresponding to the three cyclic quotient groups, labelled 2" 2 2 , 2 3 , of the central fourgroup. The class numbers of G, 2, . G, 2". G, 2,. G, and 22 • G are r, r+s, r+t, r+u, and r+s+t+u. In the third case (6. G), the central subgroup has four cyclic quotients, of orders 1, 2, 3, and 6, and so we again have a tower of four squares, corresponding to groups G, 2. G, 3. G, and 6 . G. The map tells us that a central element of order 6 is represented by 1 for r characters

(G. p):

-1 for s characters

cv or cv 2 for t characters each, r

s

This tells us that G has class number r, and that each other coset of G in G. p contains s classes of G. p. Since r - s characters of G fuse in sets of p, the class number of G. p is ps + (r - s)/p, but is easiest found from the indicator column for G. p, which reveals the number of characters for that group.

-cv or -cv 2 for u characters each,

where cv = exp(21TiI3). The class numbers of G, 2. G, 3. G, and 6. G are r, r+s, r+2t, r + s + 2t + 2u. The cohorts of characters representing the given central element by cv 2 and _cv 2 are not printed.

xxn

THE MAP OF A CHARACTER TABLE

In the general case M. G, there is a tower of squares, one for each cyclic quotient group (of order rn, say) of the multiplier. The number to the right of a square is the number of characters in any of the cohorts which realize exactly that quotient group-there are (rn) such cohorts, only one of which is printed. 5. The case G. A (trivial multiplier, arbitrary outer automorphism group) We illustrate the cases G. 4, G. 22 , G. 6, and G. S3:

(G.4):

[:]0 El r

(G.6):

(G. S3):

s

t

GB BEl ~ EJs EJ' 0 LJ

G.2

r s

G.3

the full A lr ILA§ table would have five detachments of columns, and a map:

c:JEJ EJ EJ EJ r

s

t

6. The general case M. G. A We illustrate three particular cases:

r-:'lGb, r-:lO, or LJ~ LJ~

r

s

G.2'UG . 2"

t

The 'dual' to the statement that any irreducible representation of M. G realizes only a cyclic quotient of M is the much more obvious statement that any conjugacy class of G. A is represented in some G. a, where a is a cyclic subgroup of A. Usually therefore an A lr ILA§ table contains a detachment of columns for each cyclic subgroup of the outer automorphism group A. Each 'square' of the map corresponds to a bicyclic extension rn. G. a (rn and a cyclic groups). The experienced user can often deduce information about non-bicyclic extensions, but we emphasize that 'officially'

t

However, the last three blocks are images of each other under the automorphism group, and so contain much the same information. We therefore only print one of them in full (the leader), and just the indicator and fusion columns for the other two (its followers).'In our maps, the 'squares' corresponding to follower cosets that have been suppressed in this way are 'squashed' in the way shown when we introduced this example.

~G6, LJL:J

BBs BFs B

GEJEJDD' ' , , , t

t

2'.01 _ _

2".01_7_ r

The first is the very common case 2. G. 2. The conventions about it are essentially the union of those for cases of form 2 . G and G. 2, and need little further explanation. However, it can be important to realize that any group 2 . G . 2 has an isoclinic variant, and of course the group 2. G . 2 you are investigating may not be the one that we print. For the definitions, see 'Isoclinism' (Section 7) at the end of this chapter. The A lr ILA§ tables for a group 2. G. 2 and its isoclinic variant (2 . G . 2)- have the forms

A 11" l A§ . tables give information only about bicyclic .extensions.

The first three of the above cases are the 'transposes' of the 'dual' cases 4. G, 22 • G, and 6 . G, and we shall say little more about them. In the first case we have r classes of G, s classes of G . 2 outside G, and 2t classes of G. 4 outside G. 2, but the exact class numbers of G. 2 and G. 4 are best obtained from the indicator columns for those groups. In the second case, we have r classes of G, and s, t, or u classes in the three extensions G. 2 outside G. The class numbers of these can be found from the appropriate indicator columns, but in general there is no way to read the class number of G . 22 from our table (it is not a bicyclic extension). The trouble is that a character of G that splits in (that is, extends to) each of the three extensions G. 2" G. 1" G. 2 3 may either split to give four characters of G. 2 2 or fuse to give just one, and there is no universal way to discover this from the character values. t This sort of problem does not arise in the third case, since the extending group is cyclic: a character which splits for G. 2 and G. 3 necessarily splits to give six characters for G.6. The fourth case, G. S3 introduces new phenomena since the extending group is non-abelian. There are five cyclic subgroups of S3, one of each order 1 and 3, and three of order 2, and so t When a character does extend to a group G. A (A non-cyclic) the character values we print for the various cyclic extensions should all be part of a single character for G. A. In the case G. 22 there is a simple rule which usually settles the question: if a real-valued (indicator + or -) character for G extends to characters a, {3, 'Y for each of G. 21> G .~, G. 2 3 then it extends to G. 22 if and only if an odd number of a, {3, 'Y are real-valued characters.

The orders of elements in the variant can be deduced from the statement that if g is an element of 2 . G. 2 outside 2 . G, with g2 = h, then in (2 . G. 2)- the corresponding element squares to - h, where -1 is the central element of order 2. If x, X+, x-are corresponding characters of 2. G, 2 . G. 2, and (2. G . 2)- respectively, faithful on the central element of order 2, then ind(x) = ind(x+) +ind(X-). Our second example is for. a group 3 . G. 2, in which a central element z of order 3 in 3. G is inverted by the outer automorphisms. In this case each character of 3. G in the printed cohort that represents z by w = exp(27Ti/3) fuses.in 3 . G. 2 with one from the unprinted cohort representing z by w 2 • It follows that all entries in the bottom right-hand portion of the table are zero. We therefore leave this portion blank, except for its fusion and indicator columns and lifting order rows, and represent this situation in our map by drawing only the leading two sides of the corresponding square. In such cases, the fusion column for 3 . G. 2 will contain symbols *, which indicate fusion of the unprinted characters with either their own proxies or the proxies of other unprinted characters, as indicated by the joins (see Chapter 7, Section 18).

s

xxiii

THE MAP OF A CHARACTER TABLE

The third example is for a group 2 2 • G. 3, in which the multiplier has three cyclic quotients of order 2, permuted by th~ automorphism group. The full A If ILA§ table would contain a detachment of rows for each of the three corresponding groups 2 . ',0, 2' . G, 2:' . G, and would have map:

(22 • G. 3):

c:JEJ B' B' B' r

t

We do not draw blocks 2. G. 3, 2' . G. 3, 2" . G. 3 since the corresponding groups do not exist. However, the lower three blocks in this full table are images of each other under the automorphism group, and so contain much the same information. We therefore print only one of them in full (the leader cohort), and just the lifting order rows for the other two cohorts (its followers). (The lifting order rows for triple or higher order multiple covers that have been suppressed in this way are printed in a compressed format-see Chapter 7, Section 9.) In our maps, the 'squares' corresponding to the follower cosets that have been suppressed in this way are 'squashed' in the way shown when we introduced this example. In a completely general case, the Full A If ILA§ Table (FAT) would contain a detachment of columns for each cyclic subgroup a of the outer automorphism group, and a detachment of rows for each cyclic quotient group m of the multiplier. If a transforms m into a different quotient group, there is no square on the map corresponding to m. G. a. If a acts non-trivially upon m, we print only two sides of the corresponding square, as in the case 3 . G. 2, since the corresponding portion of the character table has all character values zero. There can only be non-zero character values when a centralizes m, and we then draw a full square in our map for the portion m. G. a. In complicated cases, the table we print is usually smaller. The outer automorphism group will have various orbits on the cyclic subgroups of A or quotient groups of M, and from each orbit of the corresponding detachments of columns or rows we print just one in full, but retain only the indIcator and fusion columns, or lifting order rows, for its followers. The portions of our maps corresponding to the suppressed objects are 'squashed'. Of course in a complicated table we might have an orbit of r> 1 cosets, and also an orbit of s> 1 cohorts! In the cases we print, it happens that of the corresponding rs portions of the full A If IL A§ table all the ones that contain non-zero character values are equivalent under the automorphisms, and so we need only print one such portion in full. For an example of a case when these rs portions include several non-zero orbits under the automorphisms, see the parts labelled x and y in our map for OH3). There are occasions when cyclic upward and downward extensions G. a and m. G exist, and a centralizes m, but there is no corresponding bicyclic extension m . G. a. They are indicated in our maps by a 'broken box'

r-i iii.G.a

L.-J

which contains the name of a rather larger extension that does exist. We defer further explanations until we have explained the notion of isoclinism. 7. Isoclinism The commutator [x, y] = x-1y-1xy of two elements x, y of a group G is unaffected when we multiply x or y by an element of the centre Z(G) of G, and of course its values lie in the derived group or commutator subgroup G' = [G, G] of G (since this is defined as the group generated by all such commutators). We can therefore regard the commutator function as a map from G/Z(G)x G/Z(G) to G'. Two groups G and H are said to be isoclinic if they have 'the same' commutator map in this sense; in other words if there are isomorphisms G/Z(G)->H/Z(H) and G' -> H' which identify the two commutator maps. An equivalent and more concrete definition is to say that G and H are isoclinic if we can enlarge their centres so as to get isomorphic groups. This means that each of G and H can be embedded in a larger group K in such a way that K is generated by Z(K) and G, and also by Z(K) and H. Philip Hall introduced isoclinism in connection with the enumeration of p-groups. It is also of vital importance in the theory of projective representations and the Schur multiplier. Indeed, any irreducible representation of G can be converted to one of H of the same degree, for which the representing matrices are scalar multiples of those that represent G. To see this, restrict the representation to Z(G), when we obtain a set of scalar matrices to which we can adjoin others which extend this representation to one of Z(K), and hence extend the original representation of G to one of K. The desired representation of H is now obtained by restriction. Thus the dihedral and quaternionic groups of order 8 are isoclinic: ±

the

matrices

±

(~ ~),

±

(~ _~),

±

(~ ~),

(~ -~) represent the former, and by multiplying the last two

pairs of matrices by the scalar matrix

(~ ~)

we obtain a

representation of the latter. A similar phenomenon occurs with any faithful representation of a group H = 2 . G. 2. The representing matrices will be various matrices ±A (for elements of 2 . G), and ±B (for the remaining elements of H), and there is an isoclinic group K, (also of structure 2. G. 2) represented by the matrices ±A and ±iB. A group of structure 3 . G. 3 is one of three isoclinic variants, in which the elements in the three cosets of the subgroup 3 . G are represented by matrices w"A,w"B,wnC coriA, ewnB, e 2 w"C

wnA,

g2

w "B, ew"C,

where w = exp(21ri/3), e = exp(21ri/9). A perfect group (one for which G = G') has a unique central extension M. G realizing any quotient M of its Schur multiplier. Thus for example there is a unique proper cover 2 . An of the alternating group An for n;;;' 5. For more general groups G, the appropriate groups M. G are only unique up to isoclinism. Thus for example, the symmetric groups 5" (n;;;. 5) have two double covers, 2+5" and T5", which can be converted into each other by multiplying the elements that lie above odd permutations by a new central element i, whose square is the central element of order 2 in either group. The one we tabulate is 2+5", in which the transpositions of 5" lift to involutions. There is a sense in which the different portions of an A If IL A§ character table (or the squares of its map) correspond to isoclinism classes of bicyclic extensions, rather than to the individual extensions whose characters we choose to print. Changing to a different group in the isoclinism class merely multiplies the associated portions of a table by various roots of unity (and affects their headings). It can happen that upward and downward cyclic extensions

THE MAP OF A CHARACTER TABLE

xxiv

When such a case arises in this A. If IL A.§, we draw a broken box, for example:

G. a and m. G exist, and the automorphisms of a centralize m, but no corresponding bicyclic extension m. G. a exists. The relevant isoclinism class of extensions is well-defined, but happens to contain no group of the 'expected' order Iml.IGI.lal. We illustrate with the case G = A 6 , which has three cyclic automorphic extensions G. 2 , = S6, G. 2, = PGL 2 (9), and G. 2 3 = the 'Mathieu' group M lO • The Schur double cover 2 . G, is categorically associated with G, and so any automorphism of G can be lifted to an automorphism of 2. G. Now it happens that G. 2, and G. 2, are split extensions, from which it follows that there do exist groups 2 . G. 2, and 2 . G . 2, of the required type. To see this, take an involution x extending G to the appropriate G. 2,. Then the automorphism g ---> x- 1 gx of G lifts to an automorphism eX> say, of 2. G, and the split extension (2. G): 2 of 2. G by this automorphism is the desired group 2. G. 2, (i = 1, 2). However, the group G. 23 is a non-split extension of G. We can generate it by G together with an element x of order 4 with x 2 = y in G. It turns out that the automorphism ex of 2. G obtained by lifting the automorphism g ---> x- 1 gxinterchanges the two preimages of y in 2 . G. It is therefore impossible to achieve this by an element of order 4 (in a putative group 2. G. 2,) which squares to either preimage of y.

11 4.G.2 L~ in the appropriate place on the map, containing the name of a larger extension that does exist. In our example, this is a group 4 . G. 2 which can be obtained as the split extension of 2 . G by the automorphism ex of order 4 just described. (There is also an extension of structure 2 2 • G. 2 in this isoclinism class, but we prefer the bicyclic one.) It is slightly easier in this example to see that no enveloping group of structure 2 . G. 2 2 can exist. This is because the outer automorphism of S6 interchanges the two isoclinism classes of groups 2S6 • (It interchanges elements of cycle-types 2 ' 14 and 23 , and just one of these lifts to an involution in either 2S6 , and so neither of these groups can possess an automorphism of the required type.) We remark that the calculations needed to verify our assertions about this example can be performed in rL 2 (9).

Guide to the sections of Chapter 7 § 1. Column width and type markers ~;

§2. Centralizer orders

....

60 p power

§3. Power maps §4. p' parts §5. Class name row

pI part 1nd

§6. The rest of the detachment of rows for G.

The rest of the detachment of rows for 2~G

~

§8. Possible further detachments for downward extensions m':' G And possible lifting order rows for follows of these

~

lA

@

@

@

4 A A 2A

3 A A 3A

5 A A 5A

x, x,

+ +

3

-1

o -b5

x,

+

3

-1

0

x. x,

+

4

0

+

5

{ 1nd

1 2

5 A A B* fus 1nd

@

@

6 A A 2B

2 A A 4A

3 AB AB 6A

0

0

0

++

+

*

++

-1

0

0

++

4

3 6

5 10

2

0

-1

b5

*

x,

2

0

-1

*

b5

x.

4

0

-,

-1

X9

6

0

2

5 fus 1nd 10

0

8 8

6 6

0

0

0

00

0

0

13

00

0

~~

.,s-

~ ~'1>

'=>.

~~ ~

<;j "'. ~

~~

1

0

'---r--'

{ §19. Proxy classes and characters

------>

'-----v--'

•

'"<=

-8 -8

.a

.sa ~ r£'" u

"::s

S

8

'">

~

.<8'

." ,,0

-1

2

<= ....

..$''1> i:;.~

0

-1

<= <= S S ::s ::s

$'v 0"" C'> ~ f!

§18. The portion of table for a general bicyclic extension m -: G ~a

* -b5

-1

r

@

@

-1

§7. Lifting order rows for 2: G ~

§9.

@

'"....

"S c:J ......... ::s .£ <= ....

"u

~

~t>

'" ." '" ..c: U'" <=..c:

~

"'0

<= '"

~

" ......... 1i)e2~t>

~-ggS<=6' ""'

::S.-

~ ..... <'d ...... {/}

~..c:o"" u u '"

--'" _u~

N

.
Wo Wo

Wo

fi'"

<= . "S c:J.... ~ ..c: 0 <'I. g ..... c:J ~ '" . " S <= <'I ." " ::s." ..c:-<= ~

... = o~;>8~ (l;$

•

0<'1

" "",'" ..=~::sC3"O

0 ~~t> 0"::8

.... . u _ I-! '"

.....

,

.0

k

•

f-<

-

0u~ '"

....

1l.....""'-0 !:l

(/)

~

~{/}

g .. . '"

::s

<= 0 <= -0)-2

" S '" '" -..c:<="::B :SuS."

" , -0 ..c:::._

>- <= <= '" :0'- e " '" .- "'0 ::s 8 "'<=~ ~ 0 .... ~.,,'"

~

"

p.<=u~

0....0 ~.5.E~t~ "0 ._ 0 ......

=(/)- . . .

::s '" 0 u ..... l£~8~c:J
-

V)

0

Wo

Wo

t-=

Wo

7 How to read this ATLAS: character tables The Guide describes the parts of a typical A lr ILA§ character table. We shall first describe it row-by-row, and then columnby-column. Much of the complexity of this description arises from the need to consider groups with arbitrary Schur multiplier and outer automorphism group--the reader who is not interested in complicated cases should find that the first few sections suffice.

1. Column width and type markers The first row contains the symbol @ at the head of a column that later contains character values of some conjugacy class, and the symbol ; for one that contains fusion or indicator information. If the table is read mechanically, this row can be used to determine the widths of the columns, which may be necessary since occasionally the entries in adjacent columns abut. The information in any column is right-justified.

2. Centralizer orders The second and third rows give the order of the centralizer in G of the typical element of each class. Since the numbers are sometimes too large to fit in the appropriate portion of a single row, two rows are provided, the top one containing the leading digits of large numbers. The numbers given are the orders of centralizers in the base group G, even for columns that refer to classes in a larger group G.a.

3. Power maps The kth powers of the elements of a given class form another class. The resulting power maps between the classes can be found by repeated use of the prime power maps, the particular cases when k is prime. The fourth row gives the tag letters (see Section 5, 'Class names') of the classes that contain the powers gP, gq, g', ... of g, where p < q < r < ... are the distinct prime divisors of the order of g. Thus if g has order 60, an entry ABC means that g2 is in class 30A, g3 in class 20B, g5 in class l2C. If k is a number prime to the order of g, the class of gk can be found by applying the algebraic conjugacy operator *k to the class of g. It can happen that the pth power of an element of an automorphic extension G. a does not lie in any of our chosen generator cosets (see Section 16). In such cases the tag letter given is that of the proxy class. We do not give complete power map information for groups m. G and m. G. a. A good deal can be deduced from the orders of elements in those groups, and the power maps for the corresponding groups G and G. a.

4. p'-parts If 'IT is any set of primes, we can uniquely write

where 'IT contains all the prime divisors of the order of g('IT), but none of those of the order of g('IT'). The elements g('IT) and g('IT'), which are certain powers of g, are respectively called the 'IT-part and 'IT'-part of g. When 'IT contains a single prime p, we write g(p) and g(p') for these, and call them the p-part and p'-part of g. The 'IT-parts for general sets 'IT can be found by repeated use of the p'-parts. The fifth row of the table gives the tag letters for g(p'), g(q'), g(r'), ... , where p < q < r <. .. are the distinct prime divisors of the order of g. We do not give complete information on the p' parts for multiple covers m. G.

s.

Oass names

The conjugacy classes that contain elements of order n are named nA, nB, nC, .... The alphabet used here is potentially infinite, and reads ~B,~~~~~~~~K~~~~~~R

S, T, U, V, W, X, Y, Z, AI, Bl, ... , A2, B2, .... The class name row contains the following information:

The headings 'ind' and 'fus' for indicator and fusion columns; 'Master' class name entries, of form nX, meaning just that the column refers to a conjugacy class nX; ,Slave' class name entries, of form Y * k (or Y ** k, Y **, Y *), which refer to a class n Y, and also give the information that n Y can be obtained from the most recent 'master' class nX by applying the algebraic conjugacy operator *k (or **k, **, *). A 'master' class and the immediately subsequent 'slave' classes complete an algebraically conjugate family of classes. The algebraic conjugacy operators are defined on classes as follows: (nX).k contans the kth powers of elements of nX; (nX)••k contains the (-k)th powers of elements of nX; (nX)-- contains the inverses of elements of nX; and (nX). is the class other than nX containing elements of order n that are powers of elements of nX, when this class is unique. It is to be understood that k is prime to n. The values of characters on these classes are the images of their values on nX under the appropriate algebraic conjugacies.

6. Character values, for characters of G The typical row of the character table consists mostly of character values, together with indicator and fusion information that is described later. Usually the character values are ordinary integers, but certain algebraic irrationalities can also arise. In such cases we either print the A lr ILA§ name for the desired irrationality in full, or just an algebraic conjugacy operator by which it can be obtained from a nearby entry in the same row. To be precise, this nearby entry can be any entry for a class in the same algebraically conjugate family as the desired one that is printed in full. The complete A lr ILA§ system of names for algebraic irrationalities is given later (Section 10). The following should

CHARACTER TABLES

XXVI

suffice for simple cases:

rN=../N iN=i../N

zN = exp(27ri/N)

yN = 2 cos(27r/N)

~(-l+../N) (N== 1, mod 4) bN= ~(-l+i../N) (N==-I, mod 4)

{

mN = 2i sin(27r/ N),

where of course i denotes ../-l. Any of these algebraic numbers can be expressed as a linear combination of nth roots of unity for some n. If k is prime to n, the algebraic conjugacy operator *k acts on such numbers by replacing these roots of unity by their kth powers. The operator **k is an abbreviation for *(-k), and ** is the complex conjugation operator, which is *(-1). The operator * is defined for a quadratic number field, and replaces r + s../N by r - s../N (r, s rational, ../N irrational).

7. Lifting order rows for a group m. G

The detachment of rows for a generator cohort for which we print character values is headed by m lifting order rows as described in the previous section. Each subsequent row really represents (m) algebraically conjugate characters, the printed one being that from the generator cohort, which acts as proxy for the others. Whenever a row of an A If "- A§ table is used to represent M;;,,2 algebraically conjugate characters, the entry in its indicator column is + M, oM, -M according as the indicators (see Section 13) of these characters are +, 0, - (they are necessarily the same). So for m ;;"3, the characters we print for the generator cohort of a group m. G all represent (m) = M . characters, and are headed by the entry oM in the indicator column (their values include multiples of the non-real mth roots of unity). We illustrate all this by describing the generator cohort of characters for 3A 6 • The appropriate portion of the A If "- A§ table reads: ind

If G has a (cyclic) multiple cover m. G, then g will be the image of m distinct elements go, g" ... , gm-' of m. G. There will be some divisor d of m such that g, will be conjugate (in m. G) to gj just when i and j are congruent modulo d, so that go, g" ... , &i-I will represent the distinct conjugacy classes lying

above the class of g. For any such extension m . G for which characters are printed, we give m lifting order rows which contain the orders of the elements go, g" ... , &i-, (and by implication, the number d). There is a slightly different convention for cases when the characters are not printed (Section 9).

8. Character values for a group m. G We first describe the case m = 2. Let 0 be the central element of order 2. Any irreducible representation of 2 . G must represent o by I or -I, where I is the appropriate identity matrix, and in the former case we regard the representation as being one of G. In the latter case, the corresponding row contains the indicator of X, followed by its particular values X(go), the remaining values being deducible since X(g,) = -X(go) for these characters. Thus the character table of 2A s is: class lA o lA. 2 order 1

+ + + + +

1 3 3 4 5 2 2 4 6

1 3 3 4 5 -2 -2 -4 -6

2A o 4 1 -1 -1 0 1 0 0 0 0

3Ao 3 1 -0 0 1 -1 -1 -1 1 0

3A, 6

1 0 0 1 -1 1 1 -1 0

5A o 5 1 -bs -b! -1 0 bs

b! -1 1

SA, 10

1 -bs

-b! -1 0 -bs -b! 1 -1

5Bo 5 1 -b! -bs -1 0 b! bs -1 1

5B, 10

1 -b! -bs -1 0 -b! -bs 1 -1

In general, an irreducible character X of a central extension M. G of G restricts to M to give a scalar multiple of some linear (that is, degree 1) character X' of M. The set of X yielding a particular X' on M will be called a cohort of characters. The values of X' on M will be all the roots of unity whose order divides some m, and so the characters of the cohort can be regarded as characters of some cyclic extension m. G, a quotient of M. G. For each such cyclic extension m. G, we choose a particular cohort of characters X for which X'(O) = wI, where 0 generates the central cyclic subgroup of m. G, and w is a particular primitive mth root of unity. The chosen cohort will be called the generator cohort for m. G. A character X of m . G from a cohort with X'(O) = w,I, where w, is another primitive mth root of unity, is algebraically conjugate to a character from the generator cohort, which we shall call its proxy. (The precise rules relating characters to their proxies will be defined in Section 19.)

1 3 3

2 6 6

3

3

x"

02

3

-1

0

0

x"

02

3

-1

0

0

x"

02

6

2

0

0

x"

02

9

0

0

x"

02

15

0

0

-1

4

5 15 15

5 15 15

-b5

•

12 12

• -b5 0

-1

-1

-1

0

0

The entries in the lifting order rows give us the number of conjugacy classes (17) of 3A6 and the orders of the elements they contain. The characters of this cohort, if written out at length, would look like: class of g order of g values of X'4: XIS:

X'6: X17: X's:

lAo lA, 1 3 3 3w 3w 3 6w 6 9w 9 15 15w

4A o 4A. 4A2 5A o 4 12 12 5 ,;; -b 1 w s ,;; -b! 1 w 1 0 0 0 ,;; -1 1 w -1 -w -ill 0

lA 2 3 3,;; 3,;; 6,;; 9,;; 15,;; SA, 15 -bsw

2Ao 2 -1 -1 2 1 -1

2A, 6 -w -w 2w w -w

5A2 15 -b s';;

-b~w

-b!,;;

w -w 0

-w 0

,;;

2A2 3A o 3Bo 6 :? 3 -ill 0 0 -ill 0 0 2,;; 0 0 ,;; 0 0 -ill 0 0

5Bo 5 -b! -b s 1 -1 0

5B, 15 -b!w -bsw w -w 0

5B2 15 -b!,;; -b s';; ,;; -ill 0

where w = exp(27ri/3) is a primitive cube root of unity. The alternative cohort can be obtained by applying any algebraic conjugacy operation that takes w to ill, and the complex conjugacy operator ** is the nicest choice. The trivial cohort consisting of the 7 characters for G itself, supplemented by these two cohorts of 5 characters each for 3 . G, gives the entire set of 17 characters for 3 . G.

9. FoHower cohorts A rather larger table than the one we print is what we call the Full Atlas Table (FAT), which would contain a full detachment of rows corresponding to the generator cohort of each downward cyclic extension m. G of G. But for complicated tables the FAT table would be too fat, since there can be several cyclic covers m. G that are equivalent under the action of Aut( G). We thin it out by selecting only one (leader) extension m. G for each fanrily equivalent under Aut(G). The detachment of rows for the leader extension is printed in full, and followed by only the lifting order rows for the remaining (follower) extensions equivalent to it. To save space, the lifting order rows are printed in a slightly different format, needing only two rOWS. The top row, headed

CHARACTER TABLES

'and' gives the order of the canonical inverse images go of eleIP-ents g in G. (We always choose go to have the smallest ord~r of any inverse image of g.) The bottom row, headed 'no'. giv,,?s the number of distinct conjugacy classes of m. G represented by the inverse images of g (when this number exceeds I).

xxvii

Thus the A If L A§ table for As contains the 2 x 2 block of entries SA -bS

X2 X3

*

SA SB -b , which abbreviates -b~, X2 s * -bS X3 -b! -bs SB

or, in full, SA -!(-1+v'S) -!(-I-v'S)

10.; The A If LA§ notation for algebraic numbers Note. For greater legibility in this Introduction, we use subscripts and superscripts as in ordinary mathematical practice. However, our notation for algebraic numbers has been designed so that it can all be written on one line without loss of information. Thus what we here write as y~:s appears elsewhere in our tables and text as y"24 * 6. As usual, we let i denote a fixed square root of -1. Then for suitable integers N we define:

Iw=~

ZN

= ex:p(271"iINJ = z, say

(N~lmod2)

YN

= Z + zn = 2 cos(271"1 NJ

(N~lmod3)

XN=Z+Z"+z""

Zl4

(N~lmod4)

WN =

ZfS

(N~lmod5)

VN

=

Z

(N=l mod 6)

UN

=

z+Z ..

N-,

L z"

,=1

(n=n,=-l)

N-'

cN=l

L z"

(n = n3)

because in this instance * indicates the algebraic conjugacy that changes the sign of v'S, and we have bs=!(-I+v'S) by Gauss's theorem. In general, when only the name (*k, **k, **, or *) of an algebraic conjugacy appears as a character value entry, it is to be applied to an entry in the same row whose value is given in full, and which refers to a class in the same algebraically conjugate family as the desired one. Such families can be detected by examining the class name row (Section S). The conjugacy *k replaces nth roots of unity by their kth powers, whenever k is prime to n, and we have

!"'1

**k =*(-k)

N-,

dN=l

L

z+ zn + z"z+ Z,,3

(n = n.)

**=*(-1),

+ zn + Zll" + zn3 + zn4

(n = ns)

while * is only applied to a quadratic number field, and takes r+sv'N to r-sv'N (r, s rational, v'N irrational).

+ zn" + ... + zn s

(n = no)

(N~lmod7)

tN = z+ zn + zn" + .. . +Z"6

(n = n,)

(N= 1 mod 8)

SN = Z+

z"+ zn + ... + zn?

(n = ns)

1"'1

N-'

eN=!

fN=~

L

,-,

N-'

L z"

1'-'1

&J=~

N-'

L

zt?

!""1

N-'

hN=i

SB -!(-I-v'S) -!(-1+v'S),

L Zf ,-,

S

iN = i.JN

2

TN =.jN

mN = IN = kN

=

Z Z -

zn = 2(sin 27r/N)i

z" + Z,,2 - zn

z-z" +

iN=Z-Zn+

-

3 s

zn

7

zn

(n=n2=-I)

2qN+3&5-4&7+&9

means 2qN+3q~S-4qt'+qt"

(n = n4)

4qN&3&5&7-3&11

means 4(qN + qt' + q~S + qt') - 3q~"

(n = n6)

4qN* 3&5+&7

means 4(qt' + q~S) + qt'.

(n

=

n8).

It follows from a famous theorem of Gauss that

b _ {!(-1+v'N) N- !(-I+iv'N)

The ampersand convention On rare occasions, we have used another convention to abbreviate some very complicated irrationalities. If qN denotes any of our atomic irrationalities (e.g. y~), then linear combinations of algebraic conjugates of qN are abbreviated as in the following examples:

if

N= I, mod 4 if N=-I, mod 4.

The numbers n2, n3, ... , n8 are defined as follows: nk is the smallest integer in absolute value that has multiplicative order exactly k modulo N, the positive value being chosen when nk and -nk have the same order. (This definition ensures that n2=-1.) Further algebraic numbers

(Only much simpler examples actually occur!) To explain the 'ampersand' syntax in general we remark that' &k' is interpreted as q~\ where qN is the most recently named atomic irrationality, and that the scope of any premultiplying coefficient is broken by a + or - sign, but not by & or *k. The algebraic conjugations indicated by the ampersands apply directly to the atomic irrationality qN, even when, as in our last example, qN first appears with another conjugacy *k. 11. A note on algebraic conjugacy

-I, S, -S, 7, -7, 11, ... ,

Our algebraic conjugacy operators *k operate on a wide variety of concepts, always defined with respect to some number n prime to k: Algebraic numbers in the field of nth roots of unity-this is the primitive notion: the action was defined in Section 10. Elements g of order n: *k takes g to gk. The corresponding conjugacy classes: *k takes the class of g to that of gk. Cosets C of G in an upward extension of G, where Cn = G: *k takes C to C k • Characters defined in the field of nth roots of unity: * k acts on their values Cohorts C of characters faithful exactly on a cyclic quotient of order n of the multiplier: *k takes the cohort representing a given central element by w to that representing it by Wk.

Y~=Z24+Z~, y~=z24+z21, Y~=Z24+Z~, ....

We now turn to the column by column description of an A If L A§ table.

YN, YN,

y~,

... , XN, XN, ... , iJv, iN, ... ,

are obtained on replacing nk in the above definitions by n~, nk' ... , where

are all the numbers of multiplicative order exactly k modulo N, chosen in the order of preference

I, -I, 2, -2, 3, -3, 4, -4, .... Thus the numbers of exact order 2 modulo 24 are, in this order,

and so we have The irrational numbers that appear in A If L A§ character tables are named in terms of the atomic irrationalities we have just defined. However, when several neighbouring numbers in a row are algebraically conjugate, we often print only some of them in full, and for the others indicate only the algebraic conjugacy by which they can be obtained from their neighbours.

12. The character number column(s) In this column the rows containing characters are numbered x" X2, X3, ... for easy reference. In wide tables this column is repeated at the left and right edges of the separate pages.

xxviii

CHARACTER TABLES

The reader is warned that in complicated situations a single row may represent several characters.

of our example means that X. splits to yield two characters X~ and X~ of G. 2 = Ss, with the same values as X. on elements of As, and otherwise

13. The indicator column for G, and for m. G

X~(2B)=2,

X~(4A)=0,

X~(6A)=~l,

The Frobenius-Schur indicator function for a character X of a group H may be defined as

X~(2B)=-2,

X~(4A)=0,

x~(6A)= 1.

ind X =

I~I h~H X(h

Both X~ and X~ have indicator +.

2

The fusion case The other possibility is that two characters Xm and Xn of G may fuse to give a single character Xm,n of G. 2, with values

).

For an irreducible character X we have ind X = 0 if X takes any non-real value ind X = + 1 if X is afforded by a representation written over the reals ind X = -1 for other real-value characters. The indicator column uses the symbols 0, +, - for these three cases. The rows corresponding to a triple or higher order multiple cover m. G of G each represent c/>(m) distinct characters related by algebraic conjugation. The character values are printed only for one of these characters, which serves as proxy for the others. A symbol 02 or 04 or ... indicates that the row in being used in this way to represent 2 or 4 or ... characters, all having indicator o.

Xm.n(g) = Xm(g) + Xn(g), for elements of G Xm,n(g) = 0, for elements of G. 2 outside G.

We indicate this case by drawing a fusion join between dots against Xm and Xn in the fusion column, and then continuing only the first of these two rows into the G. 2 detachment, with the indicator of Xm.n in the indicator column, and the values (all 0) of Xm,n on elements of G. 2 outside G in the remaining columns of the G. 2 detachment. Any class of G on which Xm and Xn take distinct values fuses with the class on which those values are taken in the other order to give a single class of G . 2. Thus from our example we read

X2 X3

14. Columns for classes of G The first detachment of columns has a column for each conjugacy class of G. Against the typical character Xn of G it gives the value Xn (g) for g in the given class. The notation for algebraic irrationalities has been explained. Thus from the 3A column of our example As, we read the five character values

X, (g) = 1

ind lA 2A 3A 5A + 3 -1 0 -b5 + 3 -1 0

*

X.(g) = 1

giving the values of As characters at an element of order 3.

If G has a multiple cover m. G, then g will be the image of m distinct elements go, g" ... , gm-l of m. G. The rows of the table corresponding to characters faithful on m. G give character values only at go, since for these characters we have Xn (g,) = ",'Xn(gO), where'" is a primitive mth root of unity. (It follows that Xn(g.) = 0 unless there are m distinct conjugacy classes g, above g.) Thus in our example

-1}

X6(gO) = we read X7(gO) = 1 { X8(gO) = 0

1}

{X6(gl) = and deduce that X7(g,) = -1 , X8(g,) = 0

for the two inverse images go and g, of an element g of class 3A.

15. The detachment of columns for a group G. 2 There are simple relations between the characters of G and G.2.

The splitting case The first possibility is that a character X of G may extend to G. 2. It then necessarily does so in two ways, giving two characters XO and X' of G. 2, whose values on elements of G. 2 outside G are negatives of each other. In this case, we put the splitting symbol (:) in the fusion column, the indicators (+, -, or 0) of both XO and X' in the indicator column, and then the values of XO (only) in the columns for classes of G. 2 outside G. Thus the row

x.

ind lA 2A 3A 5A 5B fus ind 2B 4A 6A + 4 0 1 -1 -1 : ++ 2 0 -1

-b5

ind lA 2A 3A 5AB = 5A U5B 2B 4A 6A 6 -2 0 l=-bs-b~ 0 0 0

X2,3 +

The complete character table for Ss can now be read from our table. It is X~

xl X2,3

Xs(g) =-1

*

fus ind 2B 4A 6A + 0 0 0

and deduce that these two characters fuse to give a single character of Ss with values

X,(g) =0 X3(g) = 0

5B

X~ X~ X~ X~

ind lA 2A 3A 5AB 2B 4A 6A 1 1 1 1 1 + 1 1 1 1 -1 -1 -1 + 1 1 0 1 0 0 0 + 6 -2 -1 -1 1 2 0 + 4 0 1 -1 -2 0 1 + 4 0 0 1 -1 1 + 5 1 -1 -1 -1 -1 5 1 1 0 +

We remark that certain other symbols can appear in the fusion and indicator columns for a group m. G. a. They are explained in Section 18.

16. The detachment of columns for a group G. a,

a;;'

3

For an automorphic extension G. a, where a;;' 3, we choose in G. a one particular coset of G that generates G. a, and call this the generator coset for G. a. We do not print character values for the classes of G. a lying in cosets that generate G. a but are not the chosen generator coset, since each such class is an algebraic conjugate of a corresponding class in the generator coset, which we call its proxy class. The exact rules relating classes to their proxies will be described later. Classes lying in cosets that do not generate G. a are regarded as pertaining to smaller extensions G. a' (a'
The splitting case If a character X of G extends to G. a, it does so in a ways, giving a characters Xo, X" ... , X·- l of G. a, whose values at elements of the generator coset are related by

X'(g) = ",'XO(g),

for a primitive ath root of unity",. Such cases are indicated by the splitting symbol (:) in the fusion column, followed by the indicators for Xo, X" ... , X·- l (in that order) in the indicator

xxix

CHARACTER TABLES

coljlIl1ll, and then the values of XO (only) in the remaining coljlIl1lls of the G. a detachment.

2 . G. 2 appropriate to our example G = As: 1Ao lA, 2A o 3Ao 3A, 5ABo 5AB, 2B o 4Ao 4A, 6Bo 6B, 1 2 4 3 6 5 10 2 8 8 6 6

class:

Th~ fusion case It san happen that a characters Xm, Xn> •.• , Xp of G may fuse to give a single character Xm.n.....p of G. a, with values Xm + Xn + ... + XP 'on G and 0 elsewhere. In this· case, the fusion column coritains a line joining dots against those characters, and only the' earliest of the corresponding rows is continued further into the' G . a detachment of columns, by the indicator of X m. n. ...• p and its values (all 0) on the classes of the generator coset for G. a.

order:

+ + + + + + +

0

0

Mixed cases If a = a,a2 is composite, it can happen that each of a2 particular characters of G may extend, as in the splitting case, to give a, . characters of G. ab and that the resulting a, a2 characters fuse in sets of a2 to give a, characters of G. a = (G. a,) . a2. In this case our fusion column would contain a2 joined copies of the splitting symbol (:), the earliest of which would be followed by the indicators of the a, resulting characters and their values (all 0) on the ensuing classes. In the case a = 6, a, = 3, a2 = 2 this would look like: (G)

+

(G.2) 0 ... 0

(G.6) 0 ...0

o

+.where indicates a line of character entries. We can regard this either as one character of G. 2 splitting to become three of G. 6, or as three pairs of characters of G. 3, with each pair fusing to give a single character of G. 6. Note that the fusion column always refers back to G, and that : means that there has been splitting somewhere on the way from G to G. a, while the joins similarly indicate that there has also been fusion somewhere along the way. In the case a = 4, a, = a2 = 2, the table would look like: (G)

(G.2)

(G.4)

0 0

1 1 6 4 4 5 5 4 4 4 6 6

1 1 6 4 4 5 5 -4 -4 -4 -6 -6

1 1 1 1 1 1 -2 0 0 0 1 1 0 1 1 1 -1 -1 1 -1 -1 0 -2 1 0 1 -1 0 1 -1 0 0 0 0 0 0

1 1 1 -1 -1 0 0 -1 -1 -1 1 1

1 1 1 -1 -1 0 0 1 1 1 -1 -1

0 0 0

0 -i 3

i2

0 i3 -i 3 0 0

8

12

12

0 0 0 0 0

0 0 0 i2

-iz

-h

4

8

i3

0 0

However, you are just as likely to want the characters of the isoclinic variant, also of structure 2 . G. 2. To obtain this, multiply (or divide) all the numbers in the boxed portion by i; and use the new orders of elements given below the box, and the new indicators given to the right of it. For the definition of isoclinism, and the rules for computing these new orders and indicators, see Chapter 6, Section 7. A character of m. G for which we printed the indicator symbol oN is the chosen proxy from a set of N algebraically conjugate characters, as we described in Section 8. It is possible that each of these extends to give a characters of m. G. a (see 'the splitting case' in Section 15 or Section 16). In such a case, we put the symbol 00 ••• oN (with a symbols 0) in the indicator column for m. G. a. In general a string of k indicator symbols +, -, 0 followed by a number N indicates that the printed character yields k characters of the given indicators upon multiplication by linear characters of the appropriate group, and that each of these is proxy for a family of N algebraically conjugate characters which have distinct values when restricted to the central cyclic subgroup m of m. G. When the automorphism of the extending cyclic group a acts non-trivially on m, new symbols can also appear in the fusion column for m. G. a. The symbols

+: ++ " ++ 0 ... 0 +-:++and can be regarded as two pairs of characters of G. 2, each pair fusing to give a single character of G. 4.

17. Follower cosets We have already remarked that for any (cyclic) automorphic extension G. a, the corresponding detachment of columns of the A lr ILA§ table refers only to the classes in its chosen generator coset, since the values for the other classes of G. a can be deduced from these. If we printed one such detachment for each cyclic subgroup of the automorphism group of G, we should obtain what we call the Full Atlas Table (FAT) for G. Although it is useful as a theoretical standard, the FAT table is usually too fat. We have thinned it out by selecting only one (leader) extension G. a from each family of extensions equivalent under Aut(G). The detachment of columns corresponding to the leader extension is printed in full, and followed by only the indicator and fusion columns for the remaining (follower) extensions equivalent to it.

18. The portion of an A lr IL A § table for a bicyclic extension m. G. a For a bicyclic extension m. G . a in which the central cyclic subgroup m of m. G remains central, our conventions can best be described as the union of those for m. G and G. a. The reader should be able to read off the characters of the group

appearing in this column against characters XP and Xq from a given cohort C mean that XP fuses with the character in the algebraically conjugate cohort C" whose proxy is X.. The symbol *k in the fusion column against character X. means that XP fuses with the character in C' k whose proxy is XP itself. The algebraic conjugation symbol * k in this may be replaced by any of **k, **, or *. 19. The precise rules for proxy classes and characters We have already remarked that we print values for classes in only one of the cosets of G generating a given upward cyclic extension G. a, and for characters in only one of the cohorts that realize a given cyclic quotient of the multiplier. For most purposes it is unnecessary to select any particular correspondence between the retained objects and the omitted ones, which are merely their algebraic conjugates. However, a particular correspondence is used to give a precise meaning to our power maps and fusion markers. Suppose we want to find the classes of a coset C k for which a given class in the printed coset C serves as proxy, where cn = G. If g is any element of the given class, these classes are those of certain powers gk' of g, where it is only necessary to specify the numbers k' modulo the order N of the element g. In the vast majority of cases in this A lr IL A§, n divides 12, and we specify k' by the congruences:

k' == k modulo n, k'==±1 modulo powers of 2 and 3 dividing N, k'==+1 modulo any divisor of n prime to 2 and 3.

xxx

CHARACTER TABLES

The only other case that actually appears in this AT 0... A§ is for n = 5. In this case, we specify k' by k'==k modulo 5,

k' is of order dividing 4 modulo powers of 5 dividing N, k' == + 1 modulo any divisor of n prime to 5. These conventions are the simplest ones that ensure that the algebraic conjugation *k' has the correct effect on nth roots of unity, while fixing 'irrelevant' irrationalities. Any rule that extends this to all numbers n must be to some extent arbitrary. The rule for proxy characters is precisely similar. The characters in the cohort C' k for which a given character X from the cohort C serves as proxy are the algebraically conjugate characters X' k ', where k' is determined just as above from n, the order of the cyclic quotient of the multiplier realized faithfully by X, and any number N for which the values of X lie in the field of Nth roots of unity. 20. Abbreviated character tables There are some groups whose character tables seemed to deserve compression before we felt able to justify their inclusion. In these cases, indicated by the phase 'Abbreviated character table', we have printed just one representative from each algebraically conjugate family of classes or characters. An entry of form NID in the centralizer order row indicates that the corresponding class is the only printed member of an algebraically conjugate family of D classes, each of centralizer

order N. Such a family behaves in many ways like a single N

conjugacy class with centralizer order D . The entry in the class name row consists of the order n of the elements concerned, followed by the tag letters of the first and last classes in the family. The remaining entries in the appropriate column refer only to the first class in the family. So in our table for L 3 (8) (page 74), the column beginning @

3528/6 A A

7AF refers to class 7 A, and tells us that this class is the only printed representative of the 6 classes 7 A, 7 B, 7 C, 7D, 7E, 7F, all having centralizer order 3528. A row beginning +M, oM, or -M indicates that the printed character is the only printed representative of an algebraically conjugate family of M characters, all having the appropriate indicator + or 0 or -. Thus the last row of the table for L 3 (8) begins +3, indicating that the printed character is one of an algebraically conjugate family of 3 characters, all having indicator +. The complete table, except possibly for a small amount of the power map information, can be recovered by applying all possible algebraic conjugations to the printed classes and characters.

8 Concluding remarks We have given a great deal of thought to the arrangement and printing of character tables, and here pass on our observations to potential table compilers who wish to adopt our conventions.

1. Format Tables with columns of randomly varying widths have a 'ragged' appearance which is very wearying to the reader. In our tables the only permitted widths are 4, 6, 8, ... spaces. We have taken great pains, particularly with our irrationality conventions, to ensure that most columns will fit into the standard width of 4 spaces, so that large tracts of table have a regular layout. The even-width requirement entails that a wider column is seen to be wider----<:olumns whose widths are nearly but not quite equal producea slightly disorienting effect. The rows are usually at double line spacing, but have been compressed to one-and-a-half line spacing for a few large tables. The interline spaces dramatically increase legibility, and produce a pleasingly square appearance.

2. Ordering of classes and characters Many table compilers arrange the conjugacy classes in an order which to some extent reflects the power maps. Thus a class whose elements have prime order p may be followed by classes containing elements of orders 2p, 3p, etc., having powers in the original class. The merits of such arrangements are usually more evident to the compiler than the user, who is seldom properly informed about the principles (if any) of the arrangement, and so cannot use the implied power map information. It is better to choose a simpler arrangement, and explicitly indicate the power maps. In this A lr D...A§, the conjugacy classes within a given coset are arranged firstly, by increasing succession of n, the order of their elements; secondly, for elements with the same n, by decreasing succession of N, their centralizer order; thirdly, for elements with the same nand N, by increasing succession of d, the degree of the algebraic number field generated by their character values (so that rational elements come first); and fourthly, for elements with the same n, N, and d, in a manner which seems best compatible with the p' parts, so that, other things being equal, we prefer to arrange elements of order 10 in the same succession as the elements of order 5 that are their odd parts. Again, the characters within a given cohort are arranged firstly, in increasing succession of n, their character degree; secondly, for characters with the same n, in increasing succession of d, the degree of the number field generated by their values; thirdly, for characters with the same nand d, so that characters that fuse in some automorphic extension group are close together; and fourthly, so that the resulting blocks of algebraic irrationalities have a readily visible structure.

Of course any system of this kind has the merit that if two investigators use the same system for the same group, their character tables will be almost identical. [We do not think it would be sensible to make the system so precise that It would only permit one arrangement per group.] Our system tends to put the 'important' elements first, and makes it likely that the conjugacy classes in a subgroup will be in roughly the same order as those in the containing group. Similar remarks hold for characters. The ordering by degrees is useful in discussions of the possible decompositions of some compound character, and often makes it easier to identify the restriction of a character to a subgroup.

3. Some other 'virtuous' properties

When tabulating characters of a group M. G (or M. G. A) with a large multiplier, we have always arranged that for any given g, the displayed pre-images of g in all the cyclic extensions m. G are all images of a 'universal' pre-image of g in M. G. 'Dually', we have arranged whenever possible that the prolongations of a given character to all the cyclic extensions G. a are restrictions of a 'universal' character defined on G. A. There are still some choices that are more virtuous than others. For example in a group of structure 2 2 • G . 3, in which the three involutions of the normal fourgroup are permuted by the automorphism group, only one of the four pre-images of a given element can be invariant under an outer automorphism; this should be the printed 'universal' pre-image. 'Dually', in a group of structure G. 2 2 • 3 in which the outer automorphism group is A., a character of G which extends to G. 2 2 does so in four ways, only one of which can be invariant under the additional automorphisms; this character should be the printed one. In groups with complicated multiplier and automorphism group such desiderata can conflict, so we do not regard them as obligatory (and the reader cannot rely on them). But a table which satisfies them is more useful than one that does not, since there is a higher probability that the element the reader wants is exactly that which is tabulated.

4. The reliability of A lr D.. A § information With regard to errors in general, whether falling under the denomination of mental, typographical, or accidental, we are conscious of being able to point to a greater number than any critic whatever. Men who are acquainted with the innumerable difficulties attending the execution of a work of such an extensive nature will make proper allowances. To these we appeal, and shall rest satisfied with the judgement they pronounce. Preface to the first edition of the Encyclopaedia Brittanica, 1771

Our fairly wide experience has taught us that published character tables are as likely as not to contain errors (which are usually minor). We have been able to correct errors in our own tables up to a very late date before publication, but the rate of their discovery suggests that the above quotation will become apposite a few months after the A lr D...A§ appears in print. We first briefly comment on the information not contained in the character tables. Experts will know that one of the most common errors in group-theoretical papers is for the structure of

xxxii

CONCLUDING REMARKS

a group to be slightly mis-stated. We hope and believe that our own error rate in this respect is rather smaller than the average, but since the A lr ILA§ makes very many more assertions about group structures than any other work, we are certain that they will not all be exactly correct. We can only counsel the reader who feels he may have discovered an error to check the consistency of the statement he doubts with similar statements about subgroups or with analogous cases. Although we intended at first to subject all our tables to a large battery of computer tests, we must confess that in the event the only such test that has been consistently applied to every table is the orthogonality check. (There are many other mechanical tests that could be easily applied to ordinary tables, but for which the complexity of applying them to our compound tables has defeated us.) We are confident that the overall error rate in our character tables is very low indeed, but feel that the reader should be aware of the possibility that there are occasional errors, and of their most likely form. To put the problem in perspective, we emphasize again the very large amount of information that this A lr IL A§ contains. If, for example, we were to misprint one in every thousand digits, then the MONSTER character table alone would contain about 100 errors. Our actual error rate is a few orders of magnituqe smaller! If you suspect an error, perhaps the first thing to do is to check that you have not misunderstood the A lr IL A§ conventions, which are sometimes subtler than you may think. Then for: centralizer orders: check that the given number is the sum of the squares of the moduli of the entries in its column; power maps and p' parts: check the mutual consistency of these, for instance using (x 2 )' = (x 3?, and check the congruence X(g) == X(g') modulo p, whenever g' is the p' part of g, and X takes rational values on both. (For irrational values, the modulus is 1- zpo) lifting orders: check consistency with power maps-also the above congruences; indicators: the indicator should be 0 just if some character value is non-real; also, check the formulae and

~

R(g) =~_~ ind(x). X(g)

where R (g) is the number of square roots of g: fusion: check that G. a has the same number of classes as characters, also check compatibility with indicators. We have discussed these first since our experience suggests that it is this 'peripheral' information that is most likely to be at risk. The character values themselves are much safer.t Any error that remains in them probably concerns an irrational entry, or is an incompatibility between the tables for G and G. a. An extremely valuable test which we recommend when the above simple rules of thumb have failed is to compute the skew squares

of one or more characters X, and check their inner products with selected irreducibles. One should also check the consistency of restriction maps from groups to subgroups whose tables are also known. Of course it can be a good idea to check a doubtful point against another published table, which might well be the one from which ours was originally derived. A disagreement probably means that we believed we had found and corrected an error in the source table, but of course we could be wrong, or could have inadvertently introduced some error ourselves. If you do discover and confirm an error, please write to us about it! t Any complacency we might have had in this regard was rudely shattered when the pre-publication version of the table for the outer automorphism group of the Held group was found to contain an error affecting 22 entries (but obeying the orthogonality conditions)! A similar, but much smaller, error was discovered in the table for the outer automorphism group of J3 -

5. Acknowledgements This is perhaps the place for a comment about references and the bibliography. Our entries for the individual groups contain no explicit references to sources. This is because the few lines we allot to any given construction often contain material culled from many places (sometimes barely remembered) which might have been transformed beyond recognition by our own investigations or transliteratiQns. We intend no disrespect, and beg forgiveness for any offence our policy has caused. We gratefully acknowledge here that we have used information from many published and unpublished sources that are not explicitly referenced in this volume. The bibliography on pp. 243-51 is restricted to the 26 sporadic groups, and to a number of books and papers which give information about most or all of the infinite families of simple groups. The works by Gorenstein and Davis in the latter portion are very much more extensive bibliographies. It is a pleasure to record our indebtedness to many colleagues throughout the world who have freely sent us information. We mention particularly our major sources of character tables: M. Hall, Jr Zia-ud-Din A. O. Morris J. A. Todd D. Enomoto H. N. Ward B. Srinivasan J. McKay and D. Wales D. Gabrysch D. C. Hunt J. S. Frame, A. Rudvalis, and collaborators B. Fischer, D. Livingstone, M. Thorne

[the Tits group, 6A 6 , 6A7 , ••• ] [S,2, S,3] [2S.] [the Mathieu groups, U 4 (3), ...] [G 2 (q)] [2G,(q)] [S.(q)]

[3J3 : and other assistance] 2E6 (2), ...] [many, including the Baby Monster] [many, including 0.(2)] [F.(2),

[the Monster],

and finally the Aachen 'CAS' team led by J. Neubtiser and H. Pahlings both for many original tables and for improvements, extensions, and corrections to many others. Among our group-theoretical colleagues at Cambridge who have used the A lr ILA§ and contributed tables, corrections, improvements, or criticism are David Benson, Patrick Brooke, Mike Guy, David Jackson, Gordon James, Peter Kleidman, Martin Liebeck, Nick Patterson, Larissa Queen, Alex Ryba, Jan Saxl, Peter Smith, and finally John Thompson, who has acted as our friend and mentor throughout. We owe a particular debt to Jonathan Thackray, who designed and wrote the computer system in which our compound character tables are held. Many errors were discovered and eliminated as a result of his work. We thank Leonard Soicher for his valuable work which produced the Coxeter-type presentations displayed for many groups in this A lr ILA§. The authors' names appear on the cover of this A lr IL A§ in what is both the alphabetical order and the chronological order of their involvement with the A lr ILA§ project. Here are some comments about their individual roles: Robert Curtis and I started the work by designing the A lr ILA§ format and converting many pre-existing tables into it. Curtis also made many investigations into the subgroup structure and primitive permutation representations of various groups, not all of which work has been used in this volume. Simon Norton constructed the tables for a large number of extensions, including some particularly complicated ones. He has throughout acted as 'troubleshooter'-any difficult problem was automatically referred to him in the confident expectation that it would speedily be solved. Richard Parker was responsible for the initial 'mechanization' of the A lr ILA§ project, and also did a great part of the more tedious job of entering pre-existing tables into the computer. He has also computed a large number of modular character tables, intended for a later A lr ILA§ publication.

CONCLUDING REMARKS

Robert Wilson has investigated, and often completely enumerated, the maximal subgroups of a large number of A If D...A§ groups. He has greatly increased the usefulness of this A If D.. A§ by adding this and other information, and over the last few years has cheerfully shouldered the enormous task of gathering and transforming our untidy heaps of material into a form fit for ppblication.

xxxiii

My own function was to initiate and control the entire project, to collaborate with each of the above, and (eventually) to write this Introduction. John Conway November 1984 Department of Pure Mathematics and Mathematical Statistics, University of Cambridge

THE GROUPS

Alternating group A5 ; Linear group L2 (4)

= A1(5) = U2 (5)

Linear group L2 (5) Order

=A1(4) =U2(4)

- S2(5)

= 60 = 22. 3 •5

Mu1t

- S2(4) - 03(4) -

0~(2);

= 03(5) =2

OUt

=2

Constructions Alternating

Linear (4)

S5

= G.2

: all permutations of 5 letters;

AS

=G :

all even permutations; 2.G and 2.G.2 : the Schur double covers

GL 2 (4) _ 3 x G : all non-singular 2 x 2 matrices over F.lj.;

SL 2 (4) - PGL 2 (4) Unitary (4)

= PSL 2 (4) = G;

rL 2 (4)

= (3

x G).2; PrL 2 (4)

= IL2 (4) = PIL2 (4) = G.2

= 5 x G : all 2 x 2 matrices over F 16 preserving a non-singular Hermitian form; PGU 2 (4) = SU 2 (4) = PSU 2 (4) = G G0 3(4) = PG0 3(4) = S03(4) = PS0 3 (4) = 03(4) = G : all 2 x 2 matrices over F 4 preserving a non-singular quadratic form; r0 3 (4) = pr0 3 (4) = I03 (4) = PI03 (4) = G.2 GO~(2) = PGO~(2) = SO~(2) = PSO~(2) = G.2 : all 4 x 4 matrices over F 2 preserving a quadratic form of Witt defect 1, for example xf + x1x2 + x~ + x3x4; 04(2) = G GL 2 (S) = 2.(G x 2).2: all non-singular 2 x 2 matrices over F 5 ; PGL 2 (5) = G.2; SL 2 (5) = 2.G; PSL 2 (5) = G

GU 2 (4)

Orthogona1 (4)

Orthogona1 (2)

Linear (5)

Unitary (5)

GU 2 (S)

=3

x 2.G.2 : all 2 x 2 matrices over F 25 preserving a non-singular Hermitian form;

Orthogonal (5)

00 3 (5)

=2

x G.2 : the 3 x 3 matrices over F S preserving a non-singular quadratic form;

Quaternionic

2.G is the group of those quaternions q for which the coordinates of 2q are (±2,O,O,O)A, (±1,±1,±1,±1), (O,±1,±bS,±bS*)A; these generate the icosian ring

G x 2 : symm€tries of the vectors (O,±1,±bS*)C (vertices of icosahedron), or of (±1,±1,±1),

Icosahedral

(O,±b5*,±bS)C (dodecahedron), or of (±2,O,O)C, (±1,±b5,±bS*)C (icosidodecahedron: the reflections in these generate G x 2). These vectors are obtained by deleting the real parts of the above quaternions. G is the group of symmetries of determinant 1. Presentations

= <2,3,5>; G = (2,3,5) = G3,5,5 = <X1,x2,x3Ixi=(XiXj)2=1>; = (2,4,5;3) = (2,5,6;2); G x 2 = 5 = G3,S,10; G.2 x

2.G G.2

Index

12

5

G.2

Structure

Character

10 6

Orthogona1 (2)

1a+5a

N(5AB)

1a+4a+S a

N(3A)

Linear (5)

isotropic point

Linear (4)

Orthogona1 (4)

point

point

isotropic point

°2 (4), L1(16)

minus line

0"2(4), base

plus line

duad

Orthogona1 (5)

Icosahedral

base

base

point

isotropic point

pentad axis

non-isotropic point

°2 (5), L1(25)

minus point

triad axis

@

@

@

@

@

4 3 5 5 A A A A A A A A 2A 3A 5A B* fus ind

+

@

@

623 A A AB A A AB 2B 4A 6A

++

3 -1

o -b5

°

x,

+

3 -1

x.

+

4

x,

+

5

ind

1 2

° -1

4

3 6

2

x,

2

° -1 ° -1

x.

4

o

6

° °

*

+

* -b5 -1

x,

x.

Alternating

02( 4)

60 p power p' part ind 1A

[2]

Abstract

A4

@

= G4,S,6

Specifications

Maximal sUbgroups Order

2

-1

++

° ° ° 2 ° -1 -1

++

°5 °5 fus ind

8

6

10

10

8

6

b5

*

*

° °

b5

-,

-1

2

00

o

00

0

o i2

0

i3

°

Order = 168 = 2 3 .3.7

Mult = 2

=2

Out

Constructions

Linear (2)

= PGL 3 (2) = SL 3 (2) = PSL 3(2) = G :

GL 3 (2)

all ncn-singular 3 x 3 matrices over F 2 ;

the points and lines of the corresponding projective plane can be numbered (mod 7) SO that Li = {Pi+1,Pi+2,Pi+ij}; G.2 is obtained by adjoining the duality (graph) automorphism Linear

=3 x 2.G.2

GL 2 (7)

(7)

= G.2;

PGL 2 (7) Unitary

= 2.(G

GU 2 (7) PGU 2 (7)

Orthogonal

: all non-singular 2 x 2 matrices over F 7 ;

SL 2 (7)

= 2.G;

PSL 2 (7)

=G

x ij).2 : all 2 x 2 matrices over Fij9 preserving a non-singular Hermitian form;

= G.2;

SU 2 (7)

= 2.G;

PSU 2 (7)

=G

G0 3 (1) :: 2 x G.2 : all 3 x 3 matrices over F

PG0 3 (7) Reflection

G

x

= S03(7) = PS0 3 (7) = G.2;

0 3 (7)

7

preserving a non-singular quadratic form;

=G

1\~7 whose minimal (root) vectors are

2 : syrnmetries of the lattice

(±2,O,O)S, (O,±b7,±b7)S, (±1,±1,±b1")S;

the group is generated by reflections in these vectors; G : the symmetries of determinant 1

2G : the symmetries of the set of 8 (±)vectors (where b=b7, p=i7) :

Vectors

vco=(p;O,O,O) vO=(-1;b,b,b) v1=(-1;b,O,b 2 ) v2=(-1;O,b 2 ,b) v3=(-1;2,b,O) vij=(-1;b 2 ,b,O) v5=(-1;O,2,b) v6=(-1;b,O,2);

Presentations

these generate the laminated lattice A q ,b7' whose full automorphism group is 2A 7 G = (3,3Iij,ij) = (2,3,7;ij) _ <s,Tls7 =(SijT)ij=1,(ST)3=T 2 [=1J>;

G.2

= G3,7,8 = (2,3,8;ij);

= G3,8,8

2 x G.2

Maximal subgroups

Specifications Character

Abstract

Linear (2)

2ij

1a+6a

N(2A 2 )

2ij

1a+6a

N(2A 2 )

1a+1a

N(7AB)

Order

Index

G.2

Structure

21

7:6

@

168 p power p' part ind 1A Xt

@

@

@

@

@

@

8 3 ij 7 7 A A A A A A A A A A 2A 3A ijA 7A B** fus ind

+

Linear (7)

Orthogonal (7)

Reflection

point

base

b1-congruence base

line

base

b1**-congruence base

point

@

6 3 A AB A AB 2B 6A

@

isotropic point

@

ij ij A A A A 8A B*

++

o

3 -1

o

b7

..

o

3 -1

o

..

b7

+

620

-1

-1

Xs

+

7

-1

x.

+

8

0

-1

o

ind

1 2

ij

3 6

8

7

8

1ij

X7

o

ij

0

o

x,

o

ij

0

x.

6

0

0

r2

-1

-1

o

Xw

6

0

0 -r2

-1

-1

o

0

Xl1

8

0

o

r3

o -1

-1

0

!

o

+

0

000

++

o

0

r2 -r2 -1

++

-1

++

2

-1

o

0

7 fus ind 1ij

ij

12 12

16 16

16 16

-b7

..

o

0

0

0

o ..

-b7

,0 y16

*3

o

!

6

4

*5 y16 0

0

[3]

Alternating group A6 ; Linear group L2 (9)

=A1(9) = U2 (9)

- 3 2 (9) - 03(9) - 0q(3);

= C2 (2)' = ° 5 (2)'

Derived symplectic group 34 (2)'

Mult = 6 Constructions

Al ternating

3 6 :: G.2 1 : all permutations on 6 letters; A6 :: G : the even permutations; 2.G and 2.G.21: the SChur double covers The outer autornorphism of 36 interchanges the 15 duads ab (pairs) with the 15 synthemes ab.od.er (trisections), so interchanges the 6 points a (Which could be specified by 5 duads ab, ac, ad, ae, af) with the 6 totals (sets of 5 synthemes containing all duads)

Linear

=2.(G

GL 2 (9) PGL 2 (9) rL 2 (9)

x 4).22

all 2 x 2 non-singular matrices over F 9 ;

= G.2; 3L 2 (9) - 2.G; P3L 2 (9) = G; = 2.(G x 4).22; prL 2 (9) = G.2 2 ; ~L2(9) = 2.G.2 1 ;

P~L2(9)

= G.2 1

= 5 x 2.G.2 2 : all matrices over F S1 preserving a non-singular Hermitian form; PGU 2 (9) = G.2 2 ; 3U 2 (9) = 2.G; P3U 2 (9) = G G0 3 (9) = 2 x G.2 2 : all 3 x 3 matrices over F 9 preserving a non-singular quadratic form; PG0 3 (9) = 303(9) = P30 3 (9) = G.2 2 ; 03(9) = G; r0 3 (9) = 2 x G.2 2 ; pr0 3 (9) = ~03(9) = PI0 3 (9) = G.2 2

Unitary

GU a (9)

Orthogonal (9)

Orthogonal (3)

GOii(3):: 2 x G.2 1 : all 4 x 4 matrices oyer F preserving a quadratic form of Witt defect 1, for example 3

Xl2 + x2 +2 x 3 - x4;2 PGO q (3)2 = G.2 1 ;304(3)

=~

x G; P30 q (3)

= 0q(3) = G;

by taking the 4-space of the vectors (xi) in F~ with IXi=O, modulo (111111), we see the isomorphism with 2 x 36

= PG0 5 (2) = 305(2) = P30 5 (2) = G.2 1 form; 05(2) = G.2,

Orthogonal (2)

G0 5 (2)

3ymplectic

5 4 (2) :: G.2 1

Mathieu

M,O

: all 5 x 5 matrices over F 2 preserving a non-singular quadratic

all 4 x 4 matrices over F 2 preserving a non-singular symplectic form;

= G.2 3

the stabilizer of a point in M11 , or two points in M12 ; in M24 the subgroups of M10 .2 G.2 2 have orbits as follows:

=

G (',1,10;6,6), G.2 1 (2,10;6,6), G.2 2 (2,10;12), G.2 3 3. G : the monomial symmetries of the code over F 4

Hexacode

= M10

(1,1,10;12)

= {O, 1,w ,W'}

whose 64

= 36+12+9+6+1

(Oa Oa be) (36), (be be be) (12), (00 aa aa) (9), (aa bb cc) (6), (000000)

(1)

words are obtained from

(in which a,b,c is any cyclic

permutation of 1,w,w) by bodily permuting the 3 couples, and/or reversing any 2 of them;

the words of weight 6 (total words) are permuted (modulo scalar factors) like the 6 totals of 5 disjoint synthemes

2 x 3.G : the symmetries of the (~w)vectors (~2,O,O)3, (~l,~l,~n)S, (O,~h,~h)3, (~,,~wc,~Wb)3 (where w = z3,

Reflection

b = b5, c = b5*, h = z3 - b5); the group is generated by the 45 reflections in these vectors 2.G : the symmetries of the (i,w)vectors (6,0), «i+l)wa ,wb ), (wa ,(i_l)wb ) where 6 and i are quaternions with

Quaternionic

e2

= -3,

i2

= -1,

ei

= -ie,

and 2w

= -1+6;

the group is generated by the 20 quaternionic reflections in these

vectors - these reflections have order 3

= <3,4,5;2) G.2, G

Presentations

(5,51 2 ,4)

= <X"x2,x3,X4I X{=(Xi Xj)2=1>; ; 2 x G.2 2 = G4 ,5,S

Maximal subgroups Structure

G.2 1

G.22

G.23

G.22

Character

Abstract

Alternating

6

A5

35

D20

5:4

10:4

la+5a

N(2A,3A,5A)

point

60

6

35

la+5b

N(2A,3B,5A)

total

36

10

A 5 32 :4

2 3 : DS

32 :S

2 3 : QS

32 : (2 4 ]

la+9a

N(3 2 ) = N( 3A2B2)

bisection

24

15

34

34 x 2

D16

S:2

(2 5 ]

la+5a+9a

N(2A 2 ), C(2B)

duad

24

15

34

34 x 2

la+5b+9a

N(2A 2 ), C(2C)

syntheme

Order

Index

60

Linear (9)

[4J

Specifications

Orthogonal (9)

Orthogonal (3)

Orthogonal (2)

3ymplectic

Mathieu

Hexacode

icosahedral

minus plane

0q(2) , 32 (4)

total

coordinate

icosahedral

minus plane

0q(2), 32 (4)

total

weight 6 word

point

isotropic point

isotropic point

plus plane

01;(2)

point

L2 (3)

°3(3) , base

non-isotropic point

isotropic point

point

tetrad

weight 4 word

L2 (3)

° 3 (3) , base

non-isotropic point

isotropic line

isotropic line

tetrad

2 weight 6 words

@

360 power p' part ind 1A

p

@

@

@

@

@

@

8 A

9 A

9 A

4 A

5 A

5 A

A

A

A

A

A

A

3A

3B

2A

4A

5A

@

B* rus ind

@

+

X2

+

5

2

-1

-1

0

0

++

X,

+

5

-1

2

-1

0

0

++

-1

3

+

8

0

~1

-1

0 -b5

*

+

0

0

Xs

+

8

0

-1

-1

0

*

-b5

X.

+

9

-1

-1

++

3

3

+

10

-2

000

++

2

-2

ind

1 2

4

3 6

Xs

4

0

-2

X.

4

0

XlO

8

0

Xl1

8

0

10

0

r2

0

0

10

0

-r2

0

0

1

2

3 3

6 6

X13

ind

@

24 24 4 3 A A A AB A A A AB 2B 2C 4B 6A

X,

-

@

@

@

3 BC BC 6B rus ind

0

4 5 5 A BD AD A AD BD

2D

8A

3-1

0

-1

o

-1

-1

0

0

0

0

0

0-1

0

++

2

0

0

b5

*

+

0

0

0

++

2

0

0

*

b5

++

-1

-1

-1

++

0

r2 -r2

-1

-1

++

0

0

00

0

i2 -i2

20 rus ind 20

4

16

16

10 rus ind

4 12 12

8 24 24

8 24 24

02

0

0

0

+

002

0

+

002

+

002

20 20

0

-1

-1

o

0

0

0

r3

o

0

0

0

0

-2

0

-1

-1

0

0

0

i3

0

-1

-1

0 -b5

*

o

0

0

0

0

o

0

0 y20

*3

-1

-1

0

o

0

0

o

y16

*5

0

0

o

*13 y16

0

0

8

10

3-1

o

0

XIS

02

3-1

o

0

X,.

02

6

0

0

Xl7

02

9

o

0

X18

02

15

-1

0

0

ind

1 6 3 2 3

4

3 6

3

6

5

15 15

-b5

*

5 rus ind

2

2

4

6

-1

-1

* *

-1

0

0

*

8 24 24

5 30 15 10 15 30

0

8

+

+

+

5 rus ind 30 15 10 15 30

r2

0

6 rus ind

* * * * *

* -b5

24 24

6

* -b5

15 15

2

4

8

6

6

2

8

*7 y20

+ +

12 rus ind 12

4

16 16

16 16

20 20

20 rus ind 20

*

1

*

7

X,.

02

6

0

0

0

Xw

02

6

0

0

0 -r2

X21

02

12

0

0

0

0

b5

*

*

X"

02

12

0

0

0

0

*

b5

*

8C DU

0

16 16

4

4C

0

16 16

12 12

A A

+

4

0

A A

000

12 rus ind 12

0

A A

0

6 6

3

244

0

8

0

@

+

4

o

@

++

2

00

@

B* rus ind

5 rus ind 10

02

12 12

4 A A

B* lOA

@

5 10

X14

2

A A

10

@

8 8

3

3 6

@

++

++

o

@

I

4 12 12

i2 -i2 -1

-1

16 48 48

16 48 48

04

04

7

(*) There is 110 group 2.0.21 or 6.0.2).

See the Introduction: 'Isoclinism'

[5J

R(3)'

Mult :

OUt :

3

Constructions Linear

GL 2 (8) ;; 7 x G : all non-singular 2 x 2 matrices over F 8 ;

Unitary

GU 2 (S) :: 9 x G : all 2 x 2 matrices over F 64 preserving a non-singular Hermitian form;

PGU 2 (8) ;; SU 2 (8) - PSU 2 (8) ;; G Orthogonal

G0 3 (8) ;; PG0 3 (8) - S03(8) ;; PS0 3 (8) ;; 03(8) ;; G : all 3 x 3 matrices over F 8 preserving a non-singular quadratic form; r0 3 (8) ;; pr0 3 (8) ;; !03(8) ;; P!03(8) ;; G.3

Ree

R(3) ;; 2G2 (3) ;; G.3 : the centralizer in G2 (3) of an outer automorphism of order 2; the group acts doubly transitively on 33+1 : 28 points

Presentations

G;; G3,7,9 ;;

Maximal subgroups

Order

Index

56

Specifications

Structure

G.3

Character

Abstract

Linear

Orthogonal

9

23 :7

23 :7: 3

1a+8a

N( 2A 3)

point

isotropic point

18

28

°18

9:6

1a+9abc

N(3A)

14

36

°14

7:6

1a+8a+9abc

N(7ABC)

@

@

@

504 p power pi part

ind

[6]

1A

@

8 A A 2A

x,

+

X2

+

7

-1

x,

+

7

-1

x.

+

7

-1

Xs

+

7 -1

x.

+

8

X7

+

9

x.

+

9

x.

+

9

@

@

@

@

@

°2(8), L1(64) °2(8), base

@

9 7 7 7 9 9 9 A A A A A A A A A A A A A A 3A 7A B*2 C*4 9A B*2 C*4 fus ind

@

minus point

Ree

point

plus point

@

6 2 3 A A BA A BA A 3B 6A 90

+00

0

-2

0

°0 ° °0 0 °0 -y9 *2 *4 -y9 0

0

-1

° Y7 ° *4*2 °

°1

*2

*4

y7

*2

*4 y7

-1

+00

*4

+

*2

*2

*4 -y9

-1

-1

-1

°0 ° 00 ° 0 ° °

° °

0

+00

2

0 -1

+

0

0

0

~B9

9

3

Order = 660 = 2 2 .3 .. 5.11

Mult = 2

OUt = 2

Constructions

Linear

GL 2 (11) ;; 5 x 2.G.2 : all non-singular 2 x 2 matrices over F 11 ;

Unitary

GU 2 (11) ;; 3 x 2.(G x 2).2 : all 2 x 2 matrices over F 121 preserving a non-singular Hermitian form; PGU 2 (11) ;; G.2; 3U (11) _ 2.G; P3U 2 (11) ;; G 2

Orthogonal

00 (11) ;; 2 x G.2 : all 3 x 3 matrices over F 11 preserving a non-singular quadratic form; 3 PG0 (11) ;; 30 (11) ;; P30 (11) - G.2; 0 (11) ;; G 3 3 3 3 G : the automorphism group of a system of 11 "points" and 11 "lines" which can be numbered (mod 11) so

Biplane

that Li = {Pi+1,Pi+3,Pl+9,Pi+5,Pl+41; any pair of points determines two lines; any pair of lines determines two points; G.2 is obtained by adjoining duality Mathieu

G : the stabilizer of a point in the 12-point representation of M11 ; i.e. the stabilizer of a point and a

Presentations

total in M12 ; becomes G.2 by adjoining an automorphism interchanging points and totals b 5 c d G E G5 ,5,5 E < a

Specifications

Maximal subgroups Order

Index

60

11

60

11

55

12

Structure

660

P power p' part ind 1A

@

@

@

N(2A,3A,5AB)

lcosahedral

point

1a+10b

N(2A,3A,5AB)

icosahedral

total

1a+11a

N(11AB)

point

isotropic point

1a+5ab+10bb+12ab

N(2A), N(3A)

O2(11), L1 (121)

minus point

Abstract

3 4 , D20

1a+10b

12

@

Mathieu

Character

11 : 10

11:5

Orthogonal

G.2

@

@

@

@

12

6 5 5 6 11 11 A A A AA A A A A AA A A A A 2A 3A 5A B* 6A 11A B** fus ind

A

+

o o

0

0

** b11

-2

o

0

-1

10

2

o

0

-1

+

11

-1

+

12

o

o

b5

+

12

o

o

*

ind

1 2

o

5

-1

x,

o

5

-1

+

10

+

x,

-1

b11

2B

6 5 A BB A AB 4A 10A

@

@

@

566 AB AA AA BB AA AA B* 12A B*

-1 -1

o

0

0

0

000

-1

++

0

2

0

0

-1

-1

++

0

0

0

0

r3 -r3

0

++

-1

-1

-1

-1

* o

++

2

o

b5

*

0

0

b5

o

++

2

o

*

b5

0

0

355 6 10 10

12 12

11 fus ind 22

4

8 20

20 20

24 24

24 24

**

11 22

o

6

0

0

0-b11

x,.

o

6

0

0

o

**-b11

Xu

-

10

0

-2

10

-

A A

@

+

**

x.

x"

10

@

duad

++

X2

x.

@

Linear

8

20

o

0

0

000

0

0

0

-1

-1

o

r2

0

0

r2

r2

0

o

0

r3

-1

-1

o

r2

0

0-y24

*7

10

0

o

0 -r3

-1

-1

o

r2

0

0

12

0

0

b5

*

0

o

0 y20

*3

0

0

12

0

0

*

b5

0

o

0

*7 y20

0

0

8

6

*7-y24

[7]

Out = 2

Mult = 2 Constructions Linear

=3

GL 2 (13)

= G.2;

PGL 2 (13) Unitary

x 2.(G x 2).2 : all non-singular 2 x 2 matrices over F 13 ;

=7

GU 2 (13)

=G

P3L 2 (13)

x 2.G.2 : all 2 x 2 matrices over F 169 preserving a non-singular Hermit1an form;

= G.2;

PGU 2 (13)

= 2.G;

3L 2 (13)

=2

=G

3U 2 (13) - 2.G; P3U 2 (13)

Orthogonal

00 3 (13)

Presentations

= 30 3 (13) = P30 3(13) = G.2; 03 (13) = G G = G3.7.13 = (2.3.7;6) _ (2.3.7;7) = <3.TI37=T2=(3T)6=(32T)3=1> G.2 = G3.7.12 = G3.7. 14 _ (2.4.7;3); 2 x G.2 = G4• 6.7

x G.2 : all 3 x 3 matrices over F 13 preserving a non-singular quadratic form;

PG0 3 (13)

Maximal sUbgroups

Specifications

Structure

G.2

Character

Abstract

Linear

Orthogonal

14

13:6

13: 12

1a+13a

N(13AB)

point

isotropic point

14

78

D14

D28

1a+12abc+13a+14aa

N(7ABC). C(2B)

O2(13). L1 (169)

minus point

12

91

D12

D24

1a+12abc+13aa+14aa

N(2A). N(3A)

O (13). base

1a+7ab+12abc+13aa+14a

N(2A 2 )

Order

Index

78

12

@

1092 P power p' part

ind

X,

1A

@

@

@

12 6 6 A A AA A A AA 2A 3A 6A

@

@

@

@

@

@

7 7 7 13 13 A A A A A A A A A A 7A B*2 C*4 13A B* fus ind

@

2

plus point base

@

@

@

@

@

14 6 6 6 777 A A AA AA BB CB AB A A AA AA AB BB CB 2B 4A 12A 8* 14A B*5 C*3

++

+ +

7 -1

-1

o

0

O-b13

X,

+

7

-1

-1

o

0

0

X.

+

12

0

0

0 -y7

*2

*4

-1

Xs

+

12

0

0

0

*4 -y7

*2

X.

+ 12

0

0

0

*2

X,

+

13

-1

-1

X.

+

14

2

-1

-1

0

o

o

++

X.

+

14

-2

-1

1

0

o

o

++

1

4

3

12 12

7

6

14

7 14

7 14

ind

2

[8]

_ <3.TI313=(34T37T)2=1.(3T)3=T2[=1]>;

+

0

0

0

0

0

0

0

-1

++

2

0

0

0

Y7

*2

*4

-1

-1

++

2

0

0

0

*4

y7

*2

*4 -Y7

-1

-1

++

2

0

0

0

*2

*4

Y7

-1

0

-1

-1

-1

2

-1

-1

*

*-b13

13 26

o

++

13 fus ind 26

o o

o r3 -r3

000 0

0

0

4

8 8

24 24

24 24

28 28

28 28

28 28

o

o

0

0

0

0

0

*9

X,.

6

o

0

o

-1

-1

-1 b13

X"

6

0

0

0

-1

-1

-1

X,2

12

0

0

0 -y7

*2

*4

-1

-1

o

0

0

0

*3 y28

X"

12

0

0

0

*4 -y7

*2

-1

-1

o

0

0

0

*9

X,.

12

0

0

0

*2

*4 -y7

-1

-1

0

0

0 y28

XlS

14

0

2

0

0

0

0

o o

r2

r2

r2

X"

14

0

-1

r3

0

0

0

o

r2-y24

X"

14

0

-1 -r3

0

0

0

*

* b13

*3 y28 *9

*3

0

0

0

*7

0

0

0

o r2 *7-y24

0

0

0

GEJ9 BE~} 9

7

Lz(17) Mult = 2

Out

=2

Constructions GL 2 (17) :: 2.(G x 8).2 : all non-singular 2 x 2 matrices over F

Linear

17

;

PGL 2 (17) :: G.2; SL 2 (17) - 2.G; PSL 2 (17) :: G GU 2 (17) :: 9 x 2.G.2 : all 2 x 2 matrices over F 289 preserving a non-singular Hermitian form;

Unitary

PGU 2 (17) :: G.2; SU 2 (17) _ 2.G; PSU 2 (17) :: G 00 (17) :: 2 x G.2 : all 3 x 3 matrices over F 17 preserving a non-singular quadratic form; 3

Orthogonal

Presentations

PG0 3 (17) :: S03(17) :: PS0 (17) :: G.2; 0 (17) :: G 3 3 2 9 G:: <S,TIS =T =(ST)4=(s2r)3=1> _ <s,TIS17=(S4TS9T)2=1,(ST)3=T2[=1l>

Max imal sUbgroups Order

Index

136

18

24

102

24

Specifications

Structure

G.2

Character

Abstract

Linear

Orthogonal

17:8

17: 16

la+17a

N(17AB)

point

isotropic point

S4

1a+9ab+ 16bcd+ 17a+18a

N(2A 2 )

base

102

S4

1a+9ab+16bcd+17a+18a

N(2A 2 )

base

18

136

018

036

1a+9ab+16abcd+17a+18aa

N(3A), C(2B)

16

153

016

032

1a+9ab+16abcd+17aa+18aa

N(2A)

@ 2448 p power pt part

ind

1A

@

@

@

@

@

@

@

@

@

16 9 8 8 8 9 9 9 17 17 A A A A A A A A A A A A A A A A A A A A 2A 3A 4A 8A B* 9A B*2 C*4 17A B* fUs ind

+

+

9

x,

+

9

+

16

0

+

16

+

x,

o o

-1

-1

o

0

0-b17

-1

-1

o

0

0

0

0

0

0

o

0

0 -y9

16

0

o

0

+

16

0

o

0

x,

+

17

X.

+

18

2

0

-2

X,.

+

18

-2

0

0

XII

+

18

-2

0

o -r2

ind

X"

X17

X20

@

@

@

@

@

@

base

@

minus point plus point

@

18 9 8 8 8 8 999 A B A B BA CA AA A AB A AB A A A A AB BB CB 2B 6A 16A B*3 C*7 0*5 18A B*7 C*5

++

x,

Xs

@

@

2 O2(17),

O (17), L1 (289)

-2

+

0

0

0

0

0

0

000

*-b17 -1

-1

++

2

2

0

0

0

0

-1

-1

-1

*2

*4

-1

-1

++

2

-1

o

0

0

0

y9

*2

*4

0

*4 -y9

*2

-1

-1

++

2-1

o

0

0

0

*4

y9

*2

0

*2

*4 -y9

-1

-1

++

2

o

0

0

0

*2

*4

Y9

-1

-1

-1

0

++

-1

-1

-1

0

0

0

0

++

0

0

r2 -r2

0

0

0

r2 -r2

0

0

0

++

0

0 y16

*3

*7

*5

0

0

0

r2

0

o

o

++

o

o

*5 y16

*3

*7

0

o

0

16 16

9 18

9 18

9 18

17 fus ind 34

4

12 12

32 32

32 32

32 32

36 36

36 36

36 36

o

0

r3

r3

-1

143

*

0

o

2

6

8 8

16 16

8

0-1

o

0

0-1

-1

-1 b17

8

0-1

o

0

0-1

-1

-1

-2

0

0

0

16

0

16

0

o

0

0 -y9

16

0

o

0

16

0

o

18

0

0

18

0

0 -r2

18

0

0

18

0

0 -r2

17 34

*

-1

-1

r2 -r2

32 32

o

000

000

* b17 -1

-1

o

0

0

0

0

0

r3

*2

*4

-1

-1

o

r3

0

0

0

0 *13 y36 *11

0

*4 -Y9

*2

-1

-1

o

r3

0

0

0

0 *11 *13 y36

0

0

*2

*4 -Y9

-1

-1

r3

0

0

0

0 y36 *11 *13

r2 y16

*3

0

0

0

o o

0 Y32 *13

*9

*5

0

0

0

*5 y16

0

0

0

o

0

*5 Y32 *13

*7

0

0

0

*3

0

0

0

o

0

*7

*3

0

0

0

*5-y16

0

0

0

o

0 *13

*9 *11 Y32

0

0

0

r2-y16

*5 Y32

[9J

Alternating group A 7 Mult

6

Out : 2

Constructions

Al ternating

3 7 :: G.2 : all permutations of 7 letters; A7 :: G : the even permutations; 2.G and 2.G.2 : the Schur double covers

Lattice

2A 7 :: 2.G : the symmetries of the lattice 1\ 4,b7 whose minimal vectors are obtained from ±(i7;0,O,O) ±(0;i7,O,0) t(1;x,y,z) t(-x;1,y,-z) t(-xyjl,1,l) t(ljxy,1,-1) t(xyjO,z,1) !(-Xjyz,O,-l) ±(_ljO,xy,z) ±(Ojl,-x,yz) by replacing each of x,y,z by one of b1 or b1**, and cyclically permuting the last 3 coordinates

Vectors

3A7 : symmetries o.f the 21 (w)vectors obtained from (20 00 00), (00 11 11), (01 01 wW') by bodily permuting the 3 couples, and reversing any 2 couples (see A6 (hexacode»j the lattice these generate has automorphism group 6U 4 (3).2 (see U4(3»

Unitary

3A 7 has a 3-dimensional unitary representation over F 25 (see U (5» 3

Presentations

G::
G.2:: Maximal subgroups Order

Index

360

Specifications

Structure

7'6

'68

'5

L2 (7)

'68

'5

L2 (7)

120

21

S5

72

Character

Abstract

Al ternating

S6

la+6a

N(2A,3A,38,4A.5A)

point

7: 6

la+14b

N(2A, 3B, 4A, 7AB)

S(2, 3, 7)

la+ 14b

N(2A, 3B, 4A, 7AB)

S(2, 3,7)

la+68+14a

N(2A,3A,5A), C(2B)

duad

la+6a+14ab

N(3A), N(2A 2 )

triad

3

5

x 2

3 4 x S3

('4 x 3):2

35

G.2

@@@@@@@~@

2520 p power

p' part ind lA

24

•• 2.

36

9

4

5

'2

7

@@@@@@@

7

•

H

+62300

x, x,

-1

-2

o

0

b7

n

o

10

-2

o

0

n

b7

+

14

2

:2

-1

+

14

2-1

2

+

15

•

2'

+

35 1

-1

0

4

3

o -, 0

0 -,

-3

0

-1

-1

-1

4

3 6

3 6

o

-2

-1

++

0

0

++

6

2

-1

o

0

++

It

0-2

++

5 -3

o -,

p

-,

12

o

0

H

o

0

H

7 14

° -b7 o

o -, -,

2

2

-1

5 '0

0

+0000000

o -,

8 8

4

-3

5

7 fus ind

0

o -,

o

0

-,

o

-1 -1

-1

2

4

8

6

o

o

o

o

o

o

o

o -, 10 10

24 24

°

0

0

o

0

0

0

o

r3

°

0

o

0

0

14 n

o

0-1

12 12

4

o

-2

14

o

2

-1

r2

-1

o

0

0

14

o

2

-1 -r2

-1

o

0

0

X14

20

-4

-,

o

0

-1

o

o

20

2

2

0

0

o o

-1

X1S

o o

-1

-1

o

o

o o

X16

36

o

0

0

0

o

o

o

° °

r5

1 3

2

3

345

2

2

4

6

10

XI1

0

X12 X13

-

ind

3

6 6

0-1

'2

12

'5

15

02

6

2

0

0

0

XiS

02

15

-1

o

0

-1

X19

02

15

3

0

0

X20

02

21

o

-0

X21

02

21

-3

0

0

X22

02

24

0

0

0

0

-1

X23

02

24

0

0

0

0

-1

6 3

4 '2 '2

3 6

3

8

5 30

,

6

o o

2'2'

2

-1

-1

8

3

24 24

6

° 12

* •

2

*

0

* •

°

24 24

7 fus ind

6 7 6 2' 6 2'

r6

9

•

-200*+

-1

2

6

-b7

n

o

X17

ind

[10]

-1

10

2 XIO

-1

o

ind

24 '2 6 3 5 6 A AB BC AB AB A A A A AABBCABAB 2B 2C 48 68 6C lOA 12A

'20

AAAAAAAA AAAAAAAA 3A 3B 4A 5A 6A 7A Bn fus ind

'5

'0

0

0

o

b7

n

°

n

b7

7 42 21

42

12 12 12

'4

'5

2' 42

30

'4 2' 42

Xzs;026000_r2

o -, -,

-1

Xu

02

24

0

0

0

0

-1

o

b7

..

X27

02

24

0

0

0

0

-1

o ..

b7

X28

02

36

°

0

0

0

2

4

8

6

'2 '2

2'

o

o

+

7 fus ind

X24026000r2

-1

*

*

10 10

24 24

7

L2 (19) Hult : 2

OUt :

2

Constructions GL 2 (19) :: 9 x 2.G.2 : all non-singular 2 x 2 matrices over F,g;

Linear

PGL 2 (19) = G.2; SL 2 (19) =2.G; PSL 2 (19) = G Unitary

GU 2 (19) :: 5 x 2.(G x 2).2 : all 2 x 2 matrices over F 361 preserving a non-singular Hermitian form;

Orthogonal

00 (19) :: 2 x G.2 : all 3 x 3 matrices over F 19 preserving a non_singular quadratic form; 3

PG0 3 (19) = S03(19) = PS0 3 (19) = G.2; 03(19) = G G = G3.9.9 = (2,5,9;2) = < a 5 b c 5 d I (abc)5:(bcd)5: 1 > _ <S,TIS19:(S4TS10T)2:1,(S,)3:T2[:lJ>;

Presentations

G.2:: a4,5,9; 3

3

x G:: G3,9,10 Specifications

Ma.x tmal sUbgroups

Structure

G.2

Character

Abstract

Linear

Orthogonal

20

19:9

19: 18

la+19a

N(19AB)

point

isotropic point

60

57

A5

la+18cd+20a

N(2A.3A.5AB)

icosahedral

60

57

A5

1a+ 18cd+20a

N(2A,3A,5AB)

icosahedral

20

171

D20

la+9ab+18ccdd+20abcd

N(2A). N(5AB)

°2(19), L1(361)

minus point

18

190

D18

1a+9 ab+ 18ccdd+ 19a+20abcd

N(3A), C(2B)

02(19). base

plus point

Order

Index

171

@

@

@

3420

20

9

p power

A

A

pt part

ind

lA

A

A

2A

3A

@

@

@

@

@

10 10 9 9 9 A A A A A A A A A A 5A B* 9A B*2 C*4

@

10 BA AA lOA

@

@

@

@

10 19 19 AA A A BA A A B* 19A B** fus ind

18 A A

2B

@

@

@

o

9

o

-1

-1

000

o

9

o

-1

-1

o

0

0

*

0

0

0 -b5

* -b5

0

0

0

*

0

0

0 b5

* -b5

0

0

0

b19

**

@

@

@

**

b19

*

-1

* -b5

+

0

0

0

0

0

0

0

0

0

0

-1

++

0

2

0

0

0

0 b5

*

b5

*

-1

-1

++

0

2

0

0

0

0

*

-1

-1

++

0

0

0

0

0

0 y20

*3

*9

*7

* b5

-1

-1

++

0

0

0

0

0

0 *7 y20

*3

*9

0

++

-1

-1

-1

-2

0 -b5

+

18

-2

0

+

18

2

0 -b5

+

18

2

0

x.

+

19

-1

x,

+ 20

0

2

0

0

-1

-1

_1

o

0

++

2

0

Xw

+

0-1

o

0 Y9

~

*4

0

0

++

2

0 -1

Y9

x"

+ 20

0

-1

o

0

*4

y9

*2

0

0

++

2

0

+ 20

0-1

o

0

*2

*4

y9

0

0

++

2

5 10

5 10

9 18

9 18

9 20 18 20

20 20

19 fus ind 38

4

n

20

-1

3 6

-1

-1

-1

o

ind

1 2

4

o

10

0

o

0

o

0-b19

o

10

0

o

0

o

0

x"

18

0

0 -2

-2

0

0

0

0

0 -1

x,.

18

0

0 -b5

•

0

0

0 y20

18

0

0

* -b5

0

0

0

18

0

0 -b5

*

0

0

0-y20

18

0

0

* -b5

0

0

0

20

0

2

0

0

-1

-1

-1

o

20

0-1

o

0 y9

*2

*4

Xn

20

0-1

o

0

*4

Y9

x'"

20

0-1

o

0

*2

*4

X,,

@

10 9 9 9 9 10 10 10 10 AABBACAAABA AA BA AA AABABBBCBAA BA AA BA 4A 6A 18A B*7 C*5 20A B*3 C*9 D*7

+ 18

x"

@

++

+

x.

@

19 38

* b5

-1

-1

* b5

-1

-1

o

0

0

0

*2

*4

0

0

0

0

*4 y9

*2

0

0

0

0

0 -1

*2

*4

y9

0

0

0

0

8 8

12 12

36 36

36 36

36 36

40 40

40 40

40 40

40 40

000

0

0

0

0

0

0

0

-1

o

r2

0

0

0

0

r2

r2

r2

r2

*9

*7-y40

2 -1

-1

n-b19

*3

-1

-1

o

r2

0

0

0

0 *17

*7 y 20

-1

-1

o

r2

0

0

0

0-y40 *17

*3

-1

-1

o r2

0

0

0

0

*7-y20

-1

-1

o r2

0

0

0

0 *9

0

o

0

0

r3

r3· r3

0

0

0

0

0

0

o

0 r3 *13 y36 *11

0

0

0

0

*2

0

0

o

0

r3 *11 *13 Y36

0

0

0

0

y9

0

0

o

0 r3 Y36 *11 *13

0

0

0

0

*9

*7

*7-y40 *17

*9

*7-y40 *17

[11]

L2(16) Out : 4

Mult : Constructions Linear

GL 2 (16) ;: 15 x G : all non-singular 2 x 2 matrices over F,6; PGL 2 (15) " SL 2 (15) " PSL 2 (15) " G; rL 2 (15) " (15 x G).4; PrL 2 (15) " !L 2 (15) " P!L 2 (15) " G.4

Unitary

GU 2 (16) ;: 17 x G : all 2 x 2 matrices over F 256 preserving a non-singular Hermitian form;

PGU 2 (15) " SU 2 (15) " PSU 2 (15) " G G0 3 (15)" PG0 3 (15) " S03(15) " PS0 3 (15) " 03(15) " G : all 3 x 3 matrices over F 15 preserving a non-singular

Orthogonal (15)

quadratic form; r0 3 (15) , pr0 3 (15) " !03(15) " P!03(15) " G.4 G0 4 (4) " PG0 4 (4) , S04(4) " PS04 (4) " G.2 : all 4 x 4 matrices over F 4 preserving a non-singular quadratic form of

Orthogonal (4)

Witt defect 1, for example wx~+xlx2+wx~+x3x4; 04(4) ;: G Presentation4

G "

Maximal sUbgroups Order

Index

Specifications

Structur~

G.2

G.4

Character

Abstract

Linear

2 4 :<3 xO lO)

24 : 15:4

la+15a

N(2A 4 )

point L2 (4)

50

5S

A5

A x 2 5

(A5 x2)'2

la+15a+17abc

N(2A,3A,5AB), C(2B)

34

120

0 34

17:4

17:S

1a+17abcdefg

N(17A-H)

30

135

0 30

s3 x 5:4

la+15a+17abcdefg

N(3A), N(5AB)

Orthogonal (15)

Orthogonal (4)

isotropic point

isotropic point

03(4)

non-isotropic point

°2(15),

plus line

@

@

40S0

15

P power pt part

A A

ind

1A

2A

@

@

@

@

@

@

@

@

@

@

@

@

@

@

@

-1

°

0

0

0

0

0

0-y17

+

15

-1

o

0

0

0

0

0

+

15

-1

o

0

°

0

0

+

15

-1

o

0

0

0

+

15

-1

°

0

0

+

15

-1

o

0

+

15

-1

o

+

15

-1

o

Xw

+

15

Xu

+

17

-1

2

2

-1

-1

-1

-1

+

17

2

b5

*

b5

b5

*

*

o

o

+

17

2

*

b5

*

*

b5

b5

o

+

17

-1

*

b5 y15

*4

*2

*S

+

17

-1

*

b5

*4 y15

*S

+

17

-1

b5

*

*S

*2 y15

+

17

-1

b5

*

*2

*S

X16

[12]

3

@

@

@

50 4 3 5 A A AB BB A A AB AB 2B 4A 5A lOA

@

5 AB BB B* fus ind

++

15

Xu

@

15 15 15 15 15 15 15 17 17 17 17 17 17 17 17 A A A BA BA AA AA A A A A A A A A A A A AA AA BA BA A A A A A A A A 3A 5A B* 15A B*4 C*2 O*S 17A B*4 C*2 O*S E*5 F*7 G*5 H*3 fus ind

+

x,

5

pair of isotropic points

+

x.

°2(15), base

17

minus line

L1 (255)

*4

*2

*S

*5

*7

*5

*3

0

*4-y17

*S

*2

*7

*5

*3

*5

0

0

*S

*2-y17

*4

*3

*5

*5

*7

0

0

0

*2

*S

*4-y17

*5

*3

*7

*5

0

0

0

0

*3

*5

*5

*7-y17

*4

*2

*S

0

0

0

0

0

*5

*3

*7

*5

*4-y17

*S

*2

0

0

0

0

0

0

*7

*5

*3

*5

*S

*2-y17

*4

0

0

0

0

0

0

*5

*7

*5

*3

*2

*S

*4-y17

-1

-1

-1

-1

-1

-1

-1

000

o

o

o

o

°

o

0

0

0

*2

0

0

*4

0

*4 y15

0

°

0

0

0

+

0

0

000

+

0

0

0

0

0

+

0

0

0

0

0

-1

++

4

0

0

0

++

5

o

o

o

++

3

o

o

o

o

++

3 -1

°

o

* -b5

0

0

0

+

0

0

0

0

0

0

0

0

0

0

0

+

0

000

0

0

0

0

0

° °

0

0

0

0

0

0

0

°

0

@

@

5

2

3

B A

A AB A AB

4B

SA 12A

:+0+0

+

000

@

-1 -1

-1

0

o

° -b5

0

+

0

0

0

+

0

0

0

-1

:+0+0

2

0-1

0

:+0+0

*

++

o

o

o

+

0

°

0

i

-1

Out = 2

Mult = Constructions Linear

GL 3 (3)

-

2 x G : all 3 x 3 non-singular matrices over F ; 3

SL 3 (3) - PGL 3 (3)

= PSL 3 (3) =G;

the points and lines of the corresponding

projective plane can be numbered (mod 13) so that Li = {Pi,Pi+1,Pi+3,Pi+9J; G.2 is obtained by adjoining the duality (graph) automorphism Presentation#

=<s,TIS6=T3=(ST)4=(S2T)4=(S3T)3=IS2,(TS2T)2J=1>

G

Maximal sUbgroups

Specifications

Structure

Character

Abstract

Linear

32 :2S 4

1a+12a

N(3A 2 )

point

32:2S4

1a+12a

N(3A 2 )

line

13: 6

1a+13a+16abcd+27a+39a

N(13ABCD)

L 1(27)

S4 x 2

1a+12aa+16abcd+26aa+27aa+39a

N(2A 2 ), C(2B)

0 3 (3), base

@

@

Order

Index

432

13

432

13

39

144

13:3

24

234

S4

G.2

12

@

5616 p power p' part ind 1A

@

@

48 54 A A A A 2A

3A

@

9 A A

3B

@

@

8

6

A AA A AA 4A 6A

@

@

@

@

@

@

1

+

12

4

3

+

13

-3

4

o

16

0

-2

Xs

o

16

0

x.

o

16

o X8

0

o

0

@

8 8 13 13 13 13 A A A A A A A A A A A A 8A Bn 13A Bn C*5 D*8 fus ind

+

X2

6

0 -1

24

24

A A

A A

2B

4B

@

3 BB BB 6B

@

466 A AB AB A AB AB 8C 12A B*

1

++

0

0

1

-1

-1

-1

++

0

0

o

0

0

0

++

**

*5

*8

+

0

0

0

000

** d13

*8

*5 **

+

0

0

0

0

o

-1

o

0

0

0 d13

-2

o

0

0

0

0

-2

o

0

0

0 *8

*5 d13

16

0

-2

o

0

0

0

*5

*8

+

26

2 -1

-1

2

-1

000

0

0

0

++

2

-2

-1

o

X9

o

26

-2

-1

-1

o

1 i 2 -i 2

0

0

0

0

+

0

0

0

XlO

o

26

-2

-1

-1

o

1 -i 2 i 2

0

0

0

0

Xu

+

27

3

0

0

-1

++

3

3

X,2

+

39

-1

3

0

-1

++

3-1

o -1

-1

-1

@

-3

-1

r3 -r3

o

0

0

0

0

0

0

0-1

o

0

** d13

-1

10000

o

-1

-1

[I3]

U 3 (3)

Unitary group U (3) 3

G 2 (2) , = 2A2 (3);

Derived Chevalley group G2 (2)'

=

Mult

Out = 2

Constructions

=4 x G : all j x 3 matrices PGU (3) = SU (3) = PSU (3) = G 3 3 3

Unitary

GU (3) 3

Chevalley

G2 (2)

= G.2

preserving a non-singular Hermitian form over F 9 ;

: ad joint Chevalley group of type G2 over F 2 ;

G.2 : the automorphism group of a generalized hexagon of order (2,2) consisting of 63 vertices and 63 edges, each object being incident with 3 of the-other type. G.2 : the automorphism group of the F 2 Cayley algebra, whose elements are integral Cayley numbers modulo doubles of

Cayley (2)

integral Cayley numbers (see below). A pair {x,y}

{O,1} of elements with x+Y

~

=1

is either isotropic with

x2 =y2=O (36 cases), non-isotropic with x 3 =y3: 1 (28), or mixed with x 2 =O, y2=l t or vice versa (63).

Cayley (Z)

G.2

the fixed points of 08(2) under its triality automorphism

G.2

the automorphism group of the integral Cayley algebra in which i oo =1, i n+1 =i, i n +2 =j, i n +4 =k define a

quaternion subalgebra for each n (subscripts modulo 7);

~xnin

is integral provided that all 2x n are integral and

the set of n for which xn is integral is one of (0124l, (0235), (0346l, (0156l, (00013l, (00026), (00045), (000123456l or their complements.

Alternatively, they can be generated by the even sums of Xoo' XO' ••• X6 together with 1 = ~Xn' with multiplication defined by the equations 2Xn 2 = Xn -1, 2X OXoo = 1-X 1-X 2-X 4 , 2X oo XO = 1-X 3-X 5-X 6 and their images under the coordinate permutation group L2 (7). Quaternionic

2 x G : the group generated by the 63=3+12+48 quaternionic reflections in the vectors (2,0,0)S (3), (O,a,a)S (12), and (a,1,1)S (48) and their images under diag(±1,±i,±i)S, where 2a = -1-i-/5j+k

Presentation

Maximal subgroups

Specifications

Structure

G.2

Character

Abstract

Unitary

Cayley (2)

28

3+1+2'8·2 • •

1a+27a

N(3A)

isotropic point

non-isotropic pair

168

36

L (7):2

1a+7bc+21a

N(2A,3B,4C,7AB)

96

63

2 -i.\+. "53 'l·g4"2

1a+ 14a+21a+27a

N(2A)

non-isotropic point

isotropic space

96

63

4 2 :D 12

1a+7bc+21a+27a

N(2A 2 )

base

mixed pair

Order

Index

216

Chevalley

isotropic pair

Quaternionic

14

@ 6048

p power p' part ind

lA

@

vertex

root vector

edge

base

@

@

96 108 .,,9

@

6

@

@

96

96

16

A A

A A

A A

A A

A A

A A

2A

3A

3B

4A

B**

4C

12 AA AA 6A

7

7

A

A

8

8

A A

B AB A AA

12

12 AA A A AB 7A B** 8A B** 12A B** fus ind

+

[14]

@

@

24

24

3

A A

A A

BB

2B

4D

6B

o

o

0

BB

@

@

466 C AD AD A AD AD 8C 12C D**

++

6

-2

-3

+

7

-1

o

o

o

-2

-2

2

-2

3

3

-1

7

3 -2

-2i-l

o

7

3 -2

+

14

-2

5

X7

+

21

5

3

o

x.

o

21

3

o

X9

o

21

3

o -2i-3

x"

+

27

3

o

o

Xl1

o

28

-'l

4i

-4i

o

-1

o

o

o

o

i

-i

Xu

o

28

-'l

-4i

4i

o

-1

o

o

o

o

-i

i

o

32

0

-4

-1

o

o

o

0 -b7

**

o

0

o

o

o

32

0

-'l

-1

o

o

o

o

** -b7

o

o

o

o

x,

Cayley (Z)

-1

-1

-1

0

00

2

0

o

-1

2i-l

o

0

°

i

-1 i-1-i-1

2i-l -2i-1

o

o

o

-i

i-1-1 i-1

o

o

o

0

-1

-1

-1

o

o

1

-i

-i

i

-1

o

o

-i

i

i

-i

2

2

2 -1

2i-3 -2i-3

3

2i-3

3 -1

o

-1

o

-1

o

o

-1

o

-1

o

-1

o

-3

++

+

0

o

++

2

-2

++

3 -1

o -1

o

-1

o

0

o

0

++

3

3

o

+

0

o

0

+

0

o

o

o

o

o

0

o

o

+

i3 -i3

-1

o

-1

o

o

o

0

o

o

0

o

o

0

-1

Mult : 2

Out: 2

Constructions Linear

GL 2 (23) :: 11 x 2.G.2 : all non-singular 2 x 2 matrices over F

=G.2;

PGL 2 (23) Unitary

GU 2 (23)

3'

PGU 2 (23) Orthogonal

= 2.G;

SL 2 (23)

23

;

=G

PSL 2 (23)

3 x 2.(G x 4).2 : all 2 x 2 matrices over F 529 preserving a non-singular Hermitian form;

= G.2;

= 2.G;

SU 2 (23)

=G

PSU 2 (23)

00 3 (23) :: 2 x G.2 : all 3 x 3 matrices over F 23 preserving a non-singular quadratic form;

= S03(23) = PS0 3 (23) = G.2; 0 3 (23) = G G = (2,3,11;4) = <s,Tls23:(S4TS12T)2:1;(ST)3:T2[:1J>; PG0 3 (23)

Presentations

Max imal subgroups Order

Index

253

24

24

253

S4

24

253

S4

Structure

G.2

Character

Abstract

Linear

Orthogonal

23: 11

23:22

1a+23a

N(23AB)

point

isotropic point

1a+22bcddee+24abcde

N(2A 2 )

base

N(2A 2 )

base

1a+22bbddee+24abcde

N(2A), N<3A)

22

1a+22bbddee+23a+24abcde

N(11ABCDE), C(2B)

6072

24 12 12 12 11 11 11 11 11 12 A A A AA A A A A A AA A A A AA A A A A A AA 2A 3A 4A 6A 11A B*3 C*2 0*5 E*4 12A

12

23

23

AA

A

A

11

-1

-1

-1

0

0

0

0

0

b23

o

11

-1

-1

-1

0

0

0

0

0

n

b23

+

22

-2

o

0

0

0

0

-1

+

22

2

0

0

0

0

0

0

0

+

22

-2

2

o

0

0

0

0

-1

+

22

2

o

-1

0

0

0

0

0

+

22

2

o

-1

0

0

0

0

0 -r3

+

23

-1

-1

-1

+

24

0

0

0

0 y11

plus point

22 12 12 11 11 11 11 11 12 12 12 12 A A A CB DB EB AB BB BB AB BA AA A A A AB BB CB DB EB AA AA AB AB 2B 8A B* 22A B*3 C*9 D*5 E*7 24A B*7C*11 D*5

+

24

0

0

0

+

24

0

0

+

24

0

+

24

ind

-2 -2

0

2

-1

**

+

0

0

0

0

0

0

0

0

0

0

0

0

-1

++

0

2

2

0

0

0

0

0

-1

-1

-1

-1

-1

-1

++

0

r2 -r2

0

0

0

0

0

r2

r2 -r2 -r2

-1

-1

-1

++

0

0

0

0

0

0

0

0

r3 -r3

r3 -r3

-1

-1

++

0 -r2

r2

0

0

0

0

0 y24

r3

-1

-1

++

0 -r2

r2

0

0

0

0

0

-1

-1

0

0

++

-1

-1

*2

*5

*4

0

0

++

2

0

0 y11

0

*4 y11

*3

*2

*5

0

0

++

2

0

0

0

*5

*4 y11

*3

*2

0

0

++

2

0

0

0

*2

*5

*4 y11

*3

0

0

++

0

0

0

0

*3

*2

*5

*4 y11

0

0

1 2

4

3 6

8 8

12 12

11 22

11 22

11 22

11 22

24 24

24 24

o

12

0

0

0

0

o

0-b23

o

12

0

0

0

0

o

0

-

22

0

-2

r2

0

0

0

0

0

0

r2

r2

-1

-1

o y16

-

22

0

-2 -r2

0

0

0

0

0

0 -r2 -r2

-1

-1

-

22

0

-r2 -r3

0

0

0

0

0 y24

*7

-1

-

22

0

-r2

r3

0

0

0

0

0

*7 y24

22

0

r2 -r3

0

0

0

0

0-y24

22

0

r2

O.

0

0

0

0

24

0

0

0

0 y11

*3

*2

*5

*4

0

0

o

0

24

0

0

0

0

*4 y11

*3

*2

*5

0

0

o

24

0

0

0

0

*5

*4 y11

*3

*2

0

0

24

0

0

0

0

*2

*5

*4 y11

*3

0

24

0

0

0

0

*3

*2

*5

*4 yl1

0

-

x,.

°2(23), base

*3

x"

minus point

1

++

o

-

L (529)

@@@@@@@@@@@@

AA A A B* 23A Bn fus ind

+

x'"

°2(23),

24

P power p' part ind lA

Xn

= G3,8,11

Specifications

@@@@@@@@@@@@@@

x.

G.2

r3

11 22

23 46

r3 -r3

*7 *11

*5

*7 y24

*5 *11

-1

-1

-1

-1

*3

*2

*5

*4

0

0

0

0

0

*4 y11

*3

*2

*5

0

0

0

0

0

0

*5

*4 y11

*3

*2

0

0

0

0

2

0

0

*2

*5

*4 y11

*3

0

0

0

0

++

2

0

0

*3

*2

*5

*4 y 11

0

0

0

0

23 fus ind 46

4

16 16

16 16

44 44

44 44

44 44

44 44

48 48

48 48

48 48

48 48

**

000000000000

44 44

**-b23 *3

0

0

0

0

0 y16 y16

o

*5 y16

0

0

0

0

0

-1

o

*5 y16

0

0

0

0

0 *17 y48 *19 *13

-1

-1

o

*5 y16

o·

0

0

0

0 y48 *17 *13 *19

*7

-1

-1

0-y16

*3

0

0

0

0

0

*7-y24

-1

-1

0-y16

*3

0

0

0

0

0 *11

0

*7 y44 *19

*9

*5

0

0

0

0

0

0

*5

*7 y44 *19

*9

0

0

0

0

o

0

0

*9

*5

*7 y44 *19

0

0

0

0

0

o

0

0 *19

*9

*5

*7 y44

0

0

0

0

0

o

0

0 y44 *19

*9

*5

0

0

0

0

*7

*5

*3

*3

*5 y16 y16

*5 *11 *17 y48 *5 y48 *17

[15]

-

£)

Out =

22

S5

S5

26

65

65

300

325

300

'20

'20

26

24

°24

°26

52: '2

Index

Order

: °48

: °52

I

: 52: 24

G.2,

: 08 x S3

: '3: 4

: S5 x 2

: S5 x 2

: 52 :(4XS ) 3

G.2 2 3

: Q8

x S3

: '3: 4

I

: H.2

G.2

'a+13ab+24abcdef+25aa+26aacc

: H.2 2

L2 (5)

C(20)

N(2A), N(3A)

13

7

5

(*) There is no group 2.G .2.1' See the Introduction: 'lsoclinism'

15

~12.G.21112.G.221~G'~14 L:J L(*)--l

0"2(25), base

0;;(25), L, (625)

L2 (5)

C(2C)

N('3A-F), C(2B)

point

Linear

N(5A 2 )

Abstract

Specifications

GEJEJ[~~}5

'a+'3ab+24abcdef+25a+26aacc

'a+ '3b+25a+26a

'a+ '3a+25a+26a

'a+25a

Character

: 26:4

I

: 5 2 :24:2

G.2 2

G =

Structure

Maximal SUbgroups

Presentation

G0 4 (5) = 2 x G.2 2 : all 4 x 4 matrices over F 5 preserving a quadratic form of Witt defect ';

Orthogonal (5)

PG0 4 (5) = G.2 2 ; S04(5) = 2 x G; PS0 4 (5) = 04(5) = G;

G0 3 (25) = 2 x G.2, : all 3 x 3 matrices over F 25 preserving a non-singular quadratic form; PG0 3 (25) = S03(25) = PS0 3 (25) = G.2,; 03(25) = G; r0 3 (25) = 2 x G.2 2 ; pr0 3 (25) = I0 3 (25) = PI0 3 (25) = G.2 2

PGU 2 (25) = G.2,; SU 2 (25) = 2.G; PSU 2 (25) = G

GU 2 (25) = 13 x 2.G.2, : all 2 x 2 matrices over lF 625 preserving a non-singular Hermitian form;

rL 2 (25) = (3 x 2.(G x 4).2,).22; prL 2 (25) = G.2 2 ; IL 2 (25) =2.G.2 2 ; PIL 2 (25) = G.2 2

PGL 2 (25) = G.2,; SL 2 (25) = 2.G; PSL 2 (25) = G

GL 2 (25) = 3 x 2.(G x 4).2, : all non-singular 2 x 2 matrices over F 25 ;

Mult = 2

Orthogonal (25)

Unitary

Linear

Constructions

Order = 7,800 = 23 .3.5 2 .'3

Linear group L2 (25) = A,(25) = U2 (25) = S2(25) = 03(25) = 04(5)

plus point

minus point

°3(5 )

°3(5 )

isotropic point

Orthogonal (25)

non-isotropic point

non-isotropic point

isotropic point

Orthogonal (5)

Ut "-"

N

~

~ N

-

.:::l

12

2q

7aoo p power p' part 1nd lA

0

0

1

2

-2

2

2q

2q

25

26

26

26

+

+

+

+

+

+

+

X8

X9

XIO

Xll

XI2

X13

X,.

XIS

2 -r2

-1 -r2

-1 -r2

r2

-1

o

o

o

o

o

o

o

2q

26

26

26

26

26

26

X23

X24

X2S

X26

X27

X28

X29

-1

0

2 r2

r2

0

o

2q

0

0

1

1

1

1

-1

-1

-1

o

o

o

2q

X2I

-1

o

o

o

2q

2

5 10

0

0

0

0

0

o

o

o

o

o

@

-1

*q

*6

*3

o

o

o

o

o

o

o

-r2

r2

o

o -r2

r3

*7-y2q

*7

*7 y2q

-r3 y2q

r3

*7

r2

0

0

0

0

0

o

o

o

0

0-y13

o

o

o

-1

o

o

o

*5

-1

-1

0

0

0

0

0

0

*6

*q

*2

*3

-1

*2

*3

@

0

0

0

0

0

0

*q

*6

*3

-1

*6

*q

-1

_1

13 26

0

0

0

0

0

0

*2

*3

*6

*q

*2

*3

-1

-1

13 26

o

0

0

0

0

0

0

*3

0

0

0

0

0

0

++

*5

*q

*6

0

0

0

0

o·

0

:

:

13 fus 1nd 26

*3

-1

0

0

0

0

0

0

-1

0

0

0

0

0

0

0

0

0

0

0

0

o

o

o

0

@

@

@

@

@

@

0 0

o o

o y 16

*3

*5 yqa *5 *13 * 17 y qa * 19

*3-y16 *17 *11

o

*5

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

*9

*9· *7

*9 *15 *23

*7 *23 *11

0 *15 *23 *21 y52 *19

*5 *11 *17

*3-y16 yqa

*21 y52 *19 0 *23 *11 y52

o

*5 y 16

0

0

o

*5 *19 yqa *17 *13

o

o y16

*5

0

0

o

*3 y16 y16

*5 Y 16

0

0

000

o

*9

*5

o

o y52

o

*9 *15 *23 *21 y52

o *19

*5

o

o

000

*3 y16

o

o

o

r

r

[

[

00

52 rus 1nd 52

o

o

o y 16

52 52

52 52

52 52

*7 *23 *11 y52

o

52 52

o

o

I

+

o

o

o

o

0

o

o *9

52 52

qa qa

o

o

o

o

o

o

o

o

++

++

++

++

+

+

0

0

0

-1

I I I

+

++

++

++

0

0

0

-1

*5 y13

*5

*q

*6

*3

*2

o

o

o

0

*7

0 0

0 0

0

0

0

0

*3

000 0

*6

*q

*2

*3

0

*2 y13

-1

*2

*3

*5 y13

*5

-1

-1

*q

*6

*3

*2 y13

*q

*6

0

o

o

*6

*q

*2

*3

*6

*q

0

1

o

o

o

r2

-1

*5

o

*5 y13

1 -1

0

0

0

0

0

*7 *11 qa qa

o

0 y13

0

1

o

qa qa

qa qa

o

o

16 16

o

o

@

r3 -r3

*5 *11

*5 y2q

o

o

16 16

r2 -r2

r2 y2q

0 r3 -r3

o

o

q

o

o -r2

0

r2 -r2 -r2

0 r2 -r2

2

0

0

0

0

0

0

0

++

2

0

0

0

0

0

0

o

1

1

-1

++

*2

@

0

o

-1

@

++

++

-1

@. @

@

@

@

@

@

@

@

o

o

o

o

o

o

o

o

o

o

o

o

o

o

o

o

o

o

o

o

o

o

o

o

o

o

o

o o o

o

o

o

o

o

o

0

0

o

0

o

o

000

10 10

o

r5

o

o

o

-1

-1

o

o

0

o

a

o

-1

0

1

-1

0

0

12

q

o

0

0

1

-1

o

0

6

2

o

q -q

0

6 -6

0

0

0

0

o

o

I

I

+

00

00

++

++

+

+

+

+

++

o

o

o

o

o

o

15

0

I I I I 1 I I

02

02

02

02

02

02

02

20 fus 1nd 20

o

1

-1

-1

0

0

0

0

1

o

0

1

-1

1-1

1

-1

-1

1

0

5

0

0

o

5

1

q

q

5

0

0

0

5

1

q 6 6 5 5 120 120 12 12 12 12 12 12 13 13 13 13 13 13 A A A AC AD AC BD A A BB AA BA AB FB EB BB AB DB CB A A A AC AD AC BD A A AA AB AB AA AB BB CB DB EB FB 2C fus 1nd 2D qB 6B 6C lOA lOB fus 1nd BA B* 2qA B*5C*11 0*7 26A B*5 C*9 0*7 E*3F*ll

@

1

2

2

2

2

2

2

0

2B

A

A

26

@

-1

++

++

++

++

++

++

++

+

++

o

0

0

*5-y13

*2-y13

*5-y13

*5

*q

*6

-1

-1

13 26

o

o

o

o

0 0

0 0

o

-1

*5~y13

*5

*q

*6

*3

*2

o

o

000

-1

*3

*2-y13

*6

*q

*2

*3

0

0

*2-y13

*5-y13

-1

o

o

2q 2q

o

2q 2q

r3

13 26

o

o

r3 -r3

13 26

0

*5

*q

*6

*5-y13

*2-y13

*6

*q

000

-1

*6

*q

*2

*3

*5-y13

o

-1

0

0

0

0

0

*5

o

o

000

-1

12 12

@

o

0

-r3

@

o

000

1

@

000

0-y13

-1

-1

A

.;2

-1

0

o

-1

-1

-1

1

1

6A 12A

AA

12

AA

12

AA

13

@

13 13 13 13 13 A A A A A AA A A A A A A B* 13A B*5 c*q 0*6 E*3 F*2 fus 1nd

@

12

@

AA AA

@

1 -r3-y2q

-1

-1

-1

-1

-1

-1

2-3

-3

5 10

1

1

0

o

-1

~1

-1

-1

-1

3

-1

-1

X20

1

-1

-1

-1

-1

-1

-1

-2

o

o

o

2q

-

@

3-2

1

-1

XI9

X22

@

o

o

o

2q

X,8

o

o

0

12

-

X,7

a a

o

o

-2

0

2

0

0

0

0

0

0

o

0

3 6

-1

-1

-1

2

-1

1

0

0

0

O.

0

0

-1

-1

1

o

12

q

1 2

0

2q

+

X7

1nd

0

2q

+

X.

-2

0

2q

+

Xs

26

0

2q

+

X.

+

1

13

+

X3

-2

1

13

+

X2

26

1

1

+

X,

X,.

@

12 25 25 A A A A A A A A A A 2A3A qA 5A 5B

@

@

@

@

@

a

16

o

o

16

o

2q

o

2q

o

X29

X28

X27

X26

X"

X24

X23

X22

X2I

X20

X,9

X,8

X,7

XI.

XIS

X14

X13

0

0

0 1 3 -13

X,O

X9

X8

X7

X.

Xs

X4

XI2

-1

o

0

o

0

0

-1

o

0

o

0 12 -12

1

o

o

o

Xll

0

o

0

o

0

-2

-1

0

0

0

o

X,

X2

o o

o

0

XI

1

1

1

12C

on

A A qC ac

on

@

6 6 AC AC A AC AC

@

6 q q A ··AA

@

"-'"

N Ut

~

~ N

Sporadic Mathieu group M11 Out =

Mult = Constructions

l1_point

G

=- M11 : stabilizer of a point in M12 ;

the automorphism group of the Steiner system 5(4,5,11); automorphisms of the ternary Golay code C12 that fix a coordinate 12-point

G :: M11 : stabilizer of a total in M12 ;

automorphisms of the ternary Golay code C12 that fix a weight 12 word (total word); in particular, the coordinate permutations that fix the set of 22 words obtained from :(-;++-+++---+-) by cyclic permutations of the last 11 coordinates

---v •

a

Presentations

Hax imal subgroups Order

Index

120

b 5 c

d

Specifications

Structure

Character

11

M1O ;; A6'2

660

12

1~~

120

Abstract

'11-point

12-point

1a+10a

point

total

L2 (11)

1a+11a

total

point

55

M :2;; 32:Q8.2

1a+10a+~~a

N(3A 2)

duad

quadrisection

66

55

1a+10a+11a+44a

N(2A,3A,5A)

hex ad

duad

MS:5 3 - 2~S4

1a+10a+11a+44aa+55a

N(2A)

triad

tetrad

165

9

@@@@@@@@@@

7920 p power

pi part

ind

x,

x.

lA

~8

18 8 5 6 8 8 " 11 A A A A A A A AA A AAAAAAA A A 2A 3A ~A 5A 6A 8A Boo 11A B** A

+ +

10

2

2

0-1

o

10

-2

o

0

o

10

-2

o

0

+

11

3

2

-1

o

16

0

-2

o

16

0

-2

x, 55

-1

-1

-1

i2 -i2

-1

-1

-1

-1

i2

o

-1

-1

0

o

0

0 b11

0

o

0

0

o

0

o

+

0

-i2

-1

x. x"

o

-1

o

0

o

-1

-1

o

10

0

**

** b11

0

0

o

0

-1

Hult = 2

Out = 6

Constructions Linear

GL 2 (27) :: 13 x 2.G.2 : all non-singular 2 x 2 matrices over F 27 ;

PGL 2 (27) ;; G.2; SL 2 (27) ;; 2.G; PSL 2 (27) ;; G rL 2 (27) ;; (13 x 2.G.2).3; prL2 (27) ;; G.6; !L2 (27) ;; 2.G.3; P!L2 (27) ;; G.3 Unitary

GU 2 (27) :: 7 x 2. (G x 2).2 : all 2 x 2 matrices over F 729 which preserve a non-singular Hermitian form;

PGU 2 (27) ;; G.2; SU 2 (27) _ 2.G; PSU 2 (27) ;; G· Orthogonal

Presentation

00 (27) :: 2 x G.2 : all 3 x 3 matrices preserving a non-singular quadratic form over F 27 ; 3

PG0 3 (27) ;; S03(27) ;; PS0 3 (27) ;; G.2; 03(27) ;; G; r0 3 (21) ;; 2 x G.6; Pr0 3 (21) ;; !03(27) ;; P!03(27) ;; G.6 G =-

Maximal sUbgroups Order

Structure

28

351

D28

26

318

D26

12

[18)

Index

819

A~

Specifications

G.2

G.6

13:6 A~ x

3

Character

Abstract

Linear

Orthogonal

point

isotropic point

°2(27),

28:6

N(2A), N(7ABC)

26:6

N(13A-F), C(2B)

°2(27), base

N(2A 2), C(3C)

L2 (3)

s~

x 3

L1(129)

minus point plus point

°3(3), base

~

:£

2

-1

0

26

27

28

28

28

+

+

+

+

+

+

+

+

+

+

x,

X.

x,

x,

x,

Xw

x"

x"

XI)

X\4

XIS

X16

0

0

o

o

o

7

14

0

3

6

**

o o

o

4

0

0

0

28

28

28

1 2

14

14

26

26

+

+

ind

o

o

x"

x"

x"

x"

0

0

0

0

0

0

0

0

26

28

28

28

28

28

28

-

-

-

-

x"

x,.

x"

x"

X29

X30

X31

Xu

26

-

0

26

x"

0

26

-

x"

0

26

-

x"

-2

-1

-1

03

0

0

0

0

0

0

0

0

0

0

0

0

0

0

o

o

o

o

o

0

0

0

0

*2

*6

*5

*3

*3

*4

0

0

0

0

0

0

0

0

*3

0

0

0

0

0

0

0

13 26

*6

*5

*2

03

*4

0

0

0

0

0

0

0

13 26

*5

*2

*6

*5

*5

*2

*6

*4 y13

*2

*2

*6

*5

0

0

0

0

0

0

0

0

*3

*6

*5

*2

0

0

0

0

0

0

0

0

*3

*3

*4

*5

*2

*6

0

0

0

0

0

*3

*6

05

*2

0

0

0

0

0

0

0

13 26

*4 y13

*3

*4

*5

02

*6

0

14 BA CA

0

o

03

0

0

03

03

0

0

0

0

0

0

0

0

0

0

0

09 y28

0

°3

°9

0

0

0

0

0

0

03

09

09 y28

09-y28

03

0

09-y28

0

:

:

:

:

@

@

@

@

@

@

@

@

@

@

@

@

@

++

++

++

*3 *5 *6 *2 52 52

o o

o o 8 8

*3

*4

0

0

0

0

0

0

52 52

*3 52 52

0 0

r2 r2 r2 r2 0

o o o o o

0

0

0

0

0

0

0

0

0

*7

0

0

0

0

0

0 *21 *15 *19 y52 *23

o

0

56 56

0

0

0

0

0

0

0

56 56

0

0

0

0

0

0

-1

0

56 56

0

0

0

0

0

0

-1

o o 0

o

*9 *15 *17 *23

*7

*5

*9

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

*9

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

*9 y56

*9 y56 *25

0 *17 *23 *15 *25

0 *23 *15 *17

0 *15 *17 *23 y56 *25

*9 y56 *17 *23 *15

o

o

0

0

0

-1 -1 0

2

_00

o

-z3

o

o

o

o

o

o

-z3 -z3**

18

0 -z3**

0

18

2

-00

ind

++

: -00-00

9 fus

o

000

9

o

++

o

o

0

12

o

0

2

o

0

:+00+00

++

++

o

o

x,

o

o

0

0

0

0

o

o

r2

0

0

0

24 24

0

0

-1

0

0

12

0

0

0

0

x"

x'o

X29

x"

x"

X26

x"

x"

X23

x"

x"

X20

X19

x"

X17

x"

X15

x"

x"

x"

XII

XIO

x,

x,

X7

X.

x,

X4

x,

+0000X2

o

o

000

+

0

3 -1

0

3 6

t

+

+00

+

o

0

o

0

+

56 fus ind 56

0

0

0

0

0

0

-1

*5 * 13 * 11

*5

*9 y56 *25 *23 *15 *17

0 *25

0

*9 y52

0 *15 *19 *21 *23

0

56 56

0

0

0

0

0

0

0 y56 *25

*9 y52 *23

*5 *11

o

*9

0 *19 *21 *15

0 y52 *23

o

*7

0

56 56

0

0

0

0

0

0

-1

_1

-1

*3

*9 *13 *11

*9 y28

Y7

*3

*2

*5 *13

*2

Y7

*3

Or2r2r2r2r2r2

0

52 52

*5 *11

0

0

0

0

0

o

0 *23

o

*9 y52 *11

*9 y52 *23

0

0

0

0

r2

0

o

0

0

r2

o 0

0000

0

*3

*4

*5

*2

*6

0

*3

*9 y28 *11

*3

*2

y7

*3

Y7

*3

*2

*2

Y7

*3

*3

'*2

Y7

0 y28

0

0

0

0

*4 y13

*4 y13

*3

*6

*5

*2

0

0

0

0

0

0

000

52 52

*5

*2

*6 y13

*2

*6

*5

0

0

0

0

0

0

Or20

0

52 52

*6

*5

*2

*4 y13

*4 y13

*3

0

0

0

0

0

0

0

0 y13

-1

0

0

000

4

2

2

2

2

2

++

++

2

++

++

++

0

0

0

0

++

0

0

0

0

++

2

0

++

0

2

0

++

0

2

0

++

z3

z3**

z3**

z3

1nd

:+00+00

98 fus

A

9A

A

2

CB AA CB CA 6B 12A

2

3 B

A

12 4 A CA A CA 3C 6A

@

@

@

3

@

@

@

000

000

+00

26 14 13 13 13 13 13 13 14 14 14 14 14 14 A A EB FB DB BB CB AB CA AA BA CA AA BA A A AB BB CB DB EB FB AA BA CA AA BA CA 2B 4A 26A B03 C09 D05E011 F07 28A B03 C09D013Eoll F05 fus ind

@

+00000000000000

++

28 fus 1nd 28

0

0

0

0

0

0

-1

Y7

°3

*2

000

28 28

0

0

0

0

0

28 28

0

0

0

0

0

0

-1

-1

y7

*3

02

0

*3

*2

*2 -y7

03

02

y7

*3

0 y28

0

0

*3

*2 -y7

0-y28

0

0

13 26

14 AA BA

14A B*3 C*5 fus ind

14 CA AA

0 -y7

0

0

*4 y13

*4 y13

y13

*2

06

*5

0

0

0

0

0

0

0

13 26

*3

*4 y13

*6 y13

*4 y13

04 y13

0 y13

02 -y7

o

03

03

02

02 -y7

02 -y7

-1

-1

03

03

-2

0

0

13 26

7 14

*6

*2

02 -Y7

-1 -y7

-1

-1

*3

o

o

*3

0

0

0

0

0

0

0

0

'14 y13

o *5

o

0

02

-2

0

0

14

7

0

0

0

0

0

0

0

0

0 y13

-1

03

-1 -Y7

-1

-1

-1

-1

-1

_1

U-b27

O-b27

6

0

0

o

o

0

0,

o

3

03

*2

02 -Y7

0

03

02 -Y7

*3

0

0

-1

-1

*3

*2

-1

-1

*2 -y7

o

0

-1

-1

*3

*2 -y7

-1 -y7

-1

-1

-1

2

-1

-1

-1

*3

-1

-1

-1

26

2

-2

26

26

-2

26

_1

-1

14 14 14 13 13 13 13 13 13 A A A A A A A A A A A A A A A A A A 7A 8°3 Co2 13A B03 C04 005 E*2 F*6

-1 -y7

-1

-2

26

+

X.

b27

**

13

o

..

x,

A

3A 'Bn

A

b27

o

x,

13

+

2A

A

28 27 27 A A A

x,

9828 P power pi part 1nd lA

@@@@@@@@@@.@@@@@@

16

14

4

2

BEJBE}

GBBBI6

"-"

N ......:t

~

r N

Mult

=2

=2

Out

Constructions GL 2 (29) =- 7 x 2.(G x 2).2 : all non-singular 2 x 2 matrices over F

Linear

PGL 2 (29)

= G.2;

= 2.G;

SL 2 (29)

PGU 2 (29) Orthogonal

=G

PSL 2 (29)

= G.2;

=G

SU 2 (29) _ 2.G; PSU 2 (29)

G0 3 (29) =- 2 x G.2 : all 3 x 3 matrices over F

29 preserving a non-singular quadratic form;

= S03(29) = PS0 3 (29) = G.2; 0 3 (29) _ G = G3,7,15 =

G

PG0 3 (29) Presentations

_ <S,1IS29,(S41S151)2,l,(S1)3,12['l]>

Max imal sUbgroups

Specifications

Structure

G.2

Character

Abstract

Linear

Orthogonal

29: 14

29: 28

1a+29a

N(29A8)

point

isotropic point

A5

la+28defg+30abc

N(2A,3A,5AB)

icosahedral

203

A5

la+28defg+30abc

N(2A,3A,5AB)

icosahedral

30

406

D30

la+28abcdefg+29a+30aabbcc

N(3A), N(5AB), C(2B)

02(29), L1 (841)

minus point

28

435

D28

la+28abcdefg+29aa+30aabbcc

N(2A), N(7ABC)

°2(29),

plus point

Order

Index

406

30

60

203

60

@

12180 P power pi part ind lA

@

@

@

@

@

@

@

@

@

@

@.

@

@

@

@

@

15

-1

o

0

-1

-1

-1

OOOO-b29*

+

15

-1

000

-1

-1

-1

o

+

28

0

+

28

0

-2 -b5

'If

0

0

0

0

0

0 -b5

+

28

0

-2

*

-b5

0

0

0

0

0

0

+

28

0

-b5

'If

0

0

0

0

0

+

28

0

* -b5

0

0

0

0

+

28

0

*

0

0

0

x"

+

28

0

* -b5

0

0

0

Xu

+

29

+

30

+

X,.

x,. X",

[20]

30 14 A A A A 2B 4A

15 15 AB BB AB AB 6A lOA

@

@

@

@

15 14 14 14 14 14 14 15 15 15 15 AB BA CA AA BA CA AA BBA CAA DBA AAA BB AA BA CA AA BA CA AAA BBA CAA DBA B* 28A B*9 C*3D*13 E*5F*11 30AB*13C*11 D*7

x,

++

+

x"

base

@@@@@@@@@@@

28 15 15 15 14 14 14 14 14 14 15 15 15 15 29 29 A AAA A A A BA CA AA BA AA BA AA A A A A A A A A A AA BA CA AA BA AA BA A A 2A 3A 5A B* 7A B*2 C*3 14A B*5 C*3 15A 8*2 C*4 0*7 29A B* fus ind

+

x"

;

00 2 (29) :: 15 x 2.G.2 : all 2 x 2 matrices over F 841 preserving a non-singular Hermitian form;

Unitary

x,

29

0

0

0

0

-2-2000000

-b5

+

0

0

0

0

0

0

0

0

0

0

0

o

0

0

0

Xz

*-b29 -1

-1

++

2

0

-1

2

2

0

0

0

0

0

0

-1

-1

-1

-1

X4

2

0

2

b5

*

0

0

0

0

0

0

b5

*

b5

*

Xs

* -b5

*

-1

-1

++

* -b5

*

-b5

-1

-1

++202*b5000000

0

*7-y15

*2

*11

-1

-1

++

2

0

-1

b5

*

0

0

0

0

0

0

*7 y15

0

0

*4

*7-y15

*2

-1

-1

++

2

0

-1

*

b5

0

0

0

0

0

0

0

0

0

*2

*4

*7-y15

-1

-1

++

2

0

-1

b5

*

0

0

0

0

0

0

0

0

0-y15

*2

*4

*7

-1

-1

++

2

0-1

*

b5

0

0

0

0

0

-1

-1

-1

-1

0

++

-1

-1

-1

-1

-1

-1

o

*b5*b5X6 *2

*4

X7

*4

*7 y15

*2

Xs

*2

*4

*7 y15

X9

0 y15

*2

*4

XIO

*7

-1

-1

-1

2

0

0

0

y7

*2

*~.

y7

*2

*3

0

0

0

0

++

0

2

0

0

0

y7

*2

*3

y7

*2

*3

o

0

0

0

Xl2

30

2

0

0

0

*3

y7

*2

*3

y7

*2

0

0

0

0

++

0

2

0

0

0

*3

Y7

*2

*3

y7

*2

o

0

0

0

Xl3

+

30

2

0

0

0

*2

*3

y7

*2

*3

y7

0

0

0

0

++

0

2

0

0

0

*2

*3

y7

*2

*3

y7

o

0

0

0

X14

+

30

-2

0

0

0

y7

*2

*3 -Y7

*2

*3

0

0

0

0

++

0

0

0

0

0

*3 y28

*9 *11 *13

*5

OOOOXIS

+

30

-2

0

0

0

*3

y7

*2

*3 -Y7

*2

0

0

0

0

++

0

0

0

0

0

*9

+

30

-2

0

0

0

*2

*3

y7

*2

*3 -Y7

0

0

0

0

++

0

0

0

0

0 y28

ind

1 2

3 6

5 10

577 10 14 14

7 14

28 28

28 28

28 28

15 30

15 30

15 30

15 30

8 8

12 12

20 20

29 58

14

0

-1

-1

-1

o

0

0

0

0

0

-1

-1

-1

-1 b29

14

0

-1

-1

-1

o

0

0

0

0

0

-1

-1

-1

-1

-

28

0

-

28

0

-2 -b5

-

28

0

-2

-

28

0

-

28

0

28

0

28

0

0

0

0

0

0

0 -b5

* -b5

0

0

0

0

0

0

* -b5

*

0

0

0

0

0

0

*7-y15

* -b5

0

0

0

0

0

0

*

0

0

0

0

0

0

* -b5

0

0

0

0

0

29 fus ind 58 *

56 56

*5 *11 *13

*9

*3 *13

*5 *11

56 56

56 56

56 56

56 56

56 56

Xu

o

0

0

0

Xl6

o

0

0

0

Xl7

60 60

60 60

60 60

60 60

OOOOXlS

* b29 -1

o

0

*

-1

-1

o

* -b5

-1

-1

* -b5

20 20

*3 y28

00000000000

-1

-2-2000000 *

-1

0

0

0

0

0

0

0

0

r3

r3

r3

r3

x'"

0

0 y20

*7

0

0

0

0

0

0 y20

*7

*9

*3

X21

o

0

0

*3 y20

0

0

0

0

0

0

*3 y20

*7

*9

X22

*7

0

0

0

0

0

0 *13 *11 *23-y60

X",

*3 y20

0

0

0

0

0

o-y60 *13 *11 *23

X24

*7

0

0

0

0

0

0 *23-y60 *13 *11

X25

*3-y20

0

0

0

0

0

0 *11 *23-y60 *13

X,.

r3

*2

*4

-1

-1

o

0

r3 y20

*4

*7-y15

*2

-1

-1

o

0

r3

*2

*4

*7-y15

-1

-1

o

0

r3-y20

O-y15

*2

*It

-1

-1

o

0

r3

3000002220000000

Or2

0

0

Or2r2r2r2r2r2

0

0

0

0

Xn

30

0

0

0

0

y7

*2

*3

*3 y28

0

0

0

0

X28

30

0

0

0

0

*3

y7

*2

*9

*9 *17 *15 *23

0

0

0

0

X29

30

0

0

0

0

*2

*3

y7 y28

*9 *25 y56 *23 *17 *15

0

0

0

0

X30

30

0

0

0

0

y7

*2

0

0

0

0

X31

30

0

0

0

0

*3

*9

0

0

0

0

X32

30

0

0

0

0

*2

*9 *25 y56

0

0

0

0

X33

-b5

-b5

*7

*9

0

0

0

0

o

r2

0

0

0 y56

*3 y28

0

0

0

0

o

r2

0

0

0 *25 y56

*9

*3

0

0

0

0

o

r2

0

0

0

*3

*3-y28

*9

0

0

0

0

o

r2

0

0

0 *15 *23 *17 y56

Y7

*2

*9

*3-y28

0

0

0

0

o

r2

0

0

0 *17 *15 *23 *25 y56

*3

Y7-y28

*9

0

0

0

0

o

r2

0

0

0 *23 *17 *15

*3

*9 *25 *15 *23 *17

*9 *25

Mult = 2

Out = 2

Constructions Linear

GL (31) :: 15 x 2.G.2 : all non-singular 2 x 2 matrices over F 31 ; 2

Unitary

GU (31) :: 2.(G x 16).2 : all 2 x 2 matrices over F 961 preserving a non-singular Hermitian form; 2

= Go2;

PGU 2 (31)

= 2oG;

=G

PSU2(31)

00 (31) :: 2 x G.2 : all 3 x 3 'matrices over F 31 preserving a non-singular quadratic form; 3

otthogonal

= S03(31) = PS03(31) = Go2;

PG0 3 (31)

~esentation# Maxim~l

SU 2 (31)

G

0 3 (11)

18

=G

= <S,TIS31=(S~TS16T)2=1,(ST)3=T2[=ll> Specifications

sUbgroups Structure

G.2

Character

Abstract

Linear

Orthogonal

31,15

31 :30

1.+31.

N<31AB)

point

isotropic point

A 5

1a+31a+30defg+32abc

N(2A,3A,5AB)

icosahedral

2~8

A 5

1a+31a+30defg+32abc

N(2A,3A,5AB)

icosahedral

32

~65

D

1a+30ddeeffgg+32abcdefg

N(2A)

O2(961), L1 (31)

minus point

30

~96

D

1a+30ddeeffgg+31a+32abcdefg

N(3A)t N(5AB), C(2B)

02(31), base

plus point

Order

Index

~65

32

60

2~8

60

32 30

5...

1

@.@$.,.@ 1~880

P power

p' part ind lA

32 A A 2A

15 A A 3A

@

@

@

@

@

@

@

@

@

@

@@@

@@@@@@@@

16 15 15 16 16 15 15 15 15 16 16 16 16 31 31 A A A A BA AA BA AA A B A B ~ A ~, A A A A AA BA AA BA A A A A A A ~A 5A B* 8A B* 15A B*2 C*4 D*7 16A B*3 C*7 0*5 31A B** fus ind

o

15

-1

0

-1

0

0

-1

-1

0

0

0

0

b31

**

o

15

-1

0

-1

0

0

-1

-1

0

0

0

0

**

b31

+

30

-2

0

-2

0

0

2

2

0

0

0

0

x,

+

30

-2

0

2

0

0

0

0

0

0

0

0 -r2

x.

+

30

-2

0

2

0

0

0

0

0

0

0

0

x,

+

30

2

0

0

0

0 -r2

r2

0

0

0

0-y16

Xii

+

30

2

o

o

o

o

r2 -r2

o

o

o

+

30

2

0

0

0

0 -r2

r2

0

0

+

30

2

0

0

0

0

r2 -r2

0

0

+

31

-1

x"

Xu

x", x",

.15

15

A AB

BB

30

A AB AB 2B 6A lOA

15 15 15 15 15 16 16 16, 16 16 16 16 16 AB BBA CAA OBA AAA ABCDABCD BB AAA BBA CM. DBA AAAAAAAA B* 30AB*13C*11 D*7 32A B*3 C*9 D*5E*15F*13 G*7H*11

++

+

x"

16

-1

-1

0

0

-1

-1

++

0

0

0

0

0

0

0

0

r2 -r2

r2

-1

-1

++

0

0

0

0

0

0

0

0 y16

r2 -r2

-1

-1

++

0

0

0

0

0

0

0

0

*3

*'7

*5

-1

-1

++

0

0

0

0

0

0

0

0 Y32

o

*5-y16

*3

*7

-1

-1

o

o

0

o

o

o

o

o

0

0

*7

*5-y16

*3

-1

-1

++

0

0

0

0

0

0

0

0

0

0

*3

*7

*5-y16

-1

-1

+~

0

0

0

0

0

0

0

0

-1

-1

-1

0

0

r2 -r2

-1

"',

••

r2 -r2

X4

*3

*7

*5

Xs

*5 y16

*3

*7

X6

*7 *11

x,

*3

*7

*5 y16

*5 y16

*3

*7

*3

*9

*5 *15 *13

*11 y32

*3

*9

*5 *15 *13

*7 *11 y32

*3

*13

*7

X8

*9

*5 *15 *13

X9

*7 *11 y32

*3

*9

*5 *15

XI0

-1

-1

-1

-1

Xli

-1

-1

-1

o

0

o

o

o

o

*

o

o

0

o

o

o

o

o

X13

*

b5

o

o

0

o

o

o

o

o

Xl4

*7 y.15

*2

*4

OOOOOOOOX1S

y15

*2

OOOOOOOOX16

-1

0

2

2

o

-1

o

o

0

o

++

2

-1

2

2

-1'

+

32

0

2

0

b5

*

00b5*b5*

o

o

0

o

++

2

2

b5

*

+

32

0

2

0

*

b5

OO*b5*b5

o

o

0

o

++

2

2

*

b5 .1,'b;5 :~ /~~

b5

*.io;;P~

+

32

0

-1

0

b5

*

0

0

*7 y15

+

32

0

-1

0

*

b5

0

0

+

32

0

-1

0

b5

*

0

0

+

32

0

-1

0

*

b5

0

ind

1 2

~

3 6

8 8

5 10

5 10

16 16

16 16

o

16

·0

o

o

o

16

0

o

-

30

0

0

r2

-

30

0

-

30

-

-1

r2 -r2

o

0

-1

r2 -r2

o

32

-1

r2 -r2

-1

·1

++

+

0

oooooooox,

0

-1

0

+00000000

-1"i -1

-1

X12

*2

*4

0

0

0

0

++

2

-1

b5

*

*4

*7 y15

*2

0

0

0

0

++

2

-1

*

b5

*4

~7

*2

*4

*7 y15

0

0

0

0

++

2

-1

b5

*

*2

*4

*7 y-15

OOOOOOOOX17

0 y15

*2

*4

*7

0

0

0

0

++

2

-1

*

b5 y15

*2

*4

*7

OOOOOOOOX18

15 30

15 30

15 30

32 32

32 32

32 32

32 32

31 fus ind 62

~

12 12

20 20

20 20

60 60

60 60

60 60

0

o

0

0

0-b31

..

00000000

o

0

o

0

0

0

0

0 y16

*3

0

0

9'

0 y32

*3

*9

*5

-1

-1

o

0

0

0

0

0

0

0 -r2

0

0

*5 y16

0

0

0

0 *11 Y32

*3

*9

-1

-1

o

0

0

0

0

0

0

0

r2

0

0-y16

*3

0

0

0

0

*7 *11 y32

*3

-1

-1

o

0

0

0

0

30

0

0 -r2

0

0

*5-y16

0

0

0

0 *13

*7 *11 Y32

-1

-1

o

0

0

0

-

30

0

0

r2

0

0 y16

*3

0

0

0

0-y32

*3

*9

*5

-1

-1

o

0

0

-

30

0

0 -r2

0

0

*5 y16

0

0

0

0 *11-y32

*3

*9

-1

-1

0

-

30

0

0

r2

0

0-y16

*3

0

0

0

0

*7 *11-y32

*3

-1

-1

o o

-

30

0

0 -r2

0

0

*5-y16

0

0

0

0 *13

*7 *11-Y32

-1

-1

-

32

0

-1

0

2

2

0

0

_1

-1

-1

-1

0

0

0

-

32

0

2

0

b5

*

0

0

b5

*

b5

*

0

0

-

32

0

2

0

*

b5

0

0

*

b5

*

b5

0

-

32

0

-1

0

b5

*

0

0

*7 y15

*2

*~

32

0

-1

0

*

b5

0

0

*~

*7 y15

32

0

-1

0

b5

*

0

0

*2

32

0

-1

0

*

b5

0

0 y15

-

15 30

6~

M

6~

6~

6~

6~

6~

6~

6~

6~

6~

6~

6~

6~

6~

6~

o

0

0

0

0

0

0

0

X19

0

y6~

*3

*9 *27 *17 *13 *25 *11

x"

0

0

*21 y6~

*3

*9 *27 *17 *13 *25

Xn

0

0

0

*3

*9 *27 *17 *13

X23

0

0

0

0

*19

*3

*9 *27 .17

XZ4

0

0

0

0

0

*15 *19

*3

*9 *27

x"

0

0

0

0

0

0

*3

*9

xu

0

0

0

0

0

0

0

*23

*7 *21 y6~

*3

x,.,

o

0

0

0

0

0

0

0

*29 *23

*7 *21 yM

x",

0

o

r3

0

0

r3

r3

r3

r3

OOOOOOOOX'"

0

0

o

0 y20

*7 y20

*7

*9

*3

OOOOOOOOX'"

0

0

0

o

0

*3 y20

*7

*9

OOOOOOOOX31

0

0

0

0

o

r3 y20

*2

0

0

0

0

o

r3

*~

*7 y15

0

0

0

0

o

r3-y20

*2

*~

0

0

0

0

o

r3

*7

31 62

60 60

**-b31

1

*3 y20

*7 *13 *11 *23-y60

*3 y20-y60 *13 *11 *23

*7 *21 y64

*7 *21 y64

*7 *21 y6~

*5 *15 *19

o

*7 *21 y6~

*5 *15 *19

0

*5 *15 *19

0

0

0

0

0

0

Xn

OOOOOOOOX:u

*7 *23-y60 *13 *11

OOOOOOOOX"

*3-y20 *11 *23-y60 *13

OOOOOOOOX"

[21]

Mult

=2

Out • 2

Constructions A1 ternating

Sa :: G.2 : all permutations on 8 letters; Aa :: G : all even permutations; 2.G and 2.G.2 : the Schur double covers

Linear

GL 4 (2) :: PGL 4 (2) :: SL 4 (2) ;: PSL 4 (2) :: G : all non-singular It x 11 matrices over F 2 ; ~ints

the

Pi and planes Qi of the corresponding projective 3-space can be numbered (mod 15) so that

= {Pi,Pi+5,Pi+10,Pi+"Pi+2,Pi+4,Pi+S};

Qi

G.2 is obtained by adjoining the duality (graph) automorphism, interchanging points and planes

6

00 (2) ;: PG0'6(2) =- 50'6(2) :: PS0'6(2) :: G.2 : all 6 x 6 matrices over F 2 fixing a quadratic form of Witt

Orthogonal

defect 0, for example x,x2 + x3x4 + x5x6; 0'6(2) ;: G;

by taking the 6-space to be the even weight vectors in F~ modulo (11111111), with form equal to half the

weight, we see the isomorphism with S8

G : the stabilizer in M24 of an octad and a point outside it; the group acts as A8 on the octad, and

Mathieu

L 4 (2) on its complement

Presentations

G.2::

_

Maximal subgroups Order

Index

2520

8

1344

15

1344

Specifications

Structure

G.2

Character

Linear

Orthogonal

A7

S7

la+7a

24 :S4'

la+14a

N( 2A 3)

15

2 3 :L (2) 3 2 3 :L (2) 3

3(3,4,8)

point

isotropic plane

la+14a

N(2A3)

3(3,4,8)

plane

isotropic plane

720

28

36

S6 x 2

1a+7a+20a

C(2C)

duad

576

35

2 4 :(3 x3 ) 3 3

(S4x34):2

la+14a+20a

N(2 4 ) • N(2A B6 ) 9

bisection

360

56

( A x3):2 5

35 x S3

18+78+20a+288

N(3A), N(2B,3A,5A)

triad

L (2):2

3

Abstract

point

14

@

@

20160 192 P power

pi part ind 1A

A

@

@

96 180

A A

A 2A 2B

A A 3A

@

@

@

@

18 16 A A

8 15 B A A A A A 3B 4A 4B 5A

@

12 AB AB 6A

@

@

@

6 7 7 BA A A BA A A 68 7A B**

0

@

15 AA AA

15 AA AA

15A Bn fus ind

2C

isotropic point

@

48

A A 20

@

48

B A 4C

@

16 B A 40

@

@

@

18 AC AC 6C

18 BC BC 6D

2

-1

@

6 BD BD 6E

0

-1

-1

0

0

2

2

2

-1

++

9

-3

3-1

000

+

0

0

0

0

0

0

++

10

2

-2

-2

000

++

-1

++

4

-1

-1

0

++

0

0

0

-1

o

0

0 b15

**

-1

o

0

0

**

b15

o

o

0

0

0

b7

**

0

0

0

0

**

b7

0

0

o

-1

0

0

0

0

2

2

+

20

4

4

5

-1

0

+

21

-3

6

0

-1

o

21

-3

-3

0

x,

o

21

-3

-3

0

x,

+

28

-4

4

x,

+

35

3

-5

5

2

x"

o

45

-3 -3

0

0

o

x"

o

45

-3 -3

0

0

o

+

56

8

0

-4

-1

0

0

+

64

0

0

4

-2

0

0

+

70

-2

-2

0

8

o

0

-1

-1

4

10

-1

-1

3 6

2

0

2

3 6

-2

0

6

124

0

5

14

2-5

-1

++

-1

o

-2

-2

-1

o

-1

-1

3

-1

+

x"

A A

-1

4

Xn

720

0

3

x,.

@

@

0

-1

2

[22]

2

7

Xl7

line

@

@

6 4 5 A AC AC A AC AC 8A lOA 12A

++

+

1nd

non-isotropic point

10

+

x"

A1 ternating

0-1

5 12 10

6 6

o

5 -3

-2

0

000

0

0

o

0

0

0

0

0

0

0

++

4

4

0

0-2

-1

o

2

0

0

-1

-1

0

0

0-1

o

-1

+

o

-1

-1

o

000

o

-1

0

o

-1

-1

++

16

0

0

0

0

++

10

-2

-4

0

15 fus ind 30

2

4

8

8

6

6

12

8 8

10 10

24 24

00

0

0

0

0

0

0

0

2i

0

0

o

0

0

0

0

0

0

000

o

0

0

7 14

7 14

15 30

-2

-2

0

0

o

0-1

+

8

0

0

-4

2

0

0

-2

0

0

o

24

0

0

-6

0

0

0

-1

0

0

b7

**

-1

-1

0

24

0

0

-6

0

0

0

-1

0

0

n

b7

-1

-1

48

0

0

6

0

0

0

-2

0

0

-1

-1

o

0

0

0

0

0

0

0

0

r6

o

56

0

0

-4

-1

o

0

o

13

0

0

o

0

0

0

0

0

0

0

0

0

o

56

0

0

-4

-1

o

0

o -13

0

0

o

56

0

0

2

2

0

0

o

0

0

0 b15

**

o

0

0

0

0

0

0

0

0

0

o

56

0

0

2

2

0

0

o

0

0

0

**

b15

64

0

0

4

-2

0

0

o

0

-1

-1

o

0

0

0

0

0

0

0

r5

0

-1

';:3 ~

1

Out:= 2xS 3

(be)2=(cde)5. d=(ab)4,1=[e,(cbaba)5]

>

c3

d

c

d

: M10

: L2 (7):2

: L2 (7):2

: L2 (7):2 : 32 :Q8 x2

2 4 :A

A6

A6

A6

L2 (7)

L2 (7)

L2 (7) 32 :Q8

21

56

S6

56

120

120

120

280

960

360

360

360

168

168

168

72

:M 1O

: M10

S5

22+4. 3• 2 ,

2 4 :A 5

21

G.2 ,

c

b

d

•

b

b3

•

.3

3

2,

1 -1 1 1

c3

b3

d2 d2

b

7:3 x 3

: 32 : 2A 4

\

: 2 4 :(3 xA ) 5 : 2 4 :( 3xAS)

G.3

d

.3 1

_1

1

1

4,

: 32:2A4 x 2

r:6 x 3

3x3 5

1

1

1

-1

42

G.2 2

z3

-1

-1

1

6

z3

-1

1

1

: 32 :QS·2

I

z3

1

1

_1

12, 12 2

: L2 (7)x2

I

: S6

: 24S5

: 24sS

z3

1

,

1

3

cohort

122+4.3.2,

G.6

1

-1

1

•

b3

c3 c

2

.3

22 2 3

6

, ,

coset

: 32 : 2S 4

7':3 x S3

: 2 4 (3xA )2 5 : 2 4 (JxA )2 S

G.3.2 2

acting as (abc)(d) on M. and as (XXIX") on cosets and cohorts.

by characters in the printed cohorts. There is an automorphism

transformed by elements in the printed cosets and represented

960

5

I

: 32 :QS·2

I

: L2 (7):2

[

: PGL 2 (9)

2 x AS

[ 22+4. 3• 2,

G.2 3

whose full automorphism group is 6U 4 (3).2 (see U4 (3»

abc := a4 = b 4 = c 4 = d3 = 1. The table shows how these are

Index

I

e

V

c 5 d

I

g3,

The multiplier Mis generated by elements a,b,c,d satisfying

\

8 b

Order

structure


stabiliZes the complex lattice 1\

12 .G.2 :: 2 3

6.G.2

=

Mult := 4 x 4 x 3

2 .3 .2, 3 3 x AS

4

: 32 : 2S 4

\ (7:3x3):2

I

I i+

G.3. 2 3

: 32:Q8.2 x 2

I

1 1

3 3 x Ss

22 +4 .3 2 .22 ,

: 32:234 x 2

7

la+208+35abc+45ab+64a

1a+20a+35c+64a

1a+208+35b+64a

1a+20a+35a+6 1Ia

1a+208+3Sc

1a+20a+35b

1a+20a+35a

1a+20a

1a+20a

Character

6

11 11;-- -

1'-1- -

It

\7:6XS 3

\

I

10

G.D'2

122·G 1 12 2'·GI

12j.G 1 12'(G 1

11 11

BB EJF EJF 6'.GI 6".G I

: L2 (7):2 x 2

I

11

2

I

0

!

0

!

!42.G.2 2

14t.G.22

12.G·2

I

0

3

!42.G.2 3 I

0

0

0

0

1

14t.G.231

12.G.2

7

6

3

I

I

N(JA 2 )

N(2A, 3A, 4C, 7AB)

N(2A, 3A ,4B, 7AB)

N(2A,3A,4A,7AB)

N(2A,3A,3A,4C,5AB)

N(2A,3A,3A,4B,SAB)

N(2A,3A,3A,4A,5AB)

N(2A 4)

N(2A 4 )

Abstract

Specifications

6

2

!lhG.2

L (2) 3

L (2) 3

L3(2)

oval

oval

oval

line

point

Linear

6

1 6,G,2 3

I

6

5

ennead

heptad

heptad

heptad

hexad

hex ad

hex ad

pentad

point

Mathieu

I

7

4'1~BBEJF,rF,r '

4i·G1

2 x Ss : A6 '22

I

11 11

BF

4{.GI 4;'.GI

1

1

11

11

BB EJF 8

G.2.i G.23'

~EJBB~~~~~GIO

2'.GI 2".GI

22+4. 3 .2 2 ,

G.i

GL (1I) i 3.G.3 : all non-singular 3 x 3 matrices over F 4 ; 3 PGL (4) i G.3; SL (4) i 3.G; PSL 3 (4) G; 3 3 G.2 etc. are obtained by adjoining the duality (graph) automorphism; 3 rL (4) • 3.G.3.22' PrL (4) • G.3.22' IL3 (4) • 3.G.22' PIL 3 (4) • G.2 2 3 3 H .S -= G.3.2 : the set-stabilizer of a triad in M24 ; M21 i G : the pointwise stabilizer; 2 21 3 the remaining 21 points form the projective plane whose lines are the 21 pentads which complete the triad to an octad

Maximal subgroups

Presentation

Lattice

Mathieu

Linear

Constructions

Order:= 20,160:= 2 6 .3 2 .5.7

Linear group L (4) i A2 (4) 3

G.2 2 G.2'j

"-"

~

~

~ W

~

0"5-3'0

0

o

+ 35

o 115 -3

63 _1

63

+

+

+

x,

X.

x,

X.

X.

x..

0

_1

0

_1

0

o

_1

_1

+

+36110000

x"

x" x..

0

0

O-b5

o

0

o

0

0

_1

_1

OOOI_b500

o

90

2

0

-2

0

0

0

_1

_1

1( 1 l 5 5 7 7 :2 :2:2:2:2

0

02

)(20

8

8

2

-1

_1

0

0

12

12

0

0

0

0

8

0

0

8

:2

b7

.

,

u

0

2'

28

2

12

12

3

-b5

o •

o

-2

80

02

02

X;lO

x"

and no:

no:

ami

+

+

02

0

11 •

2

2

2

8 8

6 :2

4

2

16 :2

:2

6 16

and no:

+2

16 :2

8 :2 8 :2

16 :2

" "

•• •• ""

16 :2

16 16

8 :2

8 :2

8 :2

16 :2

16 :2

8 :2

16 16

8

8

8 :2

8 :2

8 :2

8 :2

16 :2

6 :2

6 :2

ahd123881l5577 no: :11 :2 :4 :4 :4 :11 :11 :4

1

+

+

2

4

• •,

6 :2

6 :2

:2

00000ub7

00000b7

0

+

J 02

•

8 8

0

++0000r2-r2

2

_1

-1

o

0

0

11

8 8

6 6

0

++0000r2r2

7 fUB 1nd 28 14 28

-1

0

_1

and no:

0

0

0

o

o

-1

7 28 111 28

0

and12381l85577 no: :4 :2:11 :11 :11:4:4:11

80

64

02

X29

_1

11

o -b5

o

-2

_1

0

5 20 10 20

0

5 20 10 20

0

8

0

2

8

X28023600200

0

28

02

Xv

0

?8

02

Xu

0

20

02

X2S

2

2

•, • •

1

o

++003000

2

'od

o

_1

_1

_1

_000200

J +2

•

11

I

o

_1

_1

_1

+000000

7 fue 1nd

:2

00000Il.b7

0

o

7 28 111 28

:2

C A

se se

B A

+000000

and no:

-1

0 _.- 0

-1

8

andt238845577 no: :4 :2 :4 :2 :4 :4 :4 :4

0

0

o

-1

++

2

80

0-1

0

• -.b5

o o o

and no:

02

x~

80

0

o

5 20 10 20

5 20 10 20

~b5

:2

:2

SA

A A

~

~

~

: 000

: 000

+

~

~

(us
• • •

+000000

+

++

++

++

and1238485577 no: :4 :2:11 :2 :11:4:4:4

02

)(23

Xno2MO

)(21025602000

02

3

2

1

'nd

•• • •

:2

:2

:2

no:

and1231111115577

and1234 no: :2 :2 ~2

+

+70-2-22000000

X'9

x" x..

-4

28

lOCOO.lb?

+

..

20000b7

x"

14

2

M

0

0

9

_2-22000

++

ind

8

A A AB A A AB 2B liD 6A

72

••••••

77fusind2

0

0

b7

0

0

14

5

10

5

10

_1

• -b5

•

o o ..

11

10

+

o b7 ..

o

_1

o

-lI

o

o

0

0

o

o

0

0

o

-1

o

_1

A A

A A 7A

an (us

7

7

0

BI

0

_1

x"

28

5

0

SA

_1 -b5

3

o

2'

-1

0

4C

3-1

_1

-1

_1

_1

5

AAA AAA

x"

10

0

liB

3 -1

0

IIA

ind12311 2 2' 6 11

611

3 _1

35

+

X.

3 _1

3-1

35

+

2'

x,

11

+

20

+

3A

AAA AAA

9 16 16 16

x,

64 A A :?A

x,

pI part ind lA

p power

20160

!ef!@@f!!@@@

Linear group L3(1I) = '\2(11) ; K21 ., F1212+6 = Fll1+

o

o 0

o

0_1

o

0

3

3

•

•

0

0

b7

u

0

-1

8.

7 AC BC

-1

_1

lI_b500

-b5

o u-

o

o

o _1

o b7

-1

21A

o

BII

7 BC AC

o

o o

1SA

5 AB BB

o

o

sa

AB

BB

5

o 3

-1

3C

5

3B

60 21 11 A A BA A A BA

••• • ••• ,,, 'od

o

o

o

++

o o o

+00+00

2

-1

0

2

III

o o

3 _1

0

o o b1

0

0

0 -_1

-1

0

7 BC

b7

u

0

0

-1

-b7

o u

o

4

00

-1

8

8

o

-1

-1

8 8

0

-1

11

10 10

_1

2

2

-1

-1

o

0

0

0

+.

o

0

_1

11

b5

14

•

6

0

o

o

o

o

0_1_1 0-1

0

o

o

_1

III

-b7

_1

• • • •

02

002

_1

28

28

-b7

u

_1

I

•

•

+2

+

+

+

• • • +

111 fUB 1nd fus 1nd fus 1nd

• • • •

2

2

_1

_1

12

0

0

8 B

0

0

8 8

2 2

o

,,

o

8 8

o

o

• 8

0

0

0026000

002

: 002

2

, 12

• , •

2

10 10

o

_1

I

b5

10 20 10 20

I ++

++

++

+

,I +:

++

++

++

++

+

++

+

++

++

10 fUB 1nd fUB 1nd 10

-1

lI-bS

-1 -b5

++6000

++

++

14 fUB 1nd ruB 1nd rue 1nd

_1

o o o o

01+1

0

6

8

002

14

14 14

_1

B 14 B 28

•

12

2

12

B

0

•

002 002

0

•• •, •

-2

• •• •• • 2' ,. 2

2 2

6

002

02

002

fUB 1nd

0

o

+

+

02

fUB 1nd

•

0

-1

o

-1

b5

11

• • • • •

•

10 fUB 1nd fue 1nd 10

0

_1

b5

10 fUB 1nd rue ind 20 10 20

++0002r200

_1

0

o o

o o

o o o o

o

++002r2000

0

0

6 6

o

o

0

0

_1

+000000

2 2

o

o

5

++

8

++

2

++

++

++

+

+

++

000002100

6

++

+000000

00

u

-b7

o

0

2 2

++

+

+

00

++

+

++00_2200

++

BII fuB 1nd fUB 1nd

5 AD BD

5 BD AD

++

00

++

A

8

A A 8E 8F lOA

A

8

++

14 fUB 1nd rus 1nd ruB 1nd 14

fUB 1nd

o 14 14

-1 8 8

0

8

++

3

AD A AD 2D 6F

A

••

60

++

00

00

,I +:

++

++

A AC BC 8D 14A BU fus 1nd rue 1nd ruB 1nd

• 7 A AC

3 _1

o

7 -1

6

8 3 A AC A AC 4E 6E

•••••• 168 A A 2C

+000000

00

00

+

++

++


_1

_1

2

BD BD 6D t2A fus

3 CB CB

+00000

~o

++

+00+00

+00+00

6C

• BB BB

•••

Xv

XU

"" XU

Xv

",.

x"

",.

x"

Xv

x"

XU

x..

x" x..

x..

XU

x"

x"

x"

x"

x'"

X"

X8

'"

X.

X.

X.

x,

X2

X,

"-'"

~

~

~ u,)

~

.

15

21

115

45

02

02

02

02

02

x)~

x~

X;J.'

x~

x"

2

•

3

2

3

3 6

0

11 12

12

12 11 12 12

0

11 12

0

12

11 12

0

'-20200

••

• ••

11

15 10 15 30

5 30

_1

60..ll

90

02

X47

42

-1

21

t4

-1

21

42

u_b7

6 12

3 12

3

11

8

24

,q

~

21

011

x"

0

0

0

2

o

0

-2

-2

o

•

_1

0

o

0

-1

-1

0

o • -b5 -1 o o o o -b7 o o o o ..

o

8 :6

16 :6

16 118 Q8 16 1f8 118

%3+1

0

0

o 0

o _1

o

1

0

.7

7

7

•• 1

-b7

-1

02

olf

6

:2

:6

16

8

:6

:6

16

211616168 :3 :3 :2 :6 :6 :6

4

:3

2

:3

+

+

o

o

02

.

•

o

o

+

• +

+

+

+2

1

• • • •

•

•

•••• •• •• •• I.

•• ••

.5

15

+

02

011

02 02 02

.5

'5

'5 •5

005 '5

.5 'S .5 .5

••

+2

l. ..

+2

••

o. '5

+2 +2

••

12

111 rus 1nd fus 1nd fus ind

8 28 28 8 ,. 14 8 28 28

8 8

10 20 10 20

K~

x..

+

+

• • •

K.,

x.. x.,

'"

+

x"

x~

2

2

••

• '2

"

2 "

•

12

2

8

8

8

8 8

10 10

'5

'5

.5 .5

N

..

••••

•• •• •• ••

10 rus 1nd fus 1nd 10

'5 .5 '5 .5

10 fus ind fus ind 20 10 20

K~

x"

x~

x..

x~

x..

x"

x"

Xw

x" x..

x.,

x"

x.,

8

8

+

• •

+2

10 fus ind fus ind 10

•

+

+

+2

1 1

x"

••

2

8

x,.

+

02

fus ind

+2

••

+

02t02Xn

B' 10 fUll 1nd futt 1nd

+2

+

+

+

111 fus ind fus 1nd fus ind 111

• • • •

2 2

8 10 8 10

8F lOA 8 10

•

02

1

8E 8

.+·+X35

+

6F 6

+

02

12

+

02

• •

+ 02

2D 2

+ •

•

111 fUll ind fus ind fus ind 111

I

o o

+

•

+

•

I

02

'5

,.

111 111

111 111

•

02

111 fus 1nd full 1nd full 1nd

all

••

8

8 8

8 8

80 ll1A 8 n

02

"

"" ••

• " • " 2

6E 6

"" ••

liE 11

'5

02

2 2

2 2

2e 2

l. ""

02

•• ••

fus ind

I

+

fus 1nd

•

02

o

o

•

+

o

0_1

o

o

13

0000002 -i3

0002

0002

0000002

]"2

15

6 8 16 16 621111848 211 48 118 8 16 16 211 118 118 2Q 118 118

16 :6

8 :6

16 48 Q8 16 118 118

z3+1

'2

12

-, J 0"

and no:

811

011

0

o

o

.7

and1238845577 no: :12 :6 :4 :3 :3 :12 :12 :12 :12 :12

60

60

ot!

K~

0

0

o

" 12 12

16 :6

16 :6

8 211 2Q 8 211 211

-213+3

0

• o

o

o o

• -bS

-bS

3

o -, o _,

x63

.'8

o

••

002000'+

'''''

60 12' 12 fus 1nd

6 18

6C

•5

2 • •

6 :2

Q 6 :3 :2

11 :3

6 6

0002

0002

o

0

0

63

63

118

0

0

B.

63 fU.

o x63

z3

o

B. 21A 15 63 15 63 15 63

o

o o

'3

o

15 15 15

15'

o o o

o

0

6 • •

55

0-1

0

o

~213-3

9

3C

3 3 3

35

02

7 fus ind 811 112 28 21 8Q 111 811 21 28 42 8iI

and no:

1f8

olf

x"

0

7 811 112 28 21 811 111 811 21 28 112 84

and1238t!85517 no: :12 :6 :t! :3 :12 :3 :12 :12 :12 :12

118

011

x~

-bS

123118855 12 12 12 12 211 211 60 60 6 6 6 12 211 211 30 30 11 11 12 11 20 20 3 6 12 15 15 12 12 12 60 60 2 11 10 10 12 12 60 60 12 15 15 3 11 11 20 20 12 30 30 6 12 12 60 60

X»043600200

'0'

2 :3

and no:

02

and12388115577 no: :12 :6 :11 :3 :3 :6 :12 :12 :12 :12

.7

0" 2 :3

0

7

I

7

0"

and no:

0

-,

_1

b7

••

2 " 12 6• .12

and12381185577 no: :12 :6 :11 :3 :6 :3 :12 :12 :12 :12

0

oQ 120

X~2

0

oQQ80000

K"

0

_1

-b5

o -bS o •

0llQ80000

Kw

0

..

b7

_1

-1

•

_1

_1

0

42

28

J

7 rus ind

8Il

M M 8q ~ 15 15 21 21 20202828 3030q2q2 60 60 ~ 8q

lq

10

10

28

~

811

7

-1

15 21 M 8q

20

~

60

5

0

15 M

20

~

60

5

0

-1

M

24

8

0

01l2qOOOO

12 12

11

12 N

12

0

x..

12

6

12

-2

o o

12 3 q 6 12

2

6

11

6

11

2

12

1

12

0

oq2qOOOO

ind

2

K"

x~

02000000

02000000

002000200

snd2116888 no: :3 :3 :2 :6 :6 :6

0

8 211 21f

and12341145577 no: :6 :6 :2 :3 :3 :6 :6 :6 :6 :6

0

8 2Q 24

8 211 211

and2116888 no: :3 :3 :2 :6 :6 :6

0

8

0

and123114115577 no: :6 :6 :2 :3 .6 :3 :6 :6 :6 :6

0

O_b7u

b500

8

0

0020000r2-r2

0

0

I

12

•

8

0

o o o

6242112Q 211 24 24

6

O.

o

0020000r2r2

"

'2

•

2

o

2

6

o

-1

0

0

02

0

x....

0

o260-ll

x..~

0

0211220200

x...

002

02

1 fus ind

0

21 111 21 112

7 112

0

x.~o211220200b5'OO

15 10 15 30

5 30

-1

o

0002

'-bS

0002

0002

o

_1

-1 -b5

o o o o

0

-1

02

o o

_1

o

_1

-1

-1

l-t

-1

'od

_1

o

3

0

-1

,

,.2'

8 fus

se

_1

02

: 002

-1

0

3 -1 3

0

-1

3

2B liD 6A SA 8B 2 11 6 8 8 61221124 6 12 211 211

o

0

002

002

: 002

7 fus lnd 21 21

an

b7 ..

o

1A 7 21 21

0002

o

0

0

0

BI 5 15 15

b7

0

0

0

SA 5 15 15

o o ••

3

-1

_1

4C 11 12 12

o o

-1

-1

3

_1

liB 11 12 12

o

-1

-1

3

IIA 11 12 12

o

0

0

0

0

3A 3

x.~o2361l0000

02

•

,

'0'

X~,

2

811

02

-1

-1

X..,

63

63

02

02

x~

-3

-3

5

-1

_1

-1

2A 2 6 6

X)Il

15

15

02

1A , 3 3

Jl')2

,

VJ

"-'"

~

~

~

Mult = 2

Out

2

Constructions Unitary

GU 4(2) :: 3 x G ; all 4 x 4 matrices over F4 preserving a non-singular Hermitian form;

Orthogonal (2)

PGU 4 (2)

= SU4(2)

G06 (2)

= PG06(2)

- PSU 4 (2)

=G

_ S06(2) :: PS06 (2) :: G.2 ; all 6 x 6 matrices over F 2 preserving a non-singular quadratic form of

Witt defect 1, for example x~+X1x2+x~+x3x4+x5x6; 06(2) :: G Orthogonal (3)

G0 5 (3) :·2 x G.2 ; all 5 x 5 matrices over F PG0 5 (3)

= S05(3) = PS0 5 (3)

_ G.2; 0 5 (3)

3

preserving a non-singular quadratic form;

=G

G : the automorphism group of a generalized 4-gon of order (3,3) consisting of 40 vertices and 40 edges, each object being incident with 4 of the other type Symplectic

SP4(3) : 2.G : all 4 x 4 matrices over F 3 preserving a non-singular symplectic form;

Weyl

G.2 : the Weyl group of E6 , generated by the 36 reflections in the minimal (root) vectors of the E6 lattice; these may be taken as 11111111 ( 0,0,0.0,0,0;1,-1) (1), (1,-1,0.0,0,0;0,0) (15), (2'2'2'-2'-2'-2;2'-2) (20) (first base) 1 1 1 1 1 1 1 1 or ( ±1,±1,0,0,0;0.0,0) (20), ±(±2'±2'±2'±2'±2i 2'2'2) (even sum) (16) (second base) where the coordinates before the semicolon may be freely permuted. The coordinates are obtained from the Ea root lattice (see 08(2)) by imposing the conditions x1+x2+x3+x4+x5+x6=0=x7+xa (first base) or x6=x7=xa (second base).

6)

The Weyl group permutes the 27 vectors (which belong to one coset of the lattice in its dual E

5.1111111 5.1111111 221111 a i = (b'-5'-5'-5'-5'-5;~'-2) (6), bi = (0:'-5'-5'-5'-5'-5;-2'2) (6), Cij = (-3'-5'3;'5'5'3;0,0) (15); the fUll automorphism group of the lattice is 2 x G.2 and is obtained by adjoining negation. Reducing modulo 2 shows the isomorphism with Biflection

°6 (2).

2 x G ; the group generated by the 45 complex reflections of order 2 in the vectors (±2,O,O,0,0)C (5), (O,±w,±w,±w,±w)C (40), where w = z3. Reducing mod 8 = i3 shows the isomorphism with

°5

(3).

6 x G : the full automorphism group of the lattice these generate.

Triflection

3 x 2.G : the group generated by the 40 complex reflections of order 3 (triflections) in the (±W)vectors (8;0,0,0) (0;8,0,0) (O;wa,wb,wc ) (wa;O,wb,-W C ), where the last 3 coordinates may be cyclically permuted; the triflection in r takes v to v - (1-w)(v.r)r/(r.r), where x.y = ZXiYi. Reducing mod 2 shows the isomorphism with U4 (2); reducing mod €I shows the isomorphism with S4(3).

Schurfli

G.2 : the group of the incidence-preserving permutations of the 27 lines on the general cubic surface in projective 3-space : a i (6), b i (6), Cij (15) Ci,j = 1, ... 6, iij). These lie in threes in 45 tritangent planes (ai,bj.cij) (30), (cij,Ckl,cmn) (15) (i,j.k.l,m,n, distinct), and two lines intersect if and only if they lie in some tritangent plane. There are 36 "double sixes" such as (adbj}' in which each line of either "six" meets all but one line of the other "six".

Hesse

G.2 : the stabilizer of a bitangent in Hesse's group (see S6(2»

of symmetries of the 2a bitangents to the general

quartic curve in the complex projective plane; the bitangents may be labelled with unordered pairs from £1,2,3,4,5,6,a,b} and the 27 bitangents other than ab then correspond naturally with ai

~

ai' bj

(see S6(2»

~

b j , ij

~

Schl~fli's

27 lines:

Cij. Generating permutations may easily be obtained from Hesse's bifid maps

e.g. (a123Ib456) and (ab12j3456)(a 1)(b 2). The 45 quartets {ab,bj,ji,ia} (30) and {ab,ij,kl,mn}

(15) containing ab yield the 45 tritangent planes. Presentations

G.2 :

Maximal SUbgroups Structure

G.2

Character

Abstract

Unitary

Orthogonal (2)

27

2 4 ;A

2 4 :S 5

1a+6a+20a

N(2 4 ) = N(2A 5B10 )

isotropic line

isotropic point

36

S6

S6 x 2

1a+151>+20a

N(2B,3C,3D,4B,5A), C(2C)

S4(2)

non-isotropic point

40

31+2. • • 2A 4

31+2 : 23 4

1a+ 15a+24a

N<3AB)

non-isotropic point

U3 (2)

base

0'2(2)wr3 3

isotropic point

isotropic line

Order

Index

960 720 648

Specifications

S

648

40

33'S4

33,(S4 x2 )

1a+151>+24a

N(3 3)

576

45

2· (A 4 xA ).2 4

H.2

1a+20a+24a

N(2A)

Orthogonal (3)

[26J

3ymplectic

N<3AB4C3D6)

Weyl

Biflection

Triflection

Schl'Mfli

base

minimal vector of E~

line

minus point

root vector of E 6

double six

isotropic line

point

isotropic point

isotropic line

plus point

reflection trisection reflection

tritangent plane

@

25920 576 p power P': part irid lA

A

A 2A

648

96

A

A A

2B

648 108 A

A

54

48

8

5

A

A

A A

5A

A

A

A

A

A

B A

3A

B..

3C

3D

4A

4B

@

@

@

@

@

72

36

36

18

12

AA

BA

6A

B..

CA CA CA CA 6C D..

DA DA 6E

CB CB 6F

72 BA

AA

@

@

@

9 A A

9 12 B BA A AA

@

12

AA BA 9A Bf. 12A B** fus ind

+

o

5

-3

..

-b27

-1

2

-1

0

**

z3-1

o

5

-3

-b27

**

-1

2

-1

0

z3-1

**

+

6 -2

-3

-3

..

3

0

2

10

2

o

10

2-2

+

15

-1

x.

+

15

x.

+

x.. Xn

X,.

-2 3z3-2

2

i3 -13 -13

13

0

0

200

z3

**

** -z3 **

z3

-z3

0 -2

-1

o

**

0

-1

-1

••

-b27

-1

-1

-1

**

z3 -z3

**

z3

** **

-z3

**

3z3-2

-1

6

6

3

0

7

3

-3

-3

0

3 -1

20

4

4

2

2

5

-1

0

0

0

-2

-2

+

24

8

0

6

6

0

3

0

0

-1

2

2

2

+

30 -10

2

3

3

3

3

-2

0

0

-1

-1

-1

-1

o

30

6

2

3b27

••

-3

0

2

0

0

z3-1

**

-i3

o 30

6

2

••

3b27

-3

0

2

0

0

**

z3-1

0

0

-2z3

2

0

0

-b27

**

-1

-1

-1

3

-1

o

2

2

-1

-1

2

@

720

48

48

A

A

B

A

2C

A

A

@

16

@

18

@

18

@

@

@

@

6

4

5

6

BCCDCDD ACC DC DD

A AC FC A AC CC 8A lOA 12C

4C

4D

6G

6H

6I

000

0

0

0

0

0

o

4

2

-2

0

0

-1

2D

-2

-2

o

5 -3

0

0

-1

-1

++

5

-1

-1

o

0

++

10

2

2

2

0

0

0

0

++

4

4

0

-1

o

0

++

10

-2

13

0-1

o

I)

**

-z3

+

0

13 -13

0-1

o

0 -z3

** +

o

2-1 -1

o

o

o

40

-8

0 3i3-5-3i3-5

-2

000

o

45

-3

-3

-9z3

**

0

0

o

3z3

**

0

0

o

45

-3 -3

••

-9z3

0

0

o

**

3z3

0

+

60

-4

4

6

6

-3

-3

0

0

0

2

'2

-1

+

64

0

0

-8

-8

4 -2

0

0

-1

o

00000

+

81

9 -3

o

o

0

0

-3

-1

o

o

0

0

0

0

2

3

3

3

3

4

8

10

6 6

12

4

6 6

6

666

6 6

6

6

9 18

9 18

12 12

**-2z3

**

Xz

x.

-1

o

-1

o

0

Xs

o

0

-1

X.

0-2

o

-1

0

XlO

-4

0

o

0

-1

Xll

0

0

0

0

0

0

0

0

0

X,.

0

0

0

0

0

0

0

0

0

0

X14

+

0

0

0

0

0

0

0

0

0

0

X16

-1

o

0

0

0

o

X19

-1

o

X20

20 20

24 24

3 -1

o **

z3

0

0

o

z3

**

0

0

0

0

0

0

z3

**

0

0

0

0

0

**

z3

-1

-1

o

0

0

0

++

10

2

-2

-2

o

0

++

16

0

0

0

000

0

++

9

-3

12 fus ind 12

4

2

** -2z3 **-2z3

0

+OOOOOOOOOOX,

++

-2

5

-2

0

0-3i3-5 3i3-5

4

0

0

-8

1

++

0

40

2

+

o

-1

o

ind

@

++

o

x,

@

3-1 8 8

8

-1

2

2

-1

0

-1

-2

-2

000

12

12

6

8 8

\

o

4

0

0

b27

**

-2

2

0

-1

z3-1

**

0

0

13

0

**

z3

**

-z3

o

4

0

0

..

b27

-2

2

0-1

**

z3-1

o

0 -13

0

z3

**

-z3

**

-

20

0

0

-7

-7

2

2

2

0

0

3

3

0

0

0

0

-1

-1

-1

-1

00

0

0 212

0

0

0

0

12

o

20

0

0 3z3+8

••

2

2

2

0

0

-3z3

**

0

0

0

0 -z3

**

-z3

**

+

0

0

0

0

0

0

0

o 20

0

0

•• 3z3+8

2

2

2

0

0

**

-3z3

0

0

0

0

**

-z3

**

-z3

20

0

0 3i3+2-3i3+2

-4

-1

2

0

0

-i3

i3

0

0

i3

0

•• -z3

-1

-1

o 20

0

0-3i3+2 3i3+2

-4

-1

200

i3

-i3

0

0 -i3

0 -z3

-1

-1

36

0

0

9z3

••

0

0

2

0

3z3

**

0

0

0

0

0

0 -z3

**

o 36

0

0

••

9z3

0

0

2

0

**

3z3

0

0

0

0

0

0

**

-z3

60

0

0

-3

-3

-6

0 -2

0

0

3

3

0

0

0

0

0

0

•• -3b27

0

3

-2

0

0

**

z3-1

o

0 -i3

0

0

0

z 3 ••

3 -2

0

0

z3-1

**

0

0

0

0

0

** o

z3 0

00

0

0

0

0

0

0

0

0

i5

0

X:n

o

0

00

0

0 4i2

0

0

0

0

0

0

i2

x,.

o

o

x••

X31

o

60

0

0

x,.

o

60

0

0 -3b27

••

0

64

0

0

-8

-8

4 -2

0

0 -1

o

00000

80

0

0

8

8

2

0

0

o

o

-

-4

0

0

0

13

0

0

-1

••

-1

+OOOOOOOOOOX21

0

0 -12

X:u

0

X,.

0

+OOOOOOOOOOX,.

+OOOOOOOOOOXzs

00

0

0 212

0

0

0

0 -12

0 -12

X30

+oooaoOOOOOX31

X,.

[27]

sz (8)

5uzuki group 5z(8) ': 2 B2 (8) Order = 29,120 = 26 .5.7.13

Mult = 22

Out = 3

Constructions Sz(8)

Suzuki

=G

the centralizer in 5 4 (8) of an outer (graph) automorphism of order 2;

G : all Lt x Lt matrices overF S preserving the set of vectors (t,x,y,z) for which xy+(x s + 2 +ys+z)t=O, where s : x -7 x4 is the automorphism of Fa with s2=2; projectively this defines an oval of 8 2 +1:65 points

on which G acts doubly transitively Orthogonal

2.G has an 8-dimensional orthogonal representation over F ; it may be generated by the maps A and B S which take (x co ,x O, •.. ,x6) (subscripts mod 7) to (xlt,-xS,x2,x,,-x6,xoo,xO,x3) and (x oo ,X"X 2 'X3'Xlt'X S 'X6'X O), respectively, together with the map

C : Xoo xt

~ ~

2x oo + Xo

+

xl + x2

+

x3

Xq + Xs + x6

+

Xoo - x_ t - X3_t + X4_t + 3Xl_t + 3X2_ t

+

3X6_t

G has a 14-dimensional complex representation which we write as a 7-dimensional representation over the

Complex

Quaternion algebra with units 1,j,k,1 and complex coefficients a+bi; G is generated by

A' :(qO .... q6) -7 (-iQO.-ikQ1.-ilQ2,lq3.-ijq4,kq5.jq6) Bt :(QO, ••• Q6) -? (Q1,Q2,Q3,Q4,Q5,Q6,QO)

C'

-4iQn -7 (i+j+k+l)Q_n+(i-l)Ql_n+(i-j)Q2_n+(1-k)Q3_n+(i-k)Q4_n+(k-j)Q5_n+(j-l)Q6_n

this is connected with the fact that one of the involution centralizers in the Rudvalis group is 22 xG

Max imal sUbgroups Order

Index

Specifications

G.3

Structure

Abstract

Character 1a+64a 1a+35abc+64a+65aabbcc

base

N(13ABC)

13: 4

13: 12

20

1456

5: 4

5:4 x 3

N(5A). C(3A)

5z(2)

14

2080

D14

7:6

N(7ABC)

L (8)

@

@

@

@

@

@

@

@

@

@

@

64 16 16 5 7 7 7 13 13 13 A A A A A A A A A A A A A A A A A A A A 2A 4A Bn 5A 7A B*2 C*4 13A B*3 C*9 fus ind

@

@

20 4 4 A AA AB A AA AA 3A 6A 12A

@

1

@

4 5 AA AA AB AA B* 15A

+00

+

[28]

parabolic

560

29120 P power pt part ind 1A

o

14

-2

o

14

-2 -2i

+

35

3

-1

-1

+

35

3 -1

+

35

3 -1

+

64

o

+

65

o

y7

*2

+

65

o

*4

)(w

+

65

o

)(u

+

91

-5

-1

-1

ind

1 2

2

4

4

X12

+

40

0

°

X14

+

40

0

+

56

+

)('9

Orthogonal

52

@

)(6

Suzuki

2i -2i

-1

000

000

-1

-i

i

-1

-1

o

0

0

000

-1

i

-i

-1

o

0

0

0-c13

+

0

-1

0

0

0

-1

0

0

0

2i

*3

*9

0

*9-c13

*3

0

*3

*9-c13

-1

-1

-1

+00

*4

0

0

0

+

Y7

*2

0

0

0

*2

*4

y7

0

0

0

o

0

0

0

0

0

7 14

7 14

7 14

13 26

13 26

13 26

0

0 -y7

*2

*4

0

0

0

.4 -y7

0

0

0

o

0

0 c13

*3

*9

56

0

0

0

o

0

0

*9 c13

*3

+

56

0

0

0

o

0

0

*3

*9 c13

+

64

0

0

0

-1

-1

-1

-1

+ 104

0

0

0

-1

-1

-1

-1

0

0

0

0

o

-1

5 10

.2

and no:

1 :2

2

4

4

5 :2

7 :2

7 7 :2·:2

13 :2

13 :2

13 :2

and no:

1 :2

2

4

4

5 :2

7 :2

7 :2

13 :2

13 :2

13 :2

7 :2

o

0

0

0

0

o

o

o

-1

o

o

o

0

~[~~}

8

2'.GCI~~ 2".G I = +00

-1

-1

11

8

5

L 2 (32) Linear group L2 (32)

= A1(32) = U2 (32) = S2(32) = 03(32)

Order: 32,736: 25 .3.11.31

Mult : 1

Out : 5

Constructions Linear

= 31

GL 2 (32)

~

PGL 2 (32) Unitary

PGU 2 (32)

~

~

SL 2 (32)

= 33

GU 2 (32)

x G : all non-singular 2 x 2 matrices over F

PSL2 (3 2 )

~

G0 3 (32)

Presentations

G

Order

Index

992

33

66 62

32736 power p' part ind lA

(31 x G).5; prL 2 (32)

~ ~L2(32) ~ P~L2(32) ~

G.5

x G : all 2 x 2 matrices over F 1024 preserving a non-singular Hermitian form;

SU 2(32)

~

PSU 2(32)

G

~

33

G.5

Character

Abstract

Linear

Orthogonal

25 :31

25 :31:5

la+32a

N(2A 5 )

point

isotropic point

496

D66

33:10

la+33a-o

N(3A). N(llABCOEF)

°2(32).

528

D62

31:10

la+32a+33a-o

N(31A-O)

02(32). base

@

32 33 A A

2A

3

Specifications

Structure

@

G[~~}

F 32 preserving a

=

Max imal subgroups

p

;

= PG0 3 (32) = S03(32) = PS03(32) = 03(32) = G : all 3 x 3 matrices over non-singular quadratic form; r0 3 (32) = pr03(32) = ~03(32) = P~03(32) = G.5

Orthogonal

@

~

G; rL 2 (32)

32

@

33

@

@

@

@

33

33

33

33

A

A

A

A

@

31

@

31

@

@

@

@

@

@

@

@

@

L

1(1024)

@

minus line ·plus line

@

@

@

@

@

@

@

@

@

@

@

@

@

31

31 31 31 31 31 31 31 31 31 31 31 31 33 33 33 33 33 33 33 33 33 33 A A A A A A A A A A A A DA EA AA BA CA DA EA AA BA CA A A A A A A A A A A A A A A A A A A A A AA BA CA DA EA AA BA CA OA EA 3A 11A B*2 C*4 0*3 £*5 31A B*2 C*4 0*8£*15 F*5G*lOH*11 I*9J*13 K*6L*12 M*7N*14 0'3 33A B'2 C.4 0*8E*16F*10G*13 H*7I*14 J'5 A A

A

A

A

@

A

x,

+

x,

+

31

-1

-2

-2

-2

-2

-2

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

x,

+

31

-1

-2-Yl1

*2

*4

*3

*5

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0 -yl1

x.

+

31

-1

-2

*5-yl1

*2

.4

*3

0

0

0

0

0

0

0

0

0

0

0

0

0

0

x,

+

31

-1

-2

*3

*5-y11

*2

*4

0

0

0

0

0

0

0

0

0

0

0

0

0

x,

+

31

-1

-2

*4

*3

*5-yl1

*2

0

0

0

0

0

0

0

0

0

0

0

0

x,

+

31

-1

-2

*2

*4

*3

*5-yl1

o

0

0

0

0

0

0

0

0

0

0

x,

+

31

-1

1-yl1

*2

*4

*3

*5

0

0

0

0

0

0

0

0

0

0

x.

+

31

-1

*5-yl1

*2

*4

*3

0

0

0

0

0

0

0

0

0

x"

+

31

-1

*3

*5-yl1

*2

*4

0

0

0

0

0

0

0

0

x"

+

31

-1

*4

*3

*5-y11

*2

0

0

0

0

0

0

0

x"

+

31

-1

*2

*4

*3

*5-yl1

0

0

0

0

0

0

x"

+

31

-1

1-yll

*2

*4

*3

*5

0

0

0

0

0

x"

+

31

-1

*5-y11

*2

*4

*3

0

0

0

0

x"

+

31

-1

*3

*5-yl1

*2

*4

0

0

0

x"

+

31

-1

*4

*3

*5-yll

*2

0

0

x"

+

31

-1

*2

*4

*3

*5-yl1

0

0

x..

+

32

0

-1

-1

-1

-1

-1

x"

+

33

o

0

0

0

0

0 y31

*2

*4

*8 *15

x"

+

33

o

0

0

0

0

0 *15 Y31

*2

*4

*8 .13

x"

+

33

o

0

0

0

0

0

*8 *15 y31

*2

*4

x"

+

33

o

0

0

0

0

0

*4

*8 *15 y31

x"

+

33

o

0

0

0

0

0

*2

*4

x~

+

33

o

0

0

0

0

0

*6 *12

x"

+

33

o

0

0

0

0

0

*3

x"

+

33

o

0

0

0

0

0 *14

x"

+

33

o

0

0

0

0

0

x"

+

33

o

0

0

0

0

0 *12

x~

+

33

o

0

0

0

0

0

x'" x"

+

33

o

0

0

0

0

0 *13

+

33

o

0

0

0

0

0

x"

+

33

o

0

0

0

0

0 *11

x"

+

33

o

0

0

0

0

0 *10 *11

Cus

ind

@

@

6 2 A AA

3 AA

A AA

AA

5A lOA 15A

x,

+0000

*2

*4

*3

*5-yl1

0

*5-yl1

*2

*4

0

0

*3

*5-yl1

0

0

0

*4

0

0

0

0

*2

0

0

0

0

0

*8 *16-y33

0

0

0

0

0

0

*4

*8 *16-Y33

0

0

0

0

0

0

0

*2

*4

*8 *16-Y33 *13

0

0

0

0

0

0

0

0 -Y33

*2

*4

*8 *16 *10 *13

0

0

0

0

0

0

0

0

0

*16-y33

*2

*4

*8

0

0

0

0

0

0

0

0

0

0 *14

*5 *10 *13

*7

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0 *13

0

0

0

0

0

0

0

0

0

0

0

0

0 *10 *13

0

0

0

0

0

0

0

0

0

0

0

0

0

-1

*7 *14

*5 *10 *11

*9 *13

*2 *11

*9 *13 *9

*6 *12 *3

*5 *10 *11 *14

*9 *13

*5 *10

*7 *14

*2

*4

*8 *16

*7 *14 *15 Y31

*2

*4

*8 *13

*8 *15 y31

*2

*4

*6 *12 *3

*5 *10 *11

*3 Y31

*7

*6 *12

*4

*8 *15 Y31

*3

*2

*4

*6

*9 *13

*5 *10 *11

*9

*6 *12 *3

*5.10 *11 *14

*9 *13

*5 *10

*9 *13

*7 *14

*5 *12

*3

*7 *14

*5 *10 *11

*7

*9

*5 *10 *11

*9 *13

*5 *10

*5

+OOOX3

*3

*5-yll

*2

*4

*3

x.

*2

*4

*3

*5-yl1

*2

*4

x,

*3

*5-yl1

*2

*4

*3

*5-yl1

*2

x,

*4

*3

*5-y11

*2

*4

*3

*5-yl1

x,

*2

*4 *14 *2

*5 *10 *13

*7 *14

*7

+OOOXk

*5 *10 *13

x.

*7 *14

*5 *10

*7 *14

x"

*8 *16-y33

*2

*4

*4

*8 *16-y33

*2

x"

*2

*4

*8 *16-y33

Xl~

*5-y33

*2

*4

*8 *16

x"

*5 *10 *13

*7 *14 *16-y33

*2

*4

*8

x"

-1

-1

-1

-1

-1

-1

-1

-1

-1

-1

+0000

2

0

_1

XliI

o

0

0

0

0

0

0

000

+

0

0

0

X19

*7 *14

*5 *10 *13

*7 *14

*5 *10

*7 *14

*7 *14

Xw

0

0

0

Xn

0000000

0

0

X2"

0000000

000

x"

0

000

0

0

x"

00000

0

0

0

0

0

x"

000

0

0

0

0

000

+OOOXZ4

000

0

000

000

X2~

o

0

0

0

0

x"

0000000

000

x"

000

000

x"

0

0

0

0

0

0

+

0

000

0

*7 *14

*2

*4

*8 *15

00000

000

0

0

+OOOX29

*7 *14 *15 Y31

*2

.4

*8

o

0

000

000

0

0

x"

.8 .15 Y31

*2

*4

o

0

0

0

0

000

0

0

x"

.2

o

0

0

0

0

000

0

0

x"

000

0

0

0

000

x"

*6 *12 *3

*7 .14

*7

*6 *12

*4

*8 *15 y31

*3

*2

*4

*6

*8 .15 y31

0

0

*9 *13

*3 Y31

*5

*3

*5 *10 *13

*3

*9 *13

*4

x"

o

*6

*2

*5

*6 *12

*5 *10 *11

*9 *13

*2 *11

*7 *14

*6 *12 *3

*3

x,

-1

*8 *15 Y31 *10 *11

*6 .12 *3

*5 *12

*7 *14

*6 *12

*7 *14

*9 *13

*7 *14

*9 *13

*5 *10 *11

*8 *15 Y31 *10 *11

*6 *12 *3

+0000

0

[29J

U 3 (4)

Unitary group U (4) 3

= 2 A2 (4)

Order = 62,400 = 26 .3.5 2 .13

Out = 4

Ml1'lt

Construction

= 5 x G: all 3 x 3 matrices PGU 3 (4) = 3U 3 (4) = PSU 3 (4) = G;

Unitary

GU (4) 3

over F,6 preserving a non-singular Hermitian form;

Design

G : the automorphism group of a Steiner system 3(2,5,65) whose blocks are the sets of 5 isotropic points corresponding to vectors orthogonal to a non-isotropic vector

Specifications

Maximal sUbgroups

Structure

G.2

2 2 +4: 15

22+4: (3 xO

Order

Index

960

65

300

208

5

150

416

52 :3

39

1600

13:3

x AS

lO

)

010 x AS

3

G.4

Character

Abstract

Unitary

22+4: C3xOlO )'2

1a+64a

N(2A 2 )

isotropic point

(010 x AS)'2

1a+39ab+64a+6Sa

N(SABCO)

non-isotropic point

1a+39ab+S2abcd+64a+65a

N(5 2 )

base

N(13ABCO)

U 1 (64)

S2:0 12

S2:(4 x 3 3 )

13:6

13: 12

~BEJ22 2 6

@

@

@

@

@

62400 320 15 16 A A A pi part A A A ind lA 2A 3A 4A

300 A A

300

300

A A

A

5A

B..

p power

[30]

x,

•

x,

-

12

-4

x,

o

13

-3

25-3&2

X.

o

13 -3

x,

o

13

x,

o

x,

@

@

@

@

@

25

25

A

300 A A

A A

A A

C*2

0*3

5E

C*

..

-3

-3

2

2

*2

*3

-bS

*

**

z5-3&2

*3

*2

*

-3

*3

.2

zS-3&2

..

-b5

*

-bS

13

-3

*2

*3

zS-3&2

*

+

39

7

0-1

-3b5

-3bS

*

*

x,

+

39

7

0 -1

*

-3bS

b5+2

x,

o

52

11

o

4z5+3&2

..

-3bS

*2

*3

x"

o

52

11

o

. . 425+3&2

*3

*2

x"

o 52

4

o

*3

*2 4zS+3&2

x"

o

52

4

o

*2

*3

**

4zS+3&2

x"

+

64

0

o

4

4

4

4

x..

•

65

-1

5

5

o

o

x"

o

65

-1

5z5

..

5

*2

*3

o

o

x"

o 65

_1

5z5

*3

*2

o

x"

o

65

_1

*3

*2

5z5

x"

o

65

-1

*2

*3

..

**

x..

o 75

-5

0-1

o

o

Xw

o

75

-5

0-1

o

x"

o 75

-5

0-1

x"

o 75

-5

0-1

0

0

-3

-3

*

n

..

@

20 CA

@

@

20 OA

BA

AA

BA

10A

B**

20

CA C*7

@

4

@

@

@

@

@

@

@

13 13 13 13 15 15 15 15 A A A A OA CA AA BA A A A A AA BA CA OA OA 0*3 13A Bn C*5 0*8 15A Bn C*2 0*8 fus ind 20

AA

@

@

3

8

A

8 5 A FB

EB

A

A ES

FS

A AB

A AB

..

2B

6A

@

8A Bn 10A

@

@

5

6 3 B AB

B* fus ind

@

@

@

4

4

A

A A

A AB A liB 12A 16A 16B

x,

:+0+0

-1

-1

-1

0

0

0

0

00

0

0

+

0

0

2i -2i

0

0

0

0

:0000

0

0-i+1 i-1

Xz

..

*2

*3

0

0

0

0

z5

**

*2

*3

z5&2

*3

*2

0

0

0

0

**

z5

*3

*2

*3

*2

z5&2

**

0

0

0

0

*3

*2

z5

**

-b5

*2

*3

**

z5&2

0

0

0

0

*2

*3

**

z5

* b5+2

b5

b5

*

*

0

0

0

000

0

0

++

3

0

-1

-1

*

-b5

++OOOOX7

*

*

*

b5

b5

0

0

0

0

0

0

0

0

++

3

0

_1

_1 -b5

*

x.

b5

*

*2

*3

**

-z5

0

0

0

0

z5

**

*2

*3

+

0

0

0

000

+0000X9

b5

*

*3

*2

-z5

**

0

0

0

0

**

z5

*3

*2

bS

-zS

*2

*3

0

0

0

0

*3

*2

z5

**

b5

**

-z5

*3

*2

0

0

0

0

*2

*3

**

zS

-1

o

o

o

o

-1

-1

-1

-1

o

0

0

0

-1

-1

-1

**

* * -1

z5&2 n

..

+

0

0

4

-1

++

5-1

*2

*3

+

*3

*2 **

0

0

0 -z5

o

z5

*3

*2

0

0

0

0

** -z5

o

o

*3

*2

z5

0

0

0

0

*3

*2 -z5

5z5

o

o

*2

*3

..

** z5

0

0

0

0

*2

*3

o

o

o

o

o

o

o

0-d13

**

*5

*8

0

0

0

0

o

o

o

o

o

o

o

o

o

**-d13

*8

*5

0

0

0

0

o

o

o

o

o

o

o

o

o

o

*8

*5-d13

**

0

0

0

0

o

o

o

o

o

o

o

o

o

o

*5

*8

**-d13

0

0

0

0

+OOOOX3

0

0

0

x,

0

x"

••

0

0

x,

0

*3

z5

0

)(.

+

*2

..

60

@

-1

.. ..

5

@

@

0

0

0

x"

0

0

0

0

o

0

-1

-1

:+0+0

o

0

:+0+0

0

000

,,,

2

_1

0

0

X13

-1

_1

Xl4

+OOOOX1S

x" +

0

0

0

0

0

0

x" x"

** -z5 +

0

0

0

0

0

0

+OOOOX19

Xw

+

000

000

x" x"

M 12 Sporadic Mathieu group M12

Order

= 95,040 = 26 .3 3 .5.11

Mult = 2

Out = 2

Constructions

Golay (3) code

2G : the automorphism group of the unique dimension 6, length 12 code over F 3 of minimal weight 6 (the ternary Golay code) which is defined in several ways below. The weight distributions of the code c.lnd the cocode (words modulo the code) are 01 62x132 9 2x220 12 2x'2 and O'12x1222x13232x220, respectively. The minimal weight representatives for the cocode are unique except that words of weight 3 are congrUF:,i;:' in sets of 4 (whose supports are linked threes).

Steiner hexads

G : the

automorphis~

group of the Steiner system 3(5,6,12) of 132 special hexads from a '2-set, such

that each pentad is in just one special hexad. The hexads are the supports of the weight 6 words in the ternary Dolay code. For a group element of shape 24 ,4 or 3 3 ,3, the fixed points together with any cycle form a hexad. Below we give, in the MINIMOG array, 3 involutions of a 4-g,oup whose fixed tetrads (linked fours) are disjoint, and 4 elements of a 9-group any 2 of whose fixed triads (linked threes) form a hex ad.

~I::IIIIII::I MINIMOG

[JTI][[]][[J[IT]

The MINIMOG constructs the code in such a way that both signed and unsigned hexads are easily recognized. Each word is written over W3

j,

three rows (rowo ' row+, row_) of four digits

ea~h,

and

belongs to the code just if: sum of each column = -(sum of rowo) row+ - row_ is one of the

~tr~~de

w?rds:

0000 0+++ 0--- +0+- +-0+ ++-0 -0-+ -+0- --+0 (or in general (a

oo ;8 0 ,a 1 ,a 2 )

= (s;a,8+s,a+2s).

For example we check co 1 ',Imn sums: + + + .f0 + -

thus:

o

rowo

SU!l1:

row

-

+ + +

+

++0-1 +

.f-

o 0 + 1++0rOW :

-

-

-

+

0

A hexad is in the Steiner system just if its column distribution is 320 2 , 3 1 ,3, 230', or 2 2 12 and its Uodd-men-out" form part of a tetracode word. For example

" " "" "" D EJ D D

colunn distribution:

odd-men-out:

030 3

1 1 3 ,

2 0 2 2

1 1 2 2

" " " "

" " ""

" "

"

+ 0 ? -

- ? - +

? ? ? ?

"

+

"

0

"

+ -

(if a point in a column is the only one in (resp. not in) the hexad, it is the odd man out for that column). We find the corresponding Gblay code words using cols and tets: col-col

col+tet

tet-tet

[]+ D + +

-

+

col+col-tet

a col

Do++ [TI]++ 0+ +0

++

~

(Mbdulo the code we have (every col)

qQ_~~

-(every tet), where

a tet

DD

and

t~t~

are the obvious words of weight 3 and

4 corresponding to columns and tetracode words.) Here is a dictionary between numberings of the 12-set and the MINIMOO array:

-eo

e 3 e oo -e 2

6

309

6

3

0

9

-dO

-e5

e9

e 8 -eX

5

2

7

10

18

2

16

13

d4

-e4

e,

e6 -e7

4

8

11

4

shuffle

modulo 11 Mathieu 24

8 12

d 1 d3 . d2 d 6 d8

d,O

-d 5 -d 7 -d 9 -d 11 lexicographic

modulo 23

G : the stabilizer in M24 of a weight 12 word (dodecad) in the binary Golay code. The hexads arise from octads of 3(5,8,24) meeting the dodecad in 6 points. The outer automorphism interchanges the dodecad with its complement. The stand ard pair of dodecad s is

Q

= {Ot

quadratic residues modulo 23},

If we define

n+

= whichever

N

= {co,

non-residues modulo 23}.

n-

of n, -n is in Q

= whichever

of 8/n, -8/n is in N

we convert the modulo 23 numbering of the MOO into shuffle numberings for Q and N, thus:

0

00

1

11

2

22

'9

3

20

4

10

18

15

6

14

16

17

8

5

9

21

13

7

12

=

0+

0-

,+

r

2+

8-

2-

3+

5-

4+ 10-

5+

1-

6+

6-

7+

8+

r

9+

4- 10+ 11- ,,+

9-

[31J

M 12 Modulo 11

The ternary Golay code is a quadratio residue code, modulo 11. Letting Q ;; Woo ;; -~ei

{O,l,3,4,5,9} and N;; {oo,2,6,7,8,X=10 . the words

to,

quadratic residuesl ;;

and wi ;; !eq_i - Ien_i (over F 3 ) and their

negatives are called total words, and span the code. 2G is generated by the maps

_w

ei ~ eA(i)

Wi

ei -

wi -

wB(i)

ei ~ tee(i)

wi -

+WC( i)

ei -

wi -

wD(i)

eB(i)

eD(i)

- (i) A 1

(upper sign just on

Q)

where A(i) : i+l. B(i) : 3i. C(i) : -l/i and D: (2X)(34)(59)(67). The group

=SL 2 (11).

so

all automorphisms of the projective line are in G. The code is self-dual in the sense that !CiW i ;; 0 just when (Ci) is in the code. There is an outer automorphism E that interchanges the actions on ei and wi' and takes A -7 A- 1 , B ~ Loop

at

C -? c- 1 ;;

-e,

D ~ D.

G is generated by the multiplication maps x -? x.a in the loop {Eoo,Eo' ••• Ex} of order 12 defined by Et.Et

= Eoo = 1,

Et.Et + q

= Et +2q ,

Et.Et + n

= Et _ n

(q in Q, n in N). For each pin L2 (11) there is a

= Ep*(c)

permutation p* such that Ea.Eb = Ec ~Ep(a) .Ep(b) the permutations given above, A*

= A,

B* = B, C*

= D.

and G is generated by the p and p*. For

These results extend (with some subtlety) to

the loop of order 24 defined by Et.Et = -E oo ' Et.Et+ q = -E t +2q , Et.Et + n = Et _ n , whose right mUltiplications generate 2G.

Orthogonal (2)

G is the intersection of two subgroups A12 in 010(2), as follows: the even SUbsets of {0,1, .•. 11} modulo complementation form an orthogonal 10-space over F 2 with quadratic form q(X) :

~IXI. The reflections x

-? x - (x.r)r in the 11 duads r : {0.11 •••• {10.111

generate one subgroup 3 12 , while those in the 11 hexads with sum 21, namely r =

{1,2,3,4,5,6} {0,1,2,3,7,S} {0,1,2,4,5,9} {0,1,3,4,6,71 {0,1,2,3,5,10} {0,1,2,4,6,S} {O,2,3,4,5,1} {O,1,2,3,6,9} {O,l,3,4,5,8} {O,1,2,5,6,7} {O,l,2,3,4,11}

generate the other. The outer automorphism of 0;0(2) interchanges the generators in the order 3t~iner

given. The above hexads generate the

Shuffle

system by reflection and complementation.

G : the case n = 12 of the following construction: If n112 we define

~

to be the group of permutations of {O,l, ••• n-l} generated by the reversal and

R : t _

Mongean shuffle permutations:

n-l-t

S : t _

min{2t .2n-1-2tl.

If mn = 12, the stabilizer in G of the canonical partition 0 .•• /n ••• /2n ••• / ••• into

o

m sets of n is a SUbgroup Mm x Mn acting on one of the standard arrays displayed:

1

o o

o 1 2 3 4 5 6 7 8 9 10 11

1 7

6

2

3

4

o

5

7

8 9 10 11

8

012 345 678 9 10 11

1 2 3 6 5 4 9 10 11

2

1

3

3 2 4 5

4

7

6

8 9 11 10

The 3teiner system is the same as that given under the "Orthogonal" construction above. The outer automorphism

R -,., R, 3 -,., 3-

5 6 7 8

9 10 11

1 conjugates Mm x M to M x Mm. n n

The shuffle numbering is related to the modulo 11 numbering as follows:

Shuffle:

o

2

3

4

5

6

7

8

9

10

11

8

6

2

X

7

Modulo 11:

00

9

3

4

:;

o

(mnemonic:

00

-2

3

4

5

o -3

6 -9 -12 -15)

G x 2 : the group of permutations on {0,1,2, ••• ,23} generated by the turnaround permutations Td : qd + r --, qd + d - 1 - r

(0

i

r

< d) for each divisor d of 24

There is an outer automorphism taking Td to T24/d.(T24)d +24/d Hadamard

2G : the automorphism group of any 12 x 12 Hadamard matrix (that is, a matrix with orthogonal rows and entries ±1); the outer automorphism interchanges rows with columns. Particular definitions of Hadamard matrices are (modulo 11 ) : a ij = ±1 according as i is in Q or N

(shuffle) : a .. : (_1)[(ij+i+j-1)/13) lJ An automorphism is a monomial permutation of rows that has the same effect as some monomial permutation of columns. Lexicographic

The code is generated by 6 words, written (d O, ••• d 11 ), each the lexicographically earliest word that differs in at least 6 places from all linear combinations of its predecessors:

000000111111 Presentations

G

000011001122

000101010212

001001012021

= G3.5.5.5.5,5.5 = "

[32]

010001021201

100001022110

M 12 Max imal sUbgroups

Specifications

Structure

G.2

Character

12

M ll

L (11):2

7920

12

M

1440

66

1440

66

MlO :2 ;; A6 '2 2 MlO :2;; A '2 2 6

660

144

L ( 11) 2

L (11):2. 3

432

220

1+2: 0 3+' 8

432

220

M :3 ;; 32 :23 4 9 3 M :3 ;; 3 2 :23 4 9 3

240

396

2 x 3

192

495

M8 ·3 4 ;; 21+4' 33

192

495

4 2 :0 12

72

1320

A x 3 4 3

Order

Index

7920

2

ll

Mathieu 12

Golay (3) code

1a+ lla

point

± weight 1 cocode word

1a+llb

total

± weight 12 code word

la+ 11a+54a

N(2B. 3A. 3A, 4B. 5A)

duad

± weight 2 cocode words

la+ 11 b+54a

N(2B • 3A. 3A. 4A, 5A)

hexad pair

± weight 6 code words

1a+ 16ab+45a+66a

N(2A. 3B, 5A. 6A. 11AB)

projective line

la+11a+54a+55a+99a

N( 3A 2)

triad

± weight 9 code word

1a+11b+54a+55a+99a

N (3A 2)

linked threes

± weight 3 cocode word

(22xA5 ):2

la+16ab+45a+54aa+66a+144a

N(2A). N(2B, 3B. 5A)

2 x 6 array

M2 x M6

H.2

la+llab+54aa+55a+66a+99a+144a

N(2B)

tetrad

: H.2

1a+ 16 ab+45a+54aa+66a+99 a+ 144a

N(2B 2 )

linked fours

N(2A 2 ). NC3B)

4 x 3 array

2

5

Abstract

34 x 3

5

3

@@@@@@@@@@@@@@@

95040 240 192 A A p power p' part A A ind lA 2A 2B

54 A A

3A

36 A

32

32 B

A

B A

A

3B

4A

4B

10 12 A BA A BA 5A 6A

6 AB AB 6B

8 A A

8A

x,

+

x,

+

11

-1

3

2

-1

x,<

+

11

-1

3

2

-1

x,

o

16

4

0

-2

o

0

o

0

x,

o

16

4

0

-2

o

0

o

x"

+

45

5

-3

0

3

-1

x,

+

54

6

6

0

0

o

x,

+

55

-5

7

x·,

+

55

-5

-1

XlO

+

55

-5

-1

x"

+

66

6

2

3

0

Xl2

+

99

-1

3

Xu

+ 120

0

-8

x"

+ 144

XIS

+ 176 ind

@

@

8

10 11 11 120 AA A A A A A A A AA 8B 10A l1A B.. fus ind 2C

24

B

B A

4C

@

@

@

6 10 BC AC A BC AC 40 6C lOB 12

A

@

@

@

@

10 6 6 AC AD BC AC BD AC C. 12A 12B

6 BC AC C.

M x M 4 3

++

3

-1

o

3 -1

-1

o

-1

o -1

-1

o

0

-1

o

0

0

-1 bll

••

0

0

-1

o

-1

-1

o

0

0

0

-1

-1

2

2

-1

-1

o

3 -1

o

-1

-1

3

0

-1

-2

-2

0

3 -1

-1

_1

-1

3

o

0

o

0

o

400

-3

0

o

-1

-4

o

-4

-1

0

o

1 2

4

2 2

3 6

3

4

4

6

o

-1

-1 -1

-1

-1

o

+

0

0

++

5

-3 0

0

0

0

0

-1

o

0

-1

++

0

o

0

0

++

5

o

0

0

o

0

0

o

0

-1

0

o

-1

0

0

-1

-1

r5 -r5

o

0

++

6

2

0

-3

-1

0

:

++

:

++

o

o

0

++

4

o

2

-1

-1

++

4

o

-2

-1

-1

11 fus ind

2

4 4

8 8

6

10

10

o

0

o

o

o

-1

-1

o

o

o

1

0

o

5 10

12

6 6

8 8

8 8

20 20

11 22

22

i2

i2

0

-1

-1

00

0

2

i2

0

0

0

-i2 -i2

0

-1

-1

00

0

2 -i2

0

++

0

0

0

00

0

0 2i2

o

10

0

-2

-2

0

0

0

0

x"

o

10

0-2

-2

0

0

0

0

x,.

+

12

0

4

3

0

0

0

2

0

x,"

-

32

0

0

-4

2

0

0

2

0

X20

o

44

0

4-1

2

0

0

-1

x"

o

44

o

4 -I

2

0

0

x"

o 110

0

-6

2

2

0

x"

0

110

0

-6

2

2

x"

+ 120

0

8

3

XlS

o 160

0

0

x,.

0

160

0

0

-1

o

0

0

0

0

0

-1

-1

o

o

0

i5

0

0

-1

o

o

0 -i5

0

0

0

0

0

0

0

0

0

0

0

0

0

0 -i2

i2

0

0

0

0

0

0

0

0

-1

o

0

0

-1

-2

-2

0

0

0

0

0

0

0

0-bl1

-2

-2

0

0

0

0

0

0

0

0

0

i2 -i2

0

0

o

0

0

0

0

-1

+000000000

o

x,,,

0

•• bl1

-1

0

o

+000000000

:

o

o

o

o

-1

-1

-1

0

0

o -1

r3 -r3 0

0

o

0

24 24

12 12

12 12

i2

-1

-1

0

0 -i2

-1

-1

0

0

0

r3 -r3

0

0

0 -i2

0

0

EJEls

GE}, 's

9

0

+000000000

+

0

0

0

0

0

0

000

-1

++

0

4

0

0

0

0

0

••

+

0

00000000

"-bll

[33]

~

w

Index

50

50

50

126

175

175

175

525

2520

2520

2520

1000

720

720

720

240

= S3

3

2S 5

M10 ;; A6 '2 3

M10 ;; A6' 2 3

M10 ;; A6 '2 3

: 3 x 2S 5

32 : 2A 4

: 2S 5 • 2

I

: A6 '2 2

•• 51+2'8'2 + ••

: <3 x 2S 5 ).2

32 :2S 4

•• 5+1+2 •'24'2 •

(7: 3x3): 2

1a+21a+28b

la+21 a+28a

Character

N(2A)

N( ••• 50 ••• )

N(. .. 5C ... )

la+21a+28b+125a 1a+21a+28c+125a

N( ••• 5B ... )

N(5A)

non-isotropic point

Petersen graph

edge

co-clique

co-clique

N( ••• 5C ... ) N( ••• 50 ... )

vertex

isotropic point

Unitary

N( ... 5B ... )

Abstract

1a+21a+28a+125a

la+125a

: 51+2 : 24

3

62 :0 12 ,

G,S

51+ 2 :8

[ L2 (7):2

: S7

G.2

la+21a+28c

7: 3x3

2 6 :S 3 •

G.3

A 7

A 7 A7

Structure

Specifications

Graph

: the set stabilizer in 2Co 1 of an S-lattice of type 3 1+3 23 (see COl); the pointwise stabilizer is G;

hence G appears in Co 2 ' Co • HiS. and McL, which stabilize various sublattices of this S-lattice. 3

2 x G,S

50 sets (co-cliques) of 15 points no two of which are joined.

from {i); two totals are joined if A7 contains an element of order 5 fixing them both. There are two systems of

stabilizer is A • and the suborbits are 1. 7. 42. The vertices may be labelled by 00 (1). 0.1, ••• 6 (7). and the 7 42 totals (see A6 ) on 6-element subsets of {0.1 •••• 6}. The vertex i joins 00 and the totals on the set disjoint

PGU 3 (5) ;; G.3; SU (5) ;; 3.G; PSU (5) ;; G; 3 3 G.2 : the automorphism group of the Hoffman-Singleton graph. a rank 3 graph on 50 vertices. in which the vertex

Maximal sUbgroups

Order

Out

2 x 3.G.3 ;; GU 3 (5) : all 3 x 3 matrices over F 25 preserving a non-singular Hermitian form;

Mult = 3

Remark: G.2 is a maximal subgroup of the Rudvalis group.

Leech

Graph

Uni tary

Constructions

2 53 .7 Order - 126 .000 -_ 24 .3.

Uni tary group U (5) ;; 2A2 (5) 3

14

11

8

BEJFII13

GBEJ~~'4

G.2' G.2"

w lJl "-" ~

e

~

'CO

6

-6

-6

0

0

2

6 6

+ 126

o 126

o 126

o 144

o 144

1

3 3

Xw

x"

x"

x"

x,.

0

-4

9

84

02

02 105

X,o

X"

X"

0

0

0

o

o

o

0

0

o

o

-1

o

5

o

0

6

-6

-6

0

0

02 105

02 126

02 126

02 126

02 144

02 144

X'3

X24

X,s

X,6

X'7

-6

o

o

-1

-1

-1

-1

-6

-1

o 0 o -b7 o __

o o

o

-1

0

o

o

o

0

0

0

o

o

-1

-1

-1

-1

-1

0 0

__

-b7

0

0

i2

i2 -i2

0 -i2

0

0

0

-1

-1

-1

:0002

:0002

:0002

:0002

:0002

o

o o

o o

-55 -x'63

-55 -x'63

o

o

o o o o

0 -i2

o o o -6

o 6

0

0

0

0

0

i2

o

o

-1

-1

-1 -z3-1

z 3+ 1

o

o

0

z3+2

-z3

30 30 30

o

o

-1

-1

o

-2

o

o

30A

AAB

10 AAB

@

-1

24 24 24

0

0

i2

i2 -i2

0

-6

o

0

o

o z 3+ 1

o

-1

o

o z 3+1

0

o o i3

o

o

o

o

o

o

o

o

o

o 6

6

o

o

-z3

-1

o

o

-z3

24 24 24

63 63 63

63 63 63

15 15 15

12 12 12

o

..

o

-b7

o

0

i2 -i2

0

-2

i3

o -i2

o __

o

o -b7

o

o

o

-1

-1

-1

o

-1

o

o

-1

0

-1

0

000

0

o

@

o

o

o

0 2i3+7

0-2i3+3

0

0

6

:0002-2i3+3

:0002-2i3+3

:0002

-4 -i3-1

o

o -4i3

o

o

o o

o

o

o

o

o

o

o

-1

0-2i3-1

:0002 2i3+3

-i3

6

6 6

6 6 6

o

000

0

0

0

o 0

0

o o

-2

0

o

-1

2

-1 02

0

o

o

o

-1

o

o

0

-1

o

0

@

788 AC AA AB BC BA BB B_ 24A B_

@

-1

7 BC AC 21A

10 AB AB 15A

2

0

@

@

-1

2

o

0 2i3+3

@

12 8 BA BA BA BA 6C 12A

@

o

o

-6

-6

6

5

8

o

5

-4

240 BA BA 6B

@

-1

20000

-2

0

0

-1

3

0

0

0

-1

-2

-2

0

0

5

0

-2

-1

-2

3-2

-2

-1

-2

-2

o

-2

3

-1

-2

o

:0002 2i3+3

2

3 3 9

o -3 3

0

0

0

0

-1

0

0

o -3

ind

rUB

000

6

10 30 30

0

000

6

6

5

788 21 24 24 21 24 24

0

-1

000

+00

+00

9

o

0

o

000

0

0

i2

-1

o

+00

+00

0

-1

-3

-4

0

0

7 21 21

0

-1

0

+

+00

-00

0

-b7

__

-1

-1

o

0-1

0

0

-1

0

i 2 -i2

0

-1

0

0

0

0

-1

o

+00

A 3C

A 3B

8A Bn 10A

ind

A

rUB

21

@

A

@ 240

8 A A

@

8 10 A AA A AA

@

@

0 -i2

0

0

-1

2

6 6

6

o __

o 0 o -b7

-1

-1

-4

5 15 15

-1

-1

0

o

0

0

o

o

-1

-1

000

5 15 15

-1

-1

0

-1

-1

2

7A Bn

-4

5 15 15

-1

-1

9

0

o

o

o

48

02

X,9

o

o

o

48

02

X18

o

o

o

48

02

X17

0

21

02

X,6

5

21

0

5 15 15

4 12 12

3

02

-3

-6

-6

0

0

0

0

-2

0

0

-1

o

-1

0

-1

3

500

0

0

0

0

XIS

ind

5-1

+ 125

X9

0

-3

+ 105

9

x,

0

84

+

X7

-4

3 -2 -2

o

4

28

+

-2

0

@

3-2

X6

-2

3

o

4

28

3

0

3 -2

0

+

Xs

-4

-5

3

+

x.

3

0

o

+

X3

2

4

5

21

-

x,

28

-4

20

+

x,

@

7 A A

@

7 A A

@

126000 240 36 8 250 25 25 25 12 p power A A A A A A A AA p' part A A A A A A A AA ind 1A 2A 3A 4A 5A 5B 5C 5D 6A

@

@

@

@

@

@

@

+

+

++

++

++

++

+

++

++

00

++

1

+2

+2

+

+

+

+

02

+

+

+

ind

1

-

-

-

-

I

-

-

-

rUB

I

rUB

@

@

0

4

5

0

5

2

0

0

4

0

0

6 -6

5

5

4-4

0

4

0

6

0

0

0

-1

-1

0

0

6 120 120 A AB A A AB A ind 2B 4B 6D

@

@

@

@

@

8

0

0

0

-1

o

0

o

-1

0

rUB

++

++

+

++

10

0

0

12

0

20

0

0

-1

o 0

o -1

20

0

0

-1

1

1

*

* * *

-j

+

*

+2

+2

+

+

+

+

+

02

+

*

ind

+

+

++

++ 0

0

-1

0

00

o

0

-1

o

i5 -i5

++

ind

++

-1

0

-1

0

rUB

o

-1

0

-1

0

4 5 6 10 10 A BB AB AB AB A BB AB AB AB 8C 10B 12B 20A Bn

@

1

+2

+2

+

+

+

+

+

02

+

+

ind

+

+

++

++

++

++

++

+

++

00

++

ind

1

*

* *

-*

1

*

*

rUB

rUB

X,7

X'6

X,s

x,.

X23

X22

X21

X20

X,9

X18

X17

X16

X,S

X,.

X13

X"

Xl1

XtO

X9

Xs

X7

X6

Xs

x.

X3

X2

Xl

"-"

.--... Ut

c

t.Jj

J1

Sporadic Janko group J, Order = 175,560 = 2 3 .3.5.7.11.19

Out

Mult

Constructions G is a sUbgroup of G2 (11) : in the 7-space of pure imaginary Cayley numbers over F 11 , spanned by i t

Janko

(subscripts mod 7), G may be generated by the elements

S : it -?-2i_ t

R : it '""-?.:!:i t (- for t:O,3,5,6)

Q : it ~ i 2t

P : i t --?it+l ;

(i,_t+ i 2_t+ i 4_t) + 3Ci6_t+iS_t+i3_t).

+

In the corresponding projective 6-space there is an orbit of 1596 (isotropic) "Whitelaw points ll each lying in two out of 266 normal sex tic curves of 12 such points.

G : the automorphism group of an l1-valent graph on 266 points in which the point stabilizer is L2 (11) and

Graph

the suborbits have size 1 (distance 0), 11 (distance 1), 110 (distance 2). 132 (distance 3) and 12

(distance 4, or opposite). There are 11 involutions indicated by superscripts i,j from {0,1,2, ••• ,S,9,X}, the involution

° being

DCDCD below. In these termS the 266 points can be written P (1), pi (11), pi,j

(110), ~ (12), and Qx i (132), for x = 00,0,1, ••• ,9,X; p is joined to pi and opposite to Qx • The group is generated by the elements A: t ----7t+1, B: t ----73t, on i,j,x C : (2X)(34)(59)(67) on i,j, and (roO)(1X)(25)(37)(48)(69) on x

pt,t+q

D

,

Q

q-t

Sq-t) (pt,t-q

p-t,-t-q) (Q t+q p3q-t,6q-t) ( Q t-q Q -t-3q t' t ' -t-2q

'

P, Qoo ) ( pt , Q_t-t ) ( Q ' Qoo-t ), (t = 0,1, ••• ,9,X, q = 1,3,4,S,9) t In the 20-space over F 2 of vectors v = ~xtet + ~Ytft (subscripts mod 11) with LX t = ~Yt =

Orthogonal (2)

° and

invariant

quadratic form q(v) = !xtY t , the above generators act as follows:

A

et ----7 et+ 1

f t ----7 f h1

B

et ----7 e3t

f t --l>f 3t

C

eO ----7 eO

f O ----7f O + eO'

et ----7 e1/t

f t ----7 f 1/ t + eO + e 3 / t + e 4 / t + eS/ t

et .-..., e9/t

f t -? f 9 / t + eO + e3/t + e4/t + eS/ t + e1/t + eS/t

D : et .-..., f_ t b S c a

G=<

Presentation

f

where et = !e i (i i t) (t = 1,3,4,S,9) (t

= 2,6,7,S,10)

t .-..., e_ t

d S e I (abc)S=1, (cde)S=a

>

Remark : G is the centralizer in the DINan group of an outer automorphism of order 2.

Maximal subgroups Order

Index

660

Specifications

Structure

Character

266

L ( 11)

1a+S6ab+76a+77a

168

1045

2 3 :7: 3

N(2A3)

120

1463

2 x A 5

N(2A)

114

1540

19:6

N(19ABC)

110

1596

11:10

N(11A)

opposite points

Whitelaw point

base: {et,f t }

60

2926

06 x D 10

N(3A), N(SAB)

pentagon

non-isotropic point

point

42

4180

7:6

N(7A)

2

Abstract

Graph

Janko

Drthogonal (2)

point

sex tic curve

10-space

{et}

base : {±it} edge

point :

eb

non-isotropic point

point: (1111111)

@@@@@@@@@@@@@@@

175560 120 30 P power A A pt part A A ind 1A 2A 3A

[36]

30 A A SA

x,

+

X:

+

S6

0

2-2bS

X,

+

S6

0

2

X.

+

76

4

X,

+

76

-4

X.

+

77

X,

+

77

-3

x~

+

77

X.

30 6 A AA A AA B* 6A

7

10 A BA A AA 7A lOA

10 11 15 AA A BA BA A AA B* 11A 15A

19 19 19 A A A A A A ~* 19A B*2 C*4 15 AA BA

*

0

0

0

0

bS

*

-1

-1

-1

*-2bS

0

0

0

0

*

bS

-1

-1

-1

-1

-1

-1

-1

o

0

0

-1

-1

1

1

_1

1

1

0

0

0

2

2

-1

0

0

0

0

-1

-1

2

bS

*

0

0

bS

*

0

bS

*

-3

2

*

bS

0

0

*

bS

0

*

bS

+ 120

0

0

0

0

0

o

0

-1

o

0 c19

Xw

+ 120

0

0

0

0

0

o

0

-1

o

X"

+ 120

0

0

0

0

0

o

0

-1

o

X"

+ 133

5

-2

-2

-1

o

0

0

000

X"

+ 133

-3

-2

~b5

•

0

0

b5

•

-bS*OOO

X"

+ 133

-3

-2

* -bS

0

0

*

bS

X"

+ 209

S-1

-1

-1

-1

-1

o

-1

15

*2

*4

0

*4 c19

*2

0

*2

* -bS -1

8

*4 c19

0

0

0

o

0

0

15

Ag

Alternating group Ag Order

= 181.440 = 2 6 .34 .5.7

=

Mult

2

=2

Out

Constructions Alternating

59 ; G.2 : all permutations of 9 letters; A9 ; G : all even permutations; 2.G and 2.G.2 : the Schur double covers

Orthogonal

2G has an 8-dimensional real orthogonal representation, which may be generated (with sUbscripts modulo 7) by A : et ~ et+1' B : et ~ e2t' C : et ~ e_1/t' D : et ~ ±et (- for 0,3,5,6) and ~

E : 4e oo

-2eoo-3eO-e3-e5-e6

4eO

4en -? -2en-3e3n+eoo-e5n+e6n (n

= 3,5,6)

~

-2eO+3eoo+e1+e2+e4

= 1,2,4)

4e q ~ -2eq+3e5q-eo-e6q+e3q (q

which permute (oo.0.... 6.Zl as (0123456). (124){365). (ooO){16){23){45). (ooO){23){15){46). and (OooZ)

respectively Presentations

G = <X1, ••• x7Ixr:(XiXj)2:1); G.2

=

Maximal sUbgroups Order

Index

20160

9

5040

Specifications

Structure

G.2

Character

A 8

38

1a+8a

36

3

57 x 2

1a+8a+27a

C(2C)

duad

2160

84

(A

1a+8a+27a+48a

N(3A)

triad

1512

120

L (8):3 2

1a+35a+84a

N(2B,3B,7A,9A)

projective line

1512

120

L (8):3

1a+351>+84a

N(2B,3B,7A,9B)

projective line

1440

126

(A

1a+8a+27a+42a+48a

N(2A 2), N(2A, 3A, 5A)

tetrad

648

280

33: 54

33:(2 x 34)

1a+27a+48a+84a+120a

N(33)

32:2A4

2 3 : 25 4

216

840

@

@

7 6

x 3):2

36 x 3

3

2

5

x A ):2 4

3

5

x 34

Abstract

n

x,

+

8

x,

o

x,

o

x,

+2773900

X.

+

28

x,

+

35

-5

3

5

-1

x,

+

35

-5

3

5

x,

+

42

6

2

x"

+

48

8

Xu

+

56

x"

+

= N(3A3B4C6)

@

@

@

•

BEJl2 18

14

trisection

N(3B 2 )

1 181440 480 192 080 81 54 24 16 60 24 6 7 9 9 20 12 15 P power A A A A A A B A A A C B A B B AA AA AA pt part A A A A A A A A A A C B A A A AA AA AA ind lA 2A ~ g ~ ~ U ~ ~ M ~ ~ lOA 12A 15A +

~E}8

point

@@@@@@@@@@@@

x,

Al ternating

5(2.3.9)

@

@

@

@

@

@

040 144 240 16 A A A A A A A A fus ind 2C 2D 4C 4D

72 AC AC 6C

72 AD AD 6D

18 CC CC 6E

18 CD CD 6F

9 BD BD 6G

3

-1

0

2

_1

@

@

@

@

@

@

@

@

@

5 15

AA AA

B**

4 10 12 7 10 B AC AC AC AC A AC AC AC AC 8A lOB 12B 14A 20A

++ 4

0

o

0

5

-1

2

2

3

21

-3

-3

3

0

-1

o

21

-3

-3

3

0

-1

-1

-1

0

0

0

-1 b15 **

o

0

0

0

-1

o

-1

0

0

-1

0

-1

2

o

0

3

2

-1

-1

0

o

0

2

-1

2

-1

-1

0

o

0

0

-3

3

0

2

-3

0

-1

0

6

3

0

0

0

-2

2

-4

0

11

2

2

-2

0

-1

84

4

4

-6

3

3

0

0

x"

+ 105

5

15

-3

-3

-1

x"

+ 120

0

8

0

3

-3

0

0

0

x"

+ 162

6

-6

0

0

0

0

-2

-3

x"

+ 168

4

0 -15

-3

0

-2

0

3

o

x"

+ 189 _11

-1

x"

+ 216

4 -4

10

-1

o

-3

9

0

0

-4

0

-9

0

0

2

0

-1

-1

2

0

0

++

6

2

4

0

** b15 -1

-1

++

15

3

5 6

3

3

0

2

-2

-1

000

0

0

0

-1

-1

0

-1

-1

0

0

0

0

0

0

0

-1

0

-1

2

0

o

0

0

-1

0

-1

o

-1

0

o

-1

0

0

0

0

-1

0

0

++

14

-2

-1

0

-1

0

0

+

0

0

-1

2

0

-1

0

0

0

0

0

o

0

0

++

14

2

-4

0

2

2

0

_1

0

0

++

20

4

0

0

2

-2

0

0

-1

-1

++

14

-6

4

0

-1

3

2

-2

o

0

0

_1

0

-1

-1

++

14

-2

-6

2

2

-2

-1

-1

o

0

0

0

-1

0

0

++

35

-1

5

-1

-1

-1

0-1

000000

++

20

4

0

0

o

0

++360-6-2000000

0

0

0

_1

o

0

0

0

-1

-1

0

-1

0

0

7 14

9 18

9 18

20

24 24

0

000

o -1

1 2

4

2

3 6

3 6

3 6

8

4

5 10

12

6 6

x"

+

8

0

0

-4

-1

2

0

0

-2

0

0

2

-1

0

0

x'"

+

8

0

0

_4

-1

2

0

0

-2

0

0

-1

2

0

0

x"

o

48

0

0

6

3

0

0

0

-2

0

0

_1

0

0

0

i6

x"

0

48

0

0

6

3

0

0

0

-2

0

0

_1

0

0

0 -i6

x"

+

56

0

0 -16

2

2

0

0

-4

0

0

0

-1

-1

x'"

+ 112

0

0

4

4

4

0

0

2

0

0

0

x"

o 120

0

0

0

3

-3

0

0

0

0

i3

x,.

o 120

0

0

0

3

-3

0

0

0

0 -i3

x.,

+ 160

0

0 -20

-2

-2

O'

0

0

0

0

_1

x..

o 168

0

0

12

-3

0

0

0

-2

0

0

0

0

x,.

0 168

0

0

12

-3

0

0

0

-2

0

0

0

x,.

+ 224

0

0

-4

-1

2

0

0

4

0

0

0

0

++

14

2

-1

++

21

-3

-4

0

-4

-1

0

-1

o

4-1

-1

-1

2

2

-1

0

-3

-3

0

0

0

-1

15 30

15 fus ind 30

2

4

8

8

6

12

6

12

12 12

8 8

0

0

o

-1

-1

-1

0-1

o

-1

0

o

++6-6-40330000

-1

ind

o

0

0

2-2 0

o

0-1

o

0

0

-2

0

0

-2

-1

+00000000000000

-1

-2

-1

0

10

-1

-1

24

14 14

40 40

+00000000000000

o

0

0

0

0

0

00

0

0

0

0

0

0

0

0

0

-1

-1

00

0

0

0

0

0

0

0

0

0

2i

0

0

0

o

0

-1

-1

00

0

0

0

0

0

0

0

0

0

0

0

0

0 110

0

0

0

0

00000000000000

000000

o

0

0

0

0

0

0 b15

••

0

0

0

0

-1

-1

0

0

00

0

0

0

0

0

0

0

0

0

0

0

0

17

0

00000000000000

.. b15 000000000031000

o

0

[37]

L3 (5)

Linear group L ,S) 3

=A2(S)

Order = 312,000 = 25 .3.5 3 .31

Mul t

Out = 2

= 1

8E}0

Constructions Linear

GL (S) ;: 11 x G : all non-singular 3 x 3 matrices over F S ; 3

PGL (S) ;: SL (S) ;: PSL (S) ;: G 3 3 3

30

Specifications

Maximal subgroups

@

[38]

@

@

@

372000 480 211 P power A A pt part A A ind lA 2A 3A

480

x,

+

x,

+

30

6

x,

+

31

7

X.

A A 4A

@

480

Order

Index

Structure

G.2

Character

Abstract

Linear

12000

31

S2: GL2 (S)

51+2 :[25 ],

la+30a

N(5A 2 )

point

12000

31

S2: GL2 (S)

la+30a

N(5A 2 )

line

120

3100

S5

N(2A,3A,5B), C(2B)

0 (5) 3

96

3875

42 :3

1I 2 :D 12

N(2A 2 )

base

93

4000

3" 3

3"6

N(31ABCDEFGHIJ)

L

@

@

@

16 500

25 A A A A A A A A Bn 4C 5A 5B

@

24 AA AA 6A

8

@

43 5 .2 Ss

3

@

@

@

x 2

@

@

@

24 24 20 24 24 20 20 A B AA AB AA AA AB A A AA AA AB AA AB 8A Bn lOA 12A B** 20A Bu

@

@

24

24

@

@

A8

8A

24 AB

AA

AB

AA

24 BA AB

24A

B..

Cu7

D*7

@

@

1

(125)

@

@

@

@

@

@

@

@

@

@

@

31 31 31 31 31 31 31 31 31 31 6 '20 '20 A A A A A A A A A A A A AB A A A A A A A A A A A A AB 31A Bn C*2Dn2 E*4F**4 G*8H**8I*16J*15 fus ind 2B 4D 6fl

@

@

4

@

@

@

5

6

10

C B8 A BB

AD AD

AD

10 AD

AD 8C 'OB 12C 20C

D*

AD

x,

++

0

6

-5

625

-5

-1

0

o

000

o

o

o

o

0

-,

-1

-1

0

6

-1

-1

2

6

-1

1

0

-1

-1 1-1-1-1

-i

i

-i

61-1

6

1

-1

0

-1

-1-1-1 1-1

i

-i

i

o -1

_1

o

-1

-1

-1

_1

-1

-1

-1

-1

-1

++

0

0

0

0

0

0

0

0

0

++

o

31

-5

x,

o

31

-5

x,

o 96

0

0

o

o

0-4

o

0

0

0

0

0

0

0

o

o

o

o

x,

o

96

0

0

o

o

0-4

o

0

0

0

0

0

0

0

o

o

o

o

** x31 **2

x,

o 96

0

0

o

o

0

_4

o

0

0

0

0

0

0

0

o

o

o

o

*16 *15 x31

x,

o 96

0

0

o

o

0

_4

o

0

0

0

0

0

0

0

o

o

o

o

*15 *16

x"

o

96

0

0

o

o

0

_4

o

0

0

0

0

0

0

0

o

o

o 96

0

0

o

o

0

_4

o

0

0

0

0

000

o o

o

Xu

o

o

o

x"

o

96

0

0

o

o

0

_4

o

0

0

0

0

0

0

0

o

o

o

o

Xu

o 96

0

0

o

o

0

-4

o

0

0

0

0

0

0

0

o

o

o

o

96

0

0

o

o

0-4

o

0

000

0

0

0

o

o

o

o o

**4

x,. x"

o 96

0

0

o

o

0-4

o

0

0

0

000

0

o

o

o

o

**2

x"

+ 124

11

4

4

0

-1

-1

2

2

-1

-1

-1

x"

+ 124

4

4

4

0

-1

-1

-2

-2

-1

-1

-1

x"

o 124

11

_4

_4

0

-1

-1

21 -21

-1

-1

-1

x"

o 124

4

_4

_4

0

_1

-1

-1

-1

-1

h,

o 124

_4

-2

41

-41

0

-1

-1

-21

21

x"

o 124

-4

-2

-41

41

0

-1

x"

o 124

_4

41

-41

0

x"

o 124

_4

-4i

41

Xu

o 124

-4

41

x"

o 1211

_4

Xu

+ 125

x"

+ 155

11

-1

Xu

o 1S5

-1

_1

X~

o 155

_1

_1 -61-5

X~

+ 186

-6

61-1 -61-1 -6i-1

5-1

0

-21

21

2

0

0

_1

2

0

0

21 -21

_1

-1

-1

0

0

-1

0

-1

-1

-1

0

0

-i

-41

0

-1

-1

-1

0

0

1

-4i

4i

0

-1

-1

-1

0

0

-i

5

5

o

0

-1

-1

-1

-,

-1

5

0-1

5

0

-1

1

5

0 -1

-1

11

o

0

-1

61-5 -61-5

6

61-5 6

-2

-i

0

-1

i-i

-1

0

*4 **8

*2 **2

*8 *15 *16

*4 **4

*2 **4

*4 **4

+OOOOOOOOXIO

*2 **4 **

*4

Xu

*2 **2

+00000000X12

** x31 **2

*2

x"

*8 **8 *16 *15 x31

**

+00000000X14

**

x31

x"

*4 n8

*8 *15 *16

00000

0

0

000

++

4

-4

o

-1

o

-1

0

0

0

++

4

4

o

0

0

0

0

0

0

0

0

0

+

0

0

-i

i

-i

o

0

0

0

0

0

0

0

0

0

-1

x" -1

0

0

-1

X17

0 XIS

0

x"

o

o

o

o

0000000000

+00000000X2O

o

o

o

o

o

"7

*1

y'24 **

i

-i

*1

**7

-1

-1

-i

0

*8 **8 *16 *15 x31

x,

0

*7

i

*4 **4

+00000000X8

*8

*4 **8

*2 u2

** x31 **2

*8 u8

0

**1

-1

*8 *15 *16

X3

x,

0

*1

-1

**8

0

*8 *15 *16

*4 u4

*2 **4 **

0

+00000000X6

0

0

y'24

-1

*8 u8 *16 *15 x31

-1

*8 **8 *16 *15

0

0

**

_1

** x31 **2

-,

Or5-r5Xz

x,

*4 **8

*2 n2

0

0

0

0

-i

-1

*4 **4

*2 **4 n

0

0

i

-1

*2 **2

0

0

**1

0

**

0

o

**

0

x31

0

0

+00000000X4

i

y'24

-1

0

0

5

-i

i

-1

o

0

i

-1

-1

i-1 0

i

-i

-, -, -, -, -i

-1

0000000000

0

-i

o

y'24

0

0

0

0

0

0

0

0

x"

0000000000

+OOOOOOOOX22

0000000000

i

i

-i

o

-i

o

x"

000000000

0

+OOOOOOOOX24

o

0

0

0

0

0

0

0

0

0

x"

o

0

0

0

0

0

0

0

0

0

o

0

0

0

0

0

0

0

0

0

o

0

0

0

0

0

0

0

0

0

0000000

0

0

0

-, -,

-i

o

..

0

++

5

++

5

5

o

-1 -1

-1

-1

0

0 X26

0

x"

+OOOOOOOOX28 X~

++

6

-6

0

0

o

-1

_1

X30

M 22 Sporadic Mathieu group M22 Order = ~~3,520 = 27 .3 2 .5.7.11

Mult = 12

Out = 2

Constructions For almost all purposes, M22 is best studied as a subgroup of M24 (q.v.). Thus M22 02: G.2 is the set stabilizer of a

Mathieu

= G is

duad in M24t and M22

=G.2

M22 .2

~

the pointwise stabilizer.

the automorphism group of the Steiner system 3(3,6,22), whose 77 hex ads are derived from the oetacts of

3(5,8,24) containing the fixed duad.

G.2 : the automorphism group of a rank 3 graph on 77 points, with sUborbits 1, 16, 60, and point stabilizer 2 4 :3 6 :

Graph

(Every hexad of 3(3,6,22) is disjoint from 16 others and meets the remaining 60 in 2 points each.) Lattice

2M 22 .2

= 2.G.2

: the automorphism group of a 10-dimensional lattice with coordinates (in Z[b7]) indexed by the

triples of 6 points, modulo complementation. The lattice is defined by requiring all coordinates xabc to be fo~m

congruent modulo b7 and their sum to be divisible by 2b7**, while their sums over sets of the

S(ab.cd.ef)

=

{ace,acf,ade,adf} and S(ab) = {abc,abd,abe,abf} must be divisible by 2 and (b7)2 respectively. There is a monomial group 2 5 :S 6 generated by permutations of {a,b,c,d,e,f} and sign changes on the sets S(ab.cd.ef). With the natural norm ~!lxabcI2 the vectors of norms ~ and 5 fall into four orbits 4X, 4Y, 5X, 5Y. The vectors of 4X are the images under the monomial group of (4,0,0,0,0,0,0,0,0,0) (i7, 1, 1, 1, 1, 1, 1, 1, 1, 1) (1 on S(ab»

(1, 1,1, 1,b7**,b7**,b7**,b7**,b7**,b7**)

(b7 on S(ab), 0 on S(ab.ed.ef)

(b7,b7,b7,b7,0,0,0,0,2,-2) and fall into 77 congruence bases modulo b7**. Unitary

In a 6-dimensional vector space of vectors (ab cd ef) over F 4 with unitary norm ab+ao+cd+ca+ef+eT, the group 3M22

= 3G

is the automorphism group of a configuration of 22 isotropic vector 3-spaces. Each non-zero isotropic

vector appears in just two of these spaces. The coordinates (ab cd ef) of a typical vector in each of the 22 3-spaces are given in the HOG array below, where X,y,z are independent elements of F 4

= x+z,

y

Presentation

3.G

=< a

2

= x+y,

S

= x+y+z,

and At

= wA,

B"

= W'B

= {O,l,w,w},

xO yO zO

SX Sy SZ

SX Styt S"2"

S"X" Styt SZ

xtXt yy zZ

xO Oy Oz

SX YS ZS

SX y'S' 2"S"

S"X" ytS' ZS

xX y"Y" zZ

xX ytY' z2

Ox yO Oz

XS SY ZS

XS Sty, Z"S"

X"S" S'yt ZS

xX yy Z"Z"

xX yY ztZt

Ox Oy zO

XS YS SZ

XS Y'S' S"2"

X"SII ytSt SZ

V

= y+z,

:

X"X" yy zZ

b 5 e

X

d I (eab)3=(abe)5=(bee)5[=(bed)5J=(aeed)~=1 >

adjoin (abee)8 = 1 for G.

e

Specifications

Maximal sUbgroups

Abstract

Mathieu

Lattice

Unitary

point

5X 1-space mod b7

isotropic 3-space

hexad

4X or 5X 1-space mod b7**

isotropic 3-space

structure

G.2

Character

22

L (4)

L 3(4):22

1a+21a

5760

77

2~:A6

2~:S6

1a+21a+55a

2520

176

A 7

1a+21a+154a

heptad

2520

176

A 7

1a+21a+15~a

heptad

1920

231

2~:S5

25 :S 5

13~~

330

2 3 :L (2) 3

720

616

M10

660

672

L 2 ( 11)

Order

Index

20160

3

=A6 '2 3

N(2A 4 )

1a+21a+55a+154a

N(2A~)

duad

4Y l-space mod b7

isotropic 1-space

2 3:L (2) x 2 3 A '22 6

1a+21a+55a+99a+15~a

N( 2A 3), C(2B)

octad

4Y l-space mod b7**

isotropic 3-space

1a+21a+55a+154a+385a

(do) deead

5Y l-space mod b7**

pair of isotropic 3-spaces

L2(11):2

1a+21a+55a+154a+210a+231a

endecad

page 40

8BI2 BBII BI4.G.2 12

10

8

---------------------~

non-isotropic 1-space

,--------------------I page 41

I

BFII BFIO

\

7

! I

\ I .

BI12.G.2 12

10

[39]

M22

@

@

@

@

@

@

443520 384 36 32 16 5 p power A A A A A p' part A A A A A ind 1A 2A 3A 4A 4B 5A XI

+

1

x,

+

21

5

3

X3

o

45

-3

0

x.

o

45

-3

0

x,

+

55

7

X6

+

99

3

x,

+ 154

10

1 -2

Xs

+

210

2

3 -2

X.

+

231

7 -3

1

XIO

o 280

-8

XII

o 280

-8

XI'

+

385

ind

1 2

2 2

XI3

o

10

X,.

o

X"

+

@

@

@

@

1

o

0

b7

**

o

0

**

b7

-1

-1

3 -1

-1

0 -1

0

3 -5

3 -1

0

-1

00

3 -5

3 -1

0

-1

3

++

13

5

-1

0

0

++

15

-1

0

0

++

14

6

2

2 -1

0

++

14 -10

-2

2 -1

0

7 -9

-1

-1

0

0

0

5 -3 -3

0 6 6

o

0 -1

000

o

0

0

0

++

0

0 bl1

**

+

0

o

0

0

++

21

0

3 -1

1

0

-2

0

0

4 4

4

5 10

6 6

7 14

7 14

8 8

11 22

11 fus ind 22

2 2

2

4 4

4

2

2

0

0 -1

b7

**

0

-1

-1

00

4

0

-2

10

2

2

0

0

n

b7

0 -1

-1

00

4

0

56

-8

2

0

0

1 -2

++

0

XI6

+ 120

-8

3

0

0

0

X17

o 126

6

0

-2

o

1

0

XIS

o 126

6

0

-2

o

1

XI'

o 154

2

1

-2

o

Xzo

o 154

2

1

-2

X'I

+ 210

10

3

X"

+

2 -3

XZ3

+ 440

-8

3 6

-1

@

@

@

n

-1

0

o

0

b7

n

o

0

n

b7

-1

-1

-1

0

0 -1

-1

0

0 -1

0

1-1

0

-1

1

0

0

o

0

0

0

0

000

o

000

b11

o

0

8

10 10

0

o

0

-2

0

o

0

0

0

0

0

0

r5

12 12

14 14 -b7

14 14 n

** -b7

0

0

1

1

0

-1

-1

++

8

0

4

0 -1

0

0

o

0

0 bl1

**

o

o

o

o

o

o

o

o

o

o

0

o

0

0

I

+

-1

-1

o

0

2i

0

0

o

o

o

o

o

o

o

o

o

o

o

-1

-1

o

0 -2i

0

0

[

+

2

0

0

++

28

0

2

0

o

0

-1

0

0

2

0

0 -1

000

++

20

0 -2

0 -1

0

0

-1

-1

-1

0

0

0

-1

0

++

8

0 -4

0 -1

0

0

-1

8

5 20 10 20

677 12 28 28 14 14 28 28

11 fus ind 44 22 44

2 2

4

4

8

20 20

24 24

14 14

14 14

1

2

3

4

4

4

12

4

2

6

4

12

X2402560200

0

-1

0

0

-2

@

8 6 5 7 7 A AC AC AB BB A AC AC AB BB 8B 10A 12A 14A Bn

o

0

-1

-1

00

0

-1

-1

-1

0

0 -1

@

1

7 -1

0

-2

@

++

o

2-1

@

1

++

0

0

@

7 7 8 11 11 344 320 48 32 6 A A A A A A A A A AB A A A A A A A A A AB 7A Bn 8A 11A Bn fus ind 2B 2C 4C 4D 6B

-1

-1

@

1

0

ind

[40J

@

0

330

1

12 AA AA 6A

@

1

3

0

@

000

-1

0

0 2z8

o

0

0-2z8

02

56

0

2

0

0

Xu

02 144

0

0

0

0

-1

0 -b7

Xz,

02 144

0

0

0

0

-1

Xu

02 160

0

-2

0

0

X,.

02 160

0 -2

0

0

X~

02 176

0 -4

0

0

X31

02 560

0

0

0

0

8 11 8 44 8 22 8 44

o

Xz,

2

** bl1

1

02

**

0

0

n -b7

0

o

0

-1

-1

0-b11

o

0

-1

-1

o

1

0

0

0

o 0

0

0

0

1

1 n

*

0

*

0

1 -2

n-b11 00* -1

-1

*

8 8

6 6

0

0

0 1

M 22

ind

1A

2A

3A

1

2 6

3

3 3

6

4A

4B

5A

6A

7A B**

8A 11A B**

445 12 12 15 12 12 15

6 6 6

7 21 21

7 21 21

8

11

24 24

33 33

11 fug ind 33 33

2

0

0

-1

-1

-1

o

0

b7

**

-1

o

0

**

b7

-1

-1

0

Xn

02

21

5

0

1

Xn

02

45

-3

0

1

X~

02

45

-3

0

X3S

02

99

3

0

X~

02 105

9

0

X~

02 105

9

0

1

X~

02 210

2

0

-2

-2

X39

02 231

7

0

-1

-1

X~

02 231

-9

0

X'I

02 330

-6

0 -2

2

X'2

02 384

0

0

0

0 -1

3 6

4 12 12 4 12 12

4 12 12

ind

1

2

6 3

6 6

2

2

3 6

6 6

3 -1

-1

* * 0

0

*

**

I

o

0

0

0

1-b11

o

0

0

0

1 **-b11

0

2

0

0

0

-2

0

0 -1

3 -1

000 0

5 30 15 10 15 30

0

o

+

o

0

*

+

00*

+

6 6 6 6 6 6

7 42 21 14 21 42

8 24 24 8 24 24

11 66 33 22 33 66

11 fug ind 66 33 22 33 66

02

66

-6

0

2

0

o b7 **

0

0

0

*

0

X~

02

66

-6

0

2

0

o **

0

0

0

*

0

X.S

02 120

-8

0

0

0

o

-1

-1

*

+

X~

02 126

6

0 -2

0

o

0

00b11

**

X~

02 126

6

0 -2

0

o

1

0

0

0

X~

02 210

10

0

2

0

0

-2

0

0

0

X~

02 210

-6

0 -2

0

0

0

0

0

2i

Xso

02 210

-6

0 -2

0

0

0

0

0 -2i

XSl

02 330

2

0

2

0

0

2

XS2

02 384

0

0

0

0

-1

0 -1

-1

2 12 6 4 6 12

3 12 6 12

4 8 12 24 12 24 4 12 12

5 60 30 20 15 60 10 60 15 20 30 60

6 7 12 84 6 42 12 28 6 21 12 84 14 84 21 28 42 84

7 84 42 28 21 84 14 84 21 28 42 84

ind

1 12 6 4 3 12 2

12 3 4 6 12

0

-2

** b11

*

1

0 -1

-1

*

+

11 132 66 44 33 132 22 132 33 44 66 132

04 120

o

o

o

o

o

o

XS'

04 120

o

o

o

o

o

o

Xss

04 144

0

0

0

0 -1

0 -b7

**

0

XS6

04 144

0

0

0

0 -1

0 ** -b7

0

XS7

04 336

o

0

0

0

000

0-b11

XS8

04 336

o

0

0

0

000

o

XS9

04 384

0

0

0

0

0 -1

0 -1

2z8,-1

-1

6

8

10

12

14

14

2 2

2

4 4

4

6 6

8

10 10

12 12

14 14

14 14

2

4

8

4

6

8

20 20

24 24

14 14

14 14

+2

+

XS3

-1

4

1

*

1-2z8

4

+

0

8 24 24 8 24 24 8 24 24 8 24 24

2

+2

0

o

2

+

X~

b7

8B 10A 12A 14A B**

+2

*

*

6B

+

+

-1

4D

0

*

0 -1

4C

0

0

-1

2C

+

0

0 -1 7 42 21 14 21 42

*

2B

-1

11 fug ind 132 66 44 33 132 22 132 33 44 66 132

L

04

1

**

02

1

**

02

1

-4

-1

2

8

6

-1

** **-b11 -1

* **

-2

[41]

J2 -- HJ

Fs-

j> ;

Sporadic Hall-Janko group J 2 Order = 604,800 = 27 .3 3 .5 2 .7

Mult = 2

Out = 2

Constructions Graph

J 2 .2

=G.2

: the automorphism group of a rank 3 graph on 100 points, with suborbits 1, 36, 63, and point stabilizer

=G2 (2).

U3 (3).2 Quaternionic

2J 2

= 2.G

This graph is one of a series continuing through G2 (4) and Suz.

: a group generated by 315 quaternionic reflections. There is a diagonal sUbgroup 23+4 of right

mul tiplications d iag( g1 ,g2,g3) (gt in (±1,±i,±j,±k}) with g1 g 2g3 = ±1, and the reflecting vectors (roots) are the images under this of (2,O,O)S (3), (O,h,h)S (24), (n,1,n S (96), and (1,wc,bW)S (192) (where W=~(-1+i+j+k)=Z3, b=b5, c=b5*, h=z3-b5). The ring of left scalar mUltiplications is the icosian ring, which contains 2A 5 as a group of units (see AS). The quaternionic reflections x

<x,y> =

~XtYt.

~x

- 2<x,m>m/<m,m> preserve the quaternionic inner product

The Euclidean norm defined by N(x) = a+b when <x,x> = a+h/5 (a,b rational), converts the lattice

generated by the above vectors into the Leech lattice. Some of the subgroups are definable over proper sUbrings, often realizable by complex numbers, for example Presentations

=~(b7l.

G = ; 2'(S3 x G) G.2

=< a

=< a b

5 b c 8 d

c 8 d e l a = (cd)4 > : adjoin (bcde)10 = 1 for 2 x G, (abcdecbdc)7 = 1 for G. e

f I a=(Cd)4,(bcde)8=1 >

Maximal sUbgroups Order

Index

Structure

6048

100

U

2160

280

1920

Specifications

G.2

Abstract

Character

Quaternionic

point



decad



1a+36a+63a

3'PGL 2 (9)

3 (3):2 : 3'A6' 22

1a+63a+90a+ 126a

N(3A)

315

2~ +4 :A 5

1 •• 2- +4 • S5

1a+14ab+36a+90a+160a

N(2A)

reflection

1152

52S

22+4: (3 xS 3)

: H.2

1a+36a+63a+90a+1608+17Sa

N(2A 2 )

base

720

840

A4 x AS

: (A4xA S ):2

1a+63a+90a+126a+160a+17Sa+22Sa

N(2B 2 ), N(2A,3B,SAB)

icosahedron

600

1008

AS x D10

: (ASxD lO )'2

1a+ 14ab+90aa+ 126a+160a+22Sa+288a

N(SAB), N(2B,3A,SCD)

pentad axis

336

1800

: L3 (2):2 x 2

N(2A,3B,4A,7A), C(2C)

300

2016

L3 (2):2 S2: D12

: S2:( 4xS3)

N(S2) = N(SAB 3CD ) 3

60

10080

AS

: Ss

N(2B,3B,SCD)

3 (3 )

: U

0EJ21 BBI7 21

[42]

Graph

11

edge



vector

HJ - Fs-

J2

@

604800 P power pt part

ind

lA

@ @ @ @ @ 1 1 920 240 080 36 96 A A A A A A A A A A 2A 2B 3A 3B 4A

@

@

@

300

300

50

A

A

A

A

A

A

5A

B.

5C

@

50 A A D.

@

24 AA AA 6A

@

12 BB BB 6B

@

@

7 8 A A A A 7A 8A

@

20 BB AB lOA

@

@

20 10 AB DA BB CA B. 10C

@

@

10 12 15 CA AA BA DA AA AA D. 12A 15A

x.

+

x,

+

14

-2

2

5

-1

2

-3b5

*

b5+2

•

-1

0

0

b5

x,

+

14

-2

2

5

-1

2

•

-3b5

*

b5+2

-1

0

0

•

b5

X4

+

21

5 -3

3

0

b5+4

•

-2b5

•

-1

0

0

-1

b5

•

0

0

x,

+

21

5 -3

3

0

*

b5+4

•

-2b5

-1

0

0

-1

•

b5

0

0

x.

+

36

4

0

9

0

4

-4

-4

o

0

0

-1

-1

x,

+

63

15

-1

0

3

3

3

3

-2

-2

o

_1

-1

0

x,

+ 70 -10

-2

7

2

-5b5

•

o

o

-1

o

0 -b5

•

x,

+ 70 -10

-2

7

2

•

-5b5

o

o

-1

o

0

• -b5

x"

+ 90

10

6

9

0-2

5

5

o

o

Xn

+ 126

14

6 -9

x"

+ 160

0

4

lJi

x"

+ 175

15

-5

-5

x..

+ 189

-3

-3

0

x"

+ 189

-3

-3

x,.

+ 224

x"

+ 224

x"

+ 225 -15

5

0

x"

+ 288

4

0

-3

x,.

+ 300 -20

0 -15

x"

+ 336

16

0

1 2

2 2

4

x"

6

-2

0

-3

0

x"

6

-2

0 -3

x~

14

6

0

ind

@

@

@

@

@

@

6 15 336 48 12 AA A A B BC BA A A A BC B. fus ind 2C 4B 4C 6C

@

@

@

@

@

@

48 16 6 6 7 12 A A AB BC AC AB A A AB BC AC AB 8B 8C 12B 12C 14A 24A

@ 12 AB AB B.

x,

++

0

o

2

o

0-1

-1

o

-1

0

0

0

0

• -b5

.-1

• -b5

-1

o

0

o

0

• -b5

0

0

++

7

3-1

0

0

-1

b5

•

+

0

0

0

0

-1

•

b5

o

0

-1

-1

0

2

-1

0

0

0

0

-1

0

0

++

7

-1

+

0

0

b5

•

b5

•

0

0

0

0

0

-3

•

-3b5

•

b5+2

0

0

0

•

b5

•

b5

0

0

0

0 -4

8

-1

0 2r5-1-2r5-1

2b5

*

0-1

o

0

o

0

0

• -b5

0 -4

8

-1

0-2r5-1 2r5-1

•

2b5

0-1

o

0

o

0

0 -b5

•

o

o

o

o

0

0

3

3

_2

-2

0

0

4

o

o

o

o

0

0

-4

-4

3 3 4 664

5 10

5 10

5 10

5 10

2

-2b5

•

*

0

2

•

-2b5

_4

2

2

4

4

0

0

0

0

++

o

-1

-1

o

0

0

0

0

++

0

0

o

0

0

0

0

-2

0

0

0

0

0

6 6

12

7 14

8 20 8

20

b5-1

o

-1

0

b5-1

•

o

-1

-1

-1

0

0

0

0

0

0

-1

-1

X6

-1

0

0

0

x,

000

0

D

0

0

0

-1

0-1

o

0

0

0

-1

-1

-1

0

0

0

0

-1

o

-1

0

x" -1

Xu

0

XIZ

o

-1

-1

X13

0

0

0

X14

0

0

0

0

0

0

0

0

0

0

0

-3

-1

8

0

-2

++

6

-2

3

-1

-1

0

0

0

0

-2

-2

o

0

XIS

0

o

0

X19

o

-1

x,.

0

0

0

0

0

0

0

15 fus ind 30

4

4

8 8

12

8 8

8

12 12

24 24

28 28

24 24

24 24

o

0

0

0

0

0

0

o

0

0

0

15 30

0

0 -b5

•

-1

b5

•

0

0

0

• -b5

-1

•

b5

0

0

0

2

0

5

2

2

o

o

o

o

o

2i

0

0

0

0

-1

0

0

x,.

o

50

10

0

5

2

2

o

o

o

o

o

-2i

0

,0

0

0

-1

0

0

x"

-

56

-8

0

2

2

0

-2b5

•

•

b5-1

-2

0

0

0

0

0

b5

•

0

b5

•

x,.

-

56

-8

0

2

2

0

•

-2b5

b5-1

*

-2

0

0

0

0

0

* b5

0

*

b5

x"

-

64

0

0

-8

-2

0 2r5+4-2r5+4

2b5

o

0

0

0

0

0

b5

•

x",

-

64

0

0

-8

-2

0-2r5+4 2r5+4

•

2b5

o

0

0

0

0

0

•

b5

x"

84

4

0 -15

0

4

-6

-1

-1

x"

- 126 -10

0

-9

0

2 3r5+1-3r5+1

-2b5

•

x"

- 126 -10

0

-9

0

2-3r5+1 3r5+1

•

-2b5

x"

- 216

24

0

0

0

0

6

6

x"

252 -20

0

9

0

4

2

2

x,.

336

16

0 -6

0

0

_4

-4

x"

- 350 -10

0 -10

2-6

o

o

o

o

2

0

0

0

0

0

0

x"

- 448

0

-2

-2

-2

-2

0

0

0

0

0

0

0

o

o

0

0

0

0

-1

-1

-1

0

0

0

0

0

0

0

-1

-1

0

0

0

0

0

0

0

-1

0

-1

0

0

0

-1

-1

0

o

0

0

0

0

0

0

0

X16

0-1

0

12 12

X4

-1

0

10 10

x,

o

++

10

0

0

-1

50

-2

0

-1

o

16

0

o 10 10

r6 -r6

XZl

XZ2

x"

o

0

r2

o

0

0

0-2r2

0

0

0

0

0

0

r2

0

r2

r2

X24

OOaQXZ5

x,.

o

0

0

0

0

0

0

OOOQXZ7

x,. 000

0

000

o

0

0

0

XZ9

x",

o

0

0

0

0

0

r3

o

0

0

0

0

0

0

o

0

r3 -r3

X31

oooaX32

x" 0

0

o

0

0

0

0

0

0

o

-1

-1

o

0

0

0

0

0

r3

o

-1

-1

o

0

0

0 4r2

0

0

0

0

0

o

0

r2

0 2r2

0

0

o

0 2r2

0

-200000

0

x"

0

x"

0

+

0

-1

0

x"

0

o

0

2-2

8

0

0

4

++

0

o

0

0

0

0

2

0

0

*

2

2

0

b5+2

0

6

-1

*

0

++

0

-3b5

0

x,

0

-3

2

0

4

0

0

-3

0

o

2

0

0

++

3

o

0

-1

o

-6

-2

-1

o

0

0

-1

0 -1

0

x,

0

o

0

0

0

o

.00

0

6

-1

0

0

++

-1

-6

+

-1

0

0

0

x,

-1

o

-1

0

1

o

-1

0

•

-5

3-3

0

-b5

-5

-1

+

0

0

0

x"

o

0 -r3

r3

X35

0

o

0

r2

X36

0

0

r2

0 -r2 -r2

X37

0

0 -r2

0

x"

r7

r2

0

0

[43]

!

=C2 (4) =05(4)

=G.2

: the automorphism group of a generalised 4-gon of order (4.4). consisting of 85 vertices and 85 edges.

all 4 x 4 matrices over F 4 preserving a non-singular symplectic form;

Out = 4

Gx 2

05(4)

G0 5 (4)

=< a

=G;

5 b 5 c

d e l (abc)4=1

120

136

136

1360

8160

7200

7200

720

S6

(A5xA5 ):2

(A 5 xA 5 ):2

L2 (16):2

L2 (16):2

120

8160

: S6 x 2

: (A5 xA5 ):22

: (A xA ): 22 5 5

: L2 (16):4

: L2 (16):4

: 26 :<3xA5 ):2

26 : <3 xA5 )

85

(S6 x 2)'2

52 :[2 5 ]

117: 16 N(2C.3B.5CD.... )

1a+50a+85b

N(2C.3A.3B.4B.5E). C(2D)

N(2A.3A.5AB)2

1a+50a+85a N(2B,3B.5CD) 2

1a+34b+85b

1a+34a+85a

1a+34b+50a

S4(2)

S2(4)wr2

°4(4)

S2(16)

°4(4)

isotropic line

N(2B 2 ). N(2C 4 )

11520 N(2C.3A.5AB •••• )

point

N(2A 2 ). N(2C 4 )

: 26 :<3 xA5):2

26 : <3xA 5 ) 1a+34a+50a

85

11520

(22 x2 2+4: 3 ): 12

Symplectic

Character

Abstract

G.4

Index

Order

Structure

Specifications

G.2

°5(2)

pI us hyperplane

minus hyperplane

isotropic line

isotropic point

Orthogonal

all 5 x 5 matrices over F 4 preserving a non-singular quadratic form;

Max imal sUbgroups

>

= PG0 5 (4) = S05(4) = PS0 5 (4) = G:

G.4 is obtained by adjoining the duality (graph) automorphism. interchanging vertices and edges

each object being incident with 5 of the other tYPe

C2 (4).2

= PSP4(4) = S4(4) = G : rS 4 (4) = G.2;

SP4(4)

Mult =

Remark: G.2 is a maximal subgroup of the Held group.

Presentation

Orthogonal

Symplectic

Constructions

2 .17 Order = 979.200 = 28 .32.5

Symplectic group S4(4)

11

5

8EJEJ27 27

"-'

~

~

rJJ ..j:::..

t;

3

3

3 -13

3 -13

9

-4

-4

12

12

51

51

+

+

+

+

+

+ 85

+ 153

+ 204

+ 204

+ 204

+ 204

X6

x,

Xg

X9

XtO

x"

X12

X13

Xt4

x"

Xt6

X17

15

+ 255 -17

15 -17

15 -17

0

4

20

+ 255

+ 255

+ 256

+ 340

+ 340

x"

x"

X24

x"

Xu

x"

4

20

0

-3

15

+ 255 -17

X21

-5

4

-3

o

4

-3

0

0

o

0

-1

-1

-1

-1

0

0

0

•

•

o

o

-5

o

o

-5

-4

-4

-5

-4

-4

o

o

-5

-5b5

•

o

@

@

@

•

-1

-1

o

-1

0

-1

0

0

0

o

-1

o

0

0

-1

0

*

0

0

0

0

0

0

-1

0

0

o

-1

0

0

•

b5

0

0

0

0

0

-1

0

0

b5

•

0

0

0

0

0 -b5

o

0

0

-1

b5

•

0

0

0

0

0

.2

0

0

0

0

0

0

0

0

'3

.6

o

*6

'3

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

.6

0

0

0

0

0

0

0

0

0

0

0

0

*2 d17

*2

'3

*6

0

0

0

0

0

0

0

0

0

0

0

-1

0

0

++

++

++

+

+

+

+

+

10

10

16

0

0

0

0

0

2

-2

0

0

0

0

0

0

0

0

-2

0

0

0

0

0

0

0

0

-2

-2

2-2

0

-2

000

-2

0

@ @

0

0

-1

-1

0

0

0

0

0

-1

o

0

0

0

0

o

0

o

0

0

0

0

0

0

0

0

0

0-1

000

0

000

0

-1

o

0

-1

o

0

-1

0

0

0

0

0

0

0

0

o

0

000

000

0

0

-1

-1

0

000

0

0

0

0

0

-1

0

0

0

0

o 0

-1

o

o 0

-1

o

0

o

-2

o

0

@

2i

2

0

@

@

@

@

0

0

0

0

0

-1

0

0

0

0

0

0

0

Xw

Xt9

x"

X17

Xt6

x"

Xt4

X13

X12

x"

XI0

X9

Xg

x,

~

XS

X4

x,

x,

Xt

4

0

: +0+0

++

0

0

0

0

0

0

0

-1

x"

Xu

X2s

X24

x"

x"

+OOOOOX21

+

0

+

0

0

r2 -r2

-1 0

0

0

0

:+0+0

0

0

0

+

++

0 :+0+0

o

o o

o

++

o

1

0-i+1

4 4 4 5 E A A EF A A A EF 8C 16A B. 20A

@

o

4F

D A

20

@

:0000-1+1 i-1

:+0+0

6 5 6 8 8 A A ED AC BD A A ED AC BD 8A B•• 10E 12A 12B fus ind

@

0 -2i

18 BD BD 6D

@

-1

2

000

0

0

0

0

-3

9 -3

++

0

-3

5

+

0

2

o

o

0

000

0

2

4

o 2

o

0

@

4

0

@

5 -3

0

0

10

4

4

0

@

++

++

+

+

++

++

o

o -1

++

o

00

++

o

*3 d17

.2 d17

0 d17

0

0

0

0

0

0

o 0

0

0

0

0

0

0

0

0

0

0

-1

o

o -1

o

o

000

-1

•

b5

0

0

0

0

0

0

•

• -b5

0

o

0

0

•

• -b5

0

000

0 -1

0

o

o

•

-1

-1

0

0

0

-1

0

@

18 AD AD 6C

@

720 48 48 16 15 17 17 17 17 A A B C CB A A A A A A A A DB A A A A D. 17A B.2 C'3 D.6 fus ind 2D 4C 4D 4E

@

@

@

@

@

• -b5

0

0 -b5

0

-1

0

0

•

0

0

-1

0

• -b5

b5

-1

* -b5

0

0

0

0

0

0

0

o

-b5

*

0

0

0

0

0

•

-b5

0

0

0

0

b5

0 -b5

o

o

0

0

0

0

•

o

0

0

0

0

0

-1

-1

o 0

b5

-1

o

*

-1

0

o

•

@

15 15 AA DB BA CB B. 15C

@

• -b5

0

-1

o

• -b5

b5

-1

b5

-1

0

o

-1

-1

•

b5

•

0

000

• -b5

•

@

20 15 CB BA DB AA D. 15A

@

• -b5

0

0

-1

-1

b5 -b5

•

0

o b5

-1

o

-1

-1

0

-1

-1 -b5

o

o

o

0

o

-1

0-1

0

0

0

0

o

o

-5b5 •

0

o

o

-5b5

0

o

o

o

*

0

o

0

o

o

o

o

0

o

o

o

o

-1

-1

* 4b5+1

•

-1

-1

-3b5

* 4b5+ 1

•

-3b5

3

3

0

0-1

o

•

-5b5

o

o

o

o

•

-3b5

• 4b5+ 1 -3b5

3

@

0-1

5

o

o

o

o

-1

o

0

• 0

0

-1

-1

-2

-1

b5+4

5

@

25 12 12 20 20 20 AAABBBAAADB A AA BB AA BA CB 5E 6A 6B lOA B. 10C

@

•

3b5

o

4

-1

3

D.

•

b5+4

3

o

o

3

5

3b5

•

5

•

3b5

•

3b5

•

•

b5+4

o

o

o

b5+4

4

-1

3

-1

4

3

-1

4

3

0 4b5+1

-1

-1

-1

-1

2

-2

-2

2

0

0

-1

-1

0 -1

o

4-5

4

0

-1

-1

-1

-3

0

o

+ 225 -15 -15

X,O

0

0

0

0 -1

0

o

+ 225 -15 -15

Xt9

-1

0

o

+ 225 -15 -15

x"

3

0

0

0

0

0

3

3

0

4

0

3 -1

4-5

-5

-1

-1

-2

2

2

-2

3 -1

0

0

5

o

_4

_4

_4

_4

-7

5

5

0

0

3

3

5

+ 225 -15 -15

-4

-4

12

12

9

21

5

3

3

2

4

4

0

3

5

21

3

3

10

2

0

0

85

51 -13

51 -13

10

10

2

+ 50

-6

-6

Xs

34

10

+

34

X4

2

+

-6

x,

18 -6

+

+

A

5C

B.

5A

x,

Xt

A

A

A

300

@

A A

@

A

A

@

300

@

300

@

300

3

3

@

979200 840 840 256 180 180 32 32 P power A A A A A C C p' part A A A A A A A 1nd lA 2A 2B 2C 3A 3B 4A 4B

@

@

@

@

@

@

"-"

~

~

..j::::..

rJ).

8 6 (2)

= C3 (2)j

Symplectic group 5 6 (2)

Orthogonal group 07(2)

2 9 .3 4 .5.7

Order = 1,451,520

Mult

= 83(2)

2

Out

Constructions Symplectic

SP6(2)

Orthogonal

G0 7 (2)

= PSP6(2) = 3 6 (2) = G : all 6 x 6 matrices = PG0 7 (2) = 507(2) = PS0 7 (2) = 07(2) = G :

over F 2 preserving a non-singular symplectic form all 7 x 1 matrices over F 2 preserving a non-singular quadratic

form; Weyl

2 x G : the Weyl group of E7 ; the vectors may be taken as the vectors of the

Ea

root lattice (see 08(2»

with

X'+ ••• +X8 = 0 (first base) or x7 = x8 (second base); the minimal (root) vectors are (in the first base) (1,-1,0,0,0,0,0,0)5 (28) and (~,~,~,~,_~,_~,_~,_~)S (35), reflections in which correspond respectively to

transpositions (ab) in SS' and Hesse's bifid maps (abcd!efgh) (see below); the minimal vectors of the dual lattice E* 3 3 1 1 1 1 1 1 S . 7 are ±(~'~'-4'-4'-4'-4'-4'-4) and correspond to the 28 bltangents ab.

Reducing mod 2 shows the isomorphism with 07(2). G : there are 28 bitangents to the general plane quartic curve, which in Cayley's notation are labelled by the 28

Hesse

unordered pairs ab, ••• gh of 8 objects. The group is generated by Hesse's bifid maps such as (abcdlefgh) = (ab,cd)(ac,bd)(ad,bc)(ef,gh)(eg,fh)(eh,fg). The relations (abcd\efgh)2=1, (abcdlefgh) (abcejdfgh)=(abcejdfgh) (de) , (abcd\efgh)(abeflcdgh)=(abghlcdef)(ab)(cd)(ef)(gh)

show that the subgroup S8 of permutations of {l,2, ••• 8l has index 36, and coset representatives the identity together with the 35 bifid maps. These 35 maps together with the 28 transpositions of S8 form the conjugacy class of "Steiner transpositions"; the group preserves the 315 quartets such as {ab,cd,ef,gh} (105), {ab,bc,cd,da} (210), whose 8 points of contact lie on a conic Presentation

G x 2 "

[

Maximal SUbgroups Order

Index

51840

Specifications Symplectic

Orthogonal

Weyl

1a+27a

0;;(2)

minus hyperplane

minimal vector of

1a+35b

°6(2)

pI us hyperplane

point

isotropic point

Structure

Character

28

U4 (2):2

40320

36

3

23040

63

12096

120

10752

135

4608

315

4320

336

1512

960

8 25 :3

6

U (3):2 3 6 2 :L (2) 3 2.[2 6 ],(3 X3 ) 3 3 3 x 3 6 3 L2 (8):3

1a+27a+35b

Abstract

N(2A)

E;

bitangent osculating cubic

root vector

Steiner transposition

G2 (2)

1a+35a+84a 1a+15a+35b+84a

N(28 3 )

isotropic plane

isotropic plane

1a+27a+35b+84a+ 168a

N(28)

isotropic line

isotropic line

1a+27a+35b+l05b+168a

N(3A), N(2C,3A,3C,48,5A)

non-isotropic line

1a+70a+84a+l05b+280a+420a

N(28,38,7A,9A)

3 2 (8)

~30

Gl3 30

[46]

Hesse

conic

56(2) @ 1451520 p power pr part ind 1A

@ @ @ @ @ @ @ @ @ @ @ @ @ @ @ 23 4 1 2 040 608 536 384 160 648 108 384 192 192 128 32 30 144 144 A A A A A A A B C C B C A AA AB A A A A A A A A A A A A A AA AB 2A 2B 2C 20 3A 3B 3C 4A 4B 4C 40 4E 5A 6A 6B

Xl

+

1

X2

+

7 -5

X,

+

15

X4

+

21

Xs

+

21 -11

5

X.

+

27

15

X7

+

35

X.

+

35

X.

@ 72 BB BB 6C

@

48 AC AC 60

@ 36 CA CA 6E

@

36 CB CB 6F

@

@

@

@

@

12 7 16 16 9 CO A A 0 B CD A A A A 6G 7A 8A 8B 9A

@

@

@

@

10 24 24 12 15 AA DB DC CA AA AA AB AC BA AA 10A 12A 12B 12C 15A

1

3 -1

4-2

3

-1

0

-3

3

-1

-3

-3

6

3

0

5

-1

3

5

-3

6

3

0

-3

-3

3

7

3

9

0

0

3

-5

3

-5

3

5

-1

2

7

-1

15

11

7

3

5

-1

2

-1

5

+

56 -24

-8

8

0

11

2

2

0

4

-4

0

0

XIO

+

70 -10 -10

6

-2

-5

7

2

2

2

2

-2

0

-1

Xll

+

84

4

4 -6

3

3

4

0

0

4

0

-1

-2

Xl2

+

5

15

-3

-3

5

-1

-5

-1

0

X13

+ 105

25

-7

9

o

6

3

-3

-3

-3

Xl4

+ 105

5

17

-3

-7

0

6

3

-3

XlS

+ 120

40

-8

8

0

15

-6

0

0

-4

4

Xl.

+ 168

40

8

8

8

6

6

-3

0

0

0

Xl7

+ 189

21

-3 -11

-3

9

0

0

9

Xl.

+ 189 -51

-3

-3

9

0

0-3

Xl9

+ 189 -39

21

-3

9

0

0 -3

X20

+ 210

50

2

2

-6

15

3

0

X21

+ 210

10 -14

10

2 -15

-6

X22

+ 216 -24

24

8

0

-9

0

X23

+ 280 -40

-8

-8

8

10

10

X24

+ 280

24

8

0

-5

-8

X2S

+ 315 -45 -21

3

3

0

X2.

+ 336 -16

16 -16

0

X27

+ 378 -30

-6

2 -6

X2.

+ 405

45 -27

X29

+ 420

20

X,.

+ 512

-5

-1 7

9-3

4

20

105 -35

40

13

-3

3

-1

2 -2

o

-1

-2 -1

-1

-1

1·

-1

2

o

3

3

0 3

0

0

-1

-1 -1

0

0

0

0

-1

2 -1

2

-2

2

0

0

-1

0

0

-1

X2

0

0

0

-2

-1

0

X,

o

-1

2

0-1

X4

-1

0

-1

0

G

Xs

-1

0

0

3

0

o

0

0-1

-3

3

-2

0

0

0

-1

0

-1

-1

o

2

0

0-1

-1

0-1

0 -2

0

0

-1

1

-2

-1

-1

3

2 -1

-2

-1

-1

-1 -1

0

0

000 1

0

0

-1

-1

0

-1 -1

-1

0 -1

o

-1

-1

0

-1

X.

o

X7

0

X.

X. 0

XlO

-1

Xli

-1

0

Xl2

0

0

-1

0

-1

0

0

-1

-1

0

0

0

0

0

0

X13

-1

0

0

0

2

0

0

Xw

o

0

0

0

-1

o

0

XlS

0

0

o

-1

Xl

-2

-1

-2

1

o

0-1

3

-3

o

4 -4

2

0

-1

0

2

2

0

-1

-1

-1

0

0

0

-2

-1

-2

-2

0

0

0

-2

-2

2

2

2

-1

-1

0

0

0

0

-1

-3

-3

0

o

0

0

0

-1

-1

0

-1

-3

-3

0

000

0

o

-1

-1

3

3

0

o

0

0

0-1

o

-1

-1

-1

2

2

0

0

0

0

0

0

1 -1

0

0

0

0

0

0

-1

0

0

0

-1

0

0

0

00000X2'

o

1 -1

-3

3-1

-3 -5

-1

-2

2

2

-2

-2

0

3

6

-2

-2

-2

-2

0

0

0

-4

4

0

0

o

0

0

0

0

0

0

4

-4

0

0

0

-9

0 -5

3

3

3

-1

0

6

-6

0

0

0

0

0

0

-9

0

0

6

2

2

-2

2

0

-3

-3

-3

5

3 -4

0

0

-4

-2

2

0-2-210

3

5

2

@

2

-1

o

-1

-2

0

-1

0

0

0

1

0

-1

Xl7

o

-1

Xl.

0

-1

Xl9

-1

-1

-1

Xl.

o

Xw

o

0

X21

o

1

X22

-1

-3

-3

0

-1

0

2

-2

-2

-2

-1

-3

0

-1

-2

0

0

0

0

0

0

0

3

0

0

0

0

0

-1

-1

0

0

0

0

2

-2

-2

2

2

-2

0

0

0

0

0

-1

0

0

0

X2.

-2

3

3

0

-1

0

0

0

0

0

0

0

0

-1

-1

0

X27

o

0

0

0

0

0

0

0

-1

0

0

-4

4

o

-1

-3

-3

0

0

4 -12

4

0

-3

0

0

0

0 -16

8

-4

0

0

0

0

0

2

0

0

0

0

0

0

0

o 7 14

-1

0

0

X24

o

X2S

OOOOOOX,.

0

0

0

0

0

0

-1

0

X~

o

0

-1

0

0

0

0

-1

X,.

8

8 8

9 18

20 20

24

24

12 12

15 30

o

2

-1

0

0

0

0

0

0

0

-1

-2

X'2

ind

1 2

4

2

4

2

3 6

3 6

3 6

4 4

8

8

4

8

5 10

12

6

6 6

12

12

6

6

X31

+

8

0

0

0

0

-4

-1

2

4

0

0

0

0

-2

0

0

3

0

0

0

0

X32

+

48

0

0

0

0 -12

3

0

8

0

0

0

0

-2

0

0

3

0

0

0

0

-1

0

0

X"

o 64

0

0

0

0

4

-8

-2

0

0

0

0

0

-1

0

0

0

0

0

0

0

1

0

0

i5

0

0

0

-1

X31

X,.

o

64

0

0

0

0

4

-8

-2

0

0

0

0

0

-1

O.

0

0

0

0

0

0

o

0

-i5

0

0

0

-1

X,.

x"

+ 112

0

0

0

0

-8

-5

-2

8

0

0

0

0

2

0

0

3

0

0

0

0

0

O.

0

0

0

0

-1

2

X"

X,.

+ 112

0

0

0

0

4

4

4

8

0

0

0

0

2

0

0

0

0

0

0

0

0

0

0

o

0

0

2

-1

X,.

X"

+ 120

0

0

0

0

0

3

6

-4

0

0

0

0

0

0

0

3

0

0

0

0

-1

0

X"

X,.

+ 168

0

0

0

0 -24

-3

0

4

0

0

0

0

-2

0

0

-3

0

0

0

0

X39

+ 280

0

0

0

0 -20

1

4

-4

0

0

0

0

0

0

0

-3

0

0

0

X40

+ 448

0

0

0

0

16

16

-2

0

0

0

0

0

-2

0

0

0

0

0

X41

+ 512

0

0

0

0 -16

8

-4

0

0

0

0

0

2

0

0

0

0

X42

+ 560

0

0

0

0

20

-7

2

8

0

0

0

0

0

0

0

-3

X43

+ 720

0

0

0

0

0

-9

0 -8

0

0

0

0

0

0

0

3

1

X31

O·

-2

0

0

0

0

0

0

-2

0

0

0

0

0

0

0

2

o

0

0

-1

0

0

0

0

0

1

0

0

0

0

0

0

0

o

0

-1

0

0

0

0

-1

X41

0

0

0

0

0

0

0

-1

0

0

0

-1

0

X42

0

0

0

0

-1

0

0

0

0

0

0

o

X43

X,. 0

X~

X40

[47J

~

""

0

Index

10

45

120

126

210

945

2520

Order

181440

40320

15120

14400

8640

1920

720

= 0.2 =0 :

M10

=A6 ·23

2 4 :35

(A 6xA 4 ):2

(A 5xA ): 4 5

(A x3):2 7

38

A 9

Structure

Out = 2

x 3

: A6 ·2 2

: 3 6 x 34 5 •• 2 .3 • 5

7

3 : (3 5 x35 ):2

: 3

: 38 x 2

: S9

0.2

=

1a+35a+42a+90a+225a+252a+300a

1a+9a+35a+75a+90a

1a+35a+90a

1a+98+35a+75a

1a+9a+35a

1a+9a

Character

0.2

quinquisection

N(2 4 ) = N(2A 10B5 ). C(2D)

3(3.4.10)

tetrad

N(2A 2)

N(2B. 3C. 4C. 5B)

bisection

triad

duad

point

Al ternating

N(2A.3A.5A)2

N(3A)

C(2C)

Abstract

Specifications

all even permutations; 2.0 and 2.0.2 : the Schur double covers

: all permutations of 10 letters;

Mult = 2

= <x1 •••• x8Ixi=(XiXj)2=1>;

A10

S10

Max imal subgroups

Presentations

Alternating

Constructions

Order = 1.814.400 = 27 .3 4 .5 2 .7

Alternating group A10

GEls 24 20

GB24

o

> ~

;!5

0

-5

-2

14

_1~

16

6

16

0

19

+ 126

+ 160

+ 210

+ 224 -16

8

+ 225

+ 252

+ 288

+ 300

315

+

+ 224 -16

5

90

+

+ 350 _10

x,

X.

x,

x"

x"

x"

x"

x"

x"

x"

x"

+ 450

+ 525 -15

+ 567

x"

x"

Xu

-15

~

~

o

o -20

o -36

o

o

o

o o -20

o -36

0

o

o

o

o

0

0

0

0

0

0

o 48

64

+

+

+ 216

+ 216

+ 336

+ 336

+ 384

+ 384

+ 400

+ 432

+ 448

+ 800

x"

x"

x"

X~

x"

x"

x"

x"

x"

x"

x"

x"

x"

6~

~

0

0

48

0

0

6

6

0

0

0

20

2fl

36

0 -20

0 -24

0 -24

0

_4

4

0

-8

0

0

6

6

o

0

0

4

o

-8

x"

0

3 6

16

0

-2

0

0

-2

-1

-2

0

2

-1

-2

0

4

-3

-3

3

3

o

o

3

3

-2

3 6

0

_2

0

0

2

_1

0

0

0

o

o

0

0

0

0

0

0

0

0

0

0

0

o

o

0

o

o 6

o

0

0

0

0

0

0

-2

-8

0

4

4

6

o

0

-~

o

-~

o

o

o

-6

o

o

-6

o

o

-2

-2

-4

5 10

0

0

0

4

0

-2

2

0

-1

-1

-1

-1

-2

-2

5 10

2

0

o -3

0

-1

-1

0

0

0

-2

0

4

4

0

-5

0

-7

2

0

o 2

-1

-1

0

0

0

-1

0

2

0

-1

~

-1

-1

5

5

-5

~

0

-3

6

5

4

Y

o

0

0

0

8

3 _1

3

-2

0

0

-2

-1

2

0

-2

0

0

2

0

-2

2

0

-1

2

0

_1

~

~

12

0

-3

0

0

0

0

2

6

0

8 8

7

-1

0

0

0

14

0

0

2

-1 -1

-1

-1

0

0

0

0

0

0

o

o

0

0

0

0

0

0

0

0

0

0

0

0

o

2

0

-2

1

-1

-1

0

0

o

o o

-1

o

-1

1

1

-1

-1

o

o

o

o o

o

o

o

0

0

o

0

0

o

o

0

0 0

0

0

0

0

0

0

0

0

0

-2

2

0

-1

1

0

1

0

0

0

0

0

0

0-2

0

0

9 18

-1

1

0

1

0

0

o

_1

0

@

0 0 0

o o

o

0

0

0

0

0

0

o

o

o

0

-1

-1

0

0

2

15 30

0

0

o

0

0

0

0

0

0

o

0

0

0

0

0

0

-r6

++

++

-1

-1

-1

-1

-1

-1

-1 -1

-1

0

0

-2

0

*

0

-1

0

-1

0

*-b21

1-b21

0

-4

10

9

-5

0

70 -10

0

0

6 -10

0

10

3 -1

-8 -10

0 -16

70 -10 -10

91

5

0 -10

3

0

-4

16

4

-4

"10

++

0 0 0 0

00 00 00

0

o

0

0

0

0

0

o

o o

o

0

o o

0

2

15

0

00

+

+

+

+

++

2

63

0

0

0

0

0

o

o

o

0

0

4

-9

0

0

0

0

0

o

o

o

0

0

8

-9

0

0

0

0

0

o

o

o

0

0

8

_1

_2

0

2

_1

2

0

2

0

0

0

0

0

_2

0

_2

A A 4E

5

32

A A 40

@

~~o

1

@

++35-575

21 fus Ind 42

0

0

++

+

++

0

20

15

20 -20

64

28

55

0

0

+

0

0

-6

2

0

6

6

-8

14 -10

14

10

4

++

* -1

2

0

5

3

2E

64

++

-1

~2

21

0

0

-1

@

++

++

34

0

*_b21

l-b21

o

0

o

-4

5

14 -10

20

21

++

++

-1

-1

2D

7 -1

2C

28

o

0

0

0

0

@

1 320 920 576 A A A A A A

++

++

0

0

0

0

-1

0

o -1

-1

-1

@ ~O

++35-575

++

++

++

++

++

fus ind

-1

-1

0

-1

-1

-1

-1

0

0

0

-1

o

0

0

-1

a.

21 AA AA

@

Or6000

0

o

o

0 0

16

0

0 -16

0

0

2~

0

o 24

-1

0

0

-1

0

-1

0

0

0

0

0

_1

o

o

o

0

_1

0

0

o 0

0

_1

0

0

o -1

0

-1

o

-1

o

0

o

24 24

o o

@

_1

-1

20

0

0

0

0

0

-1

o

o

@

0

0-1

-2

0

0

18

9

0

0

o 0

0

0

0

0

0

0

-1

o -1

0

o

0

o 0

0

-1

-1

-1

0

0

-2

0

AB AB

12

@

12 15 21 BA AA AA BA AA AA AA lOA 12A 128 15A 21A 20 AA

@

0-2

0

2

-1

0

0

0

0

0

0

0

-1

0

0

0

o 0

0

0

-1

2

0

0

0

0

0

-1

0

0

0

0

0

0

0

0

0

0

-1

0

P •

0

-1

o

0

0

0

-1

0

-1

0

0

0

0

0

0

0

0

0

0

-1

-1

-1

0

o

-1

n •

O· -2

-1

-1

0

~

00020"

12

0

0

0

0

-1

3

-3

-1

-2

-1

2

2

2

0

-2

-2

-1

3

0

-2

-1

-1

2

2

3

-2

2

-3

3

3

-1

-1

0

2

2

2-1

y

32 300 25 72 72 12 21 8 9 9 BAAAABABBACCC AAAAABABBAAAA

@@@@@@@@@@

o

0

0

0

8

3

3 -1

0

0

0

-4

0

-~

0

-2

0

2

3

3

3-1

0

0

0

0

0

11

2

-3

-3

-3

-1

0

3

0

0

0

-1

-1

3

-2

0

0

3

0

+

3 6

0

0

-~

-2

3

-1

~B

~A

3-3

-3

0

-1

0

96 A A

96 A A

-3

0

0

-1

-3

3

6

3

-6

2

2

x"

2

-9

5

2 -15

0 -24

0 -24

35

21

-6

-21

~

0

15

9

14

14

0

-2

6

3

3

0

3

3

2

3

1 2

4

34

21

6

21

15

0

-6 -21

0

6

2

-4

3

2

15

1~

6

ind

-9

10

0

+ 384

x"

0

15

+ 384

75

6

x,.

x"

.•

0

0

8~

+

x,

~2

+

x,

8

-4

36

+

3

X.

11

35

+

x,

5

+

x,

9

+

X.

XI8

7

880 38~ 560 216 81 A A A A A pt part A A A A A 1nd lA 2A 28 3A 3B 3C

p power

181~~00

2

@@@@@@@@

0

0

0

0

0

o

o

o

0

0

4

-1

-1

3

2

3

-5

0

-5

5

4

-5

0

-1

4

-1

4

0

0

0

0

0

o

o

o

0

0

6

0

3

0

-1

0

3

0

2

-3

2

0

0

0

0

0

0

o

o

o

0

0

12

0

3

0

-1

-3

0

0

0

0

0

0

o

o

o

0

0

12

0

-2

3

0

-1

-3

0

3 -3

0

0

2

0

0

-3

5-2

2

5

6

~

-1 -10

2

0

2

-1

0

0

0

-1

0

-2

0

-2

2

0

-1

-2

0

-1

323607272 B AC BE AE A AC BE AE ~F 6D 6E 6F

0

0

0

0

0

o

o

o

0

0

6

0

-1

0

-1

-2

-2

0

2

-2

2

2

-1

-1

0

36 BC BC 6G

0

0

0

0

0

o

o

o

0

0

6

0

-1

0

-1

31

0

0

0

0

o

o

o

0

0

12 12

0

0

0

-1

o

0

0

-1 0

0

0

-1

0

0

o

-1

0

-1

0

9 CE CE 6I

0

0

2

0

0

-2

-1

-1

2

-1

BD BD 6H

2~

0

0

0

0

0

o

o

o

0

0

8

-1

-1

0

0

0

0

0

0

0

o

o

o

0

0

10

3

0

0

0

0

0

o

0

-1

-2

o

0

-1

-1

-1

0

0

0

0

0

0

-1

-2

o

-1

0

2

0

0

@

@

@

@

@

@

0

0

0

0

0

o

o

o

0

r5

10 10

0

0

0

0

0

0

0

0

0

o

o

o

0

0

24

0

2

_1

0

_1

o 0

_1

2

_1

_1

0

_1

_2

2

_1

0

0

0

0

0

0

0

0

-1

0

2

0

o 0

2

-1

0

0

0

0

0

o

o

o

0

0

24

0

-1

-1

0

-1

-1

2

-1

2

0

2

-2

-2

_1

2

-1

0

-1

0

x..

0

0

x"

X26

0

0

o

o

0

0

X39

X38

X37

X36

x"

X34

x"

X32

x"

X~

o x" x,.

0

0

X2S

Xu

o 30 30

X23

X22

x"

X20

0

0

0

0 i 15

0

0

o

o

o

0

0

40 40

0

0

0

0 i 10

0

17

0

o

o

o

0

0

1~

14

0

0

0

0

000XI9

0

XI7

x" -1

XI6

XI4

-1

0

x"

XI2

Xli

-1

0

XI0

X9

Xs

-1

-1

-1

x,

X6

x,

2 0

X4

x,

X2

0

-1

x,

0

0

0

-1

0

0

0

-1

0

0-1

-1

0

-1

0

0

0

-1

o

0

0

-1

0

0

7 10 15 5 36 36 30 AC BD AD BD AC AD ABD AC BD AD BD AC AD ABD lOB 10C 12C 12D14A20A30A

@

-1

0

0

-1

8 C A 8B

@@@@@@@@

> o I---"

s

= A2 (7)

PGL 3 (7)

GL (7) 3

Out

= S3

SL 3 (7)

= 3.G; PSL 3 (7)

=G

x 3.G.3 : all non-singular 3 x 3 matrices over F 7 ;

=G.3;

=2

Mult = 3

L2 (7):2

L2 (7):2

L2 (7):2

57

57

5586

5586

5586

26068

26068

32928

32928

32928

336

336

336

72

72

57

N(19ABCDEF)

L1 (43)

base

NOA), N(2A 2 )

: 57:6

°3(7)

N(2A,3A,4A,7D)

: 19:6

° 3 (7)

N(2A,3A,4A,7C)

: 57:3

°3(7 )

N(2A,3A,4A,7B), C(2B)

19: 3

line

N(7A 2 )

N(3A 2)

point

Linear

N(7A 2)

Abstract

•• 32 .Q • 8''2

1a+56a

1a+56a

Character

: 32 : 2A4

GL 2 (7).2

2 : 62 : 2,

3

•• 62 'D • 12 : 32 : 2S 4

I

I

71+

G,S

: S3 x S4

1

: L2 (7):2 x 2

2'(L2(7)x2) .2

171+2:0 x D8 ),

G.2

2 •• 6 ·S • 3

: 7 2 : GL2 (7) : 7 2 :GL 2 (7)

G.3

Specifications

(3 x A4 ):2 32 :Q8

7 2 :2'L2(7):2

72 :2'L2(7) :2

Index

Order

Structure

Maximal subgroups

Remark: The O'Nan group contains two classes of maximal subgroups isomorphic to L3 (7):2.

Linear

Constructions

Order = 1,876,896 = 25 .3 2 .7 3 .19

Linear group L (7) 3

19

10

B8FI121 2

GBB~~22

G.2' G.2"

"-'"

'-J

VJ ...-.....

~.

_.

v:

7C

7D

8A

6

~

6

-6

-6

-6

-1

o 288

o 288

+ 342

+ 342

+ 342

o 342

o 3112

o 3112

o 342 -6

7

o 288

+ 343

+ 399

+ 456

x.

x,

X,IO

Xn

x"

x"

x..

x"

x"

x" x..

x"

Xw

x"

x"

0

0

0

o

o

o

-6

-6

_6

02 342

02 342

02 342

02 342

02 399

02 399

02 456

x"

x"

x"

Xw

x..

x"

x.,

-8

-1

15

-6

2

0

6

02 342

X36

0

-1

o

o

0

-1

o

2

o

0

-1

-1

-1

-2

2

o

15

7

7

0-1

0-1

0

0-1

0

0

-1

0

-1

o

o

o

o

-1

r2 -r2

-1

-1

r2 -r2

r2

o

-1

_1

-1

-1 -r2

r2

0

o

**

-1

-1

0

1 -1

0

-1

0

-1

H

*3 m16

1 *3 **3

r2

0

-1

m16

**

*3

*3 **3

r2

** m16 **3

1 **3

1

1 m16

-1 -r2 -r2

r2 -r2 -r2

*2

*4

:0002

o o o

o

o

o

:0002-413+3

o o

o

o

o

o

o

o

o

o

o

o

o

o o

o

6

6

6

-813

:0002-413+3

:0002

:0002

:0002

6

o

6

6

o o o o

:0002

:0002

-6 -6 -6 -4i3+3 4i3+11

o o o o o

-8

-6

o

o

6

6

0

0

2

o

0

o

_1

-1

-1~z3-1

0 -13

-1 i3+1

r2

0

-1

-1

0

-1

o

r2

1

_1

z3+1

o

-1

48

48

48

19

@ 19

@

**

0

-1

0

-1

*3 **3

o

o

0

1112 *34 *61 *37x171

**2 **5 *61 *311x171 *37

*4 *37x171 **5 **2

r2

0

**

*3

0

-1

0 0

0

0 0 0 0

0

o o o o

o O'

o

0

0

0

0

0

o *37x171

o

o

o

o

o

o o o o

o o

o o

o

o

o

o

o

o

o o o

o

o

0

0

0

o

0

0

0

o o

0

0

0

o

0

0

0

.22 *43 *34 *61

Ox171 *37 *43 *22 *61 *311

o

o

0

0

0

*10x171 *37 **2 **5

o

o

o

o

0

0

0

000

0

0

o

0

0

-1

0

0

0

o

** m16 -1

++

++

0

0

7

7

0

0

2

0

7

-1

0

000

-6

-6

000

0

+

+

6

++

6

++

0

0

16

0

0

0

0

16

0

-1

0

0

r2

++

• •

+

28 fus ind fus ind

28

++

++

••

++

++ ++

•

+

+

++

•

++

++

+

++

++

•

•

•

_1

_1

0

0

0

-1

0

•

•

++

-1

-1

o

0

0

-1

0

000

0

••

•

++

•• •

X2~

X22

X21

X20

x"

XIII

x"

X16

XIS

X14

Xl3

x"

Xli

x"

x,

x, x.

x.

x,

X.

X~

X2

.2

.2

•

+

+

.2

.2

•2

02

• • • • • •

I

I

*

*

J

J

J

.2

.2

.2

•

.2

.2

x,•

.2

x"

x"

x"

Xw

x"

x"

• •

J J

x"

02

J

.2

J .2

.2

• • • • • • • • • • • •

I I

.2

x.,

x"

x..

Xw

x"

x"

x"

*+*+X36

*+*+X~S

*+*+X~4

I

I

.j

02

x"

14

o

-1

0

0

0

++

++

Xl

• •

12

o

o

0

0

0

r7-r7

000

0

-1

0

r2 _r2

0

0

0

0

0

o

-1

0

-1 -r2

-1

-1

0

0

0

0

0

++

*+*+Xu

+

8

-1

0

0

0

0

0

0

0

0

0

_1

-1

0

••

14 AB AB B* fus ind fus ind

@

+

*

*

2

6

0

0

0

0

0

0

0

0

6

0

0

0

0

4

0

0

0

0

0

0

0

0

6

0

++

+

000

000

0

o

0

48 171 171 171 171 171 171 fus ind 48 171 111 171 171 171 171 48 171 171 171 171 171 171

0

0

0

0

0

0

0

0

** x19

**

+

+

0

_1

0

o

0

0

0

0

0

0

0

0

*11

*2

*2 **2

*4 **11 x19

** x19

** **2

*4

*4 **4

0

+

B

-7

0

-1

0

*2

8

0

++

••

++

4B

2B

@ 14 AB AB 2BA

@

B 686 7 8 A A AB A AB 88 A A AB A AB 88 6D BC 12B 148 16E F*

B -8

o

0

0

0

0

0

0

0

0

-1

A A

A A

@

@

@

@

@

++

*3 **3

*3 m16

*2

*4

0

_1

000

0

-1

*2 **2 **4

**2

0

0

-1

++

@

336 336

@

-1

r2 -r2 _r2

r2

0

o

o

** m16 **3 1 **3

1

1 m16

_1

_1 -r2 -r2

_1 -z3-1

r2 -r2

r2 -r2

-1 -r2

r2

0

0

o o

2

2

-1 -r2

-1

-1

-1

-1-z3-1

0

o -13

0

0

o

o

0

2

o o

-2

o

_1

o

6

0

19

@

-1

-1

0

0

0

o o **5 o

6

-1

** m16

**

*3

0

0

0

0

o

o

:0002

o

r2

If

0

0

_1

*2 **2 **4

o

o o

o

o

0

o o

o o

o

o

o

o

o

:0002

19

@

*4 **4 x19

o o o *4 o o o *10

-1

-1

"8

"48 "4488

z3+2

-'3

42 42 42

o o

o o

o

19

@

** x19

0 **2

20000

-1

0

0

-1

0 **11

*3 **3

*3 m16

*3 **3

**3

0-1 -1

**

r2

** m16 **3

m16

0

0

0

0

r2 -r2 _r2

_ 1 -r2 -r2

;.. 1

o

o

o

o

0

0

o

o

:0002

0

o r2

000

0

o -1

0

0

0

0

o

0

0

0

0

o o

o

o o

o

19

@

0 x19

o

0

0

o

o

-1

0

o

0

o o

-1

0

0

o o

o

:0002

o

o

o

o

16

o

o

o

:0002

-1

o

o

o

_1

0

o

o

o

o

16

@

00000

o

o

o

o

16

16

14

@

AA8 ~ AD 00 M oc ~ re ~ OC ~ AA8 M 00 oc BD ~ OC ~ oc ~ FC 42A 48AB*31C*37D**5 57AB*37C**2D**5 E*4F*34 fus Ind

@

@

@

o o o o o o

o

-'3

-'3

o

o

o

12

o o

-1

-i3

6

24 24 24

24 24 24

21 21 21

12

-1

_1

12

-1

_1

r2 -r2

r2 -r2

r2

6

o

-1

-1

_1 -r2

o

o

o

0

o

o

o o

0

o

r2

2

2

-1 -r2

-1

-1

-1

000

000

000

o

:0002

0

6

6

o

0

000

o

000

000

6 6 -1113-3

0

000

2

o

BB B.

AA

16

@

0

-2

16

o o

0

9

@

o

:0002

o

0

o

_6

413+5

o

o

0

o

-6

0

o

o

0

o

-6

@

o

0

0

0

3 3

3

o

0

0

** x19

**

0

o

-6

0

o

o

o

*2

*4

2

o

6

o

:0002-4i3-3

:0002

2

o

6

-1

:0002-4i3-3

02

-2

o

-1

0

o

o

_1

0

6

0

o

o

0

0

6

0

o

o

15

0

6

0

o

o

-1

0

6

0

o

o

6

0

0

o

o

o

0

o

-1

2

7

0

6

@

12 16 14 16 BA BA AB A8 BA BA AB BA 6c 12A 21A 24A

@

-7

8

68

BA BA

672

@

7

0

3

o 6

3

o 0

3

o

6

3

3

o o

3

0

0

o

o

9

0

0

0

0

0

3C

8 -1

3B

A A

57

672

A A

@

@

0

0

0

0

0

'00

'00

'00

000

000

000

000

'00

.00

'00

000

000

000

000

000

000

•

'00

'00

'00

19 fus ind 57 57

0

0

0

0

0

0

0

0

0

*2 **2

** **2

*4

I!lI

*4 **4

0

*4 **4 x19

0

0

0

*2

0

0

51 0

19 57 57

19

0

0

0

0

0

0

0

0

0

57

*2

_4

III x19

0

0

0

0

0

0

0

0

0

*4

0

0

0

0

_1

*2 **2

.. **2

** x19

*2 **2 **4

**2

**4

0

r2

A

19

@

*4 **4

0

0

0

0

-1

*2 n2 **4

*4 **4 x19

0

-1

0

0

** **2

0

** x19

o

-1 -r2

_1

-1

o

o

x19

0

o

o

o

o o o c o o o o o o o o o o o o o o o o 2 2-1 o 000

o

o o

-1

_1

-1

-·1

-1

_1

-1

-1

-1

-1

-1

6

-1

02 342

-1

-6

-1

bs

0

0

6

02 342

X~4

-2

o

o

o

o

02 288

x"

-6

o

o

o o

02 288

x"

-6

o

o

o

o

02 288

x"

o

o

o

-6

o

o

o

02 288

Xw

o

o

o

o o o o

-6

o o

o

o

o

02 288

x"

o

o

o

o

o

o

-6

o

o

o

o

02 288

x"

o

o

o

o

o

o

o

0-2-2-25

96

o

o

-1

o o

02

o

-1

-1

o

-1

0

o

19

2

19 57

0

57

0

57

o

x"

0

0

0

0

0

0

0

000

0

0

0

0

0

0

0

51

o

24 24

o

2' 2'

16 48 48

0-2-25-2

96

02

x"

21 21

16 48 48

A

19

@

*4 **4 x19

19 57 57

-1

0

0

0

0

0

0

0

*2

0

0

0

0

_1

III x19

*2 **2 *114

*2

*4

*4 **11 x19

16 48 48

16 48 118

8

0

-1

A

19

@

*2 **2 .. 4

.. '*2

0

0

-1

** m16

**

*3

0

0

0

_1

0

14 42 112

-1

o

_1

o

r2

*3 **3

*3 m16

*3 **3

0-1

_1

-1

n

r2

** m16 **3

**3

0

0

.. x19

0 **2

0

A

19

@

000

0

0

0

o 0

-1

-1

0 **4

0

0

A

19

@

A A A A A A 19A Bue**2 0*2 E*4F**q fus ind

A

19

@

0 x19

o o o o o o o

8

2

21 21

8

o

-1

-1

r2 -r2

r2 -r2

r2

m16

0

0

0

0

0

0

0

0

r2 -r2 -r2

0

o

r2

0

0

0

0

0

0

0

0

0

0

0

0

_1 -r2 -r2

_1

-1

0

0

0

0

0

0

000

0

o

8

o

21 21

7

0

0

-1

-1

_1 -r2

r2

0

2

2

o

0

o

0

0

o

o

0

0

0

0

0

o o

o o

0

-1 -r2

-1

_1

-1

5

000

0

_1

0

o

21 21

6 6

7

0

0

-1

-1

-1

-1

-1

-1

_1

-2

-1

0

o

96

02

x"

0

-7

12 12

677

4

0

0

o

7

-1

_1

-1

-1

-1

-1

_1

-1

-1

-1

-1

-1

-1

-1

-6

-6

-6

-6

_6

-2

15

-1

0

0

0

0

0

0

0

0

0

0

0

0

5

0

-1

_1

0

0

0

0

2

2

-2

0

0

0

0

0

-1

o

-1

0

000

-1

A

A

0-25-2-2

57

02

Xu.

0

3

-3

3

0

0

0

0

0

0

0

0

0

0

9

2 6 6

-8

0

0

8-1

5-25-2.00

o

16 8

16 8

en C*3D**3

@

@

o o o o o o

57

02

X23

3 3

1

0

0

o 288

x,

ind

0

0

o 288

02880000-6

0

+ 152

-1

o

55-2-200

x,

8-1

_1

8

+ 152

o

-1

x,

8-1

3

+ 152

-7

X.

57

+

o

B* 14A 16A

+568202100000

•

x,

x,

x,

78

A

1A

7A

A

1nd

3A

A

2A

16

A

@

16

AA AA 6A

@

12 686 49 49 49 16 16 14 A A A A A A AA A A A A A A AA

A A lIA

@

16

@

1876896 672 36 A A p' part A A

p power

@

@

@

@

@

@

@

@

@

@

......:J "-'"

~

r VJ

U 4 (3) = 2 A3 (3) = 06(3)

Unitary group U4 (3)

Mult = 3 2 x 4

Order = 3.265.920 = 2 7 .3 6 ,5.7

Out

= D8

Constructions Unitary (3)

GU 4 (3) =- 4.G.4 : all 11 x 4 matrices over F g preserving a non-singular Hermitian form; PGU 4 (3)

Orthogonal (3) G0'6(3)

=G.4; SU 4 (3) = 4.G; = 2.G.(2 2 )122 : all 6

example

Unitary, (2)

x 6 matrices over F

x~ + x~ + x~ + xt + x~ + x~; PG06(3)

3

preserving a non-singular quadratic form of Witt defect 1, for

=G.(22)122;

S06(3)

=S06(3) =2.G.2,;

PS0 6 (3)

=G.2 1 ;

06(3)

=G

3.G.22 has a 6-dimensional unitary representation over F 4 (see U6 (2». This can be obtained from the representation comple~

of 6,.G.22 as a Leech

=G

PSU 4 (3)

U4 (3).D :: G.D 8 a

: the

reflection group (see below).

set stabilizer in

Co, of an S-lattice of type 2 1+ 4 32 ; the pointwise stabilizer is G (see Co,);

hence G is the point' stabilizer in the 275-point representation of the McLaughlin group.

Complex: Leech

(2x32).GoDa: the normalizer- in 2Co, of an elementary Abelian 3 2_group; (2x3 2 ).G : the centralizer, i.e. the

centralizer in 6Suz of an element of order 3. When the Leech lattice vectors are written as quaternionic vectors (Q1, •.• ,Q6) the 32_group is generated by left and right multiplication by w Reflection

= ~(-l+i+j+k)

6 1 .G.22 : the automorphism group of the 6-dimensional lattice over Z[w) (where w = z3) whose typical vector is v1+ev2

e

(with

= i3, vl in the E6 root lattice, v2 in its dual). The group is generated by the 126 reflections in the

6 x 126 minimal vectors, which are listed (modulo scalars) in four bases below. Each base renders a different maximal subgroup monomial. The standard norm in the n-base is nl~lx.1 12 , and modulo scalars there are 126, 672, 3402 . vectors of norm 2, 3, 4 respectively. Read modulo

e

these numbers become 126, 112, 126 and we obtain the

"Orthogonal (3)" construction. Read modulo 2 they become 126, 672, 567 and we have the "Unitary (2)11 construction. In the 2-base the vectors are just those whose coordinates reduce modulo' 2 to give a hexacode word (see A6 ). The monomial SUbgroup (of 6.G.2) is (2 6 x 3)A 6 , and the minimal vectors are the images under this group of (2,0,0,0,0,0) (6) and (O,l,O,l,w,w) (120). In the 3-base the monomial subgroup is (2 x 35 )3 6 and the minimal vectors are of form

(w8e,-w be,o,O,O,O)S (45) and (wa,wb,wc,wd,we,wf)S (a+b+c+d+e+f

a 0 mod 3) (81).

In the 4-base the monomial SUbgroup is (3 x 2 5 )S6 and the minimal vectors are (±2,±2,0,0,0,O)S (30) and (±8,±1,±1,±1,±l,±1)S (even signs) (96). In the 7-base the monomial group is 6 x S7 and the minimal vectors are (2+3w,-2-3w,0,0,0,0,0)S (21) and (2w,-2-w,-2-w,1,', 1,1)S (105). (Another type of 7-base is obtained by taking the complex conjugates of these numbers.) Presentations

6,.G.22

=< a

b

d e l (edef) 3=1 >;

e

V f

32 .G.(22)'33

=< a

8 bId 8 e

I

f=(ab)4=(de)4, 1=(bedf)5 >; normal 3 2 is generated by (ebaba)5 and (edede)5

f

See also page 232.

Max imal SUbgroups Structure

G.2 1

G.4

G.2 2

G.(22)122

G.2

112

34:A6

3 4 :(2XA6 )

3 4 :(2XA6 ) '2

34:S6

34:(S6 x2 )

25920

126

U4 (2)

U4 (2):2

(S6 x2 )'2

U4 (2) x 2

U4(2):2 x 2

25920

126

U4 (2)

U4(2):2

U4 (2):2

U4(2):2 x 2

20160

162

L (4)

L 3 (4):22

Order

Index

29160

162

L3 (4)

1166~

280

31+

6048

540

U (3) 3

5760

567

5760

567

2520

1296

20160

2520 • 1296

[52J

3

L (4):22

3 3 1+4 4S

2

G.(2 )'33

G.D 8

34 :M 1O

3 4 :(2xM 1O )

34:(2XA6'22)

A '22

A6'22 x 2

(A6'22 x 2)'2

L (4):2 3 3

L (4):22

L (4):2 1 3

L (4):22

3

6

3

3

31+4. 4S4 ·2

31+4.2S4:2

31+4.4S4:2 +

3 1+4 • (2S x2) 4 +

1+4 21+4 S 3+ • • 3

21+4.D12 31+4 ._

U (3) x 2 3

U (3) x 4 3

U (3):2 3

(U (3) x 4):2 3

4 2 :S6

43S4

24 :S6

U (3):2 3 42 ( x2)S4

U (3):2 x 2 3

24 :A 6

U (3):2 x 2 3 5 2 :S6

(4 2 x2 )( 2xS4)

4 3 ( 2XS 4)

24 :A 6

2 4 :S 6

5 2 :A6

5 2 :S6

H.2

H.2 2

4(S4 xS4) .22

M10 x 2

H.2 2

A6 '2 2

H.2 2

4 • 2S

4

+.

4

+

A7 A7

+

S7 S7

2520

1296

A7

2520

1296

A 7

1152

2835

2(A4 xA 4 ).4

720

4536

M,0

= A6 '2 3

720

4536

M,0

=A6 '23

4(A4 xA 4 ) .4

A6 '22 A6 '2 2

4(S4xS4) .2

H.2

2' (22x(A4XA4) .4)

U 4 (3)

8EJi BElOEJG~;~

812.00211: 120004112.G0221~1200.23In 19

page 54

page55

I

81400021111400.41140002211400023116 - - - - - - - - - - - ---:t- - - - - - - - - - - - - - - - -

.BFi 131000221~ BFi 16100.2210 6(0 FI 12(G BF! 3(01

page 56

11

I

11

I

I.

I

11

ID

I

:

15

page 57

ID

I

. I

16

12

I

------------4-----~------------

B~ 13

813200021 i 32'001

page 58

i

11

BFi 62'001 11 BFi 122'001

I

B~ 12

I

20

11:

16

I I

ID

I

page59

ID

FI9 12

16

I

I

10

I I

Specifications Characters

Abstract

1a+21a+90a 1a+35a+90a

C(2D)

1a+351>+90a

Unitary

Orthogonal

Reflection

isotropic line

isotropic point

3-base

S~(3)

non-isotropic point

norm 2 point

S~(3)

non-isotropic point

norm 11 point mod

e

1a+21a+140a 1a+21a+ 140a

1a+90a+1898

NOA)

isotropic point

isotropic line

1a+140&+1898+210a

C(2B)

non-isotropic point

U (3) 3

18+21 a+908+ 140a+315a

base

4-base

la+21a+90a+140a+315b

base

2-base

1a+21a+90a+140a+315a+729a

7-base

1a+21a+908+140a+315a+729a

7-base

la+21a+90a+l~Oa+3151>+729a 1a+21a+90a+1~Oa+3151>+729a la+35ab+90aa+l~Oa+189a+315aI>+729a+896a

N(2A)

U2 (3)wr2

C(2F)

04<3 ) 0 4(3)

O2(3) x 0~(3)

[53]

U 4 (3)

•

@@@@@@@@@@@

1 5 3265920 152 832 972 972 81 96 16 5 72 36 36 p power AAAAAAAAAABACA pI part AAAAAAAAAABACA ind. lA H D g ~ W u ~ ~ M ~ ~ X.

+

x,

+

21

5-6

x,

+

35

3

X.

7

A A

7

A A

7A BII

8

A A

27

A

27

27

A A

A A

A A

BA

8..

••••• •• • • • • ••• 7 7

•• 6

12

0118 720

AA AA

9C nn 12A fus lnd

A

A

576 A

A

A

A

2B

2C

4C

64 108 A AB A AB liD 60

8

8

_1

-1

-1

0

0

-1

o

o

0

0

-2

++

-7

0

3

0

0

-1

2

2

-1

';'1

0

++

1

-1

0

++

-1

-1

0

0

8 _1

8-1

_1

0

0

9

0

2

0

.11..11

5

o

0

o

0

0

_1

-1

_1

_1

0

o

0

0

0

0

go

10

9

X.

+

140

12

5

x,

+

189

-3

27

X.

+

210

2

21

X.

o

280

-8

10

10

000_2-2

OOOb27

x..

o

280

-8

10

10

000_2-2

o

0

0

x..

o

280

-8

10

0

0

0

x"

o

280

-8

10

315

11

-9

18

-9

0

-,

-1

315

l'

18

0

-1

-,

9

0

-2

0

0

5

3

3

-2

o

10 10

-2

0

-3

-Cl

-1

3

0

0

0

0

0

5

-1

_1

0

0

0

-2

-2

_1

o o

0

••

o -1

-9

-9

1120

11 -39

6

6-3

x..

560 -16 -34

2

2

2

0

0

-1

0

_1

o

0

0

0

2

_1

0

0

.1

2

0

0

2

2

0

0

-1

_1

2

-5

-1

_1

-2

1

-5

~

';'1

~

6

10

10

++

28

0

CC

6G

8B

lOA

126

C..

-1

_1

2

2

_1

-1

0

0

X1

1

-1

2

2

2

_1

0

0

x,

2

2

-1

2

0

0

X4

_1

-1

XJ

-3

3

0

0

..

10

2

5

-1

_1

_1

0

o

o

AB

CC AB 12' 14A

88 8B

an

o

0

X.

0

o

0

o

o

o

o

o

0

0

X9

+

0

0

o

0

0

0

0

o

0

o

o

o

o

o

0

0

Xn

-2

o

0

x"

x"

0

x"

0

0

-1

++ _21

-5

7 _1

-3

o

o o

0

0

_1

++ _21

-5

7 -1

-3

o

o

0

0

++

28

0

12

-'I

_1

_1

0

00

0

0

o

0

9

11

3

3

3

11

2

6

6

6

I;

2

6

0

0

0

1

-1

o

o

0

0

0

++ -21

o

0

0

0

0

0

_1

_1

-1

-1

0

++

0

16

5

6

6

6

7

7

8

10

6

9

12fus1nd 12

2 2

11 4

6

6

14

11<

8

18

999 18 18 18

_2

-2

-,

_1

0

-1

_1

-21-2000

2

204-7222lfOQ

56

-8

2

11

2

2

0

0

-8

2

2

11

2

0

0

o

0

0

o

0

0

x"

+

56

X~

+

70-21677_22004

X~

o

70

_2 _11

1

_2

-2

2

0

0

-2000b21

X~

o

10

-2 _11

7

-2

-2

2

0

0

-2

X~

o

70

-2 _11

-2

1

-2

2

0

0

X~

o

10

-2 _11

-2

7

-2

2

0

0

X~

+

120

-8

12

-6

-6

X~

o

210

10

21

3

3

3

2

0

0

o

0

X>.

o

210

10

21

3

3

3

2

0

0

Xn

+

1

-2

1

-2

0

0

-2

000

_2

o

0

504

-8

18

18

-9

0

0

0

-1

-2

-8

18

-9

18

0

0

0

-1

-2

540

_2 -2

n

21

o

21

21

-1

0000000

o

o

o

o

..

o

_1

0

o

0

0

_1

b21 ••

-1

.. b27

-1

X13

_100X14 XllI

Xn

o

0

0000000

8

0

o

o

o

o

o

o

0

0

)(:t'1 X~

3

o

21

21

21

liOOx).!!

0 101

-41

0

0

o

21

o

21

21

_1

-100)(:16

o o

0

o

0

o

o

o

o

o

0

o

0

o

o

0

0

o

0

0

_1

_1

-1

0

0

o

-

0

00

o o

00 -32

0

00 -32

0

0

00

0

"

0

-111

0

8

0

-1

0

-1

-1

21+2

21+2

-1-1

-1-1

-1 1+1

-3

-31

0000000

o

o

o

o o

000)(41

0

0

0

0000000

o

o

o

o

OOOX4$

002

28

0

111+11

0

o

0

o

o

41+1

1+4

1+1

002

28

0

111+4

0

o

0

:002280

111+11

0

o

0

o o

o -21+ 1 o -21+ 1

1-2

~1_2

1+ 1

: 002 _111

0

21+2

0

-5

0

0

1 1+1

-1-1

_1_1

:00260

61+6

0

-3

0

0

0:002-320 0:002_320

21+2

0

3

0

0

0

0

-2

: 002

20

0

-41-11

0

2

0

0

_1

: 002

111

0

61+6

0

5

0

0

02

0

02

0

0

-3

0

0

0

0 1+1

0

-1

0

0

0

0

0

0

X#

02

224

0

8

8 _10

_1

0

0

_1

0

0

0

0

0

0

n

X4$

02

224

0

8 -10

8

_1

0

0

_1

0

0

0

0

0

0

X46

02

224

0

8 -10

8

_1

0

0

-1

0

0

0

0

0

0

X4'

02

280

0

31

4

0

0

-3

0

0

0

0

0

X48

02

280

0

X49

02

280

X.so

02

420

x"

02

Xn

02

X53

02

)(54

02

X55

02

896

032~~~

_1

_1

_1_1

n

_1

_1

0

-1

-1

0

_1

-1-b27..

0

-1

_1

0

-b21

o

0

o

0

0

0

0

11

0

0

3

0

0

0

0

0

-3

2

0

0

-3

0

0

0

0 1+1

o

0

_2

0

0

-3

0

0

1_1_1

o

000000b7

••

000000

b70

••

0

-2

-b27

n-b27

-2 -2

-2

o

0

0

o

0

0

0

3

0

0

0

0

o

0

0

000

o -1

b1x:18

o o

0

0

••

-1-1

-6

-2

x),

Ob7

-1_1

: 002

5

Xn

-1+2

_1

_1

o

21_1

_1

_4

Xn

o

-1

8 -10

X32

0

0-1-1

_1

5

0

0

12 12 12 12

36

8

o

12 12 12 12

-1

0

-21

X30

8 8

-1

0

_21

OX)I

12 12 12 12

12 fus 1nd 12 12 12

140

-21 -21

0

12 12 12 12

36 18 36

2211

-21 -21

_1

'0 '0 '0 '0

9 36 18 36

02

-1

0

9 36 18 36

02

21+3

0

OOOX»

9

Xc

21-3

X~

o

o

36

o·

o

o

o

7 1 8 28 28 8 14 14 8 28288

o

o

o

o

_1

o

o -21+3

o

_1

-1_1_1

o

o -21-3

o

-1

9

o

o

o

-1

18

0

00

0

X4~

00

o

-3

6 12

0

o -21

o o

6 12

0

_21

0

5 6 20 12 10 6 2012

-4

-21

0

8

3

_21

0

1233334 11 11 12 12 12 12 4 2 666611 11 12 12 12 12 4

12

-1

0

0

12

0

0

0

3

o

0

0

o

-8 -8 -8

0 -21

0

0

0

0

0

-41

0

0

0

-81

0

+

640

111

0 101

-3

X~

8110

0

00

0

000000nb10

-8

00

2

0

06400-8-8-8

-8

0

2

o

Xn

-8

2

-1

X1'

_161

o

0

0

0

000000b1**0

640

_21

0

0

06400-8-8-8

0

0

3

-21

-81

0

x"

0

o

-3

-1

41

-118

0

0 -'Zr

OX12

o

0

0

00

0

540

_1

0

0

o

0

00

0

0

6

-1

-2

o

0

-8100-121

-1

6

o -21

0

0

~

0 -41

_1

0 -39

o

o

...

2 -1

X:lO

00

2

10

14 14

0

0

-8

0

111 111

0

0

10

0

12 12

0

_6

-8

o

12 12

1 -21

20 20

o

0

-11

o

12 12

6 6

o

9

0 -17

o

12 12

12 12

0

9

10

8 8

12 12

0

14 -18

10

6 6

,,

x..

0

630

-11

o

o

-810021-100

+

0

o

0

0 -41

Xn

0

o

_1

00

o

0

o

o o

0

-1

0

-1

000

000-2_2

x"

0

0

0

b7X'8

0

o

4

0· ••

o

0

3

o

o

0

-6

o

0

2

-6

o

0

0

12

••

o

o

0

0

Ob7

o

28_101

2

120

o

0

00

2

02

o

-1

-3

X41

o

0

0

0

o

0

0

0

o

o

0

-3

0

o

0

0

000X'6

0-21

4

0

o

28 101

0

2

61

00

2

2

_61

_1

0

2

OOOXIJ

0

o

0

2

2

o

0

0

-7

-3

o

2

0

0

-3

++

0

20

o

-200X'4

o o

0

12 -27

02

0

o

0

560 -16 -34

x..,

0

o

0

+

032-4-4-4

0

o

-2

o o

Xn

896

11

o

o

_1

0

o

0

-2

o

_1

1 b27

o

0004_2_2

504

0

_1

x,

0

9

123

0

0

0

_1

0

0

o

0

o

o

0

0

o

0

00 -32

032-ll_II..1I

3

0

o

896

_1

-200X6

o

b27

-1

o

-2

+

0

729 9 0 0 0 0 _ 3

o

-3

o

-8

o

o

0

9

0

0

0

111 _10

00 -32

640

_1

o

21

lnb21

-1

-2

o o

++

o

1"

18

BC BC 120

++

OOOOOOb7"O

Xn

18

AC AC

0

aOaODO.*b70

X~

72

AC AC

0-1

-8 -8

+

72

AC AC

0

o

X~

5

A

-2

o

061100-8-8-8

x"

8

A

OB

2-3

x..

,"

9

DB

0

o

o

0

2 0

-3

lb27·.0

-2_2000 2

2 0

5

x"

+

18 CC CC 6F

o

000-21_2000

x..

X~

BC 6E

X.

2

+

x..

18

ac

++

35

+

27

A 9A

x,

x" x..

[54]

••• • ••••

_1

o

0

0

-1

-1

_1

_1

0

0 161

2 ,

,

8 8

2

000_21_2100

,, ,•

621121168 12 211 24 12 8

6

6

12

12

0

••

111

111

111

14

U U

U U X~

-1

_1

)(4'

OOOX43

X#

X#

1+ 1 -21-2

0

0

x..,

0

0

x..

0

0

X49

0

0

X.so

-1

-1

XJ'

21-1

-1+2

0-1_1

o o

-3

-31

o

o

00400

o

o

o

o

o

Ob7**Xn

o

0

11

0

0

o

o

o

o

o

0**b1x53

"o

o

OOOX54

o

o

: 002 -28

0

121+12

0

-1

0

0

-1

0

o

3

: 002

0

o

0

0

0

0

0

0

vl1l0

o

0

1-2

1+1

0

0

XJS

U4(3)

, ,, , , , , , 60118 B A

fU'

10d

X.

+~o

x,

+~o

x,

++

.

4E

96B '"C A

A

" '"

118 C

16 C

A 8C

A hD

108 OE AE

12F

12 OF

9 GE

AF

DE

120 12H

@

@

@

@

5

6

7

7

AG

BC

AE

AG AC AE 20A 21lA 2eA

@

@

@

920 576 BE A A BE A A a. fus ind 2D 2E

@

96 A A IlH

118 A A 111

2S

, •• "8

AD AD 6H

6118 108 AD CD AD CD In 6J

@

@

@

@

•••

@

@

;

++

7

o

_1

o

-1

-3

0

o

0

-2

2

o

o

-1

0

o

_0

6

_2

0

•

0

-3

o

X.

_0

0

2

2

5

-1

x,

"

-2

_0

21

5

_1

X.

_0

28

o

2

_2

X.

+

o

o

0

o

o

o o

0

0

++

9

0

0

0

++

15

++

-5 -5

_1

0

_1

o

o

o

o o

-1

-1

o

-3 _1

_1

3

3·'

o

o -3

6

6

•

3

0

_1

o

-1

++30622

3

0

0

++2041;0

-7

. -7

2

2-2

o

0

++

9

9

0

0

3

_1

-1

0

-1

0

2

_1

0

0_1 _1

0

_1

o

0

++

++

0

0

++

+

x..

+~o

x..

0000

x" x..

0000

o

0

o

0

-2

0

-2

_2

281+28 -41-11

0

o

0

32

o

0

o

0

0000

32

o

0

o

0

x..

+~o

27

3

X~

_0

o

o

full

Ind

"

•• • •

-3

o

1.'

-.

-.

0

o

0

0

-1

o o

0

o ... 1 o _1

o

0-b7

o

0

00

0

0

0

0

0

++

o

0

o

_1

_1

_1

o

0

0

o

0

z3

••

o

o

o

o

o

o

o

0

0

0

0

0

o

o

o

o

0

o

o

++

75

_1

_

15

-9 -5

0

o

~1

-3

~30300

o

-3

-3

-6

0

2

-1

0

0

3

o

0

_60-!lijO

0

0

00

3 -3

0 -3

-1

++

K.

o

0

0

++

X2

000

0

0

+

-1

_1 0

2

0

0

-1

~1

-1

o

o

K.

o

o

o

o

o

o

X,o

1 :: 1 ++

Kn

x" +

0

0

0

0

0

0

0

000

2

0

0

0

0

0

0

+

+

0

0

0

0

0

0

0

0

o

0

0

0

0

0

0

O'

0

0

+

-3

o

o

++6ijOOO

-8

-8

++

81

9 -3

0

o

o

0

0

0_1

-2

0

0

0

-1

0

0 12 12

++16000

-2

-2

o 1+1

0

0

++20002

2

2

o

o

o

o

00

'"

o

o

2 313+2-3i3+2

_Il

_1

o

00

20

o

o

2-313+2 313+2

_11

_1

o

+

0

o

o o

o

o

0

0

o o o o

++0000

o

0

0

0

0

o o

o

0

0

0

++96000

6

6

o

++2ijOO_ij

6

0

++

++

++

++

16

0

-2

0

0

0

18 fus 1nd fus ind 18

2

ij

6 6

8 8

8

8 8

0

0

0

0

0

o

0

0

0

0

0

0

02r2

0

r2

o o

o

o

o

o

28 ius 1nd 28

2 2

2

11

ij •

6 6

6 6

6 6

6 6

6 6

6 6

8

10 10

12 12

00

o

0

0

0

313

-3i3

0

0

0

0

0

0

13

0 -13

13

00

6

0

3

0

0

0

-1

0

o

0

++

o

0

5

_1

3

0

0

0

o

o

o

o

18 18

00

9 -3

o

0

0

_1

_1

_1

_1

++

X's

16 _16

-1

00

X'6

0

0

+

++

o 13 o -13

-1

•• -z3

o

o

o

o

o

o

0

3-1

-1

0

0

0

++

X,9

0

0

++

X20

10 10

12 12

2ij 211

0

0

13

13-13

0

0

0

211 fus 1nd 2ij 00

X2'

0

+

Xn

OO-r2-r2

++

K..

o

o

K~

o

o

0

-1 -z3

K"

••

0

0

0

0

o o o o o o

o o

00

X~

0000

x"

0000

x"

00

x" x..

-0-0

'"

•

0

-71+21 -31+1

0

71.21

0

2

1+3-1+1

21+3

o

0

o

0

o

6

-2

0

2

_2

-3

o

o

-1-1 1+1

0

1.1

0000

211+21

1.'

0 -31-3_1_1

31+3

Xn

0000

32

o

0

o

0

x..

0000

32

o

0

o

0

x..

0000

o

o

0

o

0

8

8

8 8

8

x..

00002

x..

00002

x..

02

...

0

o

0

00002

o

-1

-1+31.' -21+3

281+28 -41_4

X~

0

0

0000

Ind

o o

_1

31+1

x" x..

fus

-2

•• • •• ••

-1

0

0

o

0

0

0

6

0

o

~1

-1

_1

00

0

0

0

0

9i3

-9i3

0

000

0

++

80

0

0

0

_10

-10

0

++6000-2

6

6

0

o

++MOOO

-8

-8

11

-2

0

0

0

_1

0

0

12 12

12 12

12 12

12 12

6 6

6 6

8

20 20

12 12

12 12

0

o

_1

o

0-b7

o

-1

00

o

o

0

v'lla

0

12 12 12 12

12

12

12 12 12

12 12 12

80 80 80 80

211 2ij 211 2ij

1 1-1

-. -.

0

2

0

61+111 -21-2

0

o

0 -31+5

1.'

0

2

0

o

o

o o

o

0

o

1-3

-11+21

o

0

16

8

-1

1+1

•

1-7

o

-1

0

0

+

0

0

0

0

++

K..

0

0

0

0

0

0

0

+

-3

0

0

o

0

0

0

++

3

0

0

0

0

_1

0

0

++

0

0

0

0

0 -13

0

0

0

_11

2

0

0

0

0

0

0

_1

-3

-3

o

0

0

0

28 28 fus 1nd 2828 28 28 28 28

_1

o

0

-1

-1

21+3 -21.'

-1

o

-1

0

0

o

o

o o o

+

•

+

•

0

I

x.. 28

-'I

0

•

0

x..

02

o

o

o

o

o

o

_1

o

o

o

o

o

o

0

o

o

x.. K"

I

00002

211-7

51-3

0

-2

0

31+2

x"

: 00002 -.211+21 -31.1

0

-2

0

3

x"

: 00002

32

o

o

K"

, 00002

32

o

o

o

o

o o

K..

00002

28

-'I

0

_.

""

00002

o

o

0

o

0

0

-. -.

o

-1

-1

o

o

o

o

1

0

-1

o

o -b7

••

o

-1

o

o ••

-b7

o

+2

0

0

0

0

++

X29

0

0

0

0

0

0

0

0

+

Ku

0

0

0

0

000

0

0

+

Kn

0

0

0

0

0

0

0

13 -13

13

00

++

0

0

0llr2

0

0

0

r2

++

++

++

0

0

02r2

0-r2

00_r2_r2

+

+

0

000

0

0

o

0

+

0

00

00

_1

++

0

00000000000

"

0

0

0

-1

o

K"

0

0

X34

X's

Ku

0

+

Xn

++000000r5000

++

Xw

0

0

r2

11 11

2

••

++

36 36

36 fus 1nd fus 1nd 36

•

• •

•

11

6 6

8 8

8

8 8

10 10

12 12

211 211

24 fus 1nd 211

•

+

+

K~

•

+

+

X4'

+

+; 11 ::

•

I

+

X.,z

02

K..

02

K.. K..

•

K. +

+ +

+2

02

I

+

02

I •

X47

x.. K~

+

+

•

•

I

+

•

I

+

2

• +

•

o _1

0

+

x..

00002

0

0

o

x..

o

0

++

K"

0

o

o

0

0

"-b7

-1

o

0

0

K"

n

1+1

1+2

+

o

Ku Kn

00

o

XI1

o

++

x.. +~o

Xl] K••

0

o

K.

o

2

0

++

o

0

0

.,

o

0

0

X6

++

o

_1

o

++

o

0

0 -13

-1

0_1

-2

0

_1

X5

0

0

0

++

o

++ -10

0

_1

K..

Xn

X~

K,

_1

n_b7

o

o

12 AF AF

00

0

o 1+1

o

12 AF AF

++

0

++2ijOOij

o

6 AJ AJ

00

0

0

o

AF

13

0

0

o

5 B AF

000

-6i3

0

o

0

613

o

o

0

0

_1

o

0

0

0

o

1+1_1_1 -21-2

0

-1

0

0

0

-1

0

0

31+3

2-1

o

-1

11+7

++ _10

o

8

0000

++

2

o o o o

28 28

x" x..

o

2

_2

2ij 2ij

o

9

o 613+ij-613+ij

110 ijO

3

10

++

00110-80

12 12

o

++

++

o

12 12

0

++

o

12 12

_2

0

o

n

•

A

02_200

++

o

8 8

2

0

o

8 8

o

0

o •• z3

0

0

0

o

0

0

0

o

-1

2

0

0

0

o

-1

0

-1

o

o

-1

o

_1

o

_6

3

-2

3

0

00

-3

0-613+ij 613+11

0

o

-0-0

8H 100 12J 2ijB Cn fus 1nd

0

-8

0

•

x"

@

16 A A 8G

2-2

++ '00

, , •••

@

ij8 A A SF

40

0

o

_1_1 1+'

o

X~

x" x..

o

@

30 -10

0

+

o

@

ij8 9 A OF A OF IlJ 6N

K.

x" ++

@

-3

x"

x" x..

;

9

0

9

_1

_2

-2

_1

x"

;

++

0

X.

;

5ij3698 5 12 6 9 9 720 BDCEDEA AD AH BI BI AH A BO CE DE A AD AH BI AD BD A 6K 61.. 6M 8E lOB 12I 12J 1eA Bn fus ind fus 1nd 2F

+

+

+

+2

+2

•

+

•

+

X"

+2

Kn

r A

o

-1

0

0

•

+

•

'+

•

+

0 w'80

0

0

0

•

+

•

+

•

+

•

Xso

x" +

K..

+

X»

[55]

U 4 (3) ind

lA

2A

3A

38

3C

3D

i1A

1

2

6 6

3 3 3

3

3114

3 3

3 3 3

12 12

x~

02

15

-1

6

3

0

0

)(57

02

21

5

3

6

0

0

Xss

02

105

9

15

3

0

0

X59

02

105

-7

15

3

0

0

X60

02

105

9 -12

12

0

0

2

liB

5A

6A

566

13

-13

0

0

-2

•

+

Xoo

•• _z3+1

0

0

• •

x..

0

0

-2

0

0

-1

o

0

0

0

-13

b7"

0

o

o

0

0

•

0

X~

o

000

0

•

0

x"

0

•

360

8 -18 -9

0

0

0

0

0

38i1

0

211

12

0

0

0

0

-1

X67

02

1120

11

33

-6

0

0

11

0

0

x~

02

630

6

9 -9

0

0

2 -2

6

6

6

0

-2

-2

-2

2

-1

2

2 -1

2..

b7

0

0

0 -1

-1

0

0

0

••

2:3-1

0

0

+

XM

0

0

13

-13

0

0

• •

x"

3

0

0

0

0

o

o

0

0 -1

•

+

x"

0

0

0

1

-1

o

o

0

0

0

•

+

x~

3

0

0

0

0

0

o

o

0

0

-1

•

+

x,.

o

-3

0

0

0

0

o

o

0

0

5

6

6

677

9

9

9

9 12

and

'3 '3 '3

no:

9

9

9

18 9

18

18

18

12 12 12 12 12

-13

0

0

-1

0 -3

0

8

5 30 15 10 15 30

02

Xu

02

84

6

0

0

4

0 -1

X74

02

120

-8

-6

15

0

0

0

0

X75

02

126 -10

18

9

0

0

2

0

6 6 6

6 6 6

6

6

7

7

8

6

6

42

42

24

6 6

6 6 6 6

21 llj 21 112

21 111 21 42

24 8 24 21j

9 18 9 18 9 18

-2

-1

_1

0

13

6 6

6-2-330020

0

-2 _2 2

-2

0

0

_1

0

o

-2 2

0

0

18 9

•

•

12 rus 1nd

x" 2

2

4

4

6

6

6

6

8

10

12

12

12

12

14

14

2 2

4 4

4 4

4

6 6

12 12

12 12

6 6

8 8

20 20

12 12

12 12

12 12

12 12

14 14

111 14

•

x"

•• -z3+1

0

0

•

x"

••

0

0

0

•

x"

2:3-1

0

o

o

0

0

2

•

x"

i3

0

0

0

•

x"

Xu

02

210

-6 -24

-3

0

0

6

0

0

0

3

0

0

0

0

-13

X17

02

270

627

0

0

0

2

0

0

3

0

0-b7

..

0

o

o

0

0

-1

•

0

X78

02

270

627

0

0

0

2

0

0

3

0

0

"-b7

0

o

o

0

0

-1

•

0

Xl?

02

336

16

-6

6

0

0

0

0

-2

-2

-2

0

0

0

-13

13

0

0

0

•

x,.

XM

02

384

0

24

12

0

0

0

0

-1

0

0

0

-1

-1

0

••

2:3-1

0

0

0

•

Xoo

X81

02

420 -12 -21

12

0

0

-4

0

0

3

0

0

0

0

0 -z3+1

••

0

0

-1

•

X~

Xn

02

630 -18

9

-9

0

0

2

0

0

-3

-3

0

0

0

0

o

o

0

0 -1

•

x"

X83

02

630-2

9-9

0

0-2

0

0

-20021

o

o

0

0

XlW

02630-29-900_200

-200_21

o

000

Xss

02

840

8

12

6

0

0

0

0

0

-Ii

2

2

0

0

0

X86

02

840

8 -42

-3

0

0

0

0

0

2

_1

2

0

0

0

z3-1

.nd

no:

1

~6

3 :2

3 :6

3 :2

4 :6

4 :3

5 :6

6 :6

6 :6

6 :6

7 :6

7 :6

8 :6

9 ,2

ind

12333 12 12 12 12 12 6 6 6 6 6 4 4 12 12 12 3 6 3 3 12 12 12 12 2 6 6 12 12 12 3 3 3 4 12 12 6 6 6 12 12 12

3 12

4 12 12 4 12 12 4 12 12 4 12 12

8

5 60 30 20 15 60 10 60 15 20 30 60

6 12 6 12 6 12 6 12 6 12 6 12

6 12 6 12 6 12

6 12 6 12 6 12

7

7 84 42 28 21 84 14 84 21 28 42 84

8

9 36 18 36 9 36 18 36 9 36 18 36

0

-1

-3

0

0

0

-3

0

0

-3

0

0 -1

•

X~

..

84

2 :6

3 :6

0 -15

6

0

6 12

0

2

M 24

~

42 28 21 84 14 84 21 28 42 24 0

0

•

••

0

0

0

•

9

9

9 12

:2

:6

9

9

36 18 36 9 36 18 36 9 36 18 36

36 18 36

•• -2:3+1

0

0

0

0

.5 02

000

.5 02

x" XM

4 :2

11 :2

4

no:

2 :2

12 12 6 8 :2:2:2:2:2

20

9 12 rus 1nd

2

8

4

4

8

4 4 11 11

6 12

:6 36 18 36

and

:6 12 12 12

2 4

12 12 12 12

12 12 12 12 -1

.5

0 1+1

13

-13

0

0

.5

02

Xoo

0

2:3-1

••

0

0

0 _5

02

x"

0

2:3-1

••

000.502

x•

0

-1

0

0

0

_1

X91

04

420

0 -21

12

0

0

2

0

0

3

0

0

X92

04

420

0

33

-6

0

0

2

0

0

-3

0

0

Xn

04

480

0 -24

6

0

0

0

0

0

0

0

0 -b1..

X~

04

480

0 -24

6

0

0

0

0

0

0

0

0

504

0

-9 -18

0

0

4

0

_1

•• -b1

13

0 .5

_1

3

0

0

0

0

0

o

o

0

0

0

0

0

0-1-1

o

o

0

0

_1

*5

02

-3

0

0

0

0

0

-13

13

0

0

-1

*5

02

000

0

0

0

0

0

.. -z3+1

o

0

0-4

123333485666778 :12:6 :12 :4 :12 :4 :12 :3 :12 :12 :6 :6 :12 :12 :12

9

,4

X. Xoo

02

3

0

x" x.

02

0

0

0

14 28 14 28

x"

0

840

14 28 111 28

02

0

04

12 12 12 12

.5

12

x.

12 12 12 12

0 Ml

24

0

12 12 12 12

0

0

0

12 12 12 12

••

384

0

8 8

40 40 40 40

0 1+1 -2:3+1

04

0

6 12

14 :2

0

Xw

0 -15 -12

6 12

,2

111 :2

0

0

840

8 8

12 :2

0

0

4

04

6 12

12 ,2

z3-1

11

Xn

211 24

12 ,2

••

0 0

04

24 24

12 ,2

0

0 0

0475602700020

6

-1

6 0

no:

0

o

21

0 -27

6

0

0

0

0

216

X95

X~

-1

120

04

XlI6

x"

0

-13

04

12

24 8 24 24 8 24 24 8 24 8

•

•• -2:3+1

x" x,.

o

XA

-4

M

0-1-1

Xll9

and

[56]

x"

13000.+

0

Xn

4 -15

x"

+

-2

_1

4 12 12

6

0000.

-2

:3 '3 '3 '3 '3 '3 :3 '3 '3

3 6

x" x,.

• • • •

0

02

3

+

2

02

11 12 12

•

_1

x~

2

x~

0

0

x~

q 12 12

12 12

0

0

4

211 211

0

0

4

21 21

0

0

_11

21 21

0

0

X690272990000_3

6 6

o

-9

6 6

14

x"

8 -18

2

14

+

360

3

12

•

02

3

12

-1

X6-<

3

12

0

0

3

12

0

0

6

10

..

0

3 6

8

-1 -z3+1

0

3

6

0

6

:3

6

0

-6

3 6

6

2

16

3 6

12£ 14A B. .

_1

336

2 6 6

120

o -1

02

1 6

C..

0

X63

'3 '3 '3

12B

0

4

ind

lOA

••

_1

no:

8B

-z3+1

0

3

0

6G

0

0

3

0

6F

6

0

-1

3

••

6£

0

-2

3

z3-1

60

3

3

2

0

4D 4

3

o 5

-2

1

0

4C 4

0 _1

0

.nd

%3-1

2C 2

0

0

0

••

2. 2

2

0

0

9 9

12 rus 1nd

2

0

0

9

9

-1

9

9115 -15 -27

9

9

2

15

02

9

_1

3

}(71

9

gc D.. 12A

2

-5 -36

0

8

B..

0

315

0

9A

3 _1

210

0

677

BA

6 6

02

Zl

an

6 6

02

756 -12

7A

15 15

)(61

02

6C

12 12

X6:

Jew

6B

x" x.

.5 02

2 .5

x" x.

02

12

and

2

8

:11 :12 :12 :12

no:

:11

:2

9

9

9

411621121168 :11 :4:2:2:11:11

40

,4

12 :4

12

,4

12

,4

12 :1I

24 :1I

24 :4

U 4 (3) 42

4F

lIG

BC

BD

12F

120 128

20A 24A 28A

Ba

2D

!us ind

2E

4H

224 6 6 12 6 6 12

X~

002

x"

002

11

3

X~

002

25

9'

X~

002

4I

4 12 12

5 -3

6H

r.*

6 6 6

6

-4z3 _1

6J

6K

6L

666

6M

6

6 6 6 666

8

10

12

12

18

18 Cus Ind

30 30

12 12

12 12

18 18

18 18

2 _1

0

0 _1

2

0

0

213+1-213+1

4

o

0

o

413-7-413-7

2 -1

0

0 -1

0

0 -2

1 -z3..

•

+

x~

-1

0 •• -z3

•

+

XS1

••

z3

•

+

x~~

•• -z3

•

+

X5'J

-1

•

+

Xro

_1 -z3..

•

+

X...

002

35

3

3

3-213-4 213-4

2

0

0

_1

0

x..

002

50

_6

-2

2-213-7 213-7

2 _1

0

0

0

0

x" x..

002

45 -3 -3

o

_6 -3

0

0

002

64

0

0

0 213+4-213+4

_2

11

0

0

0

-1

0

0

o

0

0

0

o

0

0

0

0

0

0

0

0

0

0

0

-8z3

4

-2

0

0

0

-1

l?

0

z3"

20 -12

_4

0 413-1-413-1

2

2

0

0

0

0

-1

0

_1

2

2 613+3-613+3

0

-3

0

0

0

0

_1

-1

0

-3

}(w

x..

;

I

02

o o

_4

2

o

0

0

0

-1

X~

: 002

}(67

:

X..,.

: 002

30

6

Xw

002

81

9 -3

o

o

0

0

0

0

-1

x?"

002

36 -12

4

0

9

9

0

0

0

0

0

x"

002

45

-3

5-3

-9

.nd

2

2

4

4

6

6

6

6

6

6

2

2 6 6

4 12 12

• 12 12

6

6 6

6 6 6 6 6 6

6 6

6 6

002

no: Cus ind

6 6 2

6 6

• 12 12

n

•

0

+

X6)

+2

X....

•

+

Xu

-1

•

+

X67

0

•

+

x....

0

6 6 6 6

6 6

6 6 6 6 6 6

6

6

6

x"

002

4

0

0

2 -13-2

13-2

-2

X7)

002

16

0

0

0 -13-8

13-8

-2

x"

002

40

0

0

4-413-2 413-2

-2

Xn

00236002

o

*

0

0

8

10

12

12

18

18

8 24 24

10 30 30 10

30 30

12 12 12 12 12 12

12 12 12 12 12 12

18 18 18 18 18 18

18 fus lnd 18 18 18 18 18

-1

13

_1

*

+

Xn

13

0

z3

**

*

+

Xn

**

z3

*

+

X74

0

0

0

0

+

o

0

0

3

0

0

0

o

-1

0

0

*

+ X7S

-4

_1

0

0

0

0

0

-1

-1

-1

*

+ X~

o

0

0

0

0

o

o

0

0

0

20

0

0

2 413+8-413+8

x"

02

o

0

0

0

o

0

0

1

+2

x"

x" X~

Xn

002

16

0

0

0-8z3+6

n-2-2000

x..

002

64

0

0

0

-8z3

n

002

80

0

0

0

x"

002

60

0

0

x"

02

o

0

0

0

o

X"S

002

80

0

0

0

8z3

X~

002

40

0

and 2 no::2

2

x",

8H 10C 12J 2/JB C. .

X7'

6

: 002

8G

OOO*+X"" 0

0

SF

OOOO*+X69

-1

o

6H

X"

o

X76

o

-2

1

o

_90000

lIJ

OO·+X~2

x" 64

2F

24 24

3-213-1 213-1

5 -3

••

8E lOB 12! 12J 18A 8..

:

-1

o

0

*

+

X79

0

0

z3

**

*

+

x""

0

.. -z3

*

+

x",

4

-2

0

0

0

13-4 -13-4

_4

2

0

0

0

0 -13

-2 313+6-313+6

0

3

0

0

0

0

13

o

0

0

0

0

0

0

0

0

..

2

_4

0

0

0

0

0

0 -z3

0

-4-413-2 413-2

-2

o

0

0

0

0

4

4 :2

6 6 :2:2

8

10 12 12 18 18 :2:2:2:2:2 20 60 60 20

00*+X"2 0

0

I

+2

X~

Xl<3 X~

fus Ind

• •

12 12

12 12

2 6 6

• 12 12

•

12 12

•

12 12

6

,2

6 :2

6 :2

6 :2

12 12 12 12 12 12

12 12 12 12 12 12

12 12 12 12 12 12

12 12 12 12 12 12

6

6

8

6

6

24 24

60 60

12 12 12 12 12 12

-1

12 12 12 12 12 12

**

36 36 36 36 36 36

n

*

+ X"S

z3

*

+

XIlt>

36 fus 1nd 36 36 36 36 36

x"

*7

02

**

+2

X""

X~

*7

02

**

+2

X811

X~

*7

02

••

+2

X89

Xw

*7

02

••

+2

Xoo

x",

*7

02

••

+2

X91

X~

*7

02

••

+2

X
x"

1 o.

X~

7

1 *.

+4

•

X93 X~

x"

_7

02

X~

*7

02

**

+2

X96

x"

*7

02

••

+2

X97

x"

*7

02

__

+2

X93

and no:

42IJIJ :2 :2

12 ,2

12-1212 :2 :2 :2

6 6 :2:2

8

~

12 12 ~ ~ :2:2:2:2:2

+2

~

_7

*7

.7

Xo»

[57]

U 4 (3)

ind

lA

2A

3A

38

1

2 6 6

3 3

3

3

3 Xw

x\oo XliII

02

02 02

3D

3

3

3

4A

11 12 12

45 -3 -9 45 -3 -9

0 0

0 0

126

14

-9

0

0

0

2

02

189

-3

27

0

0

0

5

X,04

02

315

11

18

0

0

0

_1

XIQ5

02

315

-5

18

0

0

0

3

X,06

02

315

XIQ?

02

630

6C

6 6 6

6 6 6

677

9

21

211

21

21

24

-2 0

0

b7

••

••

b7

9

9

4C

40

6D

610;

6F

6G

88.

lOA

12B

Cn

12D

12£ 14A

an

2

2

•

4

6

6

6

6

8

10

12

12

12

12

14

12 12

o

o

000

_1

o

000

o

000

-1

12 fus 1nd

2C

14

Xw •

+

•

0

•

0

X,oo XLOI

_1

2

2

0

0

0

o

o

0

0

_1

•

+

_1

3

0

0

0

0

.,

o

o

0

0

-1

•

+

X1
-1

0

2

2

2

0

0

o

o

0

02.

+

X'M

_1

0

-2

-2

4

0

0

_1

o

0000.

+

XIIIS

2

X,~

0

3

-1

0

-2

4

-2

0

0

-1

o

0000.+

0

2

-2

0

3

0

0

0

0

0

o

o

0

0

-1

•

+

X,oo

0

0

-2

-2

-2

-1

-1

0

o

o

0

0

0

•

+

X'M

_1

0

0

0

_1

o

o

0

0

0

•

+

X,oo

3

0

0

0

0

0

o

o

0

0

-1

•

+

X110

o

-3

0

0

0

0

o

000

•

+

4 4 5 :3 :3 :3

6 :3

8 :3

9

9

a

9 18

18

0

0

0

0

9

0

0

0

0

-3

XIIO

02

756 -12

27

0

0

0

_4

XIII

02

9115 -15 -27

0

0

0

12333

:3

02

XlP

02

126 _10.

XU 4

02

126 -10

Xus

02

126

2 -18

6

6

3

0

6 :3

6 :3

3

331i4S66

6

6

6

7

7

6

42

112

211

21 14 21

21 14 21

24 8 24

:3

1 2 3 666 3 6 3 2 2 6 3 6 3 6 6 6

XI12

6

7 :3

7 :3

9

9

9

9

9

18

18

18

4

4

6

6

6

6

~

10

12

12

12

12

111

111

2

4 4

4

4

6 6

12 12

12 12

6 6

8

4

8

20 20

12 12

12 12

12 12

12 12

111 14

14 14

12

15 10 15 30

6 6 6 6

6 6 6 6

6 6 6 6

1i2

42

211

6

0

0

2

2

2

-1

-1

0

o

o

0

0

0

•

X'l2

_4

2

0

0

0

o

o

0

0

-1

•

X,l3

-1

•

XU4

0

0

2

0

-1

0

0

0

2

0

-1

2

_4

0

0

0

o

o

0

0

-9

0

0

0

-2

0

3

0

0

0

0

21

o

o

0

0

o

o

0

0

-2

0

3

0

0

0

**

0

o

o

0

o

o

0

0

0

o

o

o

o

000

0

0

0

0

o

o

0

0

-2

-2

-1

_1

0

o

o

0

-1

2

2

0

0

0

o

o

0

6 :6

6 :6

6 :6

7 :6

7 :6

8 :6

9 :2

:2

02

270

6

27

0

0

0

2

0

0

3

0

0 -b7

02

270

6

27

0

0

0

2

0

0

3

0

0

XU9

02

504

-8 -36

0

0

0

0

0

-1

4

-2

-2

XI20

02

540

12 -27

0

0

0

4

0

0

-3

0

0

XI2'

02

630 -18

36

0

0

0

2

0

0

0

0

X122

02

720

16

18

0

0

0

0

0

0

_2

Xl23

02

1260

.JI

-9

0

0

0

-4

0

0

and

1 :6

2 :6

3 :6

3 :2

3 :2

344 :2 :6 :3

5 :6

1 12

2 12

3 12 6 6 6 4 4 12 3 6 3 12 12 12 2 6 12 12 3 3 4 12 6 6 12 12

3 12 6 12

3 12

3 12 6 12

4 12 12 • 12 12 4 12 12 4 12 12

5 60 30 20 15 60 10 60 15 20 30 60

XI24

04

Xl2S

011

216

0 -27

0

0

0

11

0

X'26

011

360

0

9

0

0

0

_11

0

X127

04

360

0

9

0

0

0

_4

X'28

011

504

0 -36

0

0

0

360900020

6

6

12

12

6

6

12

12

6

6

12 6

12

12 6 12 6

12

n

677 8 12 84 84 24 6424224 12 28 28 8 6 21 21 24 12 84 84 24 14 14 8 84 84 24 21 21 24 28 28 8 42 42 24 84 84 24

9

XlI7

0

0

-1

*

0

Xl IS

0

0

0

•

XU9

•

X,m

2

•

XI2'

0

0

•

X'22

0

-1

•

XI23

9 ,9 :2

9 12 :2

36

9

9

9

36

36

18

18

36 18

36

36

36

36

i+l

o

o

0

18

0

0

-1

_1

0

o

000

0

-3

0

0

b7

..

0

o

0

0

-3

0

0

••

b7

0

o

o o

4

0

_1

0

0

0

0

0

0

0

0

o

0

0

6

7

7

8

9

9

9

9

:6 :12 :12 :12

:4

:11

:4

027

0

0

0

2

0

3

0

0

XIJ2

011

1260

0

-9

0

0

0

-2

0

0

3

0

and

1

2

3

3

3

3

4

8

5

6

6

:12:6 :12

:4

:4

:3 :12 :12

:6

:4 :12

6 12

'0 40 40 40

12 12 12 12

12 12 12 12

12 12 12 12

12 12 12 12

111 28 14 28

14 28 111 28

6 12

211 211

24 211

6 12 6 12

8 8 8 8

.5

02

*5

02

X'2S

X,24

X,W

o

756

14 :2

Xl27

0 1+1

04

14 :2

000-2·502

0

X13l

12 :2

o

0

0

12 :2

02

0

0

0

4

12 :2

.5

0

0

-3

4

• •

12 :2

_1

o

-3

0

4

:2

0

000

-1

0

8 8

20

0

o

0

-2

2 4 2 4

6 12 12 6 8 :2:2:2:2:2

Xl26

o

4

0

11

02

0-1-1

0

0

11 :2

.5

0

0

0

12 12 12 12 12 12 12 12 12 12 12

11 :2

-1

o

0

12 fus 1nd

2 :2

0

o

45

0 -27

no:

0

0

0

and

:6

0 -1

1-1_1

5011 540

X1I6

0

-3

04

0

•

0

04

•

-1

0

X'29

0

0

-3

Xuo

XIIS

•

0

-b7

XI17

8 24 24

0-21

2

12 12 12 12

0

12

2

12 4 12 12

-9

6

12 fus ind 12

2

6

-9

0

no:

6

0

0

and

:3

30

0

0

12

12

0

-9

XIIl

12

Xll8

no:

9

21

6

2~

0

16

ind

8

6

9C D.. 12A

0

720

no:

B..

0

729

126

0

9A

0

02

go

3

0

SA

18

02

:3

3

7A en

6 -45

-5

XIOS

02

6B

X,oo

X109

X1l6

6A

_2

o

0

02

ind

SA

5 15 15

o

0

X,
no:

4B

11 12 12

364900040

"'M

and

[58]

3e

0

0

-1

12 :4: 12

.5

02

X,~

.5

02

X,m

.5

02

Xl31

*5

02

XIJ2

and

2

8

no:

:4

:2

4 :11

11

6 24 24 6 8 :4:2:2:11:4

40 :4

12

:.

12 :4

12

:4

12 :4

14 :4

14 :ll

U 4 (3)

4E

...

'0

8C

80

,,,.

12G 12H

20A ZlIA 28A

e.

2D

2E

llH

41

6H

I..

6J

6K

6L

6M

8£ lOB 121 12J lSA 81.

2F

fus ind

2 6

6 Xw

002

x"x,

02

lIJ

• 12 12

6N

6

SF

SG

8H lQC 12J 2118

en

8 10

8

8

24

24

24

30

12 12

24 24

21l

211

2ij

30

12

24

24 fus Ind 24 24

-1

-1

6-2

o

2

2

0

o

o

o

o

o o o o o

o

:t"" 2

0

o

X"'1

002

6

x,,,.,

002

9

x"'..

002

15

3

0-3

XIM

02

o

o

o

o

o

Xl
002

0

4

0

2

XU'"

002

0

0

0

0

Xu'"

002

9-3

o

3 _1

XIIl'

0026-20_2-2

X,II

002

15

,nd no:

2

q

o

0 _1

o

0

o

o

o o o

_2

0

0

0

0

0

0

0

x,,'"

fus ind

2

6 6

_1

• •

002

XII)

1

-1

+

X"'2

•

+

Xu"

•

+

Xu"

02

XIU~

1

X'w

-1

•

+

>:'07

16 -16

•

+

X"18

000*+XI09

o

•

+

XIlO

•

+

XIII

-1

_1

-1

o

-1

-1

-1

6

8

8

8

10

12

24

24

10 30 30 10 30 30

12

24

24 fus Ind

12

24

24

12 12

24 24

2.

12

24

12

24

6

12 12

8

8

24

24

,.8

21l

24

2. 8

,.2'

0002rZOr20

24

2'24

o -r2 -r2

*

o

1

02

o

o

o

o

o

o

o

02

o

o

o

o

o

o

o o o o

02

o

o

o

o

o

o

o

o

o

+

02

Xli"

XII2

x,,) X"4

Xll~ XI16

X1l7

o

o

o

Xll~

X"9

002

Xu"

: 002

XI2l

002

Xm

x...., X,IlO XLOI

o

8 24 24 X112

-1

+

+2

•

-1

-1

0

•

o

o

0

0

o

o

0

0

o

o

r5

0

o

o

o

13 -13

13

1 1

+2

X"6 +2

•

+

X,l9

•

+

XI~

+

X,21

0

0

02rZ

O-r2

0

O-r2-r2

:00200

04r2

0

0

0

0

0

0

0

0

8 :2

10 :2

12 :2

211 :2

24 :2

8 2'

10 30 30 10 30 30

12 12 12 12 12 12

24 24 24 24 24 24

24 fus 1nd

002

0

0

and no:

2

4

6 ,2

8 ,2

8

fus 1nd

2

4 12 12

6

8 2' 2' 8

8 2' 24

6 6

8 24 24

2. 2'

X,24

*7

02 02

X'2~

*7

Xm.

1

X.21

7

XI2H

*7

02

X,29

*1

2.

XlI7

X"3

00

X'2.\

X1I5

Or2r2

* *

+X,U

13 13 -13

*

+

Xl23

24 24 24 24

2.

** **

04

1.

+2

X124

+2

X12~

+.

X,~

X,21

**

+2

02

**

+2

X'29

+2

X,30

X128

X,w

'7

02

**

XUI

*1

02

**

+2

XUI

X,12

*1

**

+2

X,32

*7 *7

'7 .7

02

,nd no:

2

4

6 ,2

8 ,2

8

8 ,2

10 '2

12

24

:2

:2

24 :2

[59]

Gz(3)

Chevalley group G2 (3) Order

= q,2q5,696 = 2 6 .36 .7.13

Mult = 3

Out

= 2

Constructions Chevalley

G : the adjoint ChevalIey group of type G2 over F 3; G.2 is obtained by adjoining the graph automorphism;

G : the automorphism group of a generalized hexagon of order (3,3) consisting of 364 vertices and 364 edges, each

object being incident with 4 of the other type; G.2 is obtained by adjoining an automorphism interchanging

vertices and edges Cayley

G : the automorphism group of the mod 3 Cayley algebra of vectors

!8ni n , with an in F 3 (subscripts mod 7), where n

runs over {oo,O,1,2,3,4;'S,6} and 1 00 =1, 10+ 1=1, i n+ 2 =j, i n+ 4 =k form a quaternion sUbalgebra

14-dtmensional G.2 acts on a 14-space with base {e oo ,eO, ••• e12} and is generated by

A

et ~et+l'

D

'V-3.eoo ~ 3e oo + /3.~e1 (1 • (0)

B: et ~e3t'

C: (e 2 e 12 -e8 ) (e6 e l0 -ell) (e5 eq

-"7)

4./-3.e i .....,. /3e oo -3(eSi+eSi+2+eSi+S+eSi+4). +(eSi+ 1+eSi+3+eSi+4+eSi+7+eSi+8+eSi+9+eSi+ 10+ e Si+ 11+ e Si+ 12) 2 D : e oo -? -e oo '

e i -:.'t -e2_i-e4_i-eS_i-e6_1+e7_1+e8_1-elO_i+e11_i-e12_i

The subgroup G2 (3) = is represented by real matrices. -There are 2 x 378 images under G2 (3) of each of the vectors (4/31013) and (31/3 13 ). These vectors are fixed by the respective subgroups and of type L (3), and negated by D2 • The two orbits are interchanged by outer elements of G2 (3).2. 3

@

@

@

@

5

5

@

@

@

@

@

@

@

@

@

@

@

@

@

@

@

@

@

@

@

@

@

@

q2q5696 576 832 832 729 162 162 96 96 72 72 18 18 7 8 8 27 27 27 12 12 13 13 512 2q 27 p power A A A A A A A A A A B A D A E A A A B C C C AA B8 A A A A CB p' part A A A A A A A A A A B A D A E A A A A A A A AA BB A A A A CB 1nd lA 2A 3A 3B 3C 3D 3E qA qB 6A 6B 6C 6D 7A 8A 8B 9A 9B C.. 12A 12B 13A B* fus ind 2B qC 6E

x,

+

[60]

@

@

@

@

@

@

@

18 18 6 EB EB CC EB EB DC 6F G. . 12C

6

7 9 9 9 AB AE CE BE AB AB BB C8 D. lqA 18A 18B C..

CC DC

x,

++

x. x,

+

lq

o

x.

5

5

-2

6q

0 -8

o

M

0

x,

+

78

x.

+

91

x,

+

x.

-q

o

2

-1

2

2

-8

q

-2

0

0

0

0

0

-8

-8

q

-2

0

0

0

0

0

-2

-3

-3

-3

-3

6

2

2

-5

10

10

10

3

3 -2

91

3 -8

19

q

-2

3

-1

0

3

0

0

0

+

91

3 19 -8

q

-2

-1

3

3

0

0

0

0

-1

x.

+

10q

8

lq

lq

5

2

-1

0

0

2

2

2

-1

-1

0

0

2

-1

-1

x"

+

168

8

6

6

6

-3

6

0

0

2

2

-1

2

0

0

0

0

0

Xn

+

182

6

20

-7

-7

2

2

2

2

0

-3

0

0

0

0

0

-1

x"

+

182

6

-7

20

-7

2

2

2

2

-3

0

0

0

0

0

0

x"

+

273

-7

30

3

3

3

3

-3

2

-1

-1

-1

0

-1

x,.

+

273

-7

3

30

3

3

3

-3

-1

2

-1

-1

0-1

x"

+

qq8

0

16

16 -11

-2

-2

0

0

0

0

0

0

0

0

0

o

0 b13

x,.

+

qq8

0

16

16 -11

-2

-2

0

0

0

0

0

0

0

0

0

o

0

x" x..

+

5q6

2 -21

+

5q6

x..

+

X20

0

0

0

o

0

0

o

0

-2

o

0

-2

o

-2

-1

2

-1

-1

-1

-1

b27

••

0

0

-1

•• b27

0

0

-1

00

8

0

-1 2z3

-1

-1

00

8

0

-1

-1

0

0

++

6

-2

-3

o

0

0

0

++

7

-1

-2

o

-1

0

0

+00000000000X7

0

0

0

x.

o

0

0

0

++

8

0

-1

-1

-1

0

0

0

-1

-1

++

0

0

0

0

0

-1

-1

2

-1

0

0

+

0

0

0

0

0

-1

-1

-1

-1

2

0

0

0

0

0

0

o

0

o

00

-1

-1

-2

-1

-2

-3

-3

-2

6

-1

2

-1

-1

0

0

0

0

0

0

2

6 -21

6

-3

-3

6

-2

2

-1

-1

-1

0

0

0

0

0

0

0

728

-8

26 -28

-1

-1

-1

0

0

-2

q

o

0

0

-1

-1

-1

0

+

728

-8 -28

26

-1

-1

-1

0

0

q-2

o

0

0

-1

-1

-1

x"

+

729

9

0

0

0

0

0

-3

-3

0

0

0

0

-1

-1

0

0

x",

+

819

3

9

9

9

0

0

-1

-1

-3

-3

0

0

0

o

0

x",

+

832

0 -32 -32

-5

q

q

0

0

0

0

0

0

-1

3

3

3

q 12 12

q 12 12

6

6

6

6

1

2

3 3

6

3

3

6

8

+

0

0

0

0

0

0

0

0

0

0

0

-1

-1

0

o

0

0 13 39 39

2q 2q

8

9

9

9

12 12 12

12 12 12

+

02

27

3

0

0

0

0

0

-1

3

0

0

0

0

-1

-1

1

0

0

0

2

0

Xzs

02

27

3

0

0

0

0

0

3

-1

0

0

0

0

-1

1

-1

0

0

0

0

2

x,. x",

02

189

-3

+

-1

.. 2z3

0

0

-1 -z3

-1

0 -1

0

-1

0

0

0

X4

0

Xs

x,

0

2

-1

-1

X9

0

0

0

0

XIO

0

0

0

0

XII

0

r3 -r3

0

**

0

0

0

0

0

0

0

0

0

0

0

o

o

'0

o

o

0-b13·

02

189

o

o

o

x"

02

X29

Xu

0

0

0

0

0

0

0

0

0

0

0

XIS

0

0

0

0

0

0

0

0

0

0

0

Xl7

x.. +

0

0

0

0

0

0

0

0

0

0

0

Xl9

X20

++273000

OO..;1000Xtl

0

++

21

-3

3

0

0

0

0

0

++

8

0

-1

2

2

0

0

13 fus 1nd 39 39

2

q

6

6

6

12

12

02

0

lq

0

0

0

Xt2

-1

-1

-1

X23

18

18

18

x,. x"

1

o -3 -3

o

o

o

o

-3

o o o o o o o o

o -3 -3

o

o

o

o o

o o

351

15

0

0

0

0

0

3

3

0

0

0

0

00000

0

0

•

02

351

-9

o

0

0

0

o -1

3

o

o o

o

o

02

X30

02

351

-9

o

0

0

0

o

3 -1

o

o

o o

1

x"

02

378

-6

0

0

0

0

0

-2

6

0

0

0

0

0

0

0

.0

0

0

1

02

Xn

02

378

-6

0

0

0

0

0

6

-2

0

0

0

0

0

0

0

0

0

0

0-2

Xn

02

729

9

0

0

0

0

0

-3

-3

0

0

0

0

-1

-1

0

0

0

0

0

x,.

02

1728

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

-1

0

x..

1

X24

-1

0

• b13

0

2q 2q

-1

**

0

0 -13

x,.

000

21 21

7

0

x"

•

o

0

0

000

6

0

X3

-1

0

0

** -z3

0

0

13 -13

x.

0

0

0

13

00

6

1nd

@

1

•

02

x",

.-b13

000

2

o

o

000

o

2

o o

-2

0

x,.

+

x" x..

x,. x" Xn

-1

-1

•

+

Xn

•

+

x,.

Gz(3)andS 4 (5)

Maximal subgroups

Specifications

Order

Index

12096

351

U3O):2

12096

351

11664

364

U3 O):2 01+2x 3 2 ):23

11664

364

11232

Structure

G.2

I I

Character

Abstract

1a+ 168a+ 182b

Chevalley

Cayley

2 A20 )

minus point

1a+ 168a+ 182a 2 2 3 .O x31+ ):D8

G2 (Z)

1a+91 b+ 104a+ 168a

N<3B 2 ). N<3A)

vertex

isotropic point

01+2x 3 2 ):23 4

1a+91 c+ 104a+168a

NOA2), N<3B)

edge

null subalgebra

378

L O):2

1a+91b+104a+182b

A2 O)

plus point

11232

378

L O):2

1512

2808

L 2 (8):3

: L 2 (8):3 x 2

N(2A.3C.7A.9A). C(2B)

2G20 )

Ree

1344

3159

2 3 ·L (2) 3

: 2 3 'L 3 (2):2

N(2A 3)

1092

3888

L (13)

: L (13):2

N(2A.3D.6C.7A,13AB)

576

7371

4

3

I

3

2 2 2+1+4-.3 • •2

1a+91c+104a+182a

2 : 2+1+4 .(3 3 x3 3 )

base

N(2A)

D2 (3)

quaternion subalgebra

~B23

BFII 23

11

3ymplectic group 3 4 (5) E C2 (5) E 0 (5) 5 Order

= 4.680.000 = 26.32.54 .13

Mult = 2

OUt

=

2

Constructions 3ymplectic

= 2.G

Sp4(S)

: all 4 x 4 matrices over F S preserving a non-singular symplectic form;

P3P4(5) E 3 4 (5) E G;

C2 (S)

=G :

the automorphism group of a generalized 4-gon of order (5,5), consisting of 156 vertices

and 156 edges. each object being incident with 6 of the other type Orthogonal

G0 5 CS)

=2

x G.2 : all 5 x 5 matrices over F S preserving a non-singular quadratic form;

PG0 5 (5) E S05(5) E PS0 5 (5) E G.2: 05(5) " G

Maximal subgroups

Specifications

structure

Order

Index

30000

156

5+1+2 •'4A 5

30000

156

15600

Character

Abstract

3ymplectic

Orthogonal

1 2 •• 5++ •'43 5

1a+65b+90a

N(5AB)

point

isotropic line

5 3 :(2XA ) '2 5

: 5 3 :(4x3 ) 5

1a+65a+90a

N(5 3 )

isotropic line

isotropic point

300

L2 (25):22

: L2 (25):22 x 2

1a+65a+104b+130a

N(2B.3A.5C.5D .... ). C(2C)

3 2 (25 )

0;;(5)

14400

325

2'(~xA5)

1a+90a+ 104b+ 130a

N(2A)

3 (5)wr2 2

0;';(5)

960

4875

2 4 :A

: 2'(A xA ) .22 5 5 4 •• 2 •'3 5

720

6500

3

480

9750

360

13000

G.2

.2

5

= N(2A 5 B10 )

: D x 3 12 5

N(3A). N(2B,3A.5C). C(2D)

(22 x A ):2 5

: 0

N(2B). N(2B.3A,5D)

A6

: 36

3

x 3

N(2 4 )

5

8

x 3

5

base

U (5) 2

O (5) x 0 (5)

2

3

2

3

O (5) x 0 (5)

N(2B.3A.3B.5EF) I

I

page 62

88

34

~c:l

WD I 34 I

page 63

26

20

[61]

84 (5)

@

pI part ind

@

@

@

@

@

lA

AAAAAA

2A

2B

3A

3B

4A

4B

@

15000

A A 5A

@

@

@

15000 150 500 A A A A A A

B*

se

5D

@

@

@

@

@

~~~~M

m A A M AA m

A

A

M

AA

~

F*

M

~

x,

+

x,

+

13

5

-2

-3

-1

5b5+3

It

-2

3 -b5

x,

+

13

5

-2

-3

-1

*

5b5+3

-2

3

x,

+40-804400

-10

-10

5

x,

+

65

17

-10

-10

X.

+

65

-1

-1

15

x,

+

78

6

x,

+

78

x,

+

90

18

x"

+ 104

x"

+ 104

6C

@

600 BA AA

lOA

@

@

@

600 100 100

AA BA

DA DA

B. 10C

@

50

@

20

@

@

12

12

@

13

@ 13

@ 13

DA CA DB CB AA A A A DA CA DB AB BA A A A D. 10E 10F 12A 12B 13A B*3 C*9

x,

5

It

2-1

b5+3

It

-b5

2-1

It

0

0

0

4

-2

0

5

0

0

-1

-1

15

5

0

0

0

5

-1

0

0

-1

0

0

0

0

X2

-1

0

0

0

0

X3

0

0

0

2

2

2

2

-3

2

2 -3

-3

2

3

3

-2

3

5

-1

5

-6

0

3

2

0

5b5+ 18

It

3

3 -b5

*

3

0

0 5b5+6

6 -6

0

3

2

0

It

5b5+ 18

3

3

*

-b5

3

0

0

6

0

0

6

0

15

15

0

5

0

0

0

0

8

0

-4

5

4

0

-21

-21

4

4

-1

-1

5

2

16

4

5

_4

0

2

4

4

9

-1

-1

-1

4

x"

+ 104 -16

4

5

-4

0-2

4

4

9

-1

-1

-1

-4

x"

+ 130

26

6

4

4

-6

0

5

55500-420

x..

+ 156 -36

4

6

0

0

0

6

6

X15

+ 208 -16

0

4

-2

0

0 -10r5+8

10r5+8

-1

-2 -b5

X16

+ 208 -16

0

4

-2

0

0

10r5+8 -10r5+8

-1

-2

x"

+ 312

24

0

0

3

-4

0

20b5-3

*

-3

x"

+ 312

24

0

0

3

_4

0

*

20b5-3

x"

+ 312 -24

0

0

-6

0

0

*

Xw

+ 312 -24

0

0

-6

0

0

x"

+ 325

5

5

-5

-2

-3

x"

+ 325

5

5

-5

-2

x"

+ 390

30

-6

0

x"

+ 390

30

-6

0

x"

+ 520

40

0

4

x"

+ 520

16

4

x"

+ 520 -16

4

x"

+ 516

0

0

x"

+ 576

0

x"

+

576

0

-1

x, 0

0

0

Xs

o

0

0

X6

-1

•

-1

0

-1

0

0

0

X7

It

5b5+6

-1

0

-1

0

0

0

X8

0

3

3

o

0

-1

-1

-1

X9

0

3

3-2-2-200

o

0

0

X10

0

0

0

X11

-1

0-2

*

2

*

-b5

2

b5

-3

2

30b5+2

-3

30b5+2

*

-25b5

-3

-3

-2

-3

-2

0

*

4

0

4

-4

-4

4

4

-6

3

3-2

-1 -1

-1

-1

-1

_1

0

0000XI2

o

0

0

0

0

;(13

-1

-1

-1

-1

0

0

0

0

0

;(14

2

0-2r5+4 2r5+4 2b5

It

-1

0

0

0

0

0

0

XIS

2

2

0 2r5+4-2r5+4

*

2b5

-1

0

0

0

0

0

0

X16

*

3

0

0-2r5-1 2r5-1 2b5

*

-1

0

0

-1

0

0

0

Xl7

*

b5

3

0

0 2r5-1-2r5-1

* 2b5

-1

0

0

-1

0

0

0

Xl8

2

b5

*

6

0

0 3r5+1-3r5+1

*-2b5

o

0

0

0

0

0

XlI"

-3

2

*

b5

6

0

0-3r5+1 3r5+1-2b5

*

0

0

0

0

2

-1

-1

-5b5

*

0

0

0

0

0000X21

*

-25b5

0

0

0

0

2

-1

-1

*

-5b5

0

0

0

0

0000X22

0 25b5+15

*

0

5

0

0

-3

0

0

b5+3

*

r5 -r5

0

-1

0

OOOXn

25b5+15

0

5

0

0

-3

0

0

*

0

-1

0

OOOX24

-5

-5 -10

0

0

0

-2

0

-5

-500000

000X2S

0-2

20

20

-5

-5

0

0

4

-4

-4

4

0

2

20

20

-5

-5

0

0

_4

-1

4

4

-1

-1

-1

-1

-1

0

0

0

0

-24

-24

6-4

o

0

0

o

o

0

0

0

0

0

0

0

0

0

-24

-24

6

_4

o

0

0

o

o

0

0

0

0

0

0

0

0

-24

-24

6-4

o

0

0

o

o

0

0

0

x"

+6240-86000

24

24

4

-6

-1

-1

0

0

-2

o

0000200000X31

x"

+ 624 -48

0

0

6

0

-26

-26

-6

4

-1

-1

-6

0

0

2

2222000000X32

x"

+ 625

25

5

-5

-5

5-1

o

0

0

0

0

-5

-1

o

o

0

0

0

x"

+ 180 -36

-4

-6

0

0

0

305500002

-6

-6

-1

-1

-1

4

3 6

3 6

4 4

8

5 10

5 10

5 10

5 10

5 10

6

10

10

10 10

1 2

2

0

30

5

o

-6

-2

0

0

o

11

r5 -r5 r5

-1

-1

*

b5+3 -r5

5

ind

[62J

@

14 4680000 400 480 360 360 240 24 P power A ~ A A A B

b5+3 -r5

r5

OOOOOOXw

-1

6

12

10 10

10 10

10

6

*

OOOOX:u;

0

0

0

Xv

0

0 c13

*3

*9

X28

0

0

0

*9 c13

*3

X29

0

0

0

*3

en

X30

0

0

*9

-1

-1

o

0

0

0

0

20

24 24

12 12

13 26

13 26

13 26

x" x~

x"

-

12

0

0

0

-3

-2

0

*

5b5+2

-3

2

b5

*

-3

0

0

b5+3

*

0

0 -r5

0

0

-1

-1

-1

X3S

x"

-

12

0

0

0

-3

-2

0

5b5+2

*

-3

2

*

b5

-3

0

0

b5+3

0

0

r5

0

0

-1

-1

-1

X36

x"

-

52

0

0

4

2

0

*

5b5-8

2

2

b5

*

-3

0

0

* *

-5b5

0

0

0

0

0

-1

0

0

0

X37

x"

-

52

0

0

4

2

0

5b5-8

*

2

2

*

b5

-3

0

0

-5b5

*

0

0

0

0

0

-1

0

0

0

X38

x"

- 104

0

0

_4

5

4

0

-21

-21

4

4

-1

-1

-3

0

0

5

5

000

0

0

x"

- 156

0

0

0

6

6

0

31

31

6

6

-6

0

0

5

5000000000X40

X41

- 208

0

0

4

-2

0

0 -10r5+8

10r5+8

-1

-2 -b5

*

-6

0

0

-2r5

2r5

0

0 -r5

0

0

0

0

0

0

X41

X42

-

208

0

0

4

-2

0

0

10r5+8 -10r5+8

-1

-2

* -b5

-6

0

0

2r5

-2r5

0

0

r5

0

0

0

0

0

0

X42

x"

- 260

0

0

_4

-1

2

0 25b5+10

*

-5

0

0

0

-3

0

0

*

b5+3

0

0

r5

0

0

-1

0

0

0

X43

X«

- 260

0

0

-4

-1

2

0

*

25b5+10

-5

0

0

0

-3

0

0

b5+3

*

0

0 -r5

0

0

-1

0

0

0

X44

x.,

- 300

0

0

0

-3

-2

0

25b5

*

0

0

0

0

-3

0

0

-5b5

*00000

x.,

- 300

0

0

0

-3

-2

0

•

25b5

0

0

0

0

-3

0

0

*

-5b5

0

0

0

0

0

x"

- 312

0

0

0

3

-4

0

20b5-3

*

-3

2

b5

*

3

0

0-2r5+5 2r5+5

0

0 -r5

0

0

-1

0

0

0

X47

x"

- 312

0

0

0

3

-4

0

*

20b5-3

-3

2

*

b5

3

0

0 2r5+5-2r5+5

0

0

0

0

-1

0

0

0

X48

x.,

- 416

0

0

_4

8

0

0

16

16 -14

_4

o

0

0

o

x"

- 468

0

0

0

0

-6

0

*

-30b5+3

3

8 -b5

*

0

0

0

3r5

-3r5

0

0 -r5

0

0

0

0

0

0

Xso

x"

- 468

0

0

0

0

-6

0 -30b5+3

*

3

8

* -bS

0

0

0

-3r5

3r5

0

0

r5

0

0

0

0

0

0

XSl

x"

- 520

0

0

4

-5 -10

0

0

9

0

0

5

5

000

0

0

x"

- 576

0

0

0

x"

- 516

0

0

x"

- 576

0

x"

- 624

x"

r5

OOOX39

x., x.,

0000000000X49

4

0

-5

0

0

0

-24

-24

6-4

o

0

0

o

o

0

0

0

0

0

0 c13

0

0

0

0

-24

-24

6-4

o

0

0

o

o

0

0

0

0

0

0

0

0

0

0

-24

-24

6-4

o

0

0

o

o

0

0

0

0

0

0

0

6

0

0

-26

-26

-6

4

-1

...1

-6

0

0

-10

-10

0

·0

0

0

- 624

0

0 -12

0

0

0

24

24

4

-6

_1

-1

0

0

0

o

0000000000XS7

x"

- 624

0

0

6

0

0

0

24

24

4

-6

-1

-1

0

0

0

o

00000r60000XS8

x"

- 624

0

0

6

0

0

0

24

24

4

-6

-1

-1

0

0

0

o

o

0

0

0

0 -r6

0

0

0

0

Xw

x"

- 780

0

0

0

-6

6

0

5

5

0

10

0

0

6

0

0

-5

-5

0

0

0

0

0

0

0

0

Xoo

0

000XS2 *3

*9

XS3

0

*9 c13

*3

XS4

0

0

*3

*9

en

Xss

0

0

0

0

0

XS6

0

@

30 BB AB 15A

@

@

@

30 15 20 AB CA BA BB CA AA B* 1se 20A

@

@

20 AA

BBA

BA Bt-

AAA 30A

30

@

30 AAA BBA

B*

x,

@

@

@

15 600 720 240 A A B A A A fus ind 2C 20 4C

@

@

@

@

@

@

@

@

@

@

@

@

12

10

-b5

•

•

b5

b5

•

x,

*

-bS

b5

•

•

b5

X.

-1

-1

-1

0

0

_1

x,

-1

-1

0

0

0

-1

X.

o

x,

-b5

36

24

18

12

12

12

13

13

13

AC AC

BD BD

BA

BC

CC

AC CIn

BA

AC

BC

CC crD

6E

6F

6G

A CC DC CD CC Fe BA A CC DC CD AC DC BA 8A lOG lDH 10r 12C 2DC 24A

40

60

50

50

30

15

S* 26A B*3 C*9 30C

x, +

0

0

0

0

0

0

0

0

0

0

0

-1

++

12

-4

0

0

-4

2

0

2

0

-3

2

-1

++

13

5

-1

-1

-2

3

o

o

++

13

*Ob5*-b5

•

+

0

o

-1

0

0

0

5 -1 -5

-1

0

0

-1

0

0

0

0

0

o

3-2

0

0

0

0

o

o

o

o

++

12

0

6

2

0

0

0

0

0

2

-3

0

0

x..

o

o

o

o

++

0

8

0

0

-4

2

0

-1

-2

0

0

-2

0

++

26

6

4

0

3

3

-1

0

0

-1

-2

0

3 -1

++2664060200

-2 -1 0

-1 ·,-1

x"

o

0

0

x"

o

0

0

-1

-1

o

0

o

o

_1

-1

-1

-1

o

o

x"

*

-bS

-1

0

0

•

b5

Xt6

-b5

*

-1

0

0

b5

•

Xl'

-b5

-b5

•

*

-b5

-1

•

0

*

-b5

0

-1

-1

0

0

0

-1

-1

0

0

0

-b5

*

0

b5

x"

*

-b5

0

x"

b5

*

0

•

b5

x"

•

*

b5

•

b5

*

-b5

b5

•

0 -b5

*

•

b5

-1

x"

-1

-1

o

0

x"

-1

-1

o

0

0

0

0

0

-1

-1

-1

-1

-1

0

0

0

o

-1

-1

0

0

0

0

0

0

0

0

0

+

0

0

0

0

0

0

0

0

0

o

0

0

0

0

-1

-1

-1

o

0

0

h

X.

0

Xs X. ~

0

0

X9

x"

0

0

0

0

0

-2

XlI

o

0

0

0

0

0

X12

-1

0

0

0

0

0

0

-1

0

0

0

0

0

0

0

0

0

0

0

0

x" -1

o

X14 XIS

x.. +

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

XI7

x" 0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

-1

-1

+

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0000

0

XI9

+

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

++

0

8

0

0

-4

2

0

-1

2

0

0

-2

0

0

-1

-1

0

0

0

++

26

14

_4

0

-1

-1

-1

2

0

-1

-1

o

0

0

0

0

++

26

6

4

0

-9

-3

-1

0

0

-1

0

0

0

0

0

x"

*3

*9

0.'(28

*3

0

X29

*9-013

0

X30

o

o

0

0

0

o

o

++

48

0

0

0

0

0

0

0

0

-2

-2

0

0

0

0

O-c13

o

o

0

0

0

o

o

++

48

0

0

0

0

0

0

0

0

-2

-2

0

0

0

0

0

*9-013

x"

o

o

0

0

0

o

o

++

48

0

0

0

0

0

0

0

0

-2

-2

0

0

0

0

0

*3

x"

o

o

o

0

o

o

++

52 -12

0

0

6

0

-2

0

0

2

2

-2

0

0

0

0

0

0

0

o

0

0

0

0

0

0

0

0

0

0

0

0

r6-r6

0

0

0

o

0

0

-1

-1

-1

0

0

0

-1

OX23 b

x"

-1

OX21

x"

x..

x"

0

Xw

*

-1

0

0

-1

++26-6-62-60200

+

b5

-1

0

X.

++26-2-40_5

x" x,.

0

0

x,

0

0

x,

-b5

X~

54 (5)

@

AD AD

*

XZl

@

B AD A AD

*-b50*b5

Xw

@

16 360

X.

x..

@

H

X2

x..

@

++

0

0

0

0

25

5

-5

-1

x" -1

xu

x" 0

X32

x"

o

o

0

0

0

o

o

++

X~

o

o

-1

0

0

o

o

++26-66-2-60200

15 30

15

15

20

20

30

30

20

20

30 30

30

x"

b5

*

0

*

-b5

b5

•

00000000000000000000X3S

x"

•

b5

O-b5

*

b5

x,.

X37

b5-1

*

-1

*

b5

• •

b5

00000000000000000000X37

x"

*

b5-1

-1

b5

*

b5

•

x"

o

o

-1

-1

2

2

o

0

-1

-1

o

0 6r2

o

X~

30 fus ind

2

4

8

8

8

5-1

12

12

-1

6

12

12

8

12

8

-1 10

10

20

o o

0

0

0 X33 -1

24

40

24

24

26

26

26

60

24

40

24

24

26

26

26

60

X34

x" 0

0-4r3

0

0

r3

0

0

0

0

0

0

0

0

0

0

r2

0

0

0

0

r2

0

r3 -r3

0

0

0

r3

X39

r2

0

0

0

0

X40

r2

x"

*

-b5

-1

0

0

*

b5+2

OOOOOOOOOOOOOOOOOOOOX41

X42

-b5

*

-1

0

0

b5+2

•

x"

x"

*

-b5

b5

•

•

b5

OOOOOOOOOOOOOOOOOOOOX43

X44

-b5

•

•

b5

b5

x"

b5

*

0 -b5

*

b5

• •

x"

•

b5

*-b5

•

b5

-b5

•

•

-05

X47

-b5

0

•

0

x...

*

-b5

0

x"

-2

-2

x"

o

o

0

x"

o

o

x"

o

X~

OOOOOOOOOOOOOOOOOOOOX4S

x"

o

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

X47

x...

0

o

o

o

-1

-1

o

o

ooooooooooooooooooooxso

0

-1

-1

o

o

-1

-1

-1

-1

0

0

0-4r3

0

0-2r3

0

0

0

0

0

0

0

0

0

0

0

r3

X~

x"

-1

o

0

0

0-4r3

0

0

r3

0

0

0

0

0

0 -r3

r3

0

0

0

r3

XS2

*3

*9

0

X~

*3

0

XS4

*9 113

0

XSS

x"

o

o

0

0

0

o

o

o

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0 113

X~

o

o

0

0

0

o

o

o

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

*9 113

x"

o

o

0

0

0

o

o

o

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

*3

o

0

0

-1

o

0

0

0

0

0

0

0-2r2

0

0

0

0

0

r2

r2

0

0

0

0

X~

-2

0

0

o

o

o

0-4r2

0

0

0

0

0

0

0

0

0 2r2

r2

0

0

0

0

0

0

XS7

x"

-1

x"

o

o

x"

o

o

o

0

o

o

o

0-4r2

0 6r3

0

0

0

0

0

0

0 -r2

r2

0

0

0

0

0

r3

XS8

x" x..

o

o

o

0

o

o

o

0-4r2

0-6r3

0

0

0

0

0

0

0 -r2

r2

0

0

0

0

0 -r3

XS9

o

0 6r2

0

0

0

0 -r2

0

0

0

r2 -r2 -r2

0

0

0

Xw

-1

-1

0

0

0

0

[63]

2::

57

15

... 399

~56

-9

-9

-9

-9

o 567

o 567

o 567

567

929

929.4

15

8

1

1

-9

-9

-9

-9

-9

02 399

02 399

02 399

02 456

02 456

02 1156

02 513

02 513

02 513

02 513

02 513

02 513

02 513

02 513

02 513

02 567

02 561

02 561

02 567

02 567

02 567

x..

x",

""

X..

Xl<>

x."

X.,

X.,

x.,

Xc<

x.,

x..

X.,

x..

;(.9

x'"

X"

X"

x"

X...

K..

-9

1

1

1

1

8

8

o

o

929'7

92914

929

929.7

92914

929

929*7

29'7-7&4

<:9.4_7.1.1

29-7&1

729-7&4

15 7z9'7-7"'1

15 7;;:9.4-7&7

-3

02 189

X..

-3

.,3

02 189

29'7-7&1

02 189

-1

<:9.4-7&7

t9-1&4

o

o

..

x"

57

..

-3

_2

..

o

o

o o

..

.... .... .... .... ....

o

....

o

o

9z3

_1

_1

_1

-t9.4

0

0

_1

_1

_1

_1

_1

_1

_1

_1

_1

18 18 18

o

0

o

..

..

..

..

..

-,/:'9.4

-z9'7

-t9

z9

z9.4

o

_1

_1

_1

_1

_1

o

o

29'7

z9.4

o

..

..

o o

_1

'2

o

o

o

o

_1

o o o

o

_1

A

lI_x19

n

.2

n_x19

n

.2 n2

n2

.4

o

Y9+1

o

o

o

u

o

..

o o

o

14

n

u

119

o

..

."

o

o o

o

o o o

110

.19

y"63

o

o

119

**

.26

o

.26

o

o

.17

.26

.26

r'

0' 0'

.. .. I0'

.17

." .10

017

117

o o

o

I 0' .. I0' .... I0'

:0002

:0002

:0002

...... I0'

.10

117 y"63122

**

• 26

.26

117 y"63143

z9'7

z9

z9.4

o

z9'7

z9·4

.,

63 fus 1nd

63

63

o

o

o

.00

: 000

: 000

+00

+00

.00

63 63

11

n

o

o o

000

-00

: 000

:

.00

1nd

63

..

o

o o

.... ..

63

63 63

.19

.10

.17 y"63143

.10

y"63

.26

.26

.17

u

010

."

.10

o y"63122

n_x19

14 ..4-x19

y"63

.10

z9'7

z9.4

.,

o

o

z9'7

z9·4

.,

63 63 63

o o

14

**2

n4

**

•

'2

y"21

y"21

n2

y7

**4

12

,7

.4

.19

.2 n2

., .,

z3

y"21

117 y"63.22

12

o **

•• ••

.. "

o

z3

11

11

y"21

.10

14

,7

"

o

o

., " ••

,7

.,

o

o

o y"63143

.4 n4

o

.. n2

n_x19

12 112 114

12

.4

12

o

o o o

0

.2 ..2 n4

112

.4 114_x19

o n2 o

n

0

n-x19

0_x19

o

o

0

o

0 ..4

o o

o

o

0

o

o

o

o

.26

"

z9'7

z9.4

."

o

_1

_1

_1

63 63 63

o o

y"21

n2

n4

.,

., ....

o o

63 63 63

n

y"21

•2

14

.,

..

o

o

0

_1

_1

_1

E"

" ,.,""

cn2

.. .. .., ., ....., .,•• .. ... .. .., .,• ... ••

•• ••

y7

,7

z9'7

.,

o

0

o

0

_1

_1

_1

o

o o

o

.." .,..

o

o

o

o

_1

_1

_1

o

o o

,..""

21

"CA Fn4"" fus

21

21

21 AA

., .. ., .. ., .. .. ., ..

.,

,lA

21

.4 -y9-1

y9+1

o

_1

_1

_1 _1

_1

_1

o

9 19 19 19 19 19 19 9575757575757 9575757575757

.4

0

_1

o

-1

.4 ..4

o

o

0

_1

0

0

0

o

_1

.4 n4_x19

.2 n2 n4

.2

o

.4

o n4

o n2

o

o

A

.2 n2 114

112

o

_1

o

o

o

_1

.4 n4_x19

0

o

.2

.4

y7

'4

y7

.2

y7

.2

A

21 CA AA

z9.4

1 _y9_1

.4

A

19 A

o

.4

y9+ 1

u

lI-x19

o-x19

., •• .,

o

0

0

_1

0

0

o o

o

-1

000

" ••.2 y9+1

.4

A

19 A

" .,•• .4 -Y9-1

.,

'4

y9+1

.2

•4

1 y9+1

7 21 21

o

y7

y7

o

o

o

0

o

y7

12

y7

.4

.4

'2

.4

y7

.4

y7

.2

7 21 21

.2

y7

z9'7

.,

12

.4

.2

.4

y7

z9·4

..

n

..

..

o

u

7 21 21

o

.4

'2

z3

_1 t9.4-&1

z9

_1

.2

y7

.4

..

_1

.4

.4

y7

.2

.2

.4

.2

u

o o

_1

.4

y7

y7

.2

..

-z9

z9'7

o

..

z3

u

o

o

_1

o o

0

o

o

_1

.4 y9-&2

.,

••

o

_1

1

..

-z9.7

18 18 18

o

z3

_1

o

.,

A

19 A

C.4 19A BIICu2 D.2 E:.4r..4

A

• 19 19 19 A A A

o

.4

.,

C

.4 Y9-&2

o Y9-&2

o

.'

9A

C

27 C

1.4y7.2

.2

o

0

uz3.4y7.2

z3

z3

o

0

7A !llI2 C.4

21 A

,., ,., , , , , ,.,,

•

21 21 A ,

1 y7

z9-&7

_1

u

.. -z3

-z3

o

_1

_1

_1

_1

_1

_1

o

_1

_1

_1

o

_1

u

.. 2%3

2t3

AA

24

BA 6A B..

" AA

24

_1 z9'7-&4

5 -3

_1

o

12 12

-3 -3

12 12

-3 -3

-3

12 12

o

o

_1

_1

_1

_1

_1

_1

o

_1

_1

_1

-,

-3

lie

A

A

64

• • •

_1

_1

_1

_1

_1

_1

_1

_1

_1

_1

o

0

_1

o

_1

_1

_1

_I

_I

_1

o

_1

-3

0

4B

A

A

64

-3 -3

_2 -3

-,

0

A

4/1.

A

3e

,

A

A

•

64

81

15z3 -3

u

o

-623

-7

,,' ,., ,,' ,., ..

,.,

,

..

15z3

o

o

7

..

-(,23

-7

lA

1512

,, ,, ,..

1512

16'

02

-7

02

57

-7

6 6

02 51

"d

,

-9

o 567

0

-9

o 567

o 513

o 513

o 513

o 513

o 513

o 513

.. 513

.. 513

.. 513

.. 512

o

~56

15

.. 399

o

15

-7

-7

.. 399

133

133

133

57

o

" ...

o

",

'" ,.

,. '"x..

'" ",

'" ,,,

'" ,,,

,,, ,,, ,,,

,..

,,, ,,,

,,, ,,,

,,,

,,,"

,. ,.

,." ,."

"

"

A

1nd

2/1.

p' part

lA

A

P

power

5515776 536

1

••

""

-,

o

o

0

0

o

8

o

o

o

o

o

o

o

8

o

o

o

o o

O.

o

o

_1

,

o

o

o

o

o

o

o o

o

o

o

o

o

o

o

o

o

o

o

2z3

2z3 2z3**

_1

,

-z3 0

1

o

o

_1

0

o

1

o

12 12 12

0

o

_\

o

o

o

,

o

12 12 12

o

o

o

o

_1

z311

o z3

DB EB 12B

0

.00

.00

• 00

+00

000

o o

o

o

,

0

-3 -3

: 000

'7

o o

:0002

:0002

:0002

:0002

:0002

0

0

o

o

o

.7 :0002

:0002

,.,.,

'7

."

o o

'7

:0002

:0002

:0002

:0002

:0002

,.,., ."

:0002 :0002

01'

:0002

'7

o o

."

9.'"

01'

o

o

o

o

,.,.,

.13 8z27+7&13

.4

.13 7z27-7&13

113 8z27+7&13

'7

.,,

z27-&13

27 27 27

o

_1

.7

.7

'7

o

o

o

o

o

o

o o

o

'7

.13

_z27+&13

27 27 27

o

o

o o

o

o

_1

z9n-&2

z9-&7

y9+1

.4

.,

••

.

,, J.' 27

o

o

o

o

o

o

24

'7

'7

01'

o

o

o o

.7

01'

.27

'7

."

.27

'7

.,,

.27

24

""

.4

.4

I~

.4

.4

.4

17

.7

_y9_1

.2

.4

;<4

'7

,..

", '" '" '" ".

X»

Xn

x""

X, •

XIS

X17

,,, ".

,..

",

x,.

XLI

X,,,

x"

X•

" ",.

"

" "

'7

EA CA Cn5

..,

o

113

z27&13

o o

'"

• 27

.27

...", ...,"

.27

..,

'" ..,

-.27

",

X.,

,. ,,,

X""

X..

X.,

x...

x.,

x..

x., x.,

X.,

X""

x'" x,.

x"

x""

'" x..

", ,,, ",

o '" o '"

o

z27

n2

14

z27

.12

.4

z27

"2

.4

.4

_z27

112

113_z27+&13

n2

.4

z27&13

.. .."••, ,.,,,

"" ",

o

'7

.7

'7

•7

'7

.7

-z9n

-.,

.4

-y9-1

.,

o

-z911

-"

"7

CA FA

"3-'/:27+&13

-z27+&13 _z27110

-2t27-&13

o o

o

o o

o

o

o

113

_z27+&13

In

'7

113

z27&13

"" "

o

o

z9n

.9

z9n

., ., Z911

o

.4

.

.,

14

_y9-1

.4

.,

1SA

EA

• "

FA

14_z27+&13

o

z27-&13

.., .., .., ...,,

'"o

z27-&13

14

.., ..,

27 27 27

o

o

o o o

o

o o

o

-\

.4

•• ••

,'''

.,

o

•7

'7

"7

,,

27

.13 -2z27-&13

'4 -2z27-&13

z27_7&13

n2

,.,,., ,., ••

o

o

o

••

•• •• •• ••

••

•• ••

Y9+1

o

z9n-&5

z9_&4

"

A A

,.,

.••, .••,

7Y9+7

z9n-7&5

z9-7&4

-7

G••

•

,,

1512

.., .., .., .., .., .. ,,,., .., .., ,.,.,..••, 9.'" ." ,.,., .., ..••,

o

113

z27-7&13

..

,.,,.,,., ,

o

'7

'7

'7

•7

'7

'7

'7

'7

••

7Y9+7

.,

.4

••

-7

"7

,,

1512

1137z27-7&13

7z27-7&13

o 8z27+7&13

o

o

17

'13

z27-7&13

,.,

,.,,.,

o o

o

9z9**

,.,

9z911

9z911

,., ,.,

Z911-7&2

z9-7&7

••.,

7y9+7

,.,.,

:0002

:0002

:0002

:0002

:0002

:0002

0'

I

:0002

:0002

:0002

o

-3

000

-3

-3

-3

0

0

0

0

-1

0

: 000

: 000

: 000

: 000

: 000

:000

: 000

: 000

: 000

: 000

: :

:000

.00

.00

~

'7

-7

"

:000

0

-\

3G

,,

1512

'7

~

•

,,

57

: 000

12 fus 1nd 12 12

o

o o

o

o

_1

z3

z3n

.00

E:C FC 12C rus lnd

• • •

9 C CA A BA 9D 12'

_2

2 2%3**

,

_1

• 24 FA FA 6E

-z3 -z3**

_2

24 EA E:A 6D

_1 -z3**

_1

o o o

o

6z3

6z3 I>z3n

o

-,

3z3

3z311

-,

6C

2-2

3F

3z3 3z3"

2

3E

DA

24

A DA

A

A

A

216

216

6 6z3n

_2

_2

,

3D

,,

216

"-'"

00

~

e VJ

~

o

o

o

*55

.,17

"62

"71

-'37

u71

"62

-x171

*37 -x171

X"

x"

X..

x"

.,0

*26

.,0

x"

*37

o

o

o

o

171 171 171

.4

.. 4

126

.. t7

*55

u'7

128

*91

uS3

..53

.91

*37 _x171

*37

155

"

.,

., .,

.,

.4

.4

.4

.4

o o

"

z9..

.4

C'4

CE

"ca "

AF

".4

.,

,,9"

"

., .,

o

DI 37

88

CO

"

..

.4

"

.,

.4

.4

z9"

"

o

"

EE Aa FIZZ

. .,

z9**

"

o

., .,

OF CE GI10

"

."

."

.52

'88

u53

...

'34

'".,

,21

'" ,21

19' 19'

....,

19' 19' 19'

o

o

14

.,

19' 19'

o

."

y"63"

y"63

o

."2

o

o

AE

.4

19'

o

o

14

.4

on

."

.2

...

."

." ." ." .52 ."

o

o o

o o

o

.., o

o o

o o o

o o o

o

..29

•58

.. 53

04

o

.. 53

..29

.40

y'189

o

o

..53

.34

.52

.52

..2

.61

." ." ." .52 ."

,"

.4

o o

..2

.4

.., .., '" '" '"

o

I

1

I

I

I I I

06

06

06

06

06

06

06

06

06

189 tus ind

o

y n l>3"

y"1>3

." ."

"0

04

'00

: 000

.,

: 000

14 14

o

" " ....,, ."

.,

.,

o

000 : 000

" :1:9"

_00

.00

'"

.4

IuS tua

BG

"

.4

161y'189164

.. 29

o

o

'" ."

.52

.88

'".,

,"

o

*58y'189*64

..53

.4

..,

y'189

,"

."

.4

o

.4

uS3

"29

o

.4

y'189

."

.., ..,

'52

.61

"53

...

• 34

167y'189*64

.88

.52

.,29

...

••2

*S8y'169,64

'"

.61

uS3

."

,"

..,

o

," ." ,"

.85

." ."

uS3

"29

...'"

167y' 189.64

o

.4

.., .., ,"

o

o

19' 19' 19'

o

o

'43

..

EO

"

AF HI43

''"" '" 19' 19' ' " '", '" .. .. 42' .., "" .. , ." '" ,21 on ," .., '" ,21 ., ." ,21 ., ." ," ., ." ,"

'"''""

o

19' 19' 19'

."

o

o

1"63**

*4

'"

"

..

y"63

.4

."

'",

•4

.4

.. .,

., .,

o

o

.4

.4

o

::*25

OF

." ."" '" '43" ." on ." '4' .. , '43 ." , ." , '43 ." .., ." ." on ." '43 , "0 ..,

.,

"• 4

"

~9..

., .,

o

B"2

"

63'

AE

"

Aa

".

"

• •

Oy'189*64

o

o

o o o

171 171 171

-x19

.. 2

.,

.4

.. 4

o

o

*55.,17

o

o

o

o

o

171 171 171

-x19

*2

••2

..4

14

o

"0 -x171

*37 -x171

.,0 _x171

o

o o

o

o

o

.26

o

o

o

o

o

111 171 171

•• 4

x'"

o

o

o

o

o

171 171 171

o

..2

11 111 1"11

14

-x19

..

x" x..

x"

x.

x"

x.

x.,

x"

x..

x" x" x.

x"

x" x.

x"

x"

x.,

x..

'"

x"

x"

_2

•• '·l

X,.,

..~

.. .., .,

,ij

".

Iil

x"

_x19

..2

.2

..

_x19

.,

o

. . . .2

o

o

o

o

_~lg

",

o

-, -, -, -, -, -,

o

X"

x.,

x"

'"

x"

x"

x"

x"

x"

x"

x"

x"

x"

o

o

o

o

x"

o

o

x.

x.

x,

x.

x,

o

"4

x.

"".,

o

Cu2

19

Aa Fa FI10

x,

"'"

19

BG EO

-, -, -, -, -, -,

57'

19

EO VG

19

Fa

ca

x,

x,

19

BG

ca

19

VG Aa

• • •

18 A

2

17

17

17

17

_1

-z9+&7

_1

z9-&7

-z9

-z9

o

0

,"

o

o

o

_,

18

o

0 _z9"

o

o

2

14 z9.._&2

14

o

666 LA MA KA

18

o

o o

o

17

17

17

17

o

o

o

.00

: 000

: 000

: 000

: 000

..

00

06

06

06

06

06

06

06

06

06

I

1

I

,

I I

I

J

.2

.2

02

I I .,

,

18 fus ind tus 1nd

0

14

14

o

14

14

-00

'00

KA " HA leD E*1FnS full Ind tus ind

0 _z9"

_1

C 9N

• •

• Mill

14-z9..+&2

14

_1

91( LI?

18 •

• • " , A

2

o

o

o

16

A

o

1

0

0

o

0

o

0

_1

_1

y7

0

o

12

o

t4

8

0

0

0

14

t2

14

t4

14

o

o

y7

o

t2

18

o

0

o

0

_1

t2

18

_1

t2

....

..

,I ..

00

00

.2

.2

.2

I .,

I

J ., J .2

J

I

-j"

02


+00

.2

0002

! ! I ! !

1

:t

100

:+00+00

.00

'00

:+00+00

'00

:000000

:+00+00

18 fus 1nd tus 1nd tus

o

o

o

_1

t4

t4 -y9

t4 -y9

o

0

o -y9

o

o

o

lt4y7t2

1

_1

_1

_1

o

_1

-1

_1

0 212-212

_2

•

777999 A BB CB AB BF er Af A AB BB CB AB BB CB BA B.. 14A BI5 Ct3 laG Ht7 Xt5 tus lnd !us 1nd !us

11>

9

CB CB 6f

8 -,

0

A

, "

504

••• @

@

@

6

o

18

0

o

-,

24

o

o

o

-,

o

o _12

"

+2

+00

0002

X,

x"

,,,

x..

,,,

x""

'" ,,,

,,.

'"

,,,

,,,

'"

x"

,,, ,,,

x"

,.

x"

Xl>

,,, ,,, ,,, ,,,

x"

X"

,,,"

,.

X,

x.

X,

x.

X,

X>

x"

x" x.

x"

x"

)(""

x•

",

,,,

x"

x"

'" '" ,,,

x"

x"

,.

I ! I ! ! ! . ,. ! ! ,. I ., ! ,.

"j

'00

~2

I

1nd

:+00+00

+00

:+00+00

+00

+00

:000000

ind tus

•

:+00+00

'00

" '00 : +00+00

'00

:000000

24 fus

o

0

-, 0

12

'od

:+00+00

1nd tuS

:+00+00

I> 3 4 4 DB DF AA AA DB DB DA DB 6G lSJ 24A St7 fus

@

e

w ~ 00 '-"

g::

= 3.(3

Out=3 x 8 3

x G).32 : all 3 x 3 matrioes over F64 preserving a non-singular Hermitian form;

PGU 3 (8) =G.32; 8U3 (8) = 3.G; P8U3 (8) = G

GU 3(S)

Mult .:: 3

3 2 :2A 4

(9 x 3).8

19: 3

3648

25536

25536

25536

34048

96768

1512

216

216

216

162

57

3 2 :2A

4

3 x L2 (8) 32 : 2A 4

2 3+6 :21

513

10152

structure

Index

Order

Maximal sUbgroups

3

: 19:9

: (9x3) .Ox8 3 )

: 32:2A4 x 3

: 32 :2A 4 x 3

: 32:2A4 x 3

: 3 x L2 (8):3

: 2 3+6 :(1:3x3)

G.3,

19:9

2 G.O )1233

Index

344

2107

14749

16856

43904

Order

16464

2688

384

336

129

=8

=1

3

43:3

L2 (7):2

8 2 :8

2(L 2 (7)x4).2

7 1+2 :48

Structure

19:6

(9x3) .012

x L2 (8) 3 2 23 3 : 4

: 8

Out = 2

(9x3) .(6x8 ) 3 19:18

x L2 (8):3 3 : 3 2 :23 4 x 3

: 8

: 23+6:(7:3x83)

G.6

(19:3 x 3):2

9 :012

2

: 018 x L2 (8)

: 23+6:(7x018)

G.8 3

: 43:6

: 2 x L2 (7):2

: (2(L 2 (7)x4).2).2 2 : 8 :°12

: 7 1+2 :(6x016)

G.2

1a+343a

Character

3 (19:9 x 3):2

92 .(6x8 )

: (0'8 x L2 (8»:3

non-isotropic point base

N(43A-N)

U1(43)

I

page 65

G.3

3'

10

G.2' G.2"

4

G.6' G.6"

10

7

10

5 8 G8 58

la+133abc+399abc+512a+51jabc

1a+512a

Character

N(19ABCOEF), C(3G)

Nt3C)

Nt3C 2 ), COF)

U1 (512)

base

U (2) 3

U (2) 3 U (2) 3

Nt3C 2 ), COO) Nt3C 2 ), COE)

non-isotropic point

isotropic point

Unitary

Nt3AB)

N( 2A 3)

Abstract

Specifications

(*) There is no group 3.G.33 or 3.G.3 3'. See the Introduction:'Isoclinism'.

: 23+6:(7x018):3

isotropic point

Unitary

N(2A t 3A,4C,7B)t C(2B) 03(7)

N(2A 2)

N(2A)

N(7A)

Abstract

Specifications

page 64

GB G.32B~EJ~~B~~28 813.G.3113.G.32 ~g)~~FrrFrr 27 28 251 G.(3x8 ) 3

x G : all 3 x 3 matrices over F49 preserving a non-singular Hermitian form;

Mult

G.2 : 2 3+6 :(7X8 ) 3

PGU 3 (7) = 8U3 (7) = P8U3 (7) = G

GU 3 (7)

Maximal subgroups

Unitary

Constructions

Order = 5,663,616 = 27.3.73.43

3 x 19:9

9 2 .OX3 3 )

: (9 x L2 (8»:3

: 2 3 +6 :63:3

= 2A2 (7)

(9 x3)'( 3x83)

: L2 (8):9

: 23+6 :7:9

G.3 3

Unitary group U (7) 3

19: 3 x 3

9 2 :83

9 x L2 (8)

2 3+6 :63

G.3 2

Remark : G.3 1 1~ a maximal subgroup of the Harada-Norton group. while G.6 is a maximal subgroup of the Thompson group.

Unitary

Constructions

Order = 5,515,116 = 29 .3 4 .1.19

Unitary group U (8) = 2 A2 (8) 3

......:J "-'"

~

C VJ

0..

§

00 "-'"

~

c VJ

"X

"X

"X ":I:

''X

..x ·X

..x

":I:

". ". •'X

"X

«x

"X

"X

''X "X

'"

"X 'X

"X

'"

0

0

,

~-

0

.., "X

0

0

0

0

0

o

L-

0

o 1.f- 1.1

0

0

0

"X

"X

"X "X ''X "X

'X 'X

'X

'X

'X

'X

.x 'X

QV QY

o

o

o

o

o

L

o

o

o 0

e

Y H

110 :l8C' ';1

aY

tr1

av

"I

o o o

o

0

0

o 0

o

o

o

o o

o

o o

o

0

o o

o

o

o o

o

0

o

0

o

..

o o

o

o

,. S'

,.

,.

S'

..

,. ,.

,.

S'

o

.... .. ,.

o

o

o

o

S'

o o o

o

o o

S'

o o

o

.. ,. ,. ,. ,. ,. .. .. S'

S'

..

S.

,.

S'

,. ,.

,. ,. ,. ,.

S.

,. ,. ,. ,.

S'

,. .. ,.

,. ,. S'

,-

,. ,.

L-'1

,.

o o

o

o

..

0

0

o o 0

o

o

o

0

o 6<:'

0

o

o

o 6~'

o o

1-

0

YV

0

0

QY

o o

o

o

0

.-

o o

o o o o

o

o

o

o o

o

o

o

o

o

o o

•

o

o

o

o

o

o o

o

o o

o

o

o

o

0

0

o

0

0

o o

o

o

o

o

o

o

o o

o

o

0

o

o

o 0

,-

,-

o o

o o

o o

,-

,-

o

o

o

o

o

i

s* r*

1-

0

o

o

~

i

o o

o

0

0

o

o

1-

gZl"

'1-

0

~

i

o 0

o o

o

0

o

000

o

o 0

o

o

0

o o

o

o

o

000

o o

0

0

000

000 0

o o o

L"GO

o

,-

o

t-

0

o

o

l

0

E.

0

S'

0

o

o

o If

1+2'1- L+(:1

•

o

'G'L

e~

,,

..

•

, V

o o

o

o

o

o

o

0

'H V

o o o

o

o o

o

o

o

o

o

o

o

o

o

o

o

o

o

o

o

o

o

o

o

,.

S'

S'

o

o

o

o

o

o

o

,.,

o

S'

S'

o

o

o

o

o

o

..

,.,

S'

,.

,.,

S'

,. ,.

,

..

,

n-gZ9

,-

,-

lL-g Z 9

H

L-

e-

ee-

eee-

e-

L- eL- ee\- e-

L-

L-

L-

L-

1-

L-

L-

o

o

o

£L

£L

L-

L-

9+G',.(9-

9""2'19-

9+2.19""

£1

9"";::'19

S'

,. .. ,.,..

S'

,., ,.

S'

,.

,.

S'

IS-

IS

IS-

IS

,,-

S.

, ,.

9""2J9

9+2.19

L-

IrIlL

0

0

0

,-

0

0

0

0

0

o

o

o

o

o

o

1r9£0

~a£o

"X

"X

0

o

~e£o

"X

"X

0

00

o

°

119£ 0

~9£

~e£ 0

!rS£ 0

"X

IIS£ 0

119£ 0

·X

o

o o o

o

~9£ 0

·X

"X

"X

o

1I1r£ 0

o

119£ 0

9-

o

L-

L-9-1I11£0

1I1r£ 0

"X

9-

L-9-1I~£O

"X

·X

L-

°

1111£

Irll£ 0

Irll£

9-

L-e-III1£O

L-

L-e-III1£O

9-

Ir!r£o

9-

9-

1111£

2

1111£ 0

0

\-

1111£ 0

"X

"X

"X

"X

"X

"X

"X

"X

"X

«x

°

£1

£1

9~'2

0

LO£ +

LO£ 0

>IX

-,'X

"X

"X

"X

LO£

° ""

9-

9~20-

Sl

9""

S~

0

0

+

g;z+

"X

"X

919£99~

'X

'X

'X

"

0

VL

£Ir

.!Modd

1,.(ed ,d

1.

"

"X

"X

"X

9-

LO£ 0

10£

LO£ 0

LO£ 0

.., .,

L-

L

L-

£1

"'X

o

o

1-19-

,

o

VG

9- 21r

9-

,

S-

°

L-

,,,,-

o

IS-

,,,,-

1.-'19- 1.-19

0

L-

"

9

L-J9- L

o

1.-J9- L

1.-19

£L

L-

£-

9

L-19

2

,

G'

119

PUJ

,

999

,

V£

.., ..

91r

V V

,

V V

L-Jg- L-19

L-J9

L-J9- \-J9

,

:)11

9-

,-

,-

,-

o

, ,

61r

air , • • • "" • ••• • ""

,

lU ,1. V9

, ,

vv , vv ,

. ,- ,

9+2'.19

S'

,.

"

V V 9992

,,-

9+;::,.(9

..

,-

S'

,-

..,

•

S99;::

V

S"

V V 9992

,

,.

9+2.19""

19

,,-

,. ,.,

S'

.. ,., ,., .. ,. ,. .. ,., .. ,.,..

,., ,.,

S'

,.

S-

,-

,.

,

S992

,-

,.

9+2'.19-

9+2.19

9-219-

L-ZJ- 'l

,-

J;::

,

S' .

1;::-

,-

,

.

V

V V

L-2.1- 1-

o

,

V

•

, , ..s , ," , ," , " ,,'''' "" "" " " " " " " V

l-Z.l

l+;n- l-Z.l

82

\-

~

212-

,-

,

,

212

l

0

S'

Y

i

o

£*

0

o

o

o

,-

92

o

Y

i

e~

..

o

0

VV Y

i

vaSV:)ll(l'(I:)S'VV ~ ~ ~ ~ ~ ~ ~ ft 9~

~

£ta S':> ne: VIIZ ('Q St:) liS V9l Vlrt

'1-

1-

l+1Z- 0

L+12- i+H: L+1(:

L-l

""

,

o

L-l-

L-'-

o

L-1

t-1-

o

l-l

o o

o

o

o

••

~

;::1I:)ng vEIl lIa

•••

(:'Q

000

o o o

o

o

o

o o

o

o o o

0

o

o

o

o

o

o

o

o

o o o

o o

,-

o

o

o

0

o

0

0

o o

o

o

o

o o o

o o

o

o

o

o

0

o SII

,. '1-

o

o

o o o

o

o

o o

o

o

0

o

(*

0

o

s*

0

If

0

0

If

0

o

•

ZZ*NLZIH<:E*1LI*llLz*r9L'IllIH SlIo"II,)"I]

• •••

YVVY,YVl/YlIVVYVVV "'" " EIr EIr" fir " EIr , EIr V EIr Ell " Ell ' Ell V Ell Ell 'EllY Ell EIr"

0

s*

::IV

•••

o

gz

El

J-

av vv

,-

gz (*

o

000

'I

!iz' SII S'

!iZI (Z' S'

0

o

i

S' El 92

o o

9trg,(HI 6<:' El"

g",J:u

o o

It

..

s* ..

0

s*

(.

o

,-

gz

ay

,-

(*

S'

J-

:>11

•

,QS::l::l9:0VYOll:l:)QQV air Sir 1111 Sir air ell etr lltr

QV

6L*H6C:*DEZ'.:ISC':*3!iIlOSr.l liS yelr

o

,-

.. gz

112 gz

S'

L-J-

. L-J

L-J-

S'

L-1

.. L-J-

o

..,

o

L-J

S"

"S

" " " "• "• QV

" " "

o

1-'1-

S'

,.

,.

o

S'

po,

00

...

.,

o

o

Si:'

V V V v 9££ 9££

air

g-

o

o

o o

o

L-

o

o

0

o

o

,- , o

o

Z

o

0

L-

o 0

o 0

o

o

9

,'" """ ,

o 0

0

9

39L girL :lZL

,

V sa aY o ,ea av v

,

•••••• ••

L 4 (3) Linear group L4 (3) :: A (3) :: 0Ge3} 3

Order = 6,065,280 = 27 .36 .5.13

=

Mul t

GBBB29 BI2.G.21112.G.22112.G·23!22 29 19 23

Out = 2 2

2

Constructions

Linear

GL ll (3) :: 2.G.2, : all non-singular 4 x 4 matrices over F • 3

PGL 4 (3) :: G.2,; SL 4(3) :: 2.G; PSLlj(3) :: G Orthogonal

GO'6(3) 3' 2 x G.2 2 : all 6 x 6 matrices over F 3 preserving a non-singular quadratic form of Witt defect 0,

for example

xl

+ x~ + x~

•• ••• •• • 2

x,

+

x,

+

26

6

2

-1

-1

x,

@

@

@

@

@

@

@

@

@

+

26

6

2

_1

8

-1

1440 A

A 4A

96

32

BA A A ~

IIC

20

36

8

27

ABACAABBBCB ABACAABBBCB

B A

A A

~

72 M

72 ~

72 ~

36 W

~

M

%

@ 27

, •, , , , •••, 20

12 13 13 13 13 20 20 A AA AA BA CB A A A A AA AA A AA 8A CA AB A A A A AA AA W lOA 12A 128 120 13A Bn C*5 DJi8 20A 8.. fus 1nd

36

36

@ @ @ 5 616 16 511

@

• •,

.. •

96

110

40

16

A A

A A

B A

80 E.**

8F

9

96

A B AC OC AAACOC 2C 40 6F 6G

B

B

A

A

8B

C..

12 CB AB 24A

• 12 CC AC

B..

x,

++

8 -1 -1

_1

420

3

0

_1

-1

2

420

o

3

-1

2

0

2

-1

_1

0 -1

2

o

X.

+

39

7

12

3

3

-1

3

-1

-1

_1

_1

4

x,

+

52

8 -4

-2

7

7

-2

-10

0

2

2

_1

-1

2

-1

_1

0

x,

5

-7

11

2

11

2

-5

_1

0

2

_1

_1

2

-1

-1

2

_1

2

_1

_1

2

-1

+

65

x,

+

65

5

-7

11

11

2

2

-5

_1

0

X.

+

go

10

10

9

9

9

0

10

-2

2

0

2

-9

18

-9

0

-4

2

0

-1

2

x.

+

234

14

x..

+

234

14

-9

-9

18

0

_11

2

0

-1

Xn

+

260

0

-4 -la

17

-1

-1

10

0

-2

0

x"

+

260

0

_4 -10

-1

17

_1

10

0 -2

0

3

x"

+

260

20

11

17 -la -10

-1

o

4

0

x"

+

351

-9

15

27

0

-9

-1

-1

x"

+

390-10 _10

x"

+

1116

16

x"

+

416

16

x"

o

x"

o

Xw

+

468

0

2

2

0

_1

0

0

0

o

000

0

o

++

12

0

3

0

-2

-2

0

0

2

+

0

0

0

0

o

o

0

0

0

o

o

+

0

0

0

0

o

o

0

0

0

o

o

0

0

2 -1

-2

o

0

0

0

0

0

_1

-1

-1

-1

0

0

0

0

0

0

0

0

0

-3

2

-1

-1

0

_1

2

0

o

0

0

0

0

0

0

0

0

2

2

-16

0

0

-2

-2

0

0

0

0

_1

_1

416 -16

0 -16

2

2

2

o

0

0

2

2

0

0

0

0

_1

_1

416 -16

0 -16

2

2

2

000

-8

-4 -18

9

9

0

-10

0

2

-2

18

-9

18

0

-5

-3

-1

0

-8

0

0

2

-8

+

0

0 -16

-8

0'00000

0-2

0

0

-8

Xs

0

_1

0

12

0

0

0

_12

0

_1

0

0

0

0

0

-212 12 _12

o

0

640

X.

2i2

_1

0

640

o

0

2

_1

+

o

0

-1

0

X~

-1

0

0

_1

2

_1

2

0

0

0

o

0

0

o

0

0

0

-1

0

0

0

0

_1

0

0

0

0

-1

0

0

0

0

-1

0

0

0

0

0

0

0

15 -15

0

0

0

o

0

0

o

o

XIS

0

o

o

XI9

0

-12

12

X:w

000

o

o

0

0

212-2i2

-2i2 -12

o

12

X21

Xn

*8

*5

0

0

00

16

0-2

o

000

0

o

o x"

0

00

16

0-2

o

o

0

0

0

o

o

16

0-2

o

o

0

0

0

o

-1

_1

o x" o x"

0

o

0

0

0

x"

0

0

X~

+64000-8-8-8

o

0

0

0

0

0

0

0

0

0

o o

x"

+

729

9

9

0

0

0

0

-1

0

0

0

0

0

-1

0

0

_1

0

0

x"

+

780 -20

12

-3

6

6

-3

o

4

0

0

_2

-2

-3

0

0

0

0

0

0

0

0

x"

+

0 -16

14

-4

-4

-4

o

0

0

0

0

0

2

2

2

0

_1

_1

0

0

0

1423333 2 6 6 6 6

8 8

8

5 10

12

12

6 6

6

6

8 8

9 18

9 18

20 20

24 24

24 24

0

0

0

*8

*5 d13

0

0

0

*5

*8

**

*1 d13

0 0

0

00

-1

_1

o

++

27

-1

0

0

-3

-3

o

0

0

0

0

0

++

26

2

-1

-1

-2

-200-2

0

0

0

0

0

0

0

00

0

0

0

0

412

-412

0

0

0

12

-12

12 12

13 26

13 26

13 26

13 26

40 40

40 fus ind 40

2 2

6

6

8 8

16

6

8 8

16

6

8 8

24 24

24 24

o

0

0

0

o

11

0

0

0

0

-3

0

0

0

10

-4i2

o

o

_2

o

o

o

o

o

o

-2

o 2i2

-12

o

o

o

o

-8

10

412

o

o

-2

o

o

o

o

o

o

-2

0_212 12

o

o

o

-8

10

-8

_IU2

o

o

_2

o

o

o

o

o

o

1

-2

o -12

212

o

o

o

-8

10

-8

412

o

o

-2

o

o

o

o

o

o

1

-2

o

12-212

o

0

0

17 -10 -10

-1

1012

2

0

0

0

0

-3

0

0

12

_1

_1

0

12

12

260

0

0

17 -10 -10

_1 -1012

2

0

0

0

0

-3

0

0 -12

-1

-1

o

416

0

0 -16

2

2

2

812

0

0

o

0

0

0

0

0

_1

_1

b

o

416

0

0 -16

2

2

2

-812

0

0

o

0

0

0

0

0

_1

-1 -i5

12

12

x"

o

416

0

0 -16

2

2

2

-812

0

0

o

0

0

0

0

0

_1

-1

12

12

o

X2S

x" X29

0

++

12

0

3

0

4

o

i2 12

o

o

o

o

o

o

o

o

o

o

o

o

X>I

o

o

-12 -12

o

o

o

o

o

o

o

o

o

o

o

o

Xn

o

o

o

o

o

o

o

-1

0

0

0

0

0

0

0 -12 -12

_1

0

0

0

0

0

15 -12 -i2

0

0

0

0

0

0

0

0

0

000

i5

x"

-12 -12

"

x" 0 -1

_1

0

00

26

0

-1

-1 -12+2

0

0 w40 *19

00

0

0

0

0

0

0 *19-w1l0

00

0

0

0

0

0

0-w40 *19

00

0

0

0

0 119 w40

00

0

0

0

o

416

0

0 -16

2

2

2

812

0

0

o

0

0

0

0

0

_1

+

480

0

0

48

12

12

3

o

0

0

0

0

0

0

0

0

0

0

x.,

+

520

0

0 -20

16

16

-2

o

4

0

0

0

0

0

0

0

0

x.,

+

520

0

0

7 -20

16

-2

o

4

0

0

0

0

3

000

XM

+

520

0

0

7

16 -20

-2

o

4

0

0

0

0

3

0

0

0,-2

x..

064000-8-8-8

o

0

0

0

0

0

0

0

0

0

o

X~

064000-8-8-8

o

0

0

0

0

0

0

0

0

0

o

0

0

0

u

18

15

x.,

064000-8-8-8

o

0

0

0

0

0

0

0

0

0

o

0

0

0

18

*5 d13

11

X~

064000-8-8-8

o

0

0

0

0

0

0

0

0

0

o

0

0

0

15

*8

*1 d13

0

0

0

0

0

0

x"

12

26

X~

-1 -i5 -i2 -12

12

00

x..

12+2 -12+2

0

0

12

12_1 -12-1

12+2

0

0 -12 -12-1

o

o

0

0

0

o

o

Xl7

o

o

0

0

0

o

o

XU

0

o

o

0

0

0

o

o

Xu

0

o

000

0

o

o

o

0

0

0

o

o o

X41

12

-12

X42

o

o

12-1

X>s

X36

0

0

0

0

0

0

0

0

_1

-1

_1

-1

0

0

++11004

o

0

0

-2

0

0

0

0

0

0

00

0

0

0

0

-212

212

0

0-212

0

0

0

o

0

0

0

0

0

+

0

0

0

0

o

o

0

0

000

o

0

0

0

0

0

*5

*8

0

0

00

16

0-2

o

0

0

00

16

0-2

0

0

00

16

0-2

0

0

00

16

0-2

0

0

00 -26

0

312+2-312+2

0

0 -12

-1

_1

X49

0

0

00 -26

0

1-312+2 312+2

0

0

12

-1

-1

Xso

o

0

++

12

0

o

-40'00

_1

-1

XSI

0

0

0 d13

X*

o

780

0

0

-3

6

6

-3

1012

-2

0

0

0

0

-3

0

0 -12

0

0

0

12 12

o

x"

o

780

0

0

-3

6

6

-3 _1012

-2

0

0

0

0

-3

0

0

12

0

0

0 -12 -12

o

x"

+

1080

0

0

27

0

0

o

_4

0

0

0

0

3

0

0

0

0

0

0

0

0

o

0

*Ji d13

0

-2

0-212 212

0

o x"

0

0

o

0

o

0

x"

00

0

0

o

Xl7

000

0

X~

o

o

0

260

o

0-2

0

o

0

16

0

x"

0

0

00

0

o

o

0

0

0

208

o

o

0

0

0

o

0

0

000000

0

0

x"

-1

x"

0

0

X~

XI/)

0

0

o

o

Xl4

*8

0

o

o

_1

0

o

208

0

-1

*5

o

o

0

11

Xu

JiJi

-8

x"

0

4

o

0

-8

-8

0

0

0

+

o

4

-1

_1

o

0

_1

0 d13

o

-1

0

_1

0

208

-1

0

2

0

o

-2

+

0

-2

-8

26

-20000000

0

o

o -8

++

212

0

0

o

_1

0

_1

0

208

_1

0

_1

0

o

0

0

2

0

x"

3

0

0 -15

-2

4

_1

00

0

0

4

39

0

0

-2

4

++

o

0

0

13

2

0

0

-1

0

-1

0

0

2

0

-1

0

0

0

40

2

0

-1

0

+

26

00

-1

Xn

++

15

-1

000

x.

2

++

0

0

0

0

2

0

Xw

0

0

o

X.

x"

0

2

-1

9 -3

0

0

-2

o x,

x"

0

2-2

3

x,

-2-200000

000

1040

-3

-1

0

-1

0

-5

13

00

0

-J

0

++

0

_1

-2

-8

_1

0

2

-1

-9

_1

0

_2

18

0 0

0

_1

-8

0 0

0

_1

-1

o x, x,

0

2

0

3

o

0

-1

o

0

0

-1

0

0

0

_1

o

0

0

2

_1

o

0

_1

_1

o

-1

2

0

0

-1

3

-2

0

-1

_1

-2

0

-1

-3

2

0

-1

-1

2

+

-2

-1

2

18

_1

2

0

-8

_1

_1

2

0

0

0

0

0

0

0

16

640

0

0

10

+

0

-1

3

x"

-1

_1

2

5

_1

_1

2

5

-1

0

3

585

0

0

3

S85

0

2

2

+

0

0

39

+

0

0

0 -16

x"

_2

-1

-1

0

0

-2

o

0

Xn

ind

[68]

@

1

5 1 1 6065280 880 152 832 9114 944 81 p,power A A A A A A pi part A A A A A A ind lA 2A 2B 3A 3B 3C 3D

13

_ x~ _ x~ _ x~; PGO;(3) :: G.2 2 ; S0'6(3) :: 2 x G; PS0'6(3) :: 06(3) :: G

0

0

_1

u

d13

0

X~

X4J

XM

o

0

0

o

o

0

o

o

o

-4

0

o

o

X4S

0

0

o

o

X~

000

0

o

o

X47

0

o

o

X
0

0

L 4 (3) Maximal subgroups Q-der

Index

151632

110

@

structure

G.2 1

G.2 2

G.2 3

3 3 :L (3) 3 3 3 :L (3) 3

33:(L (3h2) 3

31+ll:( 25 I1 x2 ) ,

31+ll:(25 q x2),

31+4:(25 q x2 2 ),

la+39a

N(3A3)

point

isotropic plane

la+39a

N<3A3)

plane

isotropic plane

2(5 q x5 4 )·2

la+26bt90a

C(2D)

5 11 (3)

non-isotropic point

la+268+90a

C(2E)

5 4 (3)

non-isotropic point

qO

51840

117

Uq (2):2

518110

117

Uq (2):2

q6656

130

3 4 :2(Aq xA q ).2

2880

2106

(qxA 6 ):2

720

842q

56

@ @

@

10530

@

@

@

13 13 13 13 20 20 20 20 CC IX; BC AC BE AD BD AE AC BC CC IX; AD AE AE AD 26A B.. C!l5Du5 qOA B.. C.21D*19 fus ind

x,

++

X200000000

++

L (3):2 3

L (3):2 x 2 3

2(5qx511) .2

2(5 I1 x5 4 ).2 x 2

H.2

H.2

H.2 2

H.2

08 x S6

H.2

H.2 2

A6·22 (5 q x 5 q ):2

A6 ·22

56 x 2

A

H.2

@

@

51 51 8qO 8qO 576

A

A

B A qE

111

6

2

++

6

1q

0

_1

-1

_1

++

9

9

Xs

0

0

0

0

i2 -i2 -i2

i2

++

20 -20

0

X6

00000000

++

25 -15

-3

x, Xs

++ _15 -1

_1

_1

-1

0

0

0

0

x2

@

(5 11 x 5 q ):2

3 q x Sq x 2

, , , , , , ,

96

96 6118 6118 216 216 108 108 A A AD AE CD BE BD CE A A AD AE cO BE BD CE qF IIG 6H 61 6J 6K 6L 6M

25

2

-2 -2

q 0

0 q

-3 0

2

-2

3 -1 _1

-3

3

5

-3

2

3

_1

5

3

2

0

-1

o

0

-3

-3

3

3

2

-2

5

-5

-1

7

3

3

7

++3030622233

o

o

20

66

60

x"

++

60

20

XI300000000

++

20

20

X'40000

++

9

9

~s

0

0

0

0

0

0

0

0

X'6

0

0

0

0

-1

-1

_1

-1

++

611

i5 -i5

i5 -i5

++

611 -6q

.. *21 *19

+

0

0

0 wqO

X,9

0

0

0

0

X2Q

0

0

0

0 -i2

++ -30 -30

0

i2

i2 -i2

++

**

15

*8

0

0

0

0

d13

*8

*5

0

0

0

0

u

0

0

0

0

*8

*5 d13

Xu

*5

*8

u

d13

-8

-8

2

0

0

_2

0

2

4

0 -3

9

4

2 -2

3

2

3

6

6

2

0 -7

-7

-3

9

9

to

-2

2

2

-3

-3

0

0

0

0

-8

-8

-3

0

5

_1

3 -3

-1

-1

-3

,

0 -3

0 -1

-1

0

0

0

0

0

0

0

-1

0

0

0

-1

0

0

3

3

0

0

0

0

0

_1 -3

-3

0

0

-3

0

0

0 -3

0

0

0

-3

o

_1

o

0

0

0

-3

-3

-2

-2

0

0

o

-1

2222_2-2

o

o _1

-1

0 0

0

-1

@

@

q8 9 96 96 16 BDGBBB ADGAAA 4H 50 SH 1* 8J

, , , 8 C

8 10 C AG

A

A AG L* 10D

8K

@

@

@

10 6 12 AG CH Cl AG AH AH E* 120 24C

@ 12 CH AI DI

x,

-8

8

q -q

-2

2

0

0

0

0

-1

0

0

0

000

0

0

0

0

0

0

0

0

0

++

-1

3

-1

-3

-3

++

0

0

0-2r2 2r2

0

+

0

0

0

0

0

0

0

0

r2 -r2 0

0

0

0

0

-1

-1

0

0

0

X4

0

0

0

r2 -r2

Xs

0

0

0

-1

0

0

++

0

q

0

2

2

-2

0

0

0

0

o

0

+

0

0

0

0

0

0

0

0

0

0

0

0

0

-1

_1

o

_1

0

o

_1

0

_1

_1

..

10

2

2

2

2

0

0

0

-1

-1

_1

-1

0

++

9

0

++

10

_2

0

0

++

16

0

-2

0

0

0

0

0

-1

++

0

0

0

0

0

0

0

0

0

+

0

0

0

0

0

0

0

0

0

0

0

o

0

0

0

0

0

3

3

_2

0

-1

-1

0

0

-1

-1

-1

x..

-q-qOOOOO

-1

-1

r5-r5 0

0

0

0

0

0

0

0 XIS

15-105

3

3

-3

-3

3

3

-3

0

0

0

0

0

-1

0

0

++

0

0

0 2r2-2r2

0

6

-3

o

3 -3

3

0

3

0

0

-1

0

0

0

o

0

0

+

0

0

0

0

0

0

0

00000

OX2'

6 -6

3

0

0

-3

0

3

0

-1

0

0

0

0

o

0

0

0

0

0

0

0

0

0

0

OX23

0

0

o

0

0

0

0

0

0

0

0

0

0

0

0

0

+

+

0

0

0

0

0

0

0

C

000

0

0

0

0

0

0

0

0

0

0

0

0

+OOOOOOOOOOOOOXzs

0

0

0

0

_1

000

++

60

60

_4

q

0

0

-3

-3

-6

-6

0

0

2

2

_1

0

0

0

X2900000000

++

80 -80

0

0

0

0 -10

10

-ll

4

2

_2

0

0

0

0

0

0

0

0

0

8

S

12 12

12 12

12

12

12

12

6

6

6 6

8

20

20

12 12

2q

24

0

0

0

0

r3

0

0 -r3 -r3

o

o

o

o

o

0

0

0

o o

X"

o

o

o

o

o

o

o

X"

o

o

o

o

o

o

o

0

0

0

Xw

80 fus ind 80

X"

0

X~

0

0

x"

0

0

_1

X20

0

o

-1

r2

0

0

-1

0 -r2

0

0

_1

0

0

-3

80 80

0

0

-3

80 80

r2 -r2

+

-3

80 80

X,?

-6

9

26 26

x,s

OOOXI6

6

81

26 26

Xl3

-6

81

26 26

~

2

++

26 26

Xs

-3

0

0

~

x" ++

0

000

0

OOOOOOOOOOOOOXll

0

0

0

Xw

-1

0

0

0

o

~

0

-1

o

0

0

0

_1

-1

0

0

-1

0

0

0

_1

0

-1

0

+

x,

-1

X30

@

x,

o

-1

0

0

@

-1

0

o

0

0

_1

0

@

0

_1

0

0

_1

0

0

,

base. 03<3 )wr2

0

000 0

o

o

_1

o

°3(9)

-1

-1

0

0

0

Xw

04(3)

_1

-1

o

-1

-1

Xv

C(2G) N(2 4 ), C(2F)

0-1

0

333 -3

2

2

0

15

++

X2S

0

60 -60

x" u

0'2(3) x 04(3)

x..

++-105

X24

isotropic point

L2 (9)

.. wllO *19 *21

X2100000000

X23 d13

64

6

-1

-3

5 -3

0

6P

0

++

0

60

line

N(2A)

10

-3

XnOOOOOOOO

0

6N

N(3 )

10 12 12 12 720 9 9 B AD AE CE AF 80 8H AI A A AD AE AE 8F CO BD AE A 8G 108 10C 120 12E 12F 18A 18B fus ind 20

727298 BFCFDF BF Cr OF'

q

, , , , , , ,

q

o -3 -3

0

@

Q-thogonal

x 5q ):2 x 2

-3

++66-6624033

0

@

llnear

x 2

q

++

x,?

@

·-;l

(SII

Abstract

la+398+90a

-3

Xw

-6

0

-3

X900000000

X'S

@

6

Character

++

0

_1

96

A

0

X4

@

A A A 2D 2E 2F'

0

x,

Uq (2):2

Uq (2):2 x 2

54 x S4

@

L (3): 2 3

33:(L <3)X2) 3

151632

576

@

Specifications G.2 2

q

2

++

0

0

0

0

0

0 3r3 3r3

o

0

0

0

0

0

+

o

o

o

o

o

o

o

o

o

o

o

o

o

o

o

0

000

++

-9

3

-3

-3

-1

_1

++

10

2

-2

-2

-2

0

0

0

0

-1

x"

-1

++

0

0

0 qr2-llr2

0

0

0

0

0

0

r2 -r2 X29

36 36

36 fus ind 36

2

q

6

S

8

8 8

16

16

10

10

12 12

++

0

0

0

0

0

0

0

0

0

0

o

o

o

o

o

o

o

o

o

o

o

o

o

o

X3'

o

o

o

o

o

o

o

o

o

o

o

o

o

o

Xn

000

o

0

o

I I[

0

OOOX27

2q 211

r3-r3

2q 24 r3 X30

+

o

o

o

o

o

o

o

o

o

o

o

o

o

o

o

o

o

o

o

o

o

+

0

000

0

0

0

0

000

0

0

0

0

0

0

0

0

0

0

0

0

+

0

0

0

0

0

0

0

0

00000

0

0

0

0

0

0

0

000

+

0

0

0

0

0

0

0

..

0

0

0

0

0

0

0

0

00000

0

0

0

0

0

0

0

0

0

0

+

0

0

0

0

0

0

000000

OX»

++

0

0

0

0

0

0

0

0

o

0

300

0

0

0

0

0

0

++

0

0

0

0

0

11

0

0

0

0

0

0

OX41

++

0

0

0

0

0

0

0

0

0

0

0

r6

o

o

o

o

x"

X~

X~

o

0

X~

o

X"

o

X"

0

0

0

0

0

0

0

0

0

0 w80

0

0

0

0

0

0

.. 161 *19

X"

o

0

0

0 . " w80 *19 *61

Xw

o

0

0

0 *21 *59 w80

X~

o

0

0

0 *59 *21

o

X..

0

..

+OOOOOOOOOOOOOX3S

x" 0

0

0

0

OOX3?

x"

** w80

0

0

X~

0

0

0

0

X.,

000

0

0

0

0

0

++

0

0

0

0

0

0 6r3-6r3

o

0

0

0

0

0

0

0

0

0

0

0

0

r3 -r3

X.,

o

0

0

0

0

0

++

0

0

0

0

0

0-3r3 9r3

o

0

0

0

0

0

0

0

0

0

r3

0

0

r3

0

++

0

0

0

0

0

0 9r3-3r3

o

0

0

0

0

0

0

0

0

0

r3

0

0

0

r3

+

0

0

0

0

0

0

0

o

0

0

0

0

0

0

0

0

0

0

0

0

0

0

+

0

0

0

0

0

0

0

0

0

0

0

0

+

0

0

0

0

0

000

o

0

0

0

0

0

0

0

0

0

0

0

0

0

0

+

0

0

0

0

0

0

0

0

0

0

0

OOX4?

+

0

0

0

0

0

0

00000

0

0

0

0

0

0

0

000

+

0

0

0

0

0

0

0

0

0

0

0

0

0

0

x" X4S d13

..*5 d13

*8

0

0

0

0

*8

*5

0

0

0

0

..

0

0

0

0

X46

u

X4?

*8

*5 d13

X48

*5

*S

If

d13

0

0

0

0

x~

o

0

0

0

0

0

0

0

x~

o

0

0

0

0

0

0

0

XSI

_1

-1

-1

-1

0

0

0

0

0

r6

X42

+OOOOOOOOOOOOOX43

x" OX45 X~

x" 0

0

OX4\> X~

++

0

0

0

0

0

0 9r3 9r3

000000000

0-r3

0000

++

0

0

0

0

0

0

0

0

0

0

r3

r3-r3

x"

[69]

L s (2)

Linear group L (2) :: Alj.(2) 5

= 9,999,360

Order

= 21°.32 .5.7.31

Mult

=1

Out = 2

Constructions

Linear Presentations

GL S (2) :: PGL 5 (2) :: SL 5 (2) :: PSL5 (2) :: G : all 5 x 5 non-singular matrices over F " 2 5 2 :G = < a~c d e l (bcf)3"(abcd)6", >; adjoin (fbaCde)5", for G f

Remark: There is a non-split extension 2 S0L 5 (2) (the Dempwolff group), which is a maximal subgroup of the Thompson group.

Maximal subgroups Order

Index

322560 322560

Specifications

Structure

G.~

Character

Abstract

Linear

31

24 :L 4 (2}

21+6: L3 (2):2,

1a+30a

N(2A 4)

point

31

2 4 :L 4 (2) 26 :(3 XL (2)) 3 3 2 6 :(3 XL (2)) 3 3 31:5

Llj. (2): 2

64512

155

64512

155

155

64512

1a+30a

N(2A 4)

hyperplane

2 4+4 : (S3xS3): 2,

1a+30a+124a

N(2 6 )

line

3 3 x L (2):2 3

1a+30a+121(a

N(2 6 )

plane

N(31ABCtf;F)

L1(32)

31:10

88 27

@ 9999360 power

p

pt part ind 1A

[70J

@ @ @ @ @ @ @ @ 21 1 504 536 504 180 384 128 32 15 A A A A B A A. A A A A A A A A A 2A 28 3A 3B 1IA 4B 4C 5A

X.

+

x,

+ 30

14

6

x,

+ 124

28

12

X.

+ 155

27

-5

x,

+ 217

-7

X.

+ 280

@

24 AA AA 6A

@

12 BB BB 6B

@

@

@

@

@

@

@

@

27

13

@

@

@

@

@

@

@

@

@

@

@

@

@

@

@

@

@

@

@

@

42 42 16 12 111- 14 15 15 21 21 31 31 31 31 31 31 720 48 18 18 96 96 16 6 5 8 8 12 A A B AA AA BA AB AB BA AA A A A A A A A BACBCAAA AC BD A A AC A A A A AA AA BA AB AB AA BA A A A A A A AACBCAAA AC BD A A AC 7A Bn 8A 12A 14A Bn 15A B** 21A B** 31A B** C*5D**5E**6 F*6 fus ind 2C 4D 6C 6D 8B C* 80 lOA 128 16A B* 24A

@ 12 AB AB

B*

++

6

0

6

2

2

4

4

4

0-1

8

5

3 -5

9

7

4-7

56

8

7

-5

8

0

x,

o 315 -21

3

0

0

3

x,

o 315 -21

3

0

0

X.

o 315 -21

3

0

x"

o 315 -21

3

Xu

o 315 -21

x"

-1

0

2

0

2

0

0

0-2-20

0

0

2

-1

0

0

-1

-1

o

-1

-1

0

0

0

3

-1

-1

0

0

0

3

-1

_1

o

0

0

3

-1

-1

3

0

0

3

-1

o 315 -21

3

0

0

3

-1

Xu

o 465 -31

9

3

0

x..

o 1165 -31

9

3

0

Xu

o 1165

17 -15

3

0

x..

o 465

17 -15

3

0

x"

+ 496

48

16

-8

x"

+ 651 -21

-5

0

x..

o 651 -21

-5

0

Xw

o 651 -21

-5

x"

+ 868 -28

4

7

x"

+ 930

50

-6

6

Xn

0930 -14

-6

Xu

o 930 -14

-6

x"

+ 960

64

x"

+1024

x"

+1240

-1

0

0

0

0

o

0

-1

-1

o

-1

0

-1

-1

-1

-1

++

0

a

000

0

0

0

++

4

0

o

0

0

0

0

0

++

5

0

0

0

000

++

5 -3

_1

_1

-1

-1

0

0 -1

_1

o

0

0

0-1

o

0

0

0

0

0

0

0

o

0

0

0

0

0

0 f31

0

0

0

o

0

0

0

0

0

0

0

0

0

0

o

0

0

0

0

0

0 **6

0

0

0

0

0

o

0

0

0

0

0

0

*6 **6

-1

0

0

0

0

0

o

0

0

0

0

0

0

*5 **5 **6

-1

0

0

0

0

0

o

0

0

0

0

0

0 **5

-3

o

-1

0

b7

** -1

-b7

..

0

0 b7

**

0

0

0

0

0

0

-3

o

-1

0

**

b7

** -b7

0

0

**

b7

0

0

a

0

a

a

o o

-1

0

b7

**

b7

a

0 b7

**

a

0

0

0

a

0

_1

o ..

b7

**b700**b7000000

_1

-1

0

0

000

o

6

3

3

-1

o

-3

3

3

-1

o

o

0 -3

3

3

-1

o

_4

4

a

-2

-1

0

-6

-2

-2

a

2

-3

a

2

-2

2

-3

0

2

-2

0 -6

0

0

0

a

4

-8

8

-5

-1

o

-1

-1

0

000

-8

2

**

0

-1

-1

-1

0

0

0

0

-1

0

0

0 b15

o

0

-1

0

0

0

o

0

0

_1

0

0

_1

-1

o

0

a

o 2b7

**

0-1

a

2

0

o

0

-1

0

0

0

0

-2

0

o

0

0

0

0

-1

0

0

0

0

-8

0

0

0

-2

a

-1

** 2b7

2

2

o

-1

-1

**

*5 **5 **6

** f31 **5

o

*6 f31

*5

0

*6

*5

*6 **6

**

*5 **5

** f31 **5 *6 f31

*6 **6

**

+

0

0

-1

0

0

-1 -1

o

0

-1

o

0

2

2

0

-1

0

0 -1

0

a

0

a

0

0

0

0

-1

X6

0

~

0

0

0

0

0

0

0

0

0

0

+

0

0

0

0

0

0

0

0

0

a

0

0

x.. aXIl

+

+

0

0

a 0

a a

0

0

0

a

0

0

4

4

-2

o

0

9 -3

0

0 -3

-3

0

0

a

a

++

**

0

0

0

0

0

0

0

0

+

** b15

0

0

a

0

0

0

0

0

o

0

0

0

0

0

0

++

10

-2

0

0

0

0

a

++

0

0

a

O-b7

**

a

0

0

a

a

a

0

0

0

** -b7

0

a

0

0

0

0

o

0

-1

-1

-1

-1

-1

-1

-1

-1

-1 000

0

0

a 0

a 0

a 0

0

a

0

OX13

a

x.. aXIS x..

++

a

~

x"

0

-1

X4

x,

2

0

0

-1

X2

x,

000

o

2

r2

a

0

0

-3

r2 -r2 -r2

0

0

o

-3

0-1

0

0

000

0

2 -1

a

+

0

o

11

0

x,

o

-1

4

0

-1

-1

2

-2

0

0

-1

10

0 2r2-2r2

** f31

a

0

++

0

*5

a o

0

X.

0

0

0

0

0

o

-1 ·-1

o

0

0

0

0

0

0

0

X17

o

a

X18

0

0

XIII Xw

2

-2

0

o

0 2r2-2r2

a

0

0 -r2

2

0

0

-1

-1

X21

r2 -r2

r2

Xn

+OOOOOOOOOOOOOX23 Xu

++

0

0

++

16

0

++

10

2

0

0 1Ir2-4r2

-2 -2

0

0

0

0

-2

-2

-2

0

0

0

0

r2 -r2

X:2S

OOOOOX26

0

-1

0

0

x"

M 23

Sporadic Mathieu group M23 Order

= 10,200,960 = 27 .3 2 .5.7.11.23

Mult = 1

Out

Constructions Mathieu

For almost all purposes, M23 is best studied as the point stabilizer in M24 (q.v.). Thus, G is the automorphism group of the Steiner system S(4,7,23), whose 253 heptads arise from the octads of S(5,8,24) containing the fixed point.

G : the automorphism group of the (unextended) binary Golay code of dimension 12, length 23, and minimal weight 7, or

Golay

of its subcode of even weight words. The weight distribution of the code is 01725385061,12881212881550616253231, and that of the cocode (words modulo the code) 01123225331771. The minimal representatives in the cocode are unique (i.e. the code is a perfect 3-error correcting code). Presentation

G~

< a~d I a"(cf)2. b"(ef)3. 1"(eab)3"(bce)5"(aeCd)4"(bcef)4 > 6 f

e

Maximal subgroups Order

Index

443520

23

40320

253

40320

253

20160

506

A 8

7920

1288

5760

1771

253

40320

Specifications

Structure

Character

Mathieu

Golay

M22

la+22a

point

weight 1 cocode word

L (4):22

la+22a+230a

duad

weight 2 cocode word

heptad

weight 7 code word

la+22a+2308+253a

octad

weight 8 code word

Mll

la+22a+2308+1035a

endecad, dodecad

weight 11 or 12 code word

24:(3xA5):2

la+22a+230aa+2538+1035a

triad

weight 3 cocode word

3 2 4 :A

Abstract

N(2A 4)

1a+22a+230a

7

N(2A 4 )

N(23AB)

23:11

~17 17

@

@

@

@

@

@

@

@

@

@

@

@

@

@

@

@

@

2

10200960 688 180 32 15 p power A A A A pI part A A A A ind lA 2A 3A 4A 5A

12 AA AA 6A

14

A A A A A 7A 8** 8A

2

0

o

o o

0

b7

**

-1

0

••

b7

-1

-1

-1

0

0

0

-1

o

0

14

8

A

11

23

23

A

A

11

14

14

15

A A

A

AA

BA BA

AA AA

0

0

-1

-1

-1

-1

-1

-1

-b7

••

0

0

-1

-1

•• -b7

0

0

-1

-1

o

0

0

0

0

0

••

o

0

••

15 AA

A AA AA A A 11A Bn 14A Bn 15A Bn 23A Bn

x,

+

x,

+

22

6

4

x,

o

45

-3

0

X.

o

45

-3

0

x,

+

230

22

5

2

X.

+

231

7

6

_1

x,

o

231

7 -3

_1

o

0

-1

0

0

0

0 b15

x,

o

231

7

_1

o

0

_1

0

0

0

0

Xo

+

253

13

-1

o

0

-1

-1

x..

o

770 -14

5

_2

0

o

0

0

0

0

0

0

0

0 b23

x"

o

770 -14

5

_2

0

o

0

0

0

0

0

0

0

0

x"

o

896

0

_4

0

o

0

0

0 b11

n

0

0

-1

_1

x"

o

896

0

-4

0

o

0

0

0

**

bll

o

0

-1

-1

X14

0

990 -18

0

2

0

0

b7

**', 0

0

0

b7

n

0

0

XIS

0

990 -18

0

2

0

0

n

b7

0

0

**

b7

0

0

x"

+

'035

27

0

_1

0

0

-1

-1

-1

-1

o

0

0

0

x"

+

2024

8

-1

0

-1

_1

-1

-1

0

0

-3

2

0 -2

-2

0

-1

-1

000

•• b15

•• b23

[71]

U s(2) Unitary group U (2) ;: 2AIj (2) 5

Order = 13.685.760 = 21°. 35. 5• 11

=

Mult

Out

1

= 2

Constructions GU S (2) ;: 3 x G : all 5 x 5 matrices over FIj preserving a non-singular Hermitian form;

Unitary

PGU S (2) ;: 5U (2) ;: PSU (2) ;: G 5 S Quaternionic

~

G x 2 : the group generated by the 165 quaternionic reflections x

x _ 2<x,m>m/<m,m> in the vectors

C2,O,O,a,O)e (S), (o,w,w,w,w)c (160) and their images under the diagonal group 2x 24 +4 = {diag(±l.±j.±k,±k.±j)C}

of right multiplications, where w = ~(-1+i+j+k). and <x,y> = Ixtyt. The vectors whose coordinates are in Z[w] form a complex lattice whose automorphism group is 6 x Uq (2) (see U4 (2). "biflection tl ) . Reducing modulo 1+1 shows the isomorphism with U (2). 5

Complex Leech

G: the stabil izer of a minImal vector 1n the complex Leech lattice (see Su1.).

• ••

82 " 13685760 9411 608 P power pt part

ind

x,

A A

lA

2A

2B

10

-6

2

•

•

77760

3888

A A

A

A

A A

3A

B..

3C

• ••• • • • 1

3888 944 3211 152 3811

A A Du

A A

A A

3E

3F

A A 4A

A A 4B

•

• • • • • • • • 288

96

15

1728

1728

1296

1296

B

A

AA

DA

CA

BA B..

CA

DA

CA

DA

AB

6C

D..

6E

F..

A

A

BA AA

4e

5A

6A

"32 DA

"32 CA

288 216

lOB

108

36

FA

FB

K..

FA FA 6L

FA

FB

Mu

6N

-1

-1

o

o

2

X2

..

1.3+2

-2z3

0

X3

-2 1.3+2

..

-21.3

••

0

X4

-1

o

o

2

X5

61

-1

0

-13

13

-2

13

-13

6c

AB BB Hn

-3

-1

1.3

EA EA

CB DB

x, 4-22-2-20

.. .. ..

-5

3

3

3

o

11

-5

3

-3i3+2

313+2

31.3-1

..

2

2

3

-1

_1

13-2

-13-2

b27

X.

o

11

-5

3

3i3+2

-313+2

**

31.3-1

2

2

3

-1

-1

-13-2

13-2

**

b27

z3

x,

• •

411

12

-4

,.

-1

_1

5

2

4

4

0

_1

6

6

3

3

3

3

2

2

-3

55

23

7

10

-1

-1

3

0

2

2

5

5

5

5

-2

-2

2

-2

-2

-2

-2

-21.3

31.3-3

..

**

x,

o

55

7-1

x,

o

55

7

-1

x,

o

66

18

10

",0

o

66

18

10

91.3-6

110

-2 -10

-25

Xn

151.3-5

,.

.. .. .. .. 10

151.3-5

-2b27

7

3 _1

0

**

7

3

0

91.3-6

u

10

-2b27

_1

3b27332-22

3b27

1.3+2

2226-620

2

**

31.3-1

8

-4

6

2

2

31.3-1

**

8

-4

6

2

2

71.3+3

1.3+2

**332-22

-25

.. .. **

71.3+3

n

3

-3

-5

x,

x,

m

DB CB 6J

BB

• • ••

•• m

•

x,

** 31.3-3

7

7

0

-313-6

313-6

0

313-6

-313-6

-21.3

n

1.3-1

n

..

-1

.. 31.3-1

2

2

••

2

2 -13-2

.. 31.3-1 1.3-1

-1

u

b27

..

3

u

b27

n

b27

3

b27

u

x,

-1

13-2 -13-2 13-2

13

-13

-13

13

-1

X7

-1

Xs

xo

x"

-2

-2

-1

_1

-2

2

2

-2

-2

2

Xli

-31.3

**

-13

13

0

..

1.3-1

o

o

0

X12

..

-31.3

13

-13

0 z3-1

..

o

o

0

X13

313+2-313+2 -13-4

13-4

1.3+2

**

-1

13

-13

••

2z3

0

X\4

13-4 -13-4

**

z3+2

-1

-13

13

21.3

••

0

X15

-2

-2

..

XI2

o

110

30

6-1513-10 1513-10

Xn

o

110

30

6 1513-10-1513-10

x"

o

110 -311

6

5b27

-313+2

313+2

11

2

-2

2

-2

0

x"

o

110 -311

6

5b27

**

313+2

-313+2

11

2

-2

2

-2

0

x"

+

120

24

8

30

30

12

12

3

3

8

0

0

0

6

6

6

6

o

o

2

2

3

2

2

3

3 -1

X16

Xn

+

165 -27

5

30

30

3

3

3 -6

5

5

-3

0

6

6

-9

-9

3

3

2

2

3

-1

-1

o

o

XI7

x..

+

176 -16

16

-"

-"

0

0

0

-"

-"

2

2

2

2

4

-4

-2

-2

-1

-1

x"

o

220 -36

-4 1513+25-1513+25

513-3

**

31.3-3

13

-13

-1

XI9

Xw

o

220 -36

-4-1513+25 1513+25

-513-3 31.3-3

..

1.3-1

1.3-2

-13

13

-1

X20

Xn

o

220

-4

12

10b27

u

-b27

2

2

0

X21

x"

o

220

-4

12

**

10b27

0

X22

x"

o

264 -211

-8 -2713+3

Xu

o

264 -24

-8

Xn

[72]

A A

• 77760

320 _64

0

..

..

z3-2

,.

14

..

-31.3+7

7

4

_4

0

0

-513-3

-31.3+7

..

7

4

-4

0

0

513-3

151.3-2

-5

4

4

4

0

0

2b27

151.3-2

**

-5

4

4

4

0

0

2713+3

-6

-6

3

3

8

0

0

_1

2713+3 -2713+3

-6

-6

3

3

8

0

0

-1

-"0

X~

o

330

-6

2 451.3+60

Xv

o

330

-6

2

Xa

+

440

88

8

Xw

o

X~

..

-"0

..

n

-"

-11

-4

14

5

-4

0

0

0

(j

..

u

3b27

n

3b27

1.3-2-313+2 313+2

..

-b27

2b27

n

-b27

13-9

-13-9

u

-61.3

-13-9

13-9

8

..

9z3-3

-3

-3

10

-2

2

0

-3

-3

10

_2

2

0

llz3+7

8

..

111.3+7

u

-61.3

8 31.3

..

B

91.3-3

..

-10

8

8

17

-1

-8

0

0

0

-2

440 -40

8 1513-55-1513-55

613+8

-613+8

_1

_1

8

0

0

0

313+5

-313+5-313+5 313+5

o

440 -40

8-1513-55 1513-55

-613+8

613+8

-1

-1

8

0

0

0

-313+5

313+5 313+5-313+5

x"

o

495 -81

15

9z3

..

9

0

-1

-1

_1

0

3z3

x"

o

495 -81

15

9z3

9

0

_1

-1

-1

0

** 451.3+60

-10

-45z3

.. .. .. ..

-1151.3

n

x"

o

495

63

-9

..

X~

o

495

63

-9

-451.3

x"

+

660

84

20

30

X~

o

704

64

0-1213-52 1213-52

x"

o

704

611

0 1213-52-1213-52

X~

o

880

118

X~

o

880

118

X~

+

891

21 -21

x..

o

891

27-21

x..

o

891

27 -21

x..

o

990 -18 990 -18

16 16

6 6

.. -20b27

91.3

9

0

-1

-5

-1

0

9z3

..

9

0

-1

-5

_1

0

-15

-15

3

3

-4

4

0

0

-4b27

2

2

0

0

0-1

2

2

0

0

0-1

-451.3

30

-20b27

u

-4b27 313+7

u

-313+7

-8

-5

0

0

0

-313+1

313+7

-8

-5

0

0

0

o

.. .. .. .. ..

o

0

0

3

3

o

o

0

0

o

o

0

0

81

u

81

811.3

811.3

451.3

18z3

Xu

o

451.3

x..

+

1024

0

0

6'

6"

16

X~

o

1215 -81

-9

o

o

x.,

o

1215 -81

-9

o

o

o o

U_9

0 0

-2

..

u

-2

..

1.3-1

n

u

" 21.3

.. 21.3

u

13-3 -13-3

n

-b27 -13-3

3

-21.3

**

213

-213

••

-21.3

-" -"

o

5z3+1

z3+1I

..

" "2

-2

2

o

n

z3-1

2

2

••

-21.3

13

-13

x" Xu

;; -2z3

••

-13

13

_1

Xu,

13

-13

-1

X27

2

2

_1

Xu

2z3

2

2

2z3

u

_1

..

u

2z3

-1

21.3

2

-31.3

..

31.3

..

u

n

-3z3

**

3z3

-3

••

3z3

..

-91.3

••

-3z3

••

31.3

-3

..

3z3

n

-91.3

..

-3z3

**

31.3

u

-3

3z3

3

3

-3

-1

-8

-8

4z3

-8

-8

u

.. ..

-81.3

-81.3

-213-6

213-6

-6z3

u

u

..

41.3

-3 -31.3

.... ..

X25

-1

-1

-1

XZ9

-1

-1

_1

X30

o

o

0

X31

-31.3

o

o

0

X32

3z3

o

o

0

}(33

o

o

0

X34

-3

-3

-1

X35

o

-2z3

••

0

}(36

0

X37

2

2

3

o

o

-2

o

o

-2

o

o

••

-2z3

0

••

-21.3

13

-13

X~

0 -21.3

..

-13

13

X~

o o

o

O}(40

o

0

X41

4z3

••

-13-3

13-3

••

4z3

13-3 -13-3

0

-z3

2

-3

2

..

3z3

6

2

-z3

-3

91.3

6

13

.. -3

..

91.3

-13

o

u

o

....

u

.. ..

x..

2

z3+4

u

..

-1 z3-1 -1

213

31.3 51.3+1

3

13-3

-213

u

3 1.3-2

2

o

-1

213-6

-213-6

-6z3

-1

9

9

o

o

o

-3

-3

0

o

o

3

3-1

9z3

..

o

o

o

o

-31.3

..

0

o

o

3

3

.. .. ..

o

9z3

o

o

o

o

u

-31.3

0

o

o

o

0

X42

o

-61.3

..

3

o

o

o

0

X43

-31.3

o o

o o

o

-31.3

3

o

o

o

o

0

X44

0

o

o

o

o

0

X45

0

o

o

o

0

X46

0

o

o

o

o o

0

X47

-1

0_2-6_2

0

-3z3

u

u

-31.3

-61.3

u

18z3

-9

0

_2

-6

-2

0

16

-8

4

0

0

0

-1

o

o

o

o

o

o

000-9330

o o

o o

o o

o

o

o

o o

o

o

o

o

000-9330

o o o

U 5 (2) Maximal subgroups

Specifications

Structure

G.2

Character

Ab stract

Unitary

165

2~+6:31+2:2All

2~+6:31+2:2Sll

la+44a+120a

N(2A)

isotropic point

root vector

17760

116

3 x Ull (2)

C3 x Ull (2»:2

1a+558+120a

N(3AB)

non-isotropic point

complex lattice

46080

291

2 4 +ll : C3 xA ) 5 34 :S 5 S3 x 31+2 :2A

2ll+IJ:C3xA5):2

1a+1208+116a

isotropic line

base

3 :( 2xS5)

N(2 ll ) = N(2A SB10 ) ll N(3 ) = N(3(AB)5(CD)10El0F15)

base

S3 x 31+2: 2S4

N(3CD). N(3E)

non-isotropic line

L 2 (11 ):2

N(2B. 3F. SA. 6N. l1AB)

Order

Index

829114

9120

1408

3888

3520

660

20736

4

ll

L 2 (11)

88 47

@

@

@

@

@

@

@

16 5ll 54 21 27 11 11 B C D C D A A A A A A A A A 8A 9A an 9C Du 11A Bu

@

m BA AA 12A

@

@

@

m

72

72

AA BA Bu

FA CA

DA

12C

Du

EA

@

@

@

24

24 36 lA HC GC EA AC BC 12E 12F Gu

@

24

@

24

12H

@ 15

@

@

15

18

AA AB AB AA Iu 15A Bu

BC AA

DB

18A

@

720 fus ind

A A 2C

@

@

@

@

@

@

@

@

@

@

@

@

ll8 18 18 96 96 16 6 12 12 8 5 8 BECFCAAA AC ND A A EB EC AECFCAAA AC FD A EB EC A llD 60 6P 8B Cn 8D lOA 12J 16A Bn 24A Bu

x,

++

x,

o

x,

1 -z3

X.

X6

@

18 AD

BA Bu

x,

x,

47

13

FB EB CB

Q.laternionic

-2

_1

-1

-1

_1

-1

2

u

0

0

-i3

i3

z3-1

..

0

_1

-1

u

-z3

0

0

i3

-i3

..

z3-1

o

_1

-1 -z3

-1

-1

o

0

-2

-2

o

0

2

2

-1

n -z3 -z3

n

o

_1

-2

2

2

_1

-1

2

000 n

-z3

z3

0

00

0

0

0

+

0

0

000

..

z3

..

....

z3

..

z3

_1

-1

o

0

0

++

II

0

0

-1

-1

o

0

_1

_1

++

5

-z3

..

0

0

0

0

..

z3

+

0

•• -z3

0

0

0

0

z3

..

n -z3

z3

..

**

z3

0

0

....

0

0

n

z3

n

z3

0

0

z3

..

-2z3

X.

z3

••

z3

u

0

0

u

z3

..

-2z3

0

0

0

z3-2

-z3

..

..

-z3

z3

z3

..

o

0

_1

-1

o

0

0

0

..

0

0

0

0

u

0

0

0

0

0

0

0

0

_1

i2 -12 -i2 0

0

0

12

Xz

0

~

X.

o

x,

0 212-2i2

-2

4

11

2

-1

-3

-3

0

0

0

0

0

0

0

0

x,

000

o 0

_1

_1

o

0

X6

0

0

0

0

b

X.

x,

o

Xw

o

0

0

0

0

0

0

x"

o

-1

_1

-1

-1

o

0

3

3

o

x"

o n

2z3

.. -z3

0

0

-13

i3

z3+2

..

0

_1

-1 -z3

x"

o

2z3

.. -z3

u

0

0

i3

-i3

..

z3+2

0

_1

_1

..

-z3

0

0

0

0

x"

x..

o -z3

.. -z3

..

0

0

..

b21

z3

..

-1

_1

o

0 -z3

..

+OOOOOOOOOOOOOX14

x"

o ..

u

n

z3

-1

-1

o

0

x"

o

0

000

.. z3-2

u

_1

-1 -z3

-z3

0

0

b21

0

0

0

-1

_1

2

2

2

2

-1

o

0

0

0

o

0

0

0

0

0

2

2

-1

-1

_1

o

0

-1

-1

000

0

_1

-1

-1

o

0

o

o

o

0

0

000

_1

-1

n

z3

0

0

-2z3

z3

u

0

n

x..

o

_1

x"

o

u-2z3

Xw

0-2z3

0

..

x"

Ouz3nz300

..

••

o

z3

u

o

0 -z3

-2z3

u

z3

o

-2z3

u

z3

u

0

0

0

0

0

0

0

0

0

~

Xw

0

0

0

0 2i2-212

0

0

0 -i2

i2 -12

i2

XII

+OOOOOOOOOOOOOXI2

.. -z3

x" ++ ++

10

2

2

S -3

_1

2

2

2

o

++44_2

0

0

0

-1

o

0

-1

o -1

0

-1

-1

_1

X16

x"

0000X18

0

+OOOOOOOOOOOOOX'9

0

.. -z3

0

0

0

0

Xw

o

0

**

z3

0

0

.. -z3

+OOOOOOOOOOOOOX21

o o

0

z3

..

0

**

x"

-1

0

0

0 -z3

..

0

0

+OOOOOOOOOOOOOX23

2z3-1

o

0

0

0

**

-z3

0

0

Xu

0

0

0

0

0

0

-1

-1

-z3

..

z3

u

0

0

0

0

.. -z3

..

z3

0

0

0

0

000

0

0

0

-1

o

0

0

0

0

0 -z3

2z3-1

o

0

0

0

0

0

n

0

0

0

0

-z3

0

0

0

0

x"

n

z3

..

0

0

-2z3

u

z3

x"

0

0

0

0

0

0

2z3

••

2z3

Xu

o

0

0

0

0

0

0

..

2z3

..

x"

o

-1

-1

-1

-1

o

o

o

Xu

000

0

0

0

0

z3

u

z3

u

Xn

000

000

0

u

z3

u

z3

Xu

o

X~

o -z3

Xw

o

u ..

o

-1

-1

o

0

-2

-2

-2

-2

•• -z3

..

0

0

2z3

..

2z3

..

-z3

0

0

••

2z3

..

u

0

0

z3

-z3

0

0

o

n

0

0

o

-1

0

..

x"

-1

+

00

0

x"

-z3

000

0

0 -z3

00

0

()

0

0 4i2-4i2

0

0

0

0

0

i2 -i2

X2S

+OOOOOOOOOOOOOXu

x" ++

10

2

-2

-2

-2

0-1

o

0

Xu

..

+OOOOOOOOOOOOOX:w

. . -z3

Xw

X31

-1

o

0

0

0

0

-z3

..

-z3

..

Xn

-1

o

000

0

0

••

-z3

..

-z3

-1

.. -z3

n

x"

o

0

0

0

0

0

••

-z3

..

-z3

-1

u

-z3

..

z3

0

0

0

0

.0000000000000X33

x..

o

0

0

0

0

0

-z3

..

-z3

**

-1 -z3

**

z3

**

0

0

0

0

x..

_1

-1

0

0

.. -z3

o

0

0

0

0

2

2

-1

o

.. -z3

..

0

0

o

o

o

o

0

0

0

0

0 -z3

.. -z3

0

0

o

o

o

o

0

0

0

0

0

z3

..

0

0

o

o

o

o

0

0

0

0

0

0

0

0

0

..

z3

0

0

o

o

000

0

0

0

0

0

0

0

0

0

0

0

-3

-3

o o

-1

o

0

o

0

u

n

x"

o

X3£>

0 -z3

x"

o ..

-z3

x..

0-2z3

n

X:w

0

X40

-1

X41

_1

o o

X4Z

-1

o

0

0

0

x"

o

0

0

0

0

X«

000

0

0

0

0

X4S

0

0

-1 -z3

0

n-2z3

0 0

0

0

o

0

-1

0

-3z3

o

o

0 -z3

0

0

..

-3z3

o

o

0

0

0

..

..

-2z3

z3

o

o

o

o

o o

o

o

o

o

o

0

-2

-2

X~

o

0

0

0 bll

x"

o

0

0

0

••

.. b11

z3

u

n

000

..

z3

..

-z3

n

z3

.OOOOOOOOOOOOOX3t

++

10

-2

2

2

-2

o

0

0

_1

-1

X3S

.000000000000

OX3£>

•

0

x" 0

0

0

0

0

0

0

0

0

0

0

0

X~ x~

++

9 -3

0

0 -3 -3

-1

o

o

0

X40

0

0

n

z3

0

0

.. -z3

0

0

z3

..

0

0

x"

z3

**

0

0

0

0

0

0

.OOOOOOOOOOOOOXe

*.

z3

0

0

0

0

0

0

X«

0

0

0

0

0

-1

-1

o

0

0

0

0

0

0

0

0

0

0

.0000000000000X46

000

0

0

0

0

0

0

0

x.,

u

-2z3

.OOOOOOOOOOOOOX41

++

16

0

-2

-2

0

0

0

OOOOOX4S

[73]

L3 (8) and 2P4(2), Linear group L3 (8)

= A2 (8)

= 16,482,816 = 29 .32 .72 .73

Order

Mult

=1

Out

=6

Constructions

Linear

=7

GL 3 (S)

x G : all non-singular 3 x 3 matrices over F 8;

PGL 3 (8) " SL3 (8) " PSL3 (8) "G; rL (8) " (7 x G).2; PrL (B) "!L3(B) " P!L3(8) "G 3 3

Maximal subgroups Order

Index

225792

Specifications

Structure

G.2

G.3

G.6

Character

Abstract

Linear

73

2 6 :<7xL2 (8»

2 3+6 :72 : 2,

2 6 :<7xL 2 (B»:3

2 3+6: 72: 6,

1a+72a

N(2A 6)

point

225792

73

18+72a

N(2A 6)

line

294

56064

2 6 :<7 xL2 (8» 7 2 :S 3

2 7 :D'2

72 : OxS3)

2 7 :( 6xS3)

N(7 2 )

base

2'9

75264

73:3

73:6

73:9

73: ,B

N(73A-X)

L,(512)

,6B

9B112

L2 (7)

L (7):2 2

L 2 (7) x 3

L 2 (7):2 x 3

COB)

L (2) 3

2 6 :<7 xL 2 (B»:3

D'4 xL 2(B)

~BBEJ72

72

Abbreviated character table.

@ ,

@ 3

64828'6 584 A A 2A

p power

p' part ind lA

@

@

63

64

A A 3A

A A 4A

@

352B/6

A A

7AF

(D'4xL2(S»:3

@

@

49/2 A A 7GH

49/3

@

@

A

A

56/6 CA

A

A 9AC

AA

AA

14AF

21AF

7IK

63/3

@

63/6 FA

10

@

6

@

@

@

@

504 9 63/' B 73/24 A A8 FB A AA A A AB 63AR 73AX fus ind 28 6A

+

@

@

@

'6/2 713 9/3 A

JB

72

8

0

06

73

9

024 441

-7

0

+ 511

-1

-2

+3 511

-1

06511

-1

01851'

-1

0

-2

9

2

2

z7+9&3

b7

Y7+1

o

o

o

-1

7

o

o

-,

7

o

o

-1

7z7

o

o

-1

7z7

o

o

512

0-1

o

8

06 584

8-1

o

8z7+9&3

b7

Y7+1

o o -y9

o

o

z7&3

z7

z7

o

o

o

-1

-2

-1

-z7

A IB AB 8AB14GIl8AC fus

A BA

ind

0

0

o

0

0

*2

020000

o

**

+12

0

0

o

0

0

*2

080000

7 -2

-,

0

-1

0 -Y9

x73

**

+3

0

0

o

0

o **

+9

0

0

o

0

++

8 -1

o

o **

+3

0

0

o

*

+

0

0

000

++3

9

0

y7

_1

-1

-1

z7*3

-z7

-z7

0

9b7

b7+3

-,

o

b7

o

o

+3 657

17

0

9y7+9

_1

y7+2&3

o

Y7+1

o

o

Order

+3

o

17

= derived

o **

o o

+00

0

6

2

0

0

-1

+00

7

-1

*

+

0

0

0

0

0

*2

02

0

0

0

0

0

*2

060000

-, 0

0

+00

8

0

-1

0002

3

-1

0

+

0

0

0

*

0

Ree group R(2)t =: 2 F4 (2)1

= 17,971,200 = 2 11 .3 3 .52 .13

Mult

=1

Out = 2

Constructions Ree, Tits

2F4 (2) =: G.2 : the centralizer in F4(2) of an outer (graph) automorphism of order 2. This leads to a description of G.2 as the automorphism group of a generalized octagon of order (2,4), which has 1755 "vertices" and 2925 "edges". Each vertex is incident with 5 edges, and each edge with 3 vertices. Tits established the existence of the simple subgroup G of index 2 (2 F4 (2 n ) is simple for n> 1).

Rudvalis

2F4 (2) =: G.2 : the point stabilizer in the permutation representation of the Rudvalis group (q.v.) on 4060 points, which form a rank 3 graph of valence 1755.

Max imal subgroups Order

Index

11232

G.2

Character

1600

L (3):2 3

13:12

1a+351a+624ab

',232

1600

'0240

1755

L (3):2 3 2.(2 8 ):5:4

7800

2304

L 2 (25)

2925

22.[2 8 ):S3

6144

[74]

Specifications

Structure

1440

124BO

1440

12480

1200

14976

A '22 6 A6 '2 2 5 2 :4A

4

Abstract

Ree, Tits

Rudvalis

N(2A)

vertex

adj acent point

1a+351a+624ab 2. [2 9 ): 5: 4

1a+78a+351a+650a+675a

L2 (25)·2 22. [2 9 ):S3

1a+27ab+ 351 a+624ab+650a

52:4S4

1a+351 a+624ab+650a+675a

non-adjacent point N(2A 2)

edge

N(28,3A,3A,4C,5A)

8 2 (2)

N(28,3A,3A,4C,5A)

8 2 (2)

N(5A 2 )

6 3 4/2 BB DA AB BB DB BA 6C '8D24AB

+00+00

0

0 -r2

•

+

0

0

0

0**2

+4

D

0

0

-1

o

-1

+00+00

o =* o •

++

0

0

0

+

0

0

0

o

+3

0

0

0

+00+00

2

-1

0

**2

0

020000

*2

ind

@

+00+00

0

z7

o

IlB

9D 12A 21HI fus

0

7

-1

IB

2r2

++3

z7 -y'63*2

712

0

o

-z7

A

4 BA BA

@ @

@

0

-y9

-y9

6B

3 A

@

++

++

-2z7

3B

@

+00

o

02 657

Tits group

-1

@

'68 B A BA

BA

++

+

+

4

o

•

+

0

0

0

b7

:*

+00

0

0

0

o :.

++

0

0

0

2P4(2), and All ~822

2 12

@

@

@

@

@

@

10 1 17971200 240 536 108 192 128

A

p power

pI part 1nd lA

A

A

B A

A A

A A A 2A 2B 3A 4A

4B

@

@

64

50 12 A AB A A AB 4C 5A 6A

32

B

B

@

@

@

@

32

16

B

C

@

16 C A A A B** 8C 8D

A

8A

@

@

10 12 AA AA AA AA lOA 12A

@

@

@

@

@

@

@

@

13 13 16 16 16 16 A A A B A B A A A A A A B* 13A B* 16A Bn C*5D**5 fus iod 12 AA AA

x,

+

x,

o

26

-6

2

_1

-2

-2

2

-1

o

o

0

0

-1

o

0

x,

o

26

-6

2 -1

-2

-2

2

1·-1

o

o

0

0

-1

o

0 -12

i 2

X.

o

27

-5

3

0

3 -1

-1

2

0

-1

-1

000

-1

i

-1

i

x,

o

27

-5

3

0

3 -1

-1

2

0 -21-1

21-1

-1

-1

000

i

-1

i

-1

x,

+

78

14

-2

-3

2 -2

2

3

2

2

0

0

-1

-1

-1

0

0

0

0

x,

+

300 -20

-4

4

0-1

o

o

0

0

0

-1

-1

o

0

0

x,

+

325

-1

-1

o

X.

+

351

31

15

0

3-1

x"

o

351

-1

-9

0

x"

o

351

-1

-9

x"

+

624 -16

x"

+

x••

640 A A

4D

@

@

@

@

@

@

@

@

@

@

@

640 96 64 16 8 6 6 8 8 10 10 A B A B A AF AF A B AD AE A A A A A AF AF A A AD AE EH lIF 40 8E 8F 12C Du 16E Fu 2aA Bn X.

++

3 -4

4

5 -11

21-1 -21-1

o

5

o

0

12 -12 -12

i i+ 1-1+ 1

Xs

x,

0

0 13 -13

0

0

0

0

x,

-1

-1

o

0

Xg

-1

-1

X9

o

0

-1

-1

-1

-1

++

5

5

9

9 -3

0-1

o

o

0

0 -1

624 -16

16

3

0

0

0 -1

o

o

0

0

+

650

10

10

2

6

2 -2

0-2

2

2

0

0

0

x"

+

675

35

3

0

3

3

3

0

0

-1

-1

-1

-1

x"

+

702

30

6

0 -6

2 -2

2

0

o

o

0

x"

+

702

30

6

0 -6

2 -2

2

0

o

o

0

x"

+ 1300

20 -12

4

0 -4

0

0

0

o

02-2000000000

x"

+

1300

20 -12

4

0 -4

0

0

0

o

o -2

2

x"

+

1728 -64

0

0

0

3

0

o

o

0

0

x"

+204800-4000-20

o

o

0

0

Xn

+

o

o

0

0

0

-1

0

0

-2

0

o

0

0

o

o

3

0

-1

00

16

0

-1

0

o

_4

-3

-1

-1

-1

++

-1

o

0

0

0

i

-1

i

-1

00

-1

o

0

0

0

-1

i

-1

i

00

r3 -r3

0

0

0

0

0

0

r3

0

0

0

0

0

0

0

0

0

0

0

0

0

0

o

0

0

_1

-1

0

0

0

0

0

0

0

0

0

0

0

0 -r2 -r2

r2

+

41+5 -41+5

0

-1

-3

-1

o

0

-3

-3

-1

o

0

-1

i

-1

i

XI0

o

0

i

-1

i

-1

Xu

0

0

0

0

0

0

X12

-41+5

41+5

-3

-3

-1

o

o

0

0

0

0

x" ++

10

++ +

r2 -r2 -r2 r2

X4

0

3 -1

0

-1-1+1 1+1

o

3

0

i

-1

0

2048

0

0 -1

o

0

o

2

0 -1

0

-1

2 -2

0

-1 -r3

-1

6

0

21-1

-3

6

o

21-1 -21-1

41+1 -41+1

++

3 -1

0

00

41+1

3

-21-1

x, 00 -41+1

o

-1

aoooooaaaOQX2

i 2 -12

3

-1

o

12

10

6

2

5

5 -3

5

o

o

0

0

-2

0

0

0

0

0

0

0

0

0

X14

o

0

-1

-1

o

0

XIS

0

0

0

0

0

0

Xl6

x"

r2 +

o

0

0

0

0

0

0

0

0

o

0

_1

-1

o

0

0

0

00

161

0

0

O-b13

*

0

0

0

0

+

o

0

0

0

*-b13

0

0

0

0

o

0

0

0

0

0

0

0

0

0

0

XI8

x" -161

0

0

0

0

0

0

0

0

i

-1

Xw

OOOOOOOOOOOX21

Xn

Alternating group All

Order = 19,958,400 = 27 .34 .52 .7.11

Mult

=2

Out

=2

Constructions

Alternating

Presentations

S11

= G.2

A11

=G

: all permutations on 11 letters;

: all even permutations; 2.G and 2.G.2 : the Schur double covers

G ~ <x1, ••• x9Ixr=~XiXj)2=1>; G.2 _

Remark: 2.G is the involution centralizer in the Lyons group.

Hax imal SUbgroups

Specifications Structure

0.2

Character

11

A10

SlO

1a+10a

362880

55

S9

S9 x 2

1a+108+44a

C(2C)

duad

120960

165

(A8 x3):2

S8 x S3

1a+10a+44a+-110a

N(3A)

triad tetrad

Order

Index

1814400

Al ternating

Abstract

point

60480

330

(~xA4):2

S7 x S4

1a+108+44a+110a+165a

N(2A 2 )

43200

462

(A6 xA S):2

S6 x S5

1a+108+44a+110a+132a+165a

N(2A,3A,5A)

pentad

7920

2520

M11

11:10

1a+132a+462a+825a+1100a

N(2B,3C,4B,5B,8A,11AB)

S(4,5,11)

7920

2520

M11

1a+132a+462a+8258+1100a

N(2B,3C,4B,5B,8A,11AB)

S(4,5,11)

~E}I

G8 31

18

27

[75]

~X

0

0

0

0

0

0

0

11 L.l 0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

o

o

+

++

0000000000000000+

0

000000000000000

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

+

0

0

0

0

0

0

0

0

0

0

0

0

0

++

0

0

0

00000

0

00

0

0

0

0

000000000++

00

+

++

000000000

0

0

9>XOOOOOOOOOO HX

0

0

0

,ox OOO

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

o

0

0

0

!OJ

0000000000000

0

0

0

0

0

0

0

9£

QV

OL

8

,0

9 0

9

ZL

9

2L

1-

9

OL

Oe- 0

o

0

2-

2'-

2L

o 0

0

o

e-

0

2-

0 9

,

0 0

,0 0

2-

, IIL- 02

!O£-!O0

9£- OL

Oc

9-

c

LCQ-O

o

o L-

0

0

0

0

0

o

L-

0 0

0

0

0

0

o

l-

c-

0

c- 0

9J- 0

o

0 0

0 0

0 0

0 0

0 0

0

cOO

cOO

0

0

0

0

0

0

L-

l-

o

o

0

0

0

0

0

0

9

9

811

811

0

0

0

0

118Sl

118SL

+

+

+

9J

+

0

IIS!OL

0

IISSL

0

0

o

0

L-

cL- 1Ic- 0

L-

cl- lie- 0

0

0

0

0

0

0 0

0

0

11 0

o

cOO

0

cOO

o

+

0

0

+

0

0

000000 0

o 0

c-

OIlIlL

0

0

0

8tr 0

0

cL

Oc

0

0 c-

0

0

0

0

0

0

0

1-

0

L-

cOO

L-

1-

o

c

0

s-

8-

cL

911

cL L-O

9S- 0

9S- 0

0

0

0

0

0

0

0

0

I1hL

9c!O

0%

9L9

9L9

cL9

+

+

+

+

eOOOOOO

0

8

Str 0

11900088

1190008S+

+

0

0

c£Cl

0

9-

c£cL

O'

0

0

0

0

c-

0

0

0

0

c-

0

S-

0

o

0

9

0

S-

OL-

0

0

9!O

c

0

OL- 0

0

9!O

c

0

0

0

tr

0

o

0

cL

tr

o

0

0

0

0

L-

0

0

£-

0

o

0

0

0

o 0

0

0

0

L-

0

0

0

0

o o

0

e

+

0

0

OOeL

0

0

0

0

0

OeL-O

0

0

0

c-

0

c

ZL

0

Oe-OOO

o

0

0

0

11-

0

0 0

0

9

c-OOOOOO

0

0

o o

0

0

0

0

0

Oc- 0

0

0

0

0

0

0

9L- 0

cl- 0

0

0

L-

c-

0

0

0

0

0

0

L-

0

0

0

o 0

0

0

0 0

0

,L-

0

0

0

0

o

t-

11-

0

0

0

L-

0

0

0

g-

9 £

£9- c-

9 £

c

P01

+

SZ

OL- 05

el- LZ

"

cl

9L- 0

IIL- OL£Z

L

,

o

0

0

0

,-

o

o

0

0

9 £

0

£-

0

9

S-

,

Z-

1I-000Z-

o

9

0

0

0

0

0

Ol 5

S-

0

5

1I-000Z9

9

OL 5

o

8

Zl

o

OeOOOOO

9L

ZL

GC

".

GC

ne

9

hS- S

11 L L

9!O

0

tr

9

9

L-

£-

0

0

6

Ll

0

o

LL

0

2'L

01

112'

0

0

1Ic.

o

11-

e

+

o

c

+

L-

95- 01l5L 0

£-

tr COO

0

tr05-Z09 t-

0

+

c-

Oe£ L

0

0

c£cL

0

11-

o

n

+

0

0

Oc- OOL l

0

0

0\-

6

0

0

0

Z

9-

9-

5-

+

c

e-

5-

+

8

Z-

066

ZOO

o

066

L0

h£

0 l-

0

11£

0 0

0

11-

9

9

12'

L-

Z-

L-

cl1- 9

0 0

0

0

£-

+

0 0

0

0

5-

LL-

0

e-

o

+

0

0

L-

0

£-

£-

95- lIe6

0 L-

Z-

0

0

o

£-

5L- 5ZS

o

o

0

0

0

0

L-

0

0

L-

0

0

0

ZOO

e

0

0

0

611

£69

+

£-

LZ

L-

c

5

0

L-

£-

£-

z-

,

6

6

LZ

ZII

£9

9-

5

e

L-

e

6

6e

IIL- OLZ

!O£

ZZ

l£Z

o££

+

+

+

,,,

Ol- 11-

£

Z-

9L

9-

oz cB

o

0

Z-

0

11-

0

2'

0

l-

V V

V V OOS 96 L

96

'9L

9

9

OZ

LZ

62'

9£

LZ

LZ

L

V V

e

S-

9

9

9

9L

£- B

9

V V

92'

9

'1L pUl ~Jlld ,d JaMod d

Ol

hll

Sh

Oll

02'l

IIL- 9Zl

nL- 9ZL

+

+

+

0

+

vE

09L 001l9566L

+

0

Z

V V

5

09

L

L

V V

59!.

2-

n

2

9

5

9

0

5l- 6 0

0 0

+

L-

Sir 9-

+

o

£-

Sir 9Z

o 0

0

0

e-

2

+

0

069-91165

Z-

0

0

0

9-

0

0

11-

0

o

Z

0

h-O

11-

055

+

+

0

Z-

e

<:911

L-

o

c

05

!OS£

0

Z-

0

o

0

0

OL

lZ

nL

0

Z

o L-

0

o

Ztr c-

o

0

0

0

,-

s-

611

2-

£-

0

Z-

Z

V>

9

0

V

all

0

o

OL

0

,,-

o

5

£-

,

£- 0

e

5

V

:lll

5

099 L-

9-

0

£

Z

6

OL

L-

V

'15

L-

9\ 0

9-

Z-

S-

0

".

a9 V9 as

V V

V'I a'l '1'1 'I 9S2' 9L5 Se

v "" " " av a ", "0 ", • • • • • • • • • • •'" •" •

0

9£- to

o

0

tr

9-

0

0

0

z

0

0

0

c

0

0

Z-

l.

0

0

0

Z

6

c- e

1.-

L-

Z

o

1-

L-

£-

0

l-

0

\-

Z-

0

0

,-

0

e-

0

0

il

116!O

o

0

0

0

0

o

B

9

o o

L-

0

0

o

0

e-

6

0

0

0

0

Z

L-

0

L-

L-

'19 V

B

Vi. 39 09 :>9 v 8:) 8aya yaJas'Ia h9 9l 9£ Cl. B

£- 0

Z-ccc-Z8-0

L-

0

0

o

L-

0

"

0

0

0

L-

0

0

0

0

o

o

o

t-

L-

Z

L-

L-

0

Z-

ZOO

L-

0

0

1-

L-

0

0

0

0

0

V

,

a

o

.-

00000e-

0

0

LLq

VV

6

vv ,

VOL '16

2-

o

0

U

l-

VLL

Z-

0

L-

0

LLQ U

o

V

0

V

..a

V LL

0

91.

0

0

o

92'

0

o

L-

o L-

t-

0

0

0

Z

eL-

0

9L

0

0

0 0

L-

,0

0

0

0

0

o 0

l-

0

0

o o

Z-

e

02'

L-

O£ Sl

Z-

0 Z-

0

0

L-

0

V>L

V LL

"

••• ••• ••••

v
VV

0

o

0

0

o£ !OL

0

,-

o

0

o

Z

0

0

0

o

0

0

c c

000£-00000

o 0

o

0

Otl On

OccOOOO

0

OOc-c-OOOO

o

ell

L-

GtI

l2'

L-

PUl snJ LZ

++

0

89L ++ 9L- 82

o

o 0

0

Z-

L-

0

Z-

0

0

L-

0

o

0

o

o

o

o

0

o

++

0

++

6LL

o

0

o L-

0

o

av

'IV

8'1 511

'10e a5L

Oe

'1'1

VV LZ

'1'1

ne LZ

PUl snJ .a '1'1 '1'1

.-

OOoe-Z-O OL

L-

++

5069++

.-

IIl- 2'11

o

9n

9-

9LZ ++

e9L ++

09L ++

LL- 1.IIL ++ 0

CL

I1S

".

IIL- Oe- S

0

0

0

2

0

0

6-

2-

o

0

c-

o

6SL ++ 0

0

6 9-

0

£- 0

0

06L

8L- 291 ++

0

OL- OL

9-

0

02'

LZ

SL

++

9£

0

0"- 01- 2'

5L 0

ilL

SOL ++

2-

Z-

00S-09911++

0 +

o 0

il

0

0

il

o

o 0

0

0

++

0

0

9S

0

cL

++

8-SII++

il

++

Le

:lZ V

++

tr020 0

S

ac

0

3Z

V

S2'

90

ilL

IlL

il

'I V 'I 'I OSO OZ6 OS8 099 Ol l Z c9£

V

++

Z

Z

311 J:)

ell

0

'OL

c

0

IIZ

S

z-

e e-

!OL

IIL- OL

OL

,

,

0

0

,-

0

aJ

IH

,

LL 0

o

2'

0

!O-

0

0

2-

!O-

9-

2

112'

all V

0

.111

OL

6

'I

96

'I 'I

9

~

z£

a

c

::IV JV

o

'" aa aa

!O

o

0

£-

£-

,

0

,11-

L-

,£-

0

0

5

0

£-

o

0

£- 0

0

0

0000000000000000+

1.-

Oc.

0

2

CL OL

c0

tl2

2'-

2-

zoo

2'-

0

0

2'-

0

o

tl2

0

0

0

0

0

0

0

0

tl2

o

0

0

0

0

ilL

0

0

0

0

0

9L 8L

0

0

0

0

0

011

c-

o L-

0

c £-

0

£-

0

L-

0

0

0

0

0

0

0

L-

0

£-

0

0

2-

0

£-

0

0

2-

0 \-

0

o

0 0

0

0

o

0 0 11-

£-

c:

o 0

£-

o

L-

0

0

0

£

2'

0

c-

0

0

0

0

o

e

0

L-

L-

Z-

,-

Z-

"

L-

o

2-

0

0

0 1-

0

0

o 0

0

0

o

,

e

o

0

o

0

Z

,-

c-

o \-

0

0

0

0

ne n2'

0

0

o

0

0

0

0

0

0

0

Z-

0

e-

0

o 0

0

0

z0

0

L-

0

Z-

0

f9

, 0

)19

Z9L 09£ 09L

19 HO av :l:)

£0

19

3a

,

o c-

,

as

OJ

L-

o

'I

"

c

o

e-

2'

£

0

L-

2

c

e

e

£

e

L-

L-

aOL :lOL 'OL

av ov

o

L-

o

L-

0

0

0

0

!OL l O O

SS SS

o 0

0

0

L-

o

0

0

0

o

0

0

0

0

0

L-

0 0

1-

o

3V

"L a" ov aa "L av '"

o

!O L.l 0

0

tOXOOO£.lcOO

..x ""X 6C){

«x

0

000000000

0

9CXOOOOO ~CX

0

..:X

o£ o£

0

L-

0

0

L-

L-

0

L-

0

0 L-

o 0

VSC YIIZ aoc V'lL 0'1 av a'l ov 0'1 a'l av HV hL Cl Oc 6

0

o


0

,,'x \-

"X

0

0

0

L-

0

0

0

L-

0

0

L-

ntX

0

0


~'X

0

o

0

L-

91X

LlX

0

o

0

L-

0

0

SIXOOOO OIX L-

0

0

0

ox

"X

<x

'X

Q

ao£ '10£ DJa .lav D:la .lay Sl SL

av ov a 3a as " ov a, , " " , '" '" " "L 0" " • • • • • • • • •" • • • • • • • • • • • • • •

'·X

"'X

"X

'X

5z(32) Suzuki group 3z(32)

= 2 82 (32)

Order = 32,537,600 = 2 10 .5 2 .31.41

=

Mul t

Out = 5

1

Constructions Suzuki

G : the centralizer in 3 4 (2) of an outer (graph) automorphism of order 2 G : all 11 x 4 matrices over F 32 preserving the set of vectors {t,x,y,z} for which xy+(x s + 2 +ys+z)t=O, where s : x ~ x8 is the automorphism of F 32 with $2:2 ; projectively this defines an oval of 32 2 +1=1025 points on which G acts doubly

transiti vel y.

Specifications

Maximal subgroups

Order

Index

Structure

G.5

Character

Abstract

Suzuki

317114

1025

2 5 +5 :31

2 5+5 :31:5

1a+1024a

N(2A5)

point

164

198400

41:4

111:20

N(41A_J)

100

325376

25: 11

25: 20

N(SA)

62

524800

062

31: 10

N<31A-O}

L,(32), point pai,r

8[~~}5 35

@

@

@

@

@

@

@

@

@

@

@

@

@

@

@

@

@

@

@

@

5

@

@

@

@

@

@

@

@

@

@

@

@

@

@

@

@

@

@

@

@

1

32537600 024 64 64 25 25 25 25 25 25 31 31 31 31 31 31 31 31 31 31 31 31 31 31 31 41 41 41 41 41 41 41 41 41 41 p power A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A pT part A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A ind lA 2A 4A Bn 5A 25A B*6C*11D*16E*21 31A B.2 C.4 D*8E*16 F*5G*10H*20 I*9J*18K*25L*19 M*7N*140*28 41A B*2 C*4 D*8E*16 F*3 G*6H*12h24 "7 fus X.

+

x,

o

124

-4

x,

o

124

_4 -4i

X.

+

775

7

x,

+

775

x,

+

775

ioo

4 4 5 AA AB A A AA AB 58 lOA 20AB*11 25F 20

4

A A

BA BA

X.

+0000

41 -41

-1

-1

-1

-1

-1

-1

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

00000

-1

4i

-1

_1

-1

-1

-1

_1

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

00000

-1

_1

-1

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0 -w41

*2

*4

*8 *16

*3

*6 *12 *24

7

-1

-1

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0 *16-w41

*2

*4

*8

*7

7

-1

_1

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

*8 *16_w41

*2

*4 *24

x,

+

775

7

-1

-1

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

*4

*8 *16-w41

x,

+

775

7

-1

-1

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

*2

*4

x,

+

775

7

-1

-1

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

*3

*6 *12 *24

x"

+

775

7

-1

_1

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

*7

Xn

+

775

7

-1

-1

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0 *24

x"

+

775

7

_1

-1

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

Xn

+

775

7

-1

_1

o

0

0

0

0

0

x..

+

1024

0

0

0

-1

-1

-1

_1

-1

-1

x"

+

1025

o

0

0

0

0

0 y31

*2

*4

*8 *16

x..

+

1025

o

0

0

0

0

0 *16 y31

*2

*4

*8 *18

x" x..

+

1025

o

0

0

0

0

0

*8 *16 y31

*2

*4

+

1025

o

0

0

0

0

0

*4

*8 *16 y31

x..

+

1025

o

0

0

0

0

0

*2

*4

x"

+

1025

o

0

0

0

0

0 *25 *19

x"

+

1025

o

0

0

0

0

0 *28 *25 *19

0

0

0

0

0

0

*5 *10 *20

0

*9 *18

*5 *10

*4

*8 *16

*7 *14 *16 y31

*2

*4

0

0

0

0

0 *14 *28 *25 *19

X~

+

1025

o

0

0

0

0

0

X~

+

1025

o

0

0

0

0

0 *19

X~

+

1025

o

0

0

0

0

0

x"

+

1025

o

0

0

0

0

0 *18

X~

+

1025

o

0

0

0

0

0

X~

+

1025

o

0

0

0

0

0 *20

X~

+

1025

o

0

0

0

0

0 *10 *20

X~

+

1271

-9

-1

-1

x"

+

1271

-9

-1

-1

Xn

+

1271

-9

-1

-1

*21-w25

x"

+

1271

-9

-1

Xu

+

1271

-9

x"

+

1271

-9

*7

*7 *14 *28 *25 *19 *7 *14 *28 *25

*5 *10 *20

*8 *16 y31

*5 *10 *20

*2

*4

*8 *16 y31

*2

*4

*9 *18 *25 *19

*9 *18

*9 *18

*5 *10 *20

*9 *18

*2 *20

*9 *18

*5 *10

Xl

-1

X3

+00000X4

*3

*6 *12 *24

x,

*7

*3

*6 *12

X.

*7

*3

*6

x,

*6 *12 *24

*7

*3

X.

+OOOOOX"

*7-w41

*2

*4

*8 *16

*3

*6 *12 *24 *16_w41

*2

*4

*8

Xl(>

*7

*3

*6 *12

*8 *16-w41

*2

*4

x..

*12 *24

*7

*3

*4

*8 *16-w41

*2

x" XLJ

*6

*6-. *12 *24

*7

*3

*2

*4

*8 *16-w4l

-1

_1

-1

-1

-1

_1

-1

-1

-1

_1

+0000

4

0

0

0

_1

Xl4

o

0

0

0

0

0

0

0

0

0

+

0

0

0

0

0

Xl~

o

0

0

0

0

0

0

0

0

0

x"

o

0

0

0

0

0

0

0

0

0

x"

000

0

0

0

0

0

0

0

Xl;

0000000

0

0

0

x..

000

0

0

+OOOOOXzo

0

0

x"

0000000000

X~

o

X~

o

0

0

0

o

0

0

000

0

0

0

0

0

000

0

000

000000000

0

X~

*8 *16

000

0

0

0

0

0

0

+OOOOOX25

*2

*4

*8

o

0

0

0

000

0

x"

*8 *16 y31

*2

*4

0000000

0

X~

*4

*8 *16 y31

*2

o

*2

*4

*2

*4

*7 *14 *16 y31

*7 *14 *28 *25

*9

*5 *10 *20

*7 *14 *28 y31

*7

*7

*7 *14 *28 *25

*5 *10 *20

*8 *18

*7 *14 *28 *25 *19

*5 *19

*7 *14

*7 *14 *28 *25 *19

*5 *19

*4

0

*7 *14 *28

*9 *18

*9 *28 *25 *19

*5 *10

0

*8 *16 Y31 *10 *20

*5 *10 *20 *14 *28 *25 *19

*9 *18

0

*5 *10 *20 *14 *28 *25 *19

*2

o

0

*9 *28 *25 *19

*7 *14 *28 y31

1025

0

*9 *18 *25 *19

*5 *10 *20

*9 *18

*2 *20

0

*9 *18

+

*9 *18

0

*8 *16 y31 *10 *20

Xn

~4

0

-i

_1

*7

*2 *12 *24

*8 *16-w41

-i

*5

*8 *16 y31

0

0

0

0

0

000

0

0

000

X~

00000

0

0

000

X~

0

_1

-1

o

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0000000000

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

o

000

+OOOOOX31

*6 *11 *16

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0000000000

x"

-1

*16 *21-w25: *6 *11

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

000

0

x"

-1

-1

*11 *16 *21-w25

*6

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0000000000

Xu

-1

-1

.6 *11 .16 *21-w25

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0000000000

x"

l-w25

*6 *11 *16 *21

0

0

0

0

0

0

0

0

0

0

0

0

+0000

X~

[77]

11

91

91

91

+

02

04

19

19

19

19

-1

30

04 819

+ 910

+2 910 -10

02 910 -10

02 910 -10

9

9

19

02 819

9

19

819

+

0

0

0

-1

-1

-1

-1

-1

o

o

o

10

201-10

""

2

""

3

o

-10

10

-1

0

_1

-1

-1

-1

-1

-1

-1

-1

-b5

-b5

-b5

-2

-b5

-b5

101-9

-2

3

9

+729900

-1

-1

19

0

8i

-1

-8

016 728

0

-1

-1

-8

8

08 728

-1

0

_1

0

8

8

8

04 728

-1

-1

0

8

0

-8

04 728

-1

-1

0

0

o

o

0

o

0

o

= 22

2

o

o

o

1012-10

101

o 10r2+10

2

-21

-2

i

o 10z8-91

10

21-1

101-9

o

-1

-1

o

9

o

o

o

-212

o

o

o

o

o

2

o

-1

-1

21-1

-1

-1

-1

-1

o

b5

-1

-i

-1

_1

o

-1

o

o

o

o

o o

o

o

o

o

o

o

o

o z8**

-1

o

_1

-m20

o -1

b5

2z8

-b5

-21

-b5

-b5

o

o

o

21

o

-2

2

o

o

o

_1

o

20AD

AA

BA

80/4

@

o

o

o

o

o

x13

)(13

o

o o

-i

Z8113

o

o 1-1

3

-1

o

2

o

o

13AD

16AD

A A

A

80/4

A

@

91/4

@

-1

12AB

A,B AA

72/2

@

-i

2

-b5

-2

12_1

_1

-2

-21"2

-2

1"2+1

_1

o

o

o

-8z8

o

-b5

o

o

o

81

o

-b5

o

o

-8

o

o

o

8'8

o

o

o

8

o

-b5

o

o

o

8

o

o

o

o

o

o

b7

o

o

o

o

o

o

12-1

o

o

tQAB

AA

BA

80/2

@

1"2+1

o

21-1

-1

_1

_1

8IJ

8GH

2

C A

2

64/2

C A

@

64/2

@

14

2

A 8EF

A

64/2

@

12

o

3

o

0

o

0

o

b7

10z8-i

o

o

101-1

8i

8

+2 728

-1

0-8

024 640

8

0-8

04 6110

10

101-1

o

BAD

A

A

5760/4

@

2

-1

-9

10

A A 7AB

91/2

@

-9

2

AA 6A

72 AA

@

12

o

3

11

o

A SAS

80/2 A

@

90

10

10

0-8

+2 728

Out

~EJEJ[~~}o

-1

A 4C

64 A

@

2

0

A 4AB

A

5760/2

@

10

02 640

-9

11

10

go

+

•

9

36

3A

A

2A

A

A A

p power pi part ind lA

81

@

@

A A

@

= 1

PGL 3 (9) =- SL (9) :; PSL (9) ;; G; rL (9) :; (8 x 0).2 2 ; PrL (9) ;; IL (9) :: PIL (9) =- G.2 3 3 3 3 3 2 3

Abbreviated character table.

@

Mult

GL 3 (9) ;; 8 x G : all non-singular 3 x 3 matrices over F g ;

4 5 5 2456960 760 832

Linear

Constructions

Order = 42,456,960 = 27 .3 6 .5.7.13

Linear group L (9) ;; A2 (9) 3

'::::i 2S

12-1

i

1"2.1

'B

-1

o

020

o

-b5

b5

..

9

40 68

A BB

A BB 8K

B C A

@

0

0

0

.12

** ++2

++2

+2

11

8

8

0

0

8

-8

0

0

_1

-1

0

0

0

0

0

0

0*+ 0*+

o o

o o

++2

o

o o

++

+2

10

0

0

9

0

0

0

0

0

o

0

0

""o

0

0

0

0

0

0

000

10 -10

10

0

**

o

9

9

0

0

..

++

+8

011+

o

o 11*

0**.40000

0**.20000

0**.20000

o

o

x91**2

x13

b7*.OOQO

0**.20000

0

9

@

@

A

G

18 8/2

AD AD AD AD

18

@

-,

0

0

0

0

o

0

0

0

-1

0

_1

0

o

0

0

000

o

o

o

0

0

r2

o

0

0

o

0

0 -1

0

0

0

-1

-1

0

0

0

0

0 000

0

000

-1

0

0

0

0

o

o

o

-b5

-b5

o

o

o

0

0

-1

000

0

o

0

o

-3

3

o

o

10CD 12C 12D16EF

AB

BB

10/2

@ @

.. ++

++

*3

02

004 *3 012

:

@

0

16

0

0

0

13

12

0

0

0

0

0

-3

4

0

-2

0

0

0

3

0

0

0

0

0

@

@

o

-1

o

0

o

0

0

0

0

0

0

0

0

0

0

000

0

*3

++

++

08

04

02

02

+

*3

++

02

o

: 002

0*+

0*+

o

o

0*+

-1

*3

o o

*3

*3

11

o

o

-b5

26

0

0

26

0

0

39

27

0

0

0

0

0

-2

0

0

2

0

0

-1

3

0

0

0

0

0

-1

0

0

-1

0

0

3

0

0

0

0

0

0

_1

0

0

_1

0

0

0

0

0

0

0

0

0

0

0

0

2

0

0

-1

-1

0

0

0

0

0

@ @

0

0

0

0

0

0

0

0

0

0

1 -12

0

0

_1

0

0

-1

0-1

0

0

0

0

0

7

6

9

@

_1

-2

4F

-2

-3

6E

@

@

@

*

+

**3 012

+2

*

0

0

0

0

32

0

0

0

-11

0

0

0

-1

000

0

0

0

0

002

++

21

21

27

0

0

0

5

3

0

0

0

3

3

0

0

0

0

o

o

o

o

*

*

+

002

+

++

0

28

0

14

0

-4

0

o

_111

o

2

0

0

0

2

0

o 5-1

0

0

o

3 -1

0 0

o o

-1

000

-2

0

o

0

()

o o

0

0

o

o

0 -21-3

0

0

0

0

0

0*5020000

o

o

++

08

04 **3

o **3 o

02

o **3

0*5020000

0*+

o

o

xl 3

002

:

0

0

0

0

o

-1

o

o

0

0

0

0

0

0

0

0

0

0

o 0

0

-i 0

o

0-1

-1

o -1

0

o

0

0

o

0

o

o

000

o

o

000

000 000-b70

0

0

_1

0-1

o

o 2

-1

o

21-1

3 -1

2

C

16

@

@

\0 '-'"

~

o

_1

o

-1

o

1

o

o

o

o

o

o

o

o

o

o

o

-1-1

o

12 7/2 8/2 12/2 AN AF AD E A A AF AO A AM 8MN 80 12Fi4AB16lJ 211EF A

96/2

@

o

0

6F

A AD BD A AD BD

00273-2

++

++

••

@

96 108

@

0*5020000

o

o

-1

2D

A A

6 048

L 1 (729)

N(7AB). N03ABCD)

(D 14 xD

6 8/2 13/4 9 8 AAACBCC AE I DC A A AC BC A AE A AC 2C 4E 6C 60 8L 12E16GH 26AD fus ind

@

base

H.2 2 26 ): 3

0 (9) 3

L (J) 3

C(2C)

N (2A 2 )

"3(3)

line

point

Linear

C(2D)

N(3A 4)

N(3A 4)

Abstract

C(2B)

1a+90a

1a+90a

Character

Specifications

A6·22 x 2

L (3):2 x 2 3

U (3):2 x 2 3

rL 2 (9).2

32 +4 : 82 .22

G.2 2

b5*+0000000

o

o

0*+

o

0*+

-1

o

20EF fus ind

AD

BD

10/2

2

(7 xD26):3

H.2

A6· 22

L (J),2 3

x

rL 2 (9 ) U (3) 3

@ @ @ 5 616 48 54

(13x0 14):3

91: 6

91:3

H.2

3

82 : D12

A6 ·2 2

x2

82 :3

x2

L (3) 3

U (3):2 3

PGL (9) 2

L (3),2 3

"3(J),2

GL (9 ).2 2

3 4:rL 2 (9 )

32 +11: 82. 2

3 4 :rL 2 (9)

32 +4: 8 2 .2

3

G.2

G.2 2

G.2 1

PGL 2 (9)

L3 (3)

U (3) 3

34 : GL 2 (9)

34:GL2(9)

Structure

++0000

o

28**

_1

80 '

..

011+

o

_1

28

A

A

720 120

@

155520

273

@

110555

384

@

58958

720

91AX fus ind

AA

CB

91/24

@

7560

5616

o

o

-1

-1

y

-m20 y'40**3

b5

-2z8

-b5

-b5

z8 y' 40**3

-i

-z8

-1

-1

o o

o

o o

o

o

o o

z8*3

-i

-1

o

BOAP

80/16 HO AA

@

-i

-1

o

40AH

AA

80/8 DD

@

z8-1

1-1

o

AA 24AD

AC

72/4

@

91 7020

6048

91

466550

466550

Index

Order

Maximal subgroups

r UJ

A

A

-8

10z5

2

o

0

0

9

9

0

9

10

10

10

+ 729

+ 730

+2 730

04 730

04 730 -10

10

04 657

6

o

10z5

10z5

o

-2

o

o

016 730 -10

024 800

~

r2

o

10

0 -10

-1

1,

-1

0

o

o

-1

o

r2

o

-2

o

1

o

o

10

-2

o o

o

o

-10z5

o

o

o

r2

o

o

o

10z5

o

z5

z5

m16

o

o

0 0

o

-2z5

0

2z5

-2

2

z5

o

-z5

z51

-1

z5

o

z5*2

z5 -z5*2

m16

o

o

-10

10z5

10

o

r2

o

-1

o z5

o

-z5*2

-z5*2

-1

z5

o

b5

o

o

o -b5-2 o -m5+1

o

o

o

-1

m5*2+1

b5

-1

z5&2

o

90/4 CDA AAA 30AD

@

z5 z5-&2

z5

o

20AD

AA

80/4 CA

@

o

o

o

-1

-1

o

16AD

A A

80/4

@

o

o

-1

z5

b5*

b5

2z5*3 -z5-2&2

m5+2z5*3

-b5+1

b5

z5&2

z5&2

2

15AD

AA

90/4 DA

@

o

.

10

o

-2

9

o

0

10

0

9z5+8&2

2z5*2

2z5

2b5

b5*

m5*3-1

-z5 2

-b5

-1

2

100/4 FA EA 10KN

@

z5+2&2

-1

2

10GJ

b5 -z5-2&2

2

9z5+8&2

2

-1

-1

-8z5

o

b5

-1

b5

8m5-8

o

-8b5 -b5+ 1

3b5+1

o -4r5+12

o

b5+2

z5+8&2

9

9z5+8&2

-3

2b5

b5+2

-3

0

-b5

z5-8&2

2

-1

-8

9

o

100/4 CA

80/2 7200/4 100/2 A CA FA AA A EA lOAD 10EF 8AB

AA

@

@

9z5-8&2 -b5-2

08 730

Out = 4

@

@

b5

17

17

8z5+16&2

-8b5*

04 657

0

0

-8b5 -b5+1

0

-1

0

9

-8

04 584

17

-1

-b5

-8b5 -b5+1

+ 657

-8

04 584

17

-10

-1

24

+2 584

17

17

-8

+2 584

9-8

-7

z5-8&2

2

-b5

z5-8&2

73

04

0473

0

9-8

3

5EF

-7

A 5AD

73

4A

90 AA AA 6A

+

3B

A A

2

A

3A

A

2A

A

@

-8

lA

@

7200/4 100/2

@

72-8-900

+

ind

pI part

@ @ @ @ @ 4 7 7 2573600 200 290 81 80 p power A A A A

12

~EJB92 92

PGU 3 (9) " SU 3 (9) " PSU 3 (9) " G

Abbreviated character table.

Unitary

Mult = 1

GU 3 (9) :: 10 x G : all 3 x 3 matrices over FS 1 preserving a non-singular 'Hermitian form;

= 42,573,600 = 2 5 .36 .52 .73

Constructions

Order

Unitary group U (9) " 2 A2 (9) 3

-z5

o

o o

o

o

-x73

o y"80*21

o y"40*3

m16

++2

+8

+4

+2

+2

++2

++

++

+2

+2

++

*f +12

1*

**

*f

o **

o o

r2

** **

o

o

o

-z5

o -1

z5

o

y"40*3

o

r2

-2z5

o

-2

z5

z5

@

@

8

8

0

0

0

8

-8

0

0

9

0

A

4B

A

2B

-1

-1

0

0

0

6B

BB

o

o

0

0

0

8C

A

A

8

@

10

9

0

0

o

o

o

o

o

o

o

o

-1

o

o

o o o o o o o

o

o

o

-2

0

0

-1

o

o

o

o

0

0

0

o

o

o

-b5*

-b5*

o

o

o

10/2 FB EB lOOP

@

o o o

o o

o

10 -'ID

10

9

0

0

9

0**+20000 -1

o

o

+2

** ++2

+2

*f

++

00

@

73:6

10 2 :0 12

720 720 9 A A BB

3

A6:22 x 2

°10 x2 • A6· 22

3 2 +4: 80 : 2

0**+20000

o

o

o

z5

o

o

-1

o

o

-1

o

o

o

z5

z5

o

++

80AP fus ind

AA

80/16 GO

@

194400

219

@

73:3

70956

600

5 x 2·A 6 .22

32 +4: 80

A6 :22 10 2 :S

5913

730

59130

720

7200

58320

80/8 73/24 CB A AA A 40AH 73AX

@

G.2 G.4

Character

@

@

o

o

r2

o

0

0

-1

0

0

0

o

0

0

-1

0

o o o o o o

o

o

-1

o

o

o

o

o

-1

o

o

o

-31

:*

:*

* *

*

*

o

o

o o

o

*3

*3

*3

*

*

o a

o

-1

o

o

* o * o

-b5*

b5*

o

o

-1

o

18/2 8/2 10/2 AB A FB AB A EB 12AB16EF 20EF fus

@

73: 12

102 : (4xS 3)

(A6: 22 x2 )'2

(Dl0 x2 • A6· 22)·2

32 +4: 80 :4

... "1

0

3

@

4

@

0

0

0

0

0

000

0

3 -1

0

+6

+4

+2

+

+

++

+0+0

+0+0

+

+

+0+0

+

+

++

0

0

0

0

o

o

o

o

o

0

2

3

0

0

3

0

0

0

0

o

o

o

o

o

0

-2

3

o

o

o

o

o

0

-1

0

o

o

o

o

o

0

0

-1

000

000

-1

000

0

0

++0000

+

+0

+0+0

0000

+0+0

ind

B

24

@

N(73A-X)

@

o

o

o

o

o

o

o

o o

-1

o

o

o

o

o

o

o

y'24

6/2 AD AD 24AB

N(2A 2 ), N(5 2 )

C(2B)

N(2A), N(SABCD)

N(3A 2 )

BC C A A BC A 4C 80 12C 16G B

24

@

1a+729a

Abstract

Structure

Order

Index

Specifications

Maximal SUbgroups

U1 (729)

base

03(9)

non-isotropic point

isotropic point

Unitary

~

\0

~

c VJ

i,

HS Sporadic Higman-Sims group HS Order = 44.352.000 = 2 9 .3 2 .5 3 .7.11

Out = 2

Mult = 2

Constructions G;2 : the automorphism group of the graph of D.G.Higman and C.C.Sims. a rank 3 graph of valence 22 on 100 vertices.

Graph

Any given vertex has 22 neighbours (points). and each of the remaining 77 vertices is joined to 6 of these points, and may be labelled by the corresponding hex ad • Two of these 77 vertices are joined just i f the corresponding hexads are disjoint. The hexads form a Steiner system SO,6,22) and the point stabilizer is Aut(SO,6.22»

=-

M22 .2. The simple group G is the subgroup of even permutations. Higman

G : the automorphism group of G. Hi gm an , s "geometry" of 176 points P and 176 quadrics Q. The group acts doubly transitively on each set. and each object is incident with 50 of the other type. There are 1100 conics. each of which defines a polarity between the 8 points on it and the 8 quadrics containing it. A quadric and a point on it are polar with respect to just one conic. The intersection of tWJ quadrics is uniquely the union of tw::> conics,

which meet in two points. The duality interchanging points and quadrics gives G.2. Taking M22 as the stabilizer of

00

and 0 in M24 , the points are labelled by the octads P containing

quadrics by octads Q containing 0 but not 2-graph

00,

00

but not 0,

and P is incident with Q just if their intersection has size 0 or 4.

G : the automorphism group of a 2-graph on 176 points, that is, an assignment of parities to the triples of points, so that the sum of the parities of the four triples in any quadruple is even. Labelling the 176 points by octads as in the previous construction, a triple of octad s is even or odd accord iog as the sum of the sizes of their pair wise inter sections is 0 or 2 mod 4.

In the Leech lattice (see Co 1 ) G is the pointwise stabilizer of a triangle OAB of type 233, for instance 0= (0,0.0 22 ), A = (5,1,1 22 ), B = (1.5,1 22 ). The setwise stabilizer is G.2.

Leech

The 100 vertices V of the Higman-Sims graph are represented by lattice points with VO, VA, VB all of type 2, namely (4,4,0 22 ) (1), (1,1,_3,1 21 ) (22), (2,2,2 60 16 ) (77), V and V' being incident in the graph just i f type(VV') = 3. There are 2x176 points P with type(PO) = type(PA) = 2, type(PB) = 3, falling naturally into 176 pairs IP L =(2,O.27 0 15 ). PR=O,1._1 7 1 15 )} satisfying PL+PR-A = 0, which correspond to the points of Higman's geometry. The quadrics correspond to the pairs IQL=(O,2,27 0 15 ), QR=(1,3,_1 7 115)} obtained by interchanging the roles of A and B in the above definition. A point is incident with a quadric just if type(PiQj) is even. In the Leech lattice mod 2 the 3-spaces containing the fixed 2-space (of type 233) become points in the quotient 22-dimensional representation over F 2' They are described below by the types of their non-zero vectors. Presentation

G =-


b4c

d

~

e

I e=(fa)2, 1=(cbf)3=(fdc)5=(bcde)4 >

f

Remark: 2.G.2 is the involution centralizer in the Harada-Norton group.

Max imal SUbgroups

[80l

Specifications

Order

Index

Structure

443520

100

M22

252000

176

U (5):2

252000

176

U (5):2

40320

1100

L (4):2 1 3

: L (4):22 3

1a+22a+778+175a+825a

N(2A, 3A, 4B. 4C. 4C, 5C , 7A)

40320

1100

S8

: S8 x 2

1a+77a+154a+1758+693a

C(2C)

11520

3850

2 4 ,S6

: 25 • S6

N(2A 4 )

10752

4125

4 3 :L (2) 3

: 43 :(L (2) x 2) 3

N(2A3)

7920

5600

M11

7920

5600

M11

7680

5775

4'24 :S

2880

15400

1200

36960

3

G.2 : M22 :2 2 5 51+ :[2 )

1

3

Character

Abstract

Graph

Higman

point

1a+22a+77a

Leech 233-2224-po in t

1a+ 175 a

point

233-2233-point

1a+175a

quadric

233-2233-point

edge

233-2233-point

conic non-edge

233-2244-point 233-2244-point

N(2A, 3A.4C.5C, 6B, 8B,11AB)

233-2334-point

1

N(2A, 3A. 4C. 5C. 6B, 8C, 11AB)

233-2334-point

1 +6· S •• 2+ • 5

N(2A)

2 x A6 '22

: H.2

N(2B) , N(2A,3A.3A.5B)

5:4 x A5

: 5: 4 x S5

N(5B), N(2B,3A,5A)

5

point pair

HS

@

@

@

7

2

@

@

@

ij4352000 680 880 360 840 256 indl A

A A 2A 28

61t

@

soo

@

300

@

@

@

@

25362117 AABAAA

A

A

A

A

A llA

A 4B

A

A

lie

A 5A

5B

4 -6

2

2

-3

2

2

2

-3

2

o

4

_1

o

4

_1

_1

o

4

-1

-1

5

0

2

0

o

ppowerAAAA

p' part

@

3 A 3A

AABAA se 6A 68

A 7A

@

16

,B

BA

@

@

@

16

16

20

20

C A

C

AA

BB

8B

+

x,

+

22

x,

+

11

13

X.

+

154

10

10

-2

x,

+

154

10 _10

-10

-2

2

x,

+

154

10 -10

-10

-2

2

x,

+

175

15

11

4

15

-1

3

0

x.

+

231

7

-9

6

15

-1

-1

6

x,

+

693

21

9

0

21

5

x"

+

170

34 -10

5 -14

2

Xn

o

770 -111

10

5 -10

x"

o

770 -14

10

5 -la

x"

+

825

25

9

6 -15

x..

o

896

0

16

-lI

0

0

0

_li

-2000000

x"

o

896

0

16

-4

0

0

0

-11

-2000000

x"

+

1056

32

0

-6

0

0

0

6

_ll

o

2

_1

0

0

0

x"

+

1386

-6

18

0

6

-2

-2

11

6

o

0

0

0

0

0

-1

x"

+

1408

0

16

4

0

0

0

8

-7

-2

0

o

0

0

0

10

555 6-2

40

11 A

12 BA

0

0

0

o

0

-1

15 BA

BA

20 AA AA

", 2" Bn" _1

0

0

++

21

-2

-2

++

28

-2

0

0

0

0

0

0

0

0

2

-2

0

0

0

0

-1

o

0

o

0

-2

2

0

0

0

0

-1

o

0

0

-1

-1

_1

o

o

0

++

21

-2

0

_1

o

0

0

o

0

++

21 -11

0

0

0

0

0

0

++

63

15

o

++

o _1

_1

_1

-1

2

-1

o

-2

0

0

0

0

0

-5

0

0

o

0

0

0

o

0

0-1

o

-2

-2

-5

0

0

o

000

o

0

0

-1

0 -i5

i5

o

-5

0

o

0

0

o

0

bl1

**

0

o

0

**

blt

0

o

0

0

0

0

_1

0

0

0

0

0

0

o

0

0

_1

o

o

0

o

o

0

6

2

0

0

0

5

5

-3

0

5

0

x"

+

1925

5

_1 -35

-3

Xn

+

2520

24

x"

+

2750 -50 -10

X~

+

3200

1nd

1

2

-8

-1

050 0

-5

0

0

0

-2

-1

-1

o

_1

_1

2·0 -2

-1

9-20000

-1

o

-1

0

0

0

5

10

2

2

0

0

0

-1

_4

0

0

0

0

-5

0

2

0

2

3

,

4

5

5

5

6

2

6

4

10

10

10

12 12

6

6

_4

+

56

8

0

2

0

0

0

X~

o

176

16

o

5 161

0

o

6

x"

o

176

16

o

5-161

0

o

6

Xu

o

616

24

0

4 161

0

0

-9

-11

o

0

x"

o

616

24

0

4-161

0

0

-9

-11

o

Xw

o

924

4

0

6 241

0

0-1

4

_1

x"

o

924

4

0

6-241

0

0

_1

,

1000 -110

0

10

0

0

0

0

0

-1 161

0

0

_1

0

0

_1

o

_1

o

o

0

14

8

o

2

0

0

-1

-1

3'

o

o

o

o

0

0

-1

0

-31

o

o

o

o

0

0

1

0

0

0

0

0

-1

0

0

0

21

-1

0

0

0

0

0

-1

0

0

0 -21

_1

o

-2

0

21

0

0-1

o

0

0

0

-1

o

-2

0 -21

0

0

_1

o

0

0

0

0

0

0

2

o

0

0

0

0

_1

_1

o

0

7

2

2 -31

-1

o

0

0

0

-1

0

0

0

-1

-1

0

o

0

0

o

o

o

o

0

0

0

0

000

o

o

0

0

-6

0

0

0

++

69

5

5-3

3

0

0

0

_1

-2

-2

0

0

_1

o

0

0

0

0

0

0

0

0

0

0

0

2

-1

0

0

_1

-1

+000000000000000

X3

X.

0

X5

o

o

x,

o

x. o x,

-1

o

o

XIO

0

Xli

o

o

-1

X13

o

o

0

X14

o

o

x" ++

48

16

0

0 -24

++

6 11

0

0

0-2

o

-1

0

00000-220

o

0

0

0

0

2

-5

0

4

6

4

0

-2

0-5

-2

0

0

0-1

o

-1

2

2

0

-1

-1

_1

-1

-9

-3

0

0

-1

o

5

-1

2

-2

++

0

0

0

0

0

0

0

0

0

0

L+

o

o

o

o

o

o

o

o

o

o

0

-1

-1

x" 0

X17

_1

o

o

-1

X18

o

0

o

o

0

X'9

o

o

-1

o

o

o

o

-1

0

-1

++000000000000000

2

0

-2

x"

0

0

o o

_1

2

-1

0-1

_1

64

-1

-2

0

++

-1

x,

_1

0

20 fus 1nd 20

-1

-1

o

0

-1

+

0

20 20

10 15 AD Ace AD ACC E* 30A

_1

-5

0

15 30

10 AD AD 200

-2

0

20 -20

12 12

@

@

2

2

++

11 22

@

-2

3 -1

6

0

11 22

0

3

70 -10

0

20

o _1

0

0

10 10

10

Br Br

o

o

o

8

@

"A 20C

3

-1

8

7

'c'C

-1

-1

8

5 12 CO BE CD AE

0

5

7

30 BC BC

o

5

o

8E lOC 100 12B

, ,

o

21

o

24 AD AD 6E

-1

++

0

6C

36 AC AC_ 6D

2

0

-1

8D

@

0-2

o

0

32 B

@

0

-1

-1

32 B

@

@

6

-5

0

@

6

-5

_1

0

5

91

0

@

5 -3

5

70 -10 -10

0

@

@

x,

++

0

_1

0

++

0

4F

o

0

0

liE

'cAC

o

0

0

@

,

-1

o

0

@

40 360 B

5

0

0

020000-20

5

_1

o

_1

x"

0

-1

-1

o

++ 0

liD

96 A

++80400-11202_2_200

i5 -i5

o

@

x,

0

-2

_1

20

-1

-2

-5 -10

2'

0

-1

5 -19

..

fus 1nd

-1

0

1925

,

_1

o

@

1

,, ,, , ,

_1

0

-2

@

320 920 320

20 AA

-5

0

@

@

-2

1750 -10

x"

@

0

+

0 -16

-2

@

-2

+

24

@

3

x..

0

o

0

,,

11

@

-7

X~

0

-2

@

A AA BB A AA BC lOA lOB 11A B" 12'

x,

6 -2

@

0

-1

o

14

111

40 40

o

o

r5

-r5

X~

x" 0

Xn

o

o

0

X23

o

o

_1

X24

20 20

20 20

30 30

o

0

10 10

12

0

r5

0

0

0

o

o

0

X25

o

o

o

o

o

o

o

o

Xu

0

-lI

4

-2

0

0

0-1

4

8

6

6

6

8

8

10

x" 000000000000000

o

o

0

+000000000000000

o

o

0

X28

x" X30

x"

,

0

o

o

0

X32

+000000000000000

o

o

0

X33

00

0

0

0

0

0

0

0

0

0

0

0

0

0

17

x"

o

1232 -16

X~

o

1232 -16

0

-1-161

0

0

7

2

2

31

-1

o

0

0

0-1

o

0

0

1

-1

-1

-1

x"

1792

0

0

-8

0

0

0

-8

2

2

0

0

0

0

0

0

0

0

_1

-1

o

2

0

0

00

0

0

0

0

0

0

0

0

0

0

0

0

0

0 110

o

0

X3S

X~

1848

8

0

-6

0

0

0

-2

8

-2

0

2

0

0

0

0

-2

0

0

0

0

-1

0

0

00

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

o

o 115

X36

1980

36

0

0 241

0

0

5

0

0

0

0

_1 -21

0

0

o

0

0

0

0

-1

-1

+000000000000000

o

o

X37

0

0

_1

2i

0

0

o

0

0

0

0

0

0

0

0 bl1

**

0

0

0

0

0

0

**

bl1

0

0

0

0

x"

o

Xu

o

1980

36

0

0-241

0

0

5

0

0

x"

o

2304

0

0

0

0

0

0

4

-6

-1

o

0

o

X~

o

2304

0

0

0

0

0

0

4

-6

_1

o

0

o

0

0

x..

2520 -24

0

0

0

0

0

-5

0

0

0

0

0

0

0

0

o

00

x"

2520 -24

0

0

0

0

0

-5

0

0

0

0

0

0

0

0

o

o

X~

o

0

1

r5-rS

o-rS

r5

X~

+000000000000000

o

o

0

X39 X~

00

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

m5&3

*3

0

X41

00

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

*7

n6&3

0

X42

GEJ24 G8 24

18

18

[81J

J3 Sporadic Janko group J

3

Order = 50,232.960 = 27 .3 5 .5.17.19

Mul t = 3

Out

=2

Constructions Unitary (2)

3 .. G has a 9-dimensional unitary representation over F 4 = {O,1,w,w} with basis vectors e( z), where z = x + iy is in

F • We also write e(O) = e co ' e«1_i)n) = en' Le. 9 ,

I

j

-1+i

1+1

o

-1

-1-i

-i

or 1-i

5

2

3

4

co

o

7

6

3.G is the subgroup of the unitary group (for the form

~xnxn)

which preserves the sUbspace S of the exterior

square spanned by the 18 vectors e ne n+4 + en+2en+6' eooe O + en_,e n_ 3 , and we ne n+3 +

wen_ 1e n_2.

3.G.2 is generated by the maps (specified by thlO image of e(z»:

z:

A

wXx+YYe(z)

BZ : e(Zz)

(z~O)

2 2 :w(X-x) +(Y-y) e( z)

DZ : W«z+Z)/z)e(z+Z)

E Z

where Z = X+iY is in F , e*(t) 9

= e(it)+e(-it)+e(-t).

W(i)

= w,

Cz

e*(Zjz)

FZ

e(Zz)

(UO)

WC-i) = W. and otherwise Wet)

= 1;

the map FZ is

antil inear .. There are four orbits {O}, I, Il. ill on vectors. (e.g. O. e co ' eO+e4' e co+eO+e4) the minimal weights of representatives being 0, 1, 2, 3 respectively. The stabilizer of <e oo > in 3.G is a group C32xA6).2 ;; 3 x 3:PGL2(9) in which <EO> is normal and AZ ' BZ ' Cz map to the elements t -7 Z+t, t -7 Zt. t -7 Ut of PGL 2 (9). The stabilizer in 3.G of <e co+eO+e4> is 3 x L2 (17), in which the L2 (17) (centralizing F 1 ) is generated by D1 and C_1' whose product has order 17. All non-zero isotropic vectors are of type Il and the group preserves a symmetrical joining relation between isotropic vectors. the vectors joining v (iO) forming a 3-dimensional space J(v). We define J(U) = <J(u):u of type Il in U>. A type If vector v (e.g. v = e O+e4) is contained in five 2-spaces (projectively, lines) inside J(v) , namely two ~~ ones ' and three odd ones . There is also

a distinguished orbit of special 4-spaces, such as <wen+wen+2In=O,1,4,5>, each the disjoint union of 17 even spaces ..

3.G has an 18-dimensional complex representation writable over QCI-3,,/5). for which explicit matrices have been

Complex

computed. In characteristic 2 this becomes the representation given above on the 18-space S.

Max im al subgro ups

[82]

Order

Index

8160

6156

3420

Structure

Specifications G.2

Character

Abstract

Unitary (2)

L 2 (16):2

: L2 (16):4

14688

L2 (19 )

, 19: 18

3420

14688

2880

17442

L2 (19) 2 4 :<3xA 5 )

: 2 4 : (3xA 5 )·2

N(2A 4 )

even line

2448

20520

L2 (17)

: L2 (17) x 2

C(2B)

type ill point

2160

23256

: <3 x MlO ):2

N(3A)

type I point

1944

25840

C3 x A6 ):22 3 2 • C3x3 2 ): 8

: 3 2 .<3x3 2 ):8.2

N(3B2)

base

1920

26163

2~ +4 :A 5

1 ..• 2- +4 .. S5

N(2A)

odd line

1152

43605

22+4:<3 xS ) 3

: 22+4:(S3xS3)

N(2A 2 )

type If (isotropic) point

special 4-space

1a+323ab+324a+1140a+1215ab+1615a

J3 @

@

@

@

1 1 50232960 920 080 243 P power A A A

pi part ind

lA

A 2A

A 3A

A 38

@ 96 A

A 4A

@ 30 A

A 5A

@ 30 A

@ 24 AA

A AA 8*

6A

@

8 A

@

@

27

27

A

B A

B A

8A

9A

B*2

@

@

27 B

10 BA

A

AA

C*4 lOA

@

@

10

12

AA BA

AA AA

@

15 BA

AA

B* 12A 15A

@

15

@

@

@

@

@

17

17 19 19 A A A BA A A A A B* 17A B* 19A Bn fus ind

AA

A

2 448 A

A 26

@

48

@

@

9

48

A BB

A

A

BB

A

4B

6B

8B

@

@

@

@

@

@

16 6 9 9 9 12 A AB BB CB AB AB A AB AB BB CB AB 8C 12B 18A B*7 C*5 24A

@

12 AB AB

@

17

@

AB

17 BB BB

B* 34A

B*

AB

X.

+

x,

o

85

5

-5

4

o

0

-1

-1

o

0

o

0

0

0 b19

**

x,

o

85

5

-5

4

o

0

-1

-1

o

0

o

0

0

0

**

b19

x,

x,

+

323

3

8

_1

3 -b5

*

0

-1

-1

-1

-1 -b5

*

0 -b5

*

0

000

+ooOOQOOOOOaOQX4

x,

+

323

3

8-1

3

*

-b5

0

-1

-1

-1

-1

*

-b5

*

-b5

0

0

x,

+

324

4

9

0

4_1

-1

_1

-1

-1

-1

x,

+

646 -10

7

-2

2-2bS

o

o

0

_1

bS

*

x,

+

646 -10

7

_2

2

-1

o

o

0

_1

x,

+

816 -16

6

6

0

2

0

o

o

o

_1

-1

x..

+

1140

20

15

6

-4

0

0

-1

o

o

o

o

0

0

_1

Xn

+

1215

15

0

0

3

0

0

0

o

o

o

0

0

x.,

+

1215

15

0

0

3

0

0

0

o

o

o

0

x"

+

1615

15

-5

-5

-1

o

0

3

-1

x..

+

1920

0

0

3

0

0

0

0

0

*4 y-9-&4

x"

+

1920

0

0

3

0

0

0

0

0

*2

x"

+

1920

0

0

3

0

0

0

0

0 y9-&4

x"

+

1938

2

3

-6

-2 -b5

x"

+

1938

2

3

-6

-2

x"

+

2432

0 -16

2

0

2

2

x"

+

2754 -14

9

0

-2

-1

-1

x"

+

3078 -10

-9

0

2

-2

-2

3

4 12 12

5

5

15 15

15 15

ind

X.

++

1

3 3

2 6 6

3 3 3

o

*-1

*-2bS

o

o

0

0

o

o

-2

-1

0

0

0

+

0

0

0

0

0

0

0

0

0

0

++

0

0

0

0

0

0

0

0

0

r6

*

o

0

++

18

2

0

2

2

-1

0

0

0

-1

-1

-1

++

9

-3

0

3-1

o

0

0

0

0

O-b17

0

0

0

0

*

b17

-1

-1

++

9 -3

0

3-1

o

0

0

0

0

0

o

0

_1

o

0

0

0

0

0

++

17

-1

-1

-1

*2

0

0

0

0

0

-1

-1

++

16

0

o

0

0 -Y9

*2

*4

0

0

*4 y 9-&4

0

0

0

0

0

-1

-1

++

16

0

o

0

0

*4 -y9

*2

0

*2

*4

0

0

0

0

0

-1

-1

++

16

0

o

0

0

*2

*4 -y9

+

0

0

0

0

0

0

-1

0

0

-1

o

o

o

o

*

bS

*

-b5

0

0

0

0

0

0

o

0

-1

-1

o

0

++

16

0

-2

-1

-1

-1

++

0

4

0

0

++

18

-2

0

-4

0

19 57 57

19 fus ind 57 57

2

4

6

8

8

0

o

o

o

o

-1

o

o

o

o

0

0

-1

6

8

9

9

9

10 30 30

10 30 30

12 12 12

o

o

o

o

b5

o

o

o

o

o

o

o

o

0

-1

o

*

-b5

*

-1

-1

*

b5

*

-b5

-1

-1

o

-bS

*

-b5

*

0

0

o

o

*

-b5

*

-bS

0

0

o

o

o

*

o

o

o

0

0

-1

b5

*-2bS

o

o

o

0

0

_1

o

o

o

-1

-1

17 51 51

o

-2 -b5

x"

02

18

2

3

o

-2

X~

02

153

-7

3

0

-b5

x"

02

153-7

3

0

*

X>o

02

171

11

6

0

x"

02

171

-5

-3

o

-1-2b5

x,.

02

171

-5

-3

o

-1

x"

02

324

4

9

0

4

-1

-1

X~

02

1215

15

0

0

3

0

0

0

o

o

o

0

0

0

0

0 b17

x"

02

1215

15

0

0

3

0

0

0

o

o

o

0

0

0

0

Xn

02

1530

10 -15

o

-2

0

0

o

o

o

o

0

0

x"

02

1530

o

-2

o

o

o

o

0

0

x,.

02

2736 -16

6

0

0

0

o

o

o

-1

-1

x"

02

2754 -14

9

0

-2

-1

-1

o

o

o

o

x"

02

2907

-5

-6

0

3

2

2

o

o

o

0

0

0

x"

02

3060

20

15

0

-4

0

0-1

o

o

o

o

0

0

-1

b

02

3078 -10

-9

0

2

-2

o

o

o

o

0

0

-1

0

2

-2

-1

-1

0

-1

-1

17 51 51

0

0

0

0

b Xw

*

Xll

*-b17

X12

o

0

X13

-1

-1

XI4

0

-1

_1

XIS

0

0

-1

_1

Xl6

0

0

0

0

X17

x..

1

02

1

02

0

*

*

o o

0

*

b5

o

1

0

-1

-1

o

~

DOh

*

0

o

0

-1

0

-1

0

0 b17

-1

0

x,

-1

*

2

0

x,

-b5

-1

0

0

3

-2

-2

*

*-1

0

0

2

0

0

b5

-1

0

2

o

*-1

0

18

o

6 24 6 24

0

++

o _1

0

0

18

3

0

o

02

-b5

0

x,

0

15 15 15

-b5

0

0

15 15 15

*

0

*b5Daaa

x"

10 -15

0

0

0

o

*-1

* -b5

o

0

+

0

0

0

2-2

12

o

0

o

0

0

-1

-1

o

0

0

-1

-1

18

18

18

24

24

-1

o

-1

Xl9

0

X20

x" 34

34

x" x" X~

x" +

02

x" x"

x"

*

+

x"

*

-1

-1

*

+

x'"

0

* b17

-1

-1

*

+

x"

o

0

0

O-b19

**

o

1

0

0

0

-1

-1

o

+2

0

*

+

x" x,.

**-b19

o

x"

-1

-1

o

0

-1

-1

*

+

x"

-1

-1

o

0

0

0

*

+

x"

0

0

0

+

x"

0

* *

+

x"

o

o

~EJ21

GFI7 21

13

[83]

~

5

••

08

11

1110 -10

1110 _10

12

1221

1221

1332 _12

12

1221

1332

1332

1332

011

04

02

02

04

04

011

04

08

11140

0211

0

1332 _12

016

12

21

21

1110 -10

02

30

1110

02

0

0

0

0

0

o

0

0

-3

21

b5

o

o

0

bS

b5

121

0

O·

12

b5

12

-12

2

o

o

o

121

_1

-2

20i_l0

10i-11

2

2

0

0

_1

10

10

_10

o

0248000

10i-l

o

0

o

11

-9

-9

0248000

111

111

02

04

-9

0

0

0

_2

o

-,

o

2

6 6

15 15

12 12

12 12

6

5

0

, ,

0248000

111

6 6

2

o

_1

21

_2

o

_1

o

o o

o

11

11

11

21

_1

0

0

0

-1

_1

21

o _12

-bS

b5

bS

bS

-2

o

_1

_1

21

21

_1

-1

_1

_1

_1

O-~--lI-~

_1

_1

_1

o

-1

-1

2

o

o

o

1+ 1

_2

2

m20-1

o

o

-1

_1

_1

-3

0

o

b5

-bS

o

b5

21

o

o

r3-1

_2

-i-l -r3+1

r3-1

2

2

_1

o

o

20 60 60

22 66 66

020

-b5

bS

_1

o o

o

o

-x37

o m'~0"3

o

o

o

o

-1

_1

_1

_i

o

o

132 132

120 120

o

-1

_1

-1

-2i+l

_1

_1

_i_l

o

~~

_1

o

37 111 111

~O

Cus

12

12

12

12

12

11

21

513+6

-10i3 -1013

-10i3

o

-5i3-~

5i3+6

5i3+6

6

6 6

0002~

00016

0008

000'

ooo~

12

12

12

12

-12

12

12

12

_12

-5i3+6

-513+6

5i3+16

000'

513+6

10

-10

10z12"-1

-513--11

2

2

o

13+2

-i3

12 12 12

o

_1

_1

_2

2

2

-,

o

_1

120

"

BC BC

o

13

13

13

121

-12

12

12

121

10z12**_11

o

o

o

o

_1

513+ 16 -13-2

-513+6

_1

-13

12 12 12

_1

_1

-3i+ 1

r3-1

_1

2

12JK

BC

CC

~8/2

-z 12+21+Z 3

2z3+2

-2z3-2

o

13

o

o

o

o

o

o

-z12+z3** z12-2i_z3**

13

-z12+21+z3

z 12-2i-z 3

-2z3-2

-2z3-2

o

_z12_z3" z12-21+z3**

_1

-13

12 12 12

o

o

_1

-21+1

_i_l

-r3+ 1

-2

2

o

-21-1

_1

2

CB BA 12HI

~8/2

10z12-10z3** 2z3+2 -3z12+2i-z3

-10-2z3-2-5r3+15i-513-5_2z3-2 -10-2z3-2

12

0

0

0

213

_1

-i3

-i3

12 12 12

121

_12

12

12

121

11

101_11

-101-10

-10r3+10

10

20

10i_l

11

_10

12

6 6 6

_1

-3

2

2

-1

6C

BA BA

"

5280/2 BB BA 12EF

_12

12

12

_12

11

0 10i3+20

0

o

0

-3

0

0

_1

21

20

20

-10

o

11

-9

-10

5280 BA BA 58

14

000~-5i3+6

0002-5i3+6

0002-513+6

ooo~

ooo~

0002 -1013

0002 -10i3

02

ooo~

0002 513+6

0002 513+6

lnd

000' : 00012

-x37

o

'002 ooo~

'002

b5 _b5

.00

0002

0002

_1

_1

0

21

."

0

0

_1

A A 3C

o

o

_1

30

o

-9

-9

_10

A A 38

5280 111

0002

.002

." ."

0002

'00

_00

.00

1nd

o m' ~0"3

o

_1

o

_1

o _1 o _2i+l

o

o

o _1

_1

_1

-1_1

fus

m20

o

_1

o

o

o

o

o

_1

_1

o

~~AB

~OAH

",20

_1

-1

_1

-3

_1

-1

-1

_1

AB AA

AA

CA

~~/2

•

45

BBFI147 48 • ~0/8

G.2' G.2"

GBB~~48

-b5

b5

r3+1

12 12 12

_2

o

_1 _31+1

_1

_1

-~--lI7-~

O_~

2

1-1

o

12 12 12

_10

7-~--lI

11 33 33

-10

-10

11 33 33

_1

_1

30

~

11 33 33

10

11 33 33

_1

o -12

o

_1

24 24

,

o

-b5

b5

o

o

1440

_1

o -21+ 1

2i_l

o

b5

b5

121

0

o

1332 -12

02

'"

012

0

0

_12

12

1332

04

12

2

1-2r3_1

_2

_1

b5

b5

0

12

1332

+2

_2

21

0

b5

0

12

12

1332

+2

o

o

_1

_1

_1

_1

o

_1

2

~~

.,

37/12 BA AA A AA AA A 20AD 22A 37AL

~O/~

21

11

_1

_1

_1

_1

-~

-1

1200

AC AC

~812

-2

b5

0

bS

0

121

0

1332 -12

0

0

2

_1

_1

-1

11

0

11

-1

1 -3

lOi-l1

-2

_1

20i-l0

_1

21

_1

_1

_1

_1

-~

_1

21

21

o

o

21

o

7

7-~7-~

_1

_10

077-~--lI

_21_1

-10

_1

-1

_1

-11

2

~8/2

•

o

@

o

BA'

@

'3

AB AA 12AB

o

-3

11

M

121 121 121 BAAAAA AAAAAA lOAB llA 11B l1C 110

32~

5

@

o

o

o

1331

-3

""

_1

2

o

1

6A

5AB

A

A AA A AA

,

~0/2

~a

~O/2

1221

21

02

02

1221

1110 _10

02

2

10

1110 _10

1110 -10

2

10

30

-6

10

_10

10

-2

310

""

1110

10

10

10

10

10i_l

_1

11

370

11

2

_10

A

A 4e

,

, llAB

48

310

111

+2

02

-9

2

110 _10

111

A A 3A

A A 2A

5280/2

•

~0/2

G.2 is a maximal subgroup of Janko's group JIj

tl

preserving a non-singular Hermitian form;

PGU (11) :: G.31 51)3(11);; 3.G; PSU (l1);; G 3 3

10915680 280 144

P power pt part iod lA

HUlt

GU 3 (11) :' 4 lC 3. G.:3 : all 3 )( 3 matrice:'l over F 121

Abbreviated character table.

Remark

Unitary

Constructions

Order = 70,915,680 = 2 5 .32 ,5.,,3. 31

lklitary group U (11) ;; 2A2 (11l 3

o

bS

os

b5

os

2

o

o

15 15 15

os

o

21

-2

2

o

_1

o

o

o

_1

_1

24

24 24

2i

o

-b5

b5

bS

b5

_2

o

30

30 30

-'5

b5

'5

b5

b5

_2

o

30AB

AA'

'BB

os -2

_1

o

-1

_1

o

2~AB

"

BA

'5

o

o

15AB

AB

BB

~O/2

• ""AB

o

z3+1

z3+1

z3+1

13

13

13

13

-~3

-'3

-'3

33 33 33

o

_1

_1

-3

33'

AB

-2

-2

o

13

z3+2

-'3

-'3

66 66 66

_1

z3+1

z3+1

o

020

-bS

b5

b5

21

_1

_z 1211II-1

z3+2

-'3

132 132

132

_1

_1

1.1

r3-1

_1

-2

_1_1

o

132A8

ABF AAE

ijij/2

020

-1>5

1>5

_1

o

_1

-z121111

-z3-1

z3+1

_z 12+z31111

o z12-2i-z3**

o

o

_1

_1

120 120 120

o m'~0""3

o

o

o

o

o

o

m m m

-x37

o -x' 333"~6

-1

-1

_1 -z3-1

o

020

-b5

OS

_1

o

o

_1

_1

o

120AH

AAA

CCA

o m'40**3

o

_1

o

o

o

o

_1

111AL

66A

37/12

32:2S~

32:2A~

2 3 :°8. 2

1

+6

+~

+2

+8

+~

+2

+2

+2

+2

+2

+2

02

+2

"" +12

""

""

..

lIII

""

""

11II

11II

11II

Cus 1nd

11II

*It

11II

++2

++2

++2

"++

10

0

10

o

_1

0

12

0

o

12 -12

12

11

o 11

11

_1

o

10 _10

10 _10

10

o

10

11

o

6

0

o

o

_1

_1

-2

_1

0

o

_1

o

o

o

-2

o

0

10

o

b5

b5

o

_1

o

o

o

o

12

o

0

_1

_1

_1

o

_1

10/2 12 BB AD AB AD lOCO 12L

•

1a+ 1331a

Character

12 C A 8C

..

320 320 12 A A A8 A A AB fus ind 2B ijD 6D

1

,

111: 6

EC AC

_1

H).2

111: 3

""

_1

x

122 :0 12

(J

37: 6

40/8

3

111 +2: (5x2~: 2»

G.5

122 :5

3

3 , H

111+2: 120

G.3

(~2x3):D12

2

PGL (9)

L 2 (11):2 x 2

(2(L 2 ( 11)x2) .2).2

111+2:{5x8: 2)

C.2

AAB ,AB

o -i3-2

o

_1

60

60 60

o

.20

-b5

OS

os

2i

-1

_1

AA' 6OAO

'1£

~O/~

32 :°8

98~9~0

~0/2

37: 3

638880

111

~O/2

(~2X3):S3

72

A6

196988

360

2~6235

A6

196988

288

A6

196988

L2 {11l:2

5372~

1320

360

L 2 (11):2

5372~

1320

360

L2 {11):2

~372~

1320

11~+2:~0

St.ruct.ure

2(L 2 (11)x2).2

1332

Index

13~31

5280

532~0

Q-der

Maximal subgroups

20

-b5

b5

o

_1

22

o

o

o

_1

o

_1

_1

_1

o

_1

o

o

10/2 11 BD BB AD BB 20EF 22B

•

N(3A 2 )

24

o

o

_1

o

o

-r3

o

_1

o

2~CD

AC

QC

12/2

..2

n

+2

. .2

..2

11II

+6

+~n+~

+2

..2

..2

.. n+6

..

n

H H

.. ..

..2

H

..

++

"

j

m2

+2

n

•

•

02

+2

+2

+2

+2

""

III

""

+2

+2

• •

+2

+2 +~

III

.".

+~

+2

.. +12

.. +12

n+81111+8

..

""

n+2."+2

III

""

11II

u+2n+2

• •

n

44 Cus ind fus ind

O

-1

o

o

o

_1

I

"

fus 1nd fus ind

0:

_1

_1

111

~~CD

AO AD

22/2

• •

base U l (l331)

N(37A-L)

°3(11)

°3(11)

(1)

°3

non_isot.ropic point

isotropic point

Ulitary

N(3A). N(2A 2 )

C(2B}

N(2A), C(JB)

N{11A)

Abstract.

Specifications

"-'"

~ ~

~

d V-J

Og(2)

Orthogonal group 08(2) :: D1I (2) Order,. 174,182,400:. 2 12 .35 .52 .7

Mutt:. 2 2

Out :. 53

Constructions

G.2' G.2"

a

Orthogonal

008:(2) " PGO (2) ;; 308(2) :: psoa:(ZJ ;; G.2 : all 8 x 8 matrices over F2 preserving a non-singular quadratic form of

Weyl

2.G.2: the Weyl group of the

~EJ~~EJ53

Wltt defect 0, for example x1 x2 + x3x4 + XsX6 + x-rx8; 08:(2) :: G

standard base are

Ea

root lattice, generated by the 120 reflections in the 240 root vectors, which in the

(;tl.~l.O.O.O.O.O.O) (56)

and

(:t!.~.~.~.~.±!.~.~)

(even sign combinations) (64).

BB

Various other bases are useful. for example that described in the nCayleyn construction below. and that in which

~(~3._!6)s

the root vectors are

(811) and (l,_'.0 7 )S (36) on 9 coordinates adding to zero.

Redooing mod 2 shows the isomorphism with 0a(2). Caytey

2'.G I 2".G I

22. G : the isotopy group of the non-associative ring of Cayley integers. The real Cayley algebra is spanned by l eo = "

10 ••••• 16 with the condition that for the sets of subscripts:

{co,l,2.41, {oo,2,3.5}. £co,3,II,6J. {co,O,1l,5}, {oo,1,5,6J. too,O,2,6}, {co,O,l,3}

53

--,I

30

0

I 27

o 14

:: is a quaternion subalgebra. In this the ring of Ca,Yley integers is the lattice (a

scaled copy of the Ea root lattice) of vectors ~anin (n = 00.0••••• 6) for which all 2a n are in Z and {n:a n in Zl is one of:

O. {0.l,2.1I1. {0.2,3.5}, £O.3,1I.6}. {oo.0.II,5}. {O,l,5,61. {oo.O,2,6}, {oo,O,l.3} or their cOlllplements. There are 2110 unit Cayley nU!l'lbers. corresponding to the E8 root vectors. An isotopy is a triple of maps (A,B.C) such that xyz = 1 ~ xAyBzC

= 1. There ar-e two such isotopies (A.B.C) and

(-A.-B.C) for- each deter-minant 1 symmetr-y C of the lattice. The isotopy gr-oup can be extended to 22' G' S3 by adjoining the triality automorphism (A,B.C)

~

(B.C,A). and the duality automor-phism (A.B.C)

~

(J-1B-1J.r'A-1J.rlc-1J) wher-e J is the Cayley conjugation map x ~x. The 1sotopies are gener-ated by (Lu.~.Bu)' Where Lu : x ~ ux.

Ru :

x ~ xu.

Bu :

x ~ u- 1 xu- 1 and u ranges over the 2110 unit Cayley nWlber-s. The r-eflection

in u is -J.Bu ' Presentation

2.G.2

~

Remark: G,S3 is a maximal subgr-oup of the Fischer- group Fi22

Specifications

Maximal sUbgroups Order

Index

Structure

11151520

120

S6(2)

11151520

120

S6(2)

11151520

120

S6(2 )

[

12902110

135

2 6 :A 8

: 26:3a

12902110

135

2 6 :A 8

3

12902110

135

2 6 :A 8

18111110

,60

18111110

960

1814110

960

" " "

G.2 :S6(2)x2

,2 "'[L 3 (2)>2)

I

155520

1120

155520

1120

(3xU 4 (2»:2

I

110592

1575

21+8:(S3xS3xS3)

: H.2

15552

11200

11 3 : 23 ' S4

111400

12096

(!sxA 5 ):z2

111400

12096

(A5 XA 5):z2

14400

12096

(A5xAS): z2

"/3"2, 3

I

[2 12 .32 ]

Character

Abstract

Orthogonal

Weyl

eayley

"3(3),2 , 53

la+35a+8I1a

C(2F)

non-isotropic point

root vector, Art£i

<J .Bu>

I I

: S3 x UIj(2):2

: S3wrS 4 : SSW r2

I

<J .Lu>

la+351>+-811 b

<J.~>

la+35C'"8I1c

2

13

.3 ]

[2

la+50a+8Ila

N(2 6 )

la+50a+8I1b

N(2 6 ) = N(2A 35C28 ) N(2 6 ) = N(2A 350 28 )

maximal isotropic subspace

null subalgebra mod 2

maximal isotropic subspace

null subalgebra mod 2

1a+ 35a+84a+300&+700b

N(3A)

minus line

la+35b+-811b+- 300a+700c

N(3B)

U4 (2)

la+35C'"811<:+300a+700d

NC3C)

UII (2)

la+50&+84abe+300&+972a

N(2A)

isotropic line

N(3 11 ) = N(3A4B4C4016E12)

0'2(2) wrS 4

A2 +A 2 +A 2+A 2

N(2B,3A,5A)2

0ij(2)wr2

A4+A4

N(2C.3B.5B)2

01j(1I)

N(2D,3C.SC)2

01j(1I )

1a+50a+allc

155520

C3xUIj(2»:2

I

G'S 3

: 39

1120

C3xU 4 (2»:2

G.3

= N(2A 35B2a )

isotropic point

standard base

congruence base mod 2

1a+8I1a+175a+700b

I

I 1

3l ""2S'

: H.3

1a+841>+-175a+700c la+811<:+ 175a+700d

j3l""(25"2)

: H,S 3

: H.3

: H,S3

52

52

1

''',

1

''''

A2 +E 6

[85]

Ot(2) •

@ @ @ @ 110 46 116 46

@

@

3 71

@ @ 77 77

@ 1

@

@ 11

P power p' part ind lA

U

A A

~

A A

~

A A

~

A A

~

A A

~

A A

~

A A

A A

~

W

A A

3£

A A

4A

A A

liB

B A 4C

@

@

, , • •

@

C OD

o

liE

,

,

7

32

, 32

27

.. • • • •

21

21

20

A

A

B

D

0

D AB

A

A

A

A

A

A

7A

M

M

~

~

20 BC

AB BC ~ 10' lOB lOC

••

36 " " 15 15 cc15 BO " " "12<" '" " 158 15C 12B 12C 120 12E 12' 15'

20 144 CO CO

BA BA

144 CA CA

CA

QC

DA

AC

2" EO

BB BB

CE

CC

120

x,

+

1

1

t

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

X,

X,

+

28

-Il

11

4

11

_11

10

10

10

1

1

8

0

0

0

0

0

3

3

3

2

2

2

-2

-2

_2

5

_1

_1

_1

1

1

1

_1

0

-2

2

1

1

1

_1

_1

_1

2

2

2

_1

0

0

0

0

0

0

X,

X,l

+

35

3 11

-5

-5

3 14

5

5

_1

2

1

_1

3

_1

_1

_1

5

0

0

6

-3

-3

2

3

0

0

0

2

-2

-2

0

0

1

1

2

_1

_1

1

0

0

-2

1

1

1

0

_1

_1

_1

0

0

):4

+

35

3

-5

11

-5

3

5

111

5

-1

2

7

_1

_1

'3

_1

_1

0

5

0

-3

6

-3

1

2

3

0

0

0

-2

2

-2

0

0

1

1

_1

2

_1

0

1

0

1

_2

1

1

_1

0

_1

0

_1

0

x, x.

x_,

+

35

3

-5

-5

11

3

5

5

14

_1

2

7

-1

_1

_1

3

_1

0

0

5

-3

-3

6

1

1

3

0

::l

0

-2

_2

2

0

0

1

1

_1

_1

2

0

0

1

1

1

_2

1

_1

_1

0

0

0

_1

X,

x~

+

50

18

10

10

10

2

5

5

5

_4

5

-2

6

2

2

2

2

0

0

0

-3

-3

-3

1

0

3

3

3

1

1

1

_1

1

0

0

_1

_1

_1

0

0

0

1

1

1

-2

_1

_1

_1

0

0

0

x~

2

X7

+

811

20

20

4

II

II

21

-6

_6

3

3

II

II

II

0

0

0

II

_1

_1

5

2

2

5

_2

-2

_1

5

_1

_1

_1

1

1

1

0

0

0

0

0

0

0

_1

_1

1

_2

_2

1

1

0

0

1

_1

_1

X,

X~

+

84

20

II

20

II

4

_6

21

-6

3

3

4

II

0

4

0

0

_1

4

_1

2

5

2

_2

5

_2

_1

_1

5

_1

1

_1

1

1

0

0

0

0

0

0

_1

0

_1

_2

1

-2

1

0

1

0

_1

1

_1

X.

84

_1

X~

+

20

4

4

20

4

-6

-6

21

3

3

4

4

0

0

4

0

-1

_1

II

2

2

5

-2

-2

5

X'"

+

175 -17

15

15

15

_1

-5

-5

-5

13

4

-1

_1

_1

_1

_1

3

0

0

0

-5

-5

-5

3

3

3

Xli

+

210 -14

26

10

10

2

39 -15 -15

-6

3

6

-2

2

_2

-2

2

5

0

0

7

1

1

_1

X'l

+

210 -111

10

26

10

2 -15

-6

3

6

_2

-2

2

_2

2

0

5

0

1

7

1

1

Xl.l

+

210 _14

10

10

26

2 -15 -15

39

_6

3

6

-2

_2

_2

2

2

0

0

5

1

1

7

X..

+

300

12

20

20

20

12

8

0

0

0

0

0

0

0

0

6

6

_1

5

1

1

_1

1

0

0

0

0

0

0

_1

_1

0

-2

-2

1

1

0

0

1

_1

-1

1

X~

-2

_2

0

0

0

2

0

_1

_1

1

1

1

0

0

0

_1

_1

_1

_1

_1

_1

_1

0

0

0

X'"

1

-2

-5

1

_1

1

1

_1

0

0

0

0

0

0

1

0

0

3

-3

-3

0

_1

1

1

_1

0

0

X"

_1

1

-2

1

-5

1

_1

1

_1

0

0

0

0

0

0

0

1

0

-3

3

-3

0

1

_1

1

0

_1

0

Xll

1

1

-1

_2

6

2

2

2

3

1

-5

1

1

_1

_1

0

0

0

0

0

0

0

0

1

-3

-3

3

0

1

1

_1

0

0

_1

X"

0

0

2

2

2

0

_1

2

_2

0

0

0

0

0

0

2

2

2

_1

0

0

0

0

0

0

X,.

30

30

3

-6

-2

35

35

35

-1

-1

26

2

-2

_2

-2

2

0

0

0

-5

-5

-5

_1

_1

_1

7

1

1

1

_1

_1

_1

1

0

0

0

_1

_1

_1

0

0

0

_1

_1

_1

_1

1

1

1

0

0

0

x"

5 -19

30

30

30

12

3

-7

1

-3

-3

-3

1

0

0

0

6

6

6

2

2

2

0

3

3

3

_1

_1

_1

_1

0

_1

_1

0

0

0

0

0

0

2

2

2

2

0

0

0

0

0

0

X,~

x"

+

350

-2 -10 -10 _10

+

525

45

Xl>

+

567

Xl"

+

567

x..

+

X,.,

0

5

5

-9

39

-9

-9

-9 81

0

0

0

0

15

_1

_1

3

3

_1

7

-3

-3

9

0

0

-3

0

0

0

0

0

0

0

0

0

0

0

-1

_1

0

0

0

_1

1

1

-3

0

0

0

_1

0

0

1

0

0

Xl>

-9

-9

39

-9

-9

0

81

0

0

0

15

-1

3

_1

3

_1

-3

7

-3

0

9

0

0

-3

0

0

0

0

0

0

0

0

0

0

_1

_1

0

0

0

1

-1

1

0

-3

0

0

0

_1

0

0

1

0

x,~

567

-9

-9

-9

39

-9

0

0

81

0

0

15

-1

3

3

-1

_1

-3

-3

7

0

0

9

0

0

-3

0

0

0

0

0

0

0

0

0

.1

_1

0

0

0

1

1

_1

0

0

-3

0

0

0

-1

0

0

1

X,.

+

700

92

20

20

20

_4 _20 -20 -20

-2

7

0

8

0

0

0

0

0

0

0

_4

_4

_4

_11

_4

_4

2

-1

_1

_1

_1

_1

_1

_1

0

2

_2

1

1

1

0

0

0

0

0

0

0

0

0

0

0

0

0

x,.,

60 -20 _20

X11

Xl'

+

700

_ll

Xn

+

700

_ll -20

Xl.l

+

700

_ll _20 -20

60 -20

X,.

8110

8

X"

8110

8 _40

x...

8110

8 _110 -110

X"

972 108

60

211 _110 -40

12

55

10

10

7

4

_4

_4

II

0

0

0

0

0

0

_1

2

2

3

-2

-2

-1

_4

2

2

0

-2

_2

0

0

0

0

_2

1

1

0

0

0

_1

2

2

_1

1

0

0

0

0

0

12

10

55

10

7

4

-4

_4

0

4

0

0

0

0

0

2

_1

2

-2

3

_2

_1

2

_4

2

-2

0

_2

0

0

0

0

1

_2

1

0

0

::l

2

_1

2

_1

0

1

0

0

0

0

X>l

12

10

10

55

7

4

-4

_4

0

0

4

0

0

0

0

2

2

_1

-2

-2

3

_1

2

2

_4

-2

_2

0

0

0

0

0

1

1

-2

0

0

0

2

2

_1

_1

0

0

1

0

0

0

x>..

8 _211

0

8 -10 -10

0

2

2

_1

_1

_1

_1

3

_1

_1

_1

0

0

0

0

0

0

_1

0

0

4

-2

-2

1

0

0

0

1

0

0

X>4

8 _10

2

0

2

_1

_1

_1

_1

_1

3

_1

-1

0

0

0

0

0

0

0

_1

0

-2

4

_2

1

0

0

0

0

1

0

Xl_' X...

30

30

3

3

16

0

0

0

0

0

-5

0

8

30 -211

30

3

3

16

0

0

0

0

0

0

-5

211

8

30

3

3

16

0

0

0

0

0

0

0

36

12

0

0

0

0

0

0

8

0

0

0

0

-3

-3

50 -30 -30

-6

15

15

15

-3

6 -10

_2

2

_2

-2

_2

0

0

_6

15

15

15

-3

6 -10

_2

_2

2

-2

_2

0

0

-6

15

15

15

-3

6 _10

_2

-2

_2

2

_2

0

0

811 -211 -211

-6

-6

0

0

0

0

0

0

_1

4

4

4

-8

-6

-6

0

0

0

0

0

0

4

_1

4

-8

11

84

-6

-6

0

0

0

0

0

0

11

4

_1

-8

-8

11

36

211 -110 36

1050

58

1050

58 -30

1050

58 -30 -30

XJ,

131111

64

64

0

0

0

X.l>

1344

611

0

64

0

0 -24

1344

611

0

0

611

X,.,

39 -15

_1 -2

30

x,~

X>.

+

X",

50 -30 50

30 -211

811 -24

0 -10

-5 -10 _10 -3

8

2

2

0

-1

_1

_1

_1

_1

_1

3

_1

0

0

0

0

0

0

0

0

_1

_2

_2

4

1

0

0

0

0

0

1

0

0

0

0

0

0

0

0

0

0

0

0

0

0

-1

_2

2

0

0

0

1

1

1

0

0

0

0

0

0

0

0

0

0

Xl'

0 -17

7

7

_1

3

3

1

4

-2

-2

2

0

0

0

0

0

0

0

0

0

0

0

0

_1

_1

_1

_1

_1

1

1

0

0

0

X",

0

7 -17

7

3

_1

3

1

_2

4

-2

0

2

0

0

0

0

0

0

0

0

0

0

0

_1

_1

_1

_1

1

_1

1

0

0

0

X""

0

7

7 -17

3

3

-1

1

_2

_2

4

0

0

2

0

0

0

0

0

0

0

0

0

0

_1

_1

_1

_1

1

1

_1

0

0

0

X.'"

-8

4

0

0

-2

11

_2

_2

-2

0

0

0

0

0

0

0

0

0

_1

0

0

0

0

0

0

0

0

0

-1

1

1

X.ll

-8

0

4

0

_2

_2

4

-2

0

-2

0

0

0

0

0

0

0

0

0

_1

0

0

0

0

0

0

0

0

1

-1

1

0

0

11

-2

_2

_2

4

0

0

-2

0

0

0

0

0

0

0

0

0

_1

0

0

0

0

0

0

0

1

1

_1

x" X,..

X_,_,

+

X,.

+

11100 -72

110

110

110

-8

50

50

-11

5 -16

0

0

0

0

0

0

0

0

_6

-6

-6

-2

-2

-2

0

-3

-3

-3

1

1

1

1

0

0

0

_1

_1

-1

0

0

0

2

2

2

2

0

0

0

0

0

0

X,..

X."

+

1575 -57

15

15

15

-9

90 -45 -115

9

0

11

3

-1

_1

_1

_1

0

0

0

-6

3

3

_6

3

3

-3

0

0

0

0

0

0

0

0

1

1

0

0

0

0

0

0

2

_1

_1

_1

2

_1

_1

0

0

0

X..,

x",

+

1575 -57

15

15

15

-9 -45

9

0

11

3

_1

_1

_1

_1

0

0

0

3

-6

3

3

-6

3

-3

0

0

0

0

0

0

0

0

1

1

0

0

0

0

0

0

_1

2

-1

_1

_1

2

_1

0

0

0

X....

X_"

+

1575 -57

15

15

15

-9 -45 -45

90

9

0

11

3

_1

_1

-1

_1

0

0

0

3

3

-6

3

3

_6

-3

0

0

0

0

0

0

0

0

1

1

0

0

0

0

0

0

_1

_1

2

_1

_1

_1

2

0

0

0

X_"

X.'"

+

2100

52 -60

20

20

11

75 -60 -60

-6

3

12

-11

-11

0

0

0

0

0

0

-5

4

11

3

_4

_4

-2

1

1

1

3

_1

_1

1

0

0

0

0

0

0

0

0

0

3

0

0

0

_1

0

0

0

0

0

X_'"

x.w

+

2100

52

20 -60

20

II -60

-6

3

12

-11

0

_4

0

0

0

0

0

4

-5

4

-ll

3

_4

-2

1

1

1

_1

3

_1

1

0

0

0

0

0

0

0

0

0

0

3

0

0

0

_1

0

0

0

0

x_,.

X..,

+

2100

52

20

-6

3

12

-4

0

0

-11

0

0

0

0

4

4

-5

_4

_4

3

-2

1

1

1

_1

_1

3

1

(,

0

0

0

0

0

0

0

0

0

0

3

0

0

0

_1

0

0

0

x..,

X..

+

2240 -64

611

0

-11 -40 -40 -10

2

0

0

0

0

0

0

-5

0

0

_4

8

8

11

0

0

2

_4

2

2

-2

0

0

0

0

0

0

2

_1

_1

_1

0

0

0

0

0

0

0

0

0

1

0

0

X..

x.,

22110 -611

0

611

0

X."

22110 -611

0

0

611

X....

2268 -36

X4_'

+

20 -60

2268 -36 -36

12 -36

2268 -36 -36 -36

X4'

2835 -45

12

51 -45 -45

2835 -115 -115

x..

0

12 -36 -36

x...

X..

,..

224

...

1100

672 840

o ,

,.. ,. '" x,,,

...

1008

672

x_

x"

,,, ,,, ,,,

...

1008

o " o "

1008

o "

1296 11100 ...

11100

+

21100

o o

-2' 60

35811

,

"'00

,," 11536

o o o

60

,

...

5600

0_80

o

,,' no:

1

2

2

X'"

,,,

,,,

,,' 00'

1

,2

2

0

0

0

0

0

0

0

0

0

1

0

X.>

0

_1

0

0

0

0

0

0

0

0

0

1

X.,

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

2

_1

_1

-3

0

0

0

_1

0

0

1

0

0

x..

12

0

81

0

0

0 -12

4

0

_4

0

0

3

-2

3

0

9

0

0

-3

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

_1

2

_1

0

-3

0

0

0

-1

0

0

1

0

X4'

12

0

0

81

0

0 -12

4

0

0

_4

0

3

3

_2

0

0

9

0

0

-3

0

0

0

0

0

0

0

0

0

0

0

0

0

0

_1

_1

2

0

0

-3

0

0

0

_1

0

0

1

X....

3 -81

0

0

0

0

3

-5

3

3

_1

5

0

0

-9

0

0

3

0

0

0

0

0

0

0

0

0

0

0

_1

_1

0

0

0

1

0

0

3

0

0

0

1

0

0

_1

0

0

X.7

3

0 -81

x••

3

0

3

0

0

-9

0

0

3

0

0

0

0

0

0

0

0

0

0

_1

_1

0

0

0

0

1

0

0

3

0

0

0

1

0

0

-1

0

5

0

0

-9

0

0

3

0

0

0

0

0

0

0

0

0

_1

-1

0

0

0

0

0

1

0

0

3

0

0

0

1

0

0

_1

X.,

0

0 -40 -40 -40

14

-11

0

0

0

0

0

0

0

0

0

8

8

8

0

0

0

2

-11

_4

_4

0

0

0

0

1

0

0

_1

-1

_1

0

0

0

0

0

0

0

0

0

0

0

0

0

x_'"

0

64

-8

-8

0

0

0

0

0

0

-11

-'I

_4

0

0

0

0

0

0

0

0

0

0

0

0

0

0

1

0

0

1

1

1

0

0

0

0

0

0

0

8 -30 -30 -30

15

-3

-8

-8

0

0

0

0

0

0

0

-6

-6

-5

_2

-2

-2

3

3

3

3

1

1

1

_1

0

0

0

0

0

0

0

0

0

-2

-2

_2

_1

3

0

0

0

0

_1

1

1

0

0

0

0

0

0

0

0

0

7 111

8

8 9 9 9 10 818181810

20

20

64 0

o

5

64 0

0

0

0

-9

3

3

3

3

3

3

3

4

4

,

8

6

6

6611

4

_4

-'I

o o

_1

2

4

o o 11 -16 _16 2 2 12 o o 31 4 4 11 4 8 o o 34 -20 -20 -2 -2 16 o o 14 -40 -'I -1 2 16 o o 111 -4 -40 -1 2 16 o o 25 -20 -20 II 10 -8 o o 16 16 16 16 -2 0 o o 74 20 20 -7 2 8 o o -30 -84 24 -3 6 16 o o -30 24 -811 -3 6 16 o o 21 -60 -60 3 -6 20 o 0903636908 o 0-4536-72908 o 0-45-72 3 6 9 0 8 o o 81 0 0 0 0 24

,2

,2

_1

0

-3

2 :2

0

o o o

4

o

o o o Q o 2

o

o o o o

0

2

o o o

o o

-2

o

334

:2:2:2:2:2:2

o

03-2-2-6

-8

-2 _2

_4

0

0

1

669

0

0

0

15

o o

0000_9

o

o

o

0

0

0

12

o o o o o o

00003

o

0-5009

, , , ," , ,2

0000-9 00003

o

5

0

0

18

011_640

o o o o o o

0

0

0

-3

0-466-9

o

0

0

0 -12

555 :2

:2

:2

555 :2

:2

:2

o

2 -2

o o o o o o o o o

o o o 6

o o o o o o o o o o o

0

o o o o o o o o o

0 -3

_1

000

_1

o -3

_1

-3

o o

-3 _2

0

000 000 000

0

000

-7

003

3

000

o 6 12

:2

0

6 12

o o o

:2

12

12

6 :2

~

~

2

2

_1

_1

_1

0

2

_1

_{

_1

1

0

o o o

0-2-2-2_20

1 -2

-2

1_1

3

0

0

0

1

_1

2

2

2

1

15 15 15 303030

o 0011X" o o 1 _1 _1 X" o o 1 _1 _1 X'" o o -1 0 0 x.<7 o o _1 0 1 X" 0

Ox""

0

0

o

0

o

0

1

1

2

0

1

1

0

o

0

_1

_1

_1

1

0

o

_1

0

0

X'"

-2

0

0

o o

0

0

0

0

0

1

0

0-2_2410

o

0

1

-1

X6"

-2

0

0

o

0

o o

o o o

1

0

o o

o

0

0

000

1

0

0-24-210

o

o

_1

1

x....

-2

0

0

0

o

-2

_1

0

o

5

o o

0

0

0

2

o o

0

0

0

_1

0

0

0

_1

_4

o

0

0

o o o o o o o o o o o o o o o o o

o o o

0

C

0

0

2

0

0

2

0

0

-2

0

2 2

000

o o

o o

o o o o o o o o o

o o o o o

o o o o o o o o o o o

6

12 12 12 12 12 1212121212

1

6 6 :2

X"

0

0 _2

0

0

0

-2

0

0

2

0

0

2

0

0

2

0

0

0

0

0

-2

0

0

0

0

0

0

0

0

0

0

0

2

0

0

6

0 _1

0

1

1

1_1

0

1

1

1

1

0

0

_1

_1

2

_1

0

0-211_210

0

_1

2

_1

_1

0

0

0

0

o

0

0

0

_1

0

0

o

0

0

0

-1

0

o o o

0

o

0

0

0

0

_1

0

1

0

0

0

0

1

0

_2

2

_1

-1

0

0

-2

_1

_1

_1

0

0

0

0

0

000

0

o o o o o

0

1

1

1

0

001_2100

o o o o o

0

0011_20

0

0 _1

0

o

2

0

0

-1

0

0

o o

0

000

1

0

C

0

0 0 1

o

0

0

_1

2

-1

o

_1

_1

2

0

0

0

1

1

1

0

0

0_200000

0

0

0

o

o

2

0

0

0

0

o

o

0

_1

_1

12

6 '2

12

7 :2

8

12

12

6 :2

7 :2

8

6

0

_1

_2

:2:2

6

0

_1

o o o

-3

0

o

o o

0 0

0

00

o

-3

0

o

000

-3

0

o

3

0 0

o o

0

0

01-2-2_21

003

o o o o

o

1

0

X.,

00-21100

3

3 3

o o

X"

0

1

o o o o o o o

-6

0

_1

0

0

o o o o

o 0 -3

0

0

_1

0

0-2-2410

000

003

0

2

-1

0

o

1

_2

0 0

0

000

0 -3 0 -3

o o o o o o o o o o

0 12

0 0

o o

o

o o

o

0 12

0 0

o o o o

3 3

o

_2

o -3

0114-60 0_4_4_40

o

o 2

o

o

o o

3

o

2

o

o o

_11

3

-6

0-500-3

006

03-2-23

-16

3

-8

_2

o o

0646464_8_80

2

-3 -3

o o o

o o

o o o o o o

4 :2

o o

0

o o o

_4

0

3 :2

0-2-2-20 0500-6

o o o

-4

o o

12

009

_1

o

o o o

0

2_16

0722_9

o o

03-2-23

-4

0

o

o

-'1

-,

-16

0

0

2-6

o o o o o o o o o o

o

o

o

11

3

6

o o

0

-3

_4

6

2

20

_2

-11

4

0

11

_2

1

-5

o

6

6

11

o

15

0

6 6

-6

0 -12

60

0

12

_1

-6 -16

o

0

12

_1

0

10 -24

0

6 6

o o

o o o o o o o o

12

-8

0

6

o o o o

0

8

16

0

6

0

-2

-4

6

0

6 6

0500-6

1

-'I

-3

0

5 10

o o

8

-4

0

5 10

4

_2

8-4

0

0

o o o o

2 _2

1

-13

0 5 10

o o o

o o

o o o o o o o o

3 :2

35811

16

0

2

0

3 :2

+

o o

_1

_1

0

233 :2 :2

3360

2

_1

9

0

32110

+

_1

0

3

20

+

,.

0

0

3

o o -81 o o 20

,,, ". ,,. ,..

o -110 o "

0

0

-2

60

2800

0

0

0

-75

+

0

-2

0

o

x"

-'10

0

0

0

20

o

_2

0

_4

o o

2800

0

_4

4

o

2800

2

2

0 -12

0

+ +

_11

2

0

0 0

60

2

2

0

o o -25 20 20 o o o 95 -40 -40 o o o 60 60 60 o o o 55 -80 -80 o o o -5 1l0-140 o o o -5-1110 40 o o o 81 0 0 o o o -6 -60 -60 o o o -611 80 -64 o o o -64 -64 80

o -80 o -40

Xn

o o o o o o

2

11

0

o o

...

XM

0

0

5

o o

o

4

0

0

2

o o o

0

_4

0

o -'1 o 24 o 16 o -16 o -16

...

560

8

8

0

o

,,,

",

X~l

-11

8

81

2

o " o 32 o 56 o 16 o 16

8

-5

12

27 -'15 -45 -45 -21

o

0

0

_1

122

224

X"

". ,•.

-5

0

_1

6075

o o

0

0

3

+

"

0

-5

0

160

0 0

3

40

112

0 0

_5

0

...

0 0

3

110

+

0 0

3

0

x"

0 0

3

110

'"

2 2

3

0

,

-11 -10

3

4200 -211

.

-4 -40 _10

3

4096

X"

0 -40

0 -110 -40

0

+

X,.

0

75

0

+

2

75 -60

11 -60 -60

0

X'l

,,'

90 -115

0

3200 128

0

50

0

51

+

0

0 -211 -211

0 -81

51 -45

2835 -45 -45 -45

x_'" x"

x,_,

[86]

A A

• ••

1 1 6ll 300 300 300 728 128 128 288 288 288 216 216 216 216 72 72 12 211 E AAABACAABBC CD DA EA EA EA ~ ~ ED ~ A AA BA CA AB BC CD CA EA EA EAEBECEDEE Ilf SA 58 se 6A 6B 6C 60 6£ 6f 6G 6H 61 6J 6K R M 6N

, , , ,, ,,

1711182400 592 060 030 080 072 760 760 760 944 6118 608 512 192 192 192

000

0

0

0

0

0

2

2

2

_1

_2

2

_1

2

2

_1

2

o o

2

-4

_1

_1

o

2

_1

_1

o o o o o

2

-3

o

0

_1

2

2

-1

_1

0_1 _1

1

_4_421 -2-210

Xbl

0100XM

o

0

1

1

x""

01-2X~;

o

-2

1

x""

1

0

0

X".,

00

o

OX.,.,

o

OOOOX7I OOOOX"

3

o

0

0

_1

_4

2

_1

1

o o

OOOOX"

_1

o

_1

2

_4

-1

1

o

OOOOX"

o o o o o o

-3 2

o

0

0_1

o

0100X;6

2

2

2

0

o

o

0

o

0

0

0

0

000

0

0

o

0

0

_2

_2

o 0101X,. o o 1 1 0 X'" o o _1 _1 _1 x.., oooox., o o _1 0 0 X'l

Q

0

1_1

000

1

2

0

OOOOX74

_1

0

0

0

0

o

8 9 9 9 :2:2:2:2

20

10 :2

~

12 12 12 12 :2:2:2:2

~

12 :2

~

15 15 15 :2:2:2

8 9 9 9 :2:2:2:2

20

20

10 :2

12 :2

~

~

12 :2

15 :2

_1

X,...

_4

12 :2

2

12 :2

_1

12 :2

o

0

15 :2

0

15 :2

X"

x""

,

I\J

0

0

0

0

0

0

0

000

'"

0

~ ~

o

0

0

b-

0

0

0

0

I\J I\J

o

000

o

0

,

N

.!.

o

0

000

o

0

o

0

0

0

o

o

0

o

.t. 1-

0

0

o!.

000

o

000

!.

0

o

0

00

0

o

0

!...

o

o!.!.o

o

0

o

0

!.

0

o

0

000

0

0

0

0

o

o

o

o

o

0

00

o

"

,

0

!.

0

000

I\J

.t.

"

I

0

0

000

0

o

0

0

0

0

o

o

0

0

o

o

o

o

o

o

000

o

0

o

1 !.

0

o

o

o

00

o

0

w

I.

0

o

0

o

o

o

o

00

o

00

000

o

ONO

0

N

o

000

o

o

0

0

1:

000

0

0

~

000

o

1:

I\J

0

, I\J

o

o

~j,Uik

o

CO

,

o

o

'"

o

0

~ ~

co

, ~ ,

o~:::l~~~ o~~o(t>~ 00 ~ ~

~~

o

~ ~ ~ ~

t

t t t t

I :

: : : t

t

0

o

o

o

N

,

o

o

0

0

0

"'

0

J.,

!. !.

o

0

00

o

000

o

o

o

o o

000

o

o

o

o

o

o

o

o

o

o

o

o

o

o

o

o

o

o

o

0

0 0

0

0

0

0

0

0

_

o

o

0

o

o

o

w

0

o

o

1

Vi

0

o

o

N

0

0

000

0

0;;

o

o

N

1

0

o

;;;j

o

000

,

~ ~

t

o

+

o

0

o

0

0

0

o

o

I\J I\J

0

N

o

0

o

o 0

!.

rv !.

000

o

000

0

N

N

00

000

0

0

000

o

o

o

o

o

o

o

;;;j

;;;j

co 0

_

w

0

o

0

o

o

0

N

N

~ ~

0

0

0

0

0

~

!...

0

!...

0

0

0

0

0 0

o

0

!..

0

o

0 0

o

0

J.,

0

°

°

0

J., !.

0

0

I\J 0

o

°

o

8888

°

o

o

o

0

o!.oo

!.

o

o

1

:

o

o

o

o

o

o

o

o

o

o

o

o

o

o

0

0

0

0

0

_

o

o

o

o 0

0

00

o

o

o

o

I"

o

0

o

N

0

0

I"

0

0

0

0

,;

, ~ ,

~

,

0

0

0

0

o

o

_

I..

~

N

"

"

~

"

0

°

o

o

o

o

o

°

o

o

o o 0

o

o

o o

o

o

o o

o

o

o

o o

o o o

o

o

o

o

o o

o

o o

o

o

o

o

o

o

o

J.,

o

N

0

0

o

1

0

o

0

0

o

o

o

o

o

0

!.

o

0

o

o

o

o

o

o

o

o

o

o

o

o

o

o

o

o

o

o

o

•

. . .• •

o

o

o

o

0

o

o

0

0

0

o

o

o

o

o

o

o

o

o

o

I •

o

o

o

o

,

o

o

o

ON

o

0

ONO

o

000

o

N

o

o

0

,

N

I..

Vi 0

o

I"

0

0

:

o

N

N

,

o

o

o

o

o

0

00

o

o

o

o

o

.

0

000

o

o

o

OWN

o

o

0

,

~

rv ~

;;;

.;;

..l.,

o

0

o!:::

o

!..

~

1 0

;;; 0

0 ;;;

o

,

o

0

.!.

o

0

0

0

!.

!..

!. 0

° ° o

o

!. I\J !.

!. ~

888

o

000

0_0

°

0

0

o

o

o

o

o

0

o

o

o

o

o

o

o

o

0

0

o

J,

_

o

0

0

o

o

o

o

o

o

o

o

,

0

N

N

,

• 8

co

o

,

N

N

,

0

0

N

0

00

o

o

01"

o

o

o

o

0

000

o

o

o

o

o

o

o

o

o

o

ONO

o

000

0"

.• . .• •

o

o

o

o

o

00_

o

o

o

o

o

° $ ~ <3, '2

o

,

o

o

1-

000

o

o

t

1-

o

o

o

0

o

o

o

o

N

t

00<3

+

1

h

1

o

o

o

!.

0

0

ONO

o

o

!.

Vi

o

o

I..

o

o

o

0

~

I"

0

,

rv .;;

0

o

o

0

I"

o

o

o

o

o

o

o

••

!..

1

o

0

o

o

,

N

o

I..

•

o

o

0

0

0

0

0

!.

0

o

o

o

00

o

\$<

o

o

o

o

o

o

o

o

o

o

o

0

0

~

o

°

;;;

°

o

, c..

0

J:, ;:;

0

!...

~

Vi

o

o

o

o

o

o

••

W

o

o

•• •

o

o

o

o

o

o

o

o

o

o

o

o

o

o

o o

N

:

•

•• •

o

o

o

o

o

o

o

o

0

o

o

o

o

o

o

ON

o

o

o

o

00

o

o

N

I..

o o

. ·• .

,

0

0

0

,

,

0

00

o

o

o

o

I..

"

o

• ••

o

o

·:

:

o

o

o

o

o

o

o

o

o

.

o

o

N

0

N

,

"'I

~

~ ~

;(

C

·

",

o

~_

(0)

""

~

Cl»'"

W

""

"'l»"'I\J
W

g,:;;

:g > >>nol\J

~»­ > Cl ,",0

fi ~ ~'"

8~8t5

N > 0:;; ::r:::r:::r: ...

g;1;ll;ig

OCl<;)'"

0»>:;;

W»

""


"'10 rv""

"'00'" "CI"'l"'lCO

o

0»>(;

... >0> ~ ...

W

... » , c ~

W

W

> CllOW
~>Cll~rv""

Cl

· . · ·

? 88 [

0

0

-

rv ~

N-

,

N

g

~

Cl»CO"'''''

N

~. N~

"'1»0_""

N

@

@

36

35

50

_4

476

595 -115

714 -22

714 106

1020

+

+

+

+

+

+

+

+

+

x,

x,

x,

x.

x.

x" x..

20

60

1176

_11

-11

11

63

-2 -21

14 -21

14 _21

0

3

3

3

3

8

20 -28

+ 2142 126 -10

+ 2142 -66 -10

+ 2142 -66 -10

15

15

15

+ 2295-105

+ 2835 -45 -45

0

+ 22,95-105

+ 2835 _45 -45

64

+ 2295-105

+ 2835 -45 -45

56

+ 2176 128

+ 2835 -45 -45

40

~~

+ 2142 126 -10

0

+ 1344

+ 2856 -88

+ 2856 104

+ 3264 _64

+ 4096

x"

Xw

x"

x"

x"

x"

x"

Xv

x"

x"

Xw

x"

x"

x"

x"

+

+ 5355

x"

x"

64

90

-8

84

35

21

21

@

@

0

-2

A

3

@

@

-8

-6

-9

9

0

0

0

0

0

0

0

3

3

3

3

3

-6

11

-5

14

7

7

@

@

@

@

0

0

9

0

0

-9

-1

0

4 -12

0

0

5 -10 -16

0

-8

6

16

o

3

3

3

3

_1

0

@

@

3

0

-4

0

0

0

0

3

3

3

-1

0

1

0

_1

-1

-1

0

2

2

-2

-2

3

0

0

0

0

0

0

-1

-1

-1

_4

8

@

@

8

0

0

-4

_1

0

0

0

0

0

0

0

0

5

0

-3

2

0

-4

0

_4

-1

0

_2

0

4

-4

-2

0

0

0

0

3

3

0 -6

0

@

@

0

-1

0

0

0

0

0

-3

0

0

0

0

o

2

0

_1

0

@ @

@

@

_1

2

o

-1

0

0

-1

0

0 0

o o

0 0

0 -1

o

o

-1

2

0

0

0

0

0

0

0

0

0

0

0

0

0

*

0

0

0

0

-1

2

0

0

0

0

0

0

0

-2

* 2b5 -2

o

0

*

0

-1

0

r5

1-2b5

0 2b5

0

0-2b5

0

0

_1

o -r5

-1

0

-1

0

0

0

0

-1

0

2

0

0

0

-1

0

0

_1

0

0

0

0

0

_1

-1

-1

0

o _1

0

-1

_1

0

_1

-1

-1

_1

-1

-1

0

•

-1

0 2

o o o o -1

o o o -1

0

0

0

0

0

0

0

0

0

0

0

0

0

o 0

0 0

0 0

0

0

0

0

0

~1

o

0

0

-1

0

0

0

0

0

0

0

0 0

0

o _1

0 -1

0

000000-r56

000000r5-r5

0

o

_1

o o

o o

0

0

39

0

0

0

0

o 0

-1

0

0

0

*6

_1

0

o

0

*3

*6

*3

0

0

0

0

0

0

0

0

0

0

0

-1

0

0

0

0

*2

*3

*6

0

0

0

0

0

0

-1

27

_1

_1

0

0

0

0

0

0

0

-1

0

0

0

*

_1

_1

0

0

0

0

0

0

0

0

0

-1

39

0

0

0

0

•

b5 b5

•

0

0

0

o

-1

_1

0

0

0

0

0

0

0

0

0

0

0

-1

_1

0

0

0

0

0

0

0

* -b5

O-bS

0

0

l-b21

0-b21

0

0

0

0

0

0

*2-d17

*3-d17

*2-d17

*2

0

0

0

0

0

0

0

0

88

-2

0

0

0

0

0

0

0

0

0 *6

0

2 -1

-2

2

-1

0

o

o o

-1

0

0-d17

0

o

o

o 0

2

_1

o

-1

* 3b5+2 -1

0

0

0

_2_200000000

o

3b5+2

0

0

o 0

0

0

-2000000

0

-1

0

0

000000

-1

o

o

o

_1

o

*

0

0

0

_1

-1

++

++

++

210

84

0

-3

16

0

32

0

0

0

0

81l

_4

_4

_Il

0

0

0

9

0

0

0

10

16

2

0

-3

-3

-6

-8

11l

-8

-6

0

-6

0

15

-9

0 -16"

10

0

16 -16 -16

20

++ 315 -21

++ 280

++ 378

++ 512

++ 336

++

0

4 -20

0

000

000

++ 420

+

+

+

++ 405 -27 -15

++ 160

+

10

16

10

0

29

000

++ 210 _14 +

10

16

-3 -19

18

70 -10

0

++ 336

oH

+

++ 189

++ 189

++

++

2 -10

-6

Il

0

0

0

-3

0

0

0

2

0

-2

0

-3

-3

7

0

-6

0

0

0

9

9

15

7

0

0

6

-6

10

0

8

-6

3

0 -9

10

2 -9 0

0

0

0

0-3

0

0

0

0 -16

0

-4

-4

0

0

0

0

o

0

0

-6

-2

0

0

1

0

_2

3

0

_2

2

0

0

0

0

_4

0

0

-2

_1

0

-3

0 -3

3

_6

0 -20

0

0

-3

0

-4

2

-2

0

3

0

0

0

3

-2

-2

0

0

0

0

2

0

0

3

-2

0

0

0 -2 -4

0

-2

_1

0

0

0

0

0

0

0

-2

-1

0

0

0

0

2

2

0

0

0

-2

0

_4

0

-2

2

-4

0

0

0

-2

0

0

0

1

-2

-2

0

-1

0

0

0 -3

2 o

0

0

0

0

0

0

0

2

0

0

0

o

0

0

0

-2

0

0

0

0

o 0

0

0

0

-2

5

0

3

3

-3

0

0

0

0

1

0

0

_1

-1

0

-1

0 _1

o

0

-1

0

0

0

0

0

0

0

0

-2

2

-1

0

2

0

2

o

0

0

0

o

2

-2

0

0

_4

@

@

@ @

@

-1

0

0

0

0

0

0

0

0

0

0 _1

0

0

0

0

0

0

-1

o

0-1

000

0

2

0

0

0

0

0

0

1-1

-1

0

0

0

0

0

0

0

0

1

o

X4

x,

0

0

0-1

0

0

0

-1

-1

-1

0 0

0

0 -1

0

0 0

o

o

-1

X14

_1

o

0

Xw

x"

X'8

x"

OX'6

XIS

Xl3

-1

X'2

o

XU

XIO

0

x" 0

0

0

Xu

000X25

0

_1

0

_1

-1

_1

x"

X36

x"

)(34

000X38

0

0

0

0

_1

o 0

00000X39

o

o

o o

OOOOOX33

x"

00000X3'

x~

00000X29

x"

OOOOOX27

-1

-1

00000X23

x"

OOOOOX2I

o

0

o

x. X9

_1

0

0

_1

N

x, " - "

x,

-1

0

0

_1

0

00000

0

0

0

0 0

o 0

0

0

o

_1

0 0

o

-1

o

0

o

-1

0

0

0

_1

o

0

0

0 _1

o 001 x, ,,-......

x,

OOOOOX,

-1

o

0

0

o 0

_1

-1

0

-1

o o

_1

_1

000

-2

2

-2

0

0

0

0

0

0

0

-2

1 -1

0

-1

_1

_1

0

2

-1

-1

-1

o

1

2

-3

@

-2

2 _1

0

@

o

3 -1

2

2

@

7 9 10 12 15 48 36 12 12 BE EF DG EH AD AB CE AC coo AD AD A£ AC coo AE BF CG 8H 12E 12F 120 12H 14A 18A 20A 24A 30C

@

2 -2

000

0

0

0

0

0

0

0

0

0

_1

0

0

0

4

2

4

-1

144 BF AF 12D

@

0 -2

-2

2

o

0 -1

0

0 -1

0

0

0

3-1

0

0 _1

30 AD AD 10D

@ @

-1

0

o 0

2

_1

2

3

-2

0 -1 2

@

96 32 8 A B D A A A 8c 80 8E

@

000

0

3 -1

3

0

0

0

0

-2

0

0

3 -2

0

0

-1

1 _2

0

2

1 -1

0

0

0

o

-3

2

1

36 BE BE 6L

@

2 _1

-1

_2

72 CE CE 6K

@

0 _1

2

-2

2

-3

-3

3

_1

1 2

3 -1

6 -6

2 -15

0

2 -5

0

-3

5

2

3 -1

8

-6

0

0

4 _4

_Il

0

0

0

9

0

0

0

-6

0

2

0

-3

-3

2 -6

0

-2 9

8

6

6

o

2

2 _4

0

6

15

6

6

1 -3

2 -2

-3

3

2

6 -3

9

6

5

-2

18 -2

6

-3

2

2 _2

2

o

++ 210

-6

2

13

-6

2

2

1 -3

o

5 _1

10

10

5

6 -2

14

10

8

++353-5-5-53

++ 151l

10

-2 -10

10

6 -10

78

42

++2159

++2040

18

0

0

_1

0

0

++

@ @ @ @ @ @ @ @ @ @ @ 1451 4 3 2 2 520 608 840 304 384 384 128 160 648 144 108 A A B B A 8 B AD CD AE BD A A A A A A A AD CD AE BD fUs ind 2D 2E 4E 4F 4G 4H 4I 6G 6H61 6J

-7

o

0

_1

0

0

@

98

-1

0

_1

0

0

000

_1

o

o

@

21 30 30 AA BBBAAB AAAABBBB B* 30A B*

@

105

-1

* 3b5+2

0

@

++

-1

o

0

@

++

3b5+2

0

@

2200000000

o

0

@

2

-2

o

@

2

_1

-2

-2

-2

45 AA AA

@

17 17 17 17 21 A A A A AA A A A A AA BI 15C 17A B*2 C*3 D*6 21A

90 AB AB

@

2

_1

-2

-2

90 A8 A8 15A

@

0

0

0

0

0

o

0

0

0

0

_1

_1

2

0

_1

0

0 0

_1

0

0

-1

_1

_1

0

0

0

0

0

-1

_1

2

0

0

1 _1

_1

0 -2

0

0

0

0

0

0

0

0

0-1

0

0-1

0-1

2

-1

0 -1

-1

1

-1

-1

o _1

0

-1

_1

_1

0-1

o

0

0

0

0

0

0

0-1

_1

o

2

-2

2

0

0 -1

0

0

-2

-1

0

1

r5 -1"5

-1

0

-2

0

0

0

-1

0

-2

0

_1

S* 10C 12A 128 12C

0

0

0

0

0 -2

o

0

0

0

0

0

o 0

0

0

0

_1

-1

_1

_1

0

0

0

0

0

0

0

0

-1

_1

0

0

0

-1

@

000

0 -1

0

2-1

-2

_1

2

-2

0

0

400

2

0

0

_1

-1

-1

-1

o

0

0

-1

0

0

0

0

0

_1

0-1

0

0

0

-2

-1

0

o 0

0

o

0

0

o

0-1

0

-1

-1

0

0

0

0

0

-1

0

_1

0 -1

0

0

1 _1

0

0

0

0

0

0

0

0

0

o 0

0

0

0

0

-2

2

-2

2

_1

0 -1

0

0

o

-2

2

0

_1

2

9A lOA

0

0

@

9 60 60 20 48 211 12 C AA AA AB AA CC DB A AA AA A8 AA AC CB

o

0

8B

B A

32

@

o

0

BA

A A

32

@

2

_1

7A

A A

0 _2

0

0

0

0

-1

_1

0

_1

6F

BC BC

@

0

2

-1

_1

0

0

0

0

0

0

0

2

0 _1

0-1

0-1

2-1

3 _1

_4

2 -1

0

0

0

0

3

3

-2

o

0 -1

-1

-1

-2

o

-2

-2

o

-2

-2

o

3 -1

3 -3

3

7

5 -1

3 -3

-1

4

-5

0

-3

-5

0

6

-3

-3

-3 -6

0

3

-9

2

2

0

2

_1

_1

2

6E

BB BB

3

-2

-5

-5

0

_1

1 _4

-4

0

0

0

0

0

0

_1

0

5 -3

0

2

2

2

_1

2

60

CA CA

_1

_4

0

0

0 -6

-4

2

2

2

2

-2

-11

5

2

6C

AB AB

5 _2

-1

-5

5

11

6

2

2

4

5

_2

4 _1

-1

6B

BA BA

5-1

0 -3

0 _2

0

2

-1

_1

-1

3

0

2

3 -1

3 _1

3 -1

-8

3

3

3

3

11

11

0

_2

-2

-2

-2

-4

0

3 -1

0

-6

-6

3 -8

0

0

0

0

0

0

'1

2

2

0

0

-6

12

0

-2

3 -1

3 _1

-1

-9

4

2

2

-1

11

11

11

0 _4

0

0

0

2 -2

-5

0

0

-3

0

0

If

_1

10

2

SA

40

6A

A AA A AA

C A

3 _1

2

-9

2 -6

2

-4

0

_2

0

0

0

6

-6

2

0 -5

0 -5

0

0

_4

6

-2

8

0

0

8

0

a

~

8

Ij

0

11

3 _4

-6

6

A

7 _1

-2

A

3 -10

_1

8

6

6

5 -7

6

6

6

6

0

9

6

10

-11

2

-3

9

3 -3

3

6 -3

A

21 -12

64

-8 -70

12

0

0 -36

8 -84

42

0

0

0

0

-9 -45

-9 -45

-9

0

15

15

75 -45 -21

40

60 -20

4760 -110

+ 4284

x"

b

x"

1428

0

-2 -21

64

+ 1190

x"

x~

70 -50 -10

4 63

15 _41

+

x"

1071

15 -111

+ 1071

21

x"

31 -17

1071 111

+

21

15

x"

-1

12

10 _21

-6

1071 -81

31

56

14

3 -35

12

12

42

-6

36

21

6

13

A

+

60

311

20

13 -11

4

20

20

3

2

-A

x..

60

5

114

357

204

204 -20

84

11

+

19

x.

51

10

+

2

x,

311

lA

A

+

x"

@

B A A A A A A A A A 2A 28 2C 3A 3B 3C 4A 4B lie

A

x,

ind

ppower pI part

+

x"

@

81146360111

197406720 320 080 072 480 080 648 536 536 192 64 t80 576 360 288 12 72 24 21

@

x,

~

00

0 8(2) and 3D 4 (2)

=2DlI (2)

Orthogonal group 0a(2)

Order = 197.1106,720 = 2 12 .311 .5.7.17

Mul t = 1

Out = 2

Constructions

a

Orthogonal

= PGO a (2) = SOa(2) = PSOa(2) =G.2

GO (2)

: all 8 x 8 matrices over F 2 preserving a quadratic form of Witt defect 1.

=G

for example x1x2 + x3xll + x5x6 + x¥ + x7x8 + xij; 0a(2) Presentation

See page 232.

Max imal subgroups

Specifications

Order

Index

1658880

119

2 6 :U ll (2)

11151520

136

2580118

765

1811320

1071

120960

1632

8160

211192

4320

45696

'68

11750110

Structure

G.2

Character

Abstract

Orthogonal

: 26 :U lI (2):2

la+311a+811a

N(2 6 ) = N(2A27B36)

isotropic point

S6(2)

:S6(2)x2

la+51a+811a

C(2D)

non-isotropic point

2 3 +6:(L (2)X3) 3 21+8 :(3 3 xA5)

: 2 3+6:(L (2)x3 ) 3 3 : 21+8 :<3 3 x35)

1a+811a+201lb+1I76a

N(2A 3 )

isotropic plane

la+311a+811a+1I76ab

N(2A)

isotropic line

(3 x A8 ):2

:S3 xS 8

la+811a+1I76ab+595a

N(3A)

°i(2)x06:(2)

L 2 (16):2

: L2 (16):4

N(2e. 3S. 5A. 15AB. 17ABCD)

04'(4)

(S3 xS 3xA5):2

: (S3xS3):2 x S5

N(3 2 ) = N(3A2B2)' N(2B.3A.5A)

Oq(2) xOij'(2)

L2 (7)

: L2 (7):2

N(2C. 3C, 40. 7A)

Tr1ality twisted group 3011 (2) Order = 211,341.312 = 2 12 .34 .7 2 .13

Mult = 1

<Xlt = 3

Constructions Steinberg

G : the centralizer in 0'8(8) of an outer automorph1sm of order 3. This leads to a description of G.3 as the automorphism group of a generalized hexagon of order (2.8). consisting of 819 "verticesll and 2457 "edges". Each vertex is incident with 9 edges. and each edge is incident with 3 vertices.

Jordan

G.3 is a subgroup of the compact Lie group F4cnn. the automorphism group of the 27-dimens1onal exceptional Jordan algebra. This consists of 3 x 3 Hermitian matrices

[

~C

b

5]

B

A

c

C

A

(a,b.cIA.B.C)

(a,b,c real)

over the real Cayley algebra with units i oo ' i O' ••• i 6 • in lOhich i oo = 1. i n+ 1 """7 i. i n+ """7 j. i n+ 4 ....., k 2 generate a quaternion subalgebra. G.3 is the automorphism groUp of the set of 819 images (roots) of the

0]

",,"potents[, ° o

0

0

[0 ° 0: (3)

000

~

0

~

[ks' , 'J ,:2 'Its'4, 'ijs,'It

(48)

Jis

~

011 2 2

(768)

~

under the group [2 11 ]. S3 generated by the maps taking (a,b.colA.B,C) to (a,b.cliAi.iB,CO, (b,c.a!B,C,A) and (a,c,bIJr.-C'.-g)

(*)

where i = :tin and s = ~!in. This group extends by (i O i 1 ••• 16 ) to a maximal subgroup of G. and by (i 1 i 2 1 11 )(i 3 i 6 i 5 ) to a maximal SUbgroup of G.3. The simple group G is generated by the llreflections" y ~ y + 1I«y.r)r - y x r) in the roots, where y x z is the Jordan product ~(yz + zy) and y.z is the natural inner product corresponding to Norm(y) = !(Norm(Yij». There are 2457 sets of 3 mutually orthogonal roots

(~).

and

each root is in just 9 bases. By restricting i in (*) to be :t;l, we obtain the SUbgroup L2 (7), while by restricting it to be ±1 or ti o ' we obtain the subgroup U (3). The normalizers of these groups are maximal subgroups of G, and 3 they are related to the complex and quaternionic reflection groups 2 x L (2) and 2 x U (3) respectively. 3 3 Remark: G.3 is a maximal subgroup of Thompson's group.

Maximal subgroups ll-der

Index

258048

8'9

86016

2457

Specifications Structure

G.3

Character

Abstract

steinberg

Jordan

1a+26a+324a+1I68a

N(2A)

vertex

root

la+324a+4683+1664a

N(2A 2 )

edge

base

G2 (2)

quaternionic

21+8 : L2(8)

: 21+8: L2 (8): 3

2 2 .[2 9 ]:(7 XS ) 3

: 22.[2 9 ]:<7:3xS ) 3 : U (3):2 x 3 3

COC) N(3A)

12096

171172

U (3): 2 3

3024

69888

S3 x L2 (8)

: 3

2352

89856

(7xL 2 (7»:2

: (7: 3xL2(7»:2

N(7ABC)

163072

31+

: H.3

N(3B)

1176

179712

7 2 :2A

N(7 2 )

2'6

978432

32 :2A 4

: 72 :<3 X2A 4 ) : 32 :2A 4 x 3

NOB 2 ), C(3D)

52

4064256

13:4

: 13: 12

N(13ABC)

1296

2 • 2311

1I

3

x L2 (8):3

88 35

complex

35

14

[89]

@

@

468 -12

637 -35

1053 -99

1274 154 -14

1664 128

1911 119

1911 119

1911 119

90

90

2106

2106

2106

2184-120

2457-135

2457-135

2457-135

2808 -72

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

x,

x,

x"

Xu

x"

x"

x"

x"

x"

x"

x"

x"

x"

x"

x"

Xu

0

-9

0

9

0

0

14

14

3822

3822

3969 -63 -15

3969 -63 -15

3969 -63 -15

4096

5096 168

+

+

+

+

+

+

+

x"

x"

x"

x"

x"

x,.

0

14

3822

+

-8

0

6

6

6

-7

-8

0

0

0

0

-9

-9

@

@

16

16

16

5

5

5

-8

-2

-2

-2

7

7

7

0

-7

-8

0

0

0

9

9

9

-6

-6

-6

0

0

0

-3

-3

-3

-8

6

6

6

7

7

7

0

0 -16

0

-7

-7

-7

6 -14

6 -14

0

-3

17

8

3

3

3

0

5

8

0

-2

14 -10

29

-7

0

11

11

11

8

13

0

8

6

6 -14

0

0

0

0

9

-9

0

9

0

0

0

3

3

3

-3

x"

8

5

0

15

0

0

0

2808 -72

8

8

8

18

18

18

0

0

-9

9

0

0

0

0

3

7

-2

-1

8 -10

14

0

+

x"

@

7 -11

0

9

9

-9

0

-3

5

12

7

7

9

x"

2808 -72

90

63

63

7

0

+

351

63

12

x,.

x"

x"

351

+

x,

351

324

+

X.

36

273 -15

+

x,

-6

7

-'I

196 -28

+

X.

-7

7

-'I

20

52

+

-1

x,

2

+

-6

+

26

@

@

@

@

0

-1

-1

-1 0

o

*4

*2

*2

*4 6y7-3&4

*4 6Y7-3&4

7

o

o

*2

7

0

0

o

o

0

0

o

0

0

0

0

-1

2

2

2

o 0

0 0 0

0 0

o

o

o

o

o

8

o

o

o

o

o

o

8

o

o

o

o

o

o

8

o 0

0

0

o

o

o 0

0

0

o

*4 8y7+3&4

*2

0

0

0

-1

-1

*2

o

o 2

2

-2

o

o -2

o

o

*2

@

0

0 0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

-1

0

0

0

2

-2

-1

0

0

0

o o

0

o

*3

0

*3

*3

*9

0

0

*9 013

*9 013

013

*40000

*4 y9-&4

*4 y9-&4

*2

0

o

o

o 0

0

o

o

o o

0

o

o

o

o

0

o

o

o

o

0

o

o

o

o

0

0

o

o

o

o

o

0

o

o

o 0

0

0

o

0

o

0

o

o

*4

0

o

0

0

0

0

0

0

0

-1

0

0

0

0

o

o

@

0

0

0

*2

*2

*4 Y9-&2

-1

-1

-1

0

0

0

0

0

0

-1

0

0

o

*4 Y9-&2

0 Y9-&4

0

0

0

*4

*2

o

-1

*2

7y7

*4

*4 8y7+3&4

-1 8Y7+3&4

-1

_1

0

0

0

_1

0

0

0

0

-1

o

*2

-1

*2

-1

-1

-1 y9-&2

-1

0

0

-1

7y7

*2

0

0

0

0

-1

_1

-1

0

0

-1

*2

o

-1

0

0

0 6Y7-3&4

0

0

0

o

o

o

0

*4

0

o

o

o

0

7

-2

-2

-2

-2

0

0

o

o

o

o

-2

o

3 -4

3

3

-1

0

-1

0

o

o

o

-1

*4

0

-1

o

o

o

-1

6

6

6

0

o

0

0

o

o

o

7y7

0

0

o

o

o

o

0

o

o

o

-1

0

o

o

-1

o

o

2

-1

o

o

y7+3&4 2

@

@

12 13 13 13 B AB A A A A A A A BB C*4 12A 13A B*3 C*g 54

@

o

2

A B*2

B

54

@

*4 -2

-2

-1

2

9A

A

54 B

@

*2

*4

2

_1

-2

2

0

32 B A 8B

@

*2

*4

*2

2

9

9

9

y7+3&4

0

7

7

7

2

0

o

o

o

0

3 -2

-2

3

5

3

5

@

3

5

A

B*2

A

7A

@

49 32 A A A A A A C*4 70 8A

1176

@

Y7+3&4

0

2

-1

-1

-1

1176 A

@

1176 A

@

o

-3

0

0

0

-1

-1

-1

2

0

-3

0

0

0

0

3

-1

2

3

0-3

0

-2

-2

-2

0

0

0

0

2

2

2

-1

_1

-1

0

2

-3

-3

0

-1

-1

-1

0

0

0

2

258 3 1 3 1 048 072 512 648 584 536 64 72 24 AAABBAAB A A A AAAABAAB A A A 2A 2B 3A 3B 4A 4B 4C 6A 6B

x,

pt part 1nd lA

211341312 p power

@

x,

~

0

0

0

0

0

-1

AA

0

0

-1

BA

0

0

-1

0

0

0

0

0

0

*2

*4

y7

.4

y7

*2

28 BA

Y7

y9

*2

*4

-1

0

0

0 0

0

0

0

o 0

0

0

0

0

0

,1

0

0

000

*2

*4

y7

0

0

0000000

o o

0

-1

-1

-1

3

5

-1

3

4

3

-1

-1

3

-2

-1

3

-1

0

0

0

0

_1

0

o

-1

-1

0

-1

0

0 -1

0-1

o

-1

-1

-1

0

X6

Xs

X4

x,

x,

+

+

+00

+00

+00

+00

+00

0

0

8

14

27

7

27

0

0

8

5

0

-2

0

-1

3

0

0

8

-2

0

0

0

-2

3 -3

7

3

0

0

0

0

2

0

0

0

-1

-1

0

0

0

0

0

2

3

-1

-1

0

0

0

2

-3

-1

-1

0

0

0

2

-1

-1

3

0

0

0

-1

0

0

0

0

0

-1

0

0

0

0

o

-1

o

-1

0

0

o

0

-1

-1

0

0

0

0

-1

XIS

XI4

XI3

x"

Xn

x"

x,

x,

0

0

0

0

2

0

0

0

2

0

0

0

0

0

0

-2

0

0

-2

0

0

0

0

0

0

0

0

0

0

0

0

0000000

0

0

-2

0

0

0

x,.

X,s

Xu

x"

X22

X21

+

0

0

0

0

0

0

0

0

0

0

0

0

0

0

OX28

0

0

0

-6

o

0

0

0

6

x" +

0

0

+

+

42

+00

x"

x"

Xt8

*4

y7

*2

-y7

*2

*4

-1

*4

-y7

*2

-2

0

0

0

0

0

o

0

X34

+005620_2_22_42000000X3S

0

o

0

o

0

o

-8

o o o

64

+00

o o

x"

x"

o 0

0

0

0

2

o o

0

0

-1

2

2

-1

X31

o

0

3

-1

-1

-1

2

x,

o

o

0

@

o o 0

@

x"

0

0

@

o

o

0

@

x"

o o

0

@

o

o

0

0

0000000

o o

y9

0

o 0000000

o

o

*4

0

0

*2

*4

y7

*4

-y7

*2

-1

*2

*4

-y7

o *2

*2

0

@

x"

o o

Y9

0

@

o

o *4

*4

@

o

o

o

o

3

0

+0027

@

+00000000000000X7

-3

3

21

-1

-1

0

+00

-2

-2

-1

7

7

8

+00

+00

+00

+00

@

6 7 8 8 4 48 24 18 96 48 32 CBOBBCB CB CA EC DA DC BB DA CBDBACB CB CA DC DA DC CB CA 60 6E 90 12B 12C 12D 12£ 180 210 24A 24B

@

x"

o o

*2

y7

*2

*4

*2

*4

-y7

-1

-y7

*2

*4

o

@

o

o

o

@

o

o

o o

o

o

o

o o

o

o

o

*4 *2

o Y9

-1

o

-1

28

*4 -y7-1

*2

@

12 096 216 192 A A CA AA A A CA CA C*3 fus ind 3C 30 6C

@

*4 -y7-1

-1

o

-1

28 CA BA B*5

@

o o

o

*4 -Y7

*2

*4

Y7

*2

*4

0

0

o

*2

*4 -y7

*2

*4

y7

*2

0

0

o

0

0

0 -y7

0

0

0

0

0

0

o

0

0

0

0

0

0

0

0

0

o

0

0

0

0

0

0

0

0

0

o

*4

-y7

*2

-y7

*2

*4

-1

2y7+&4

*2

*4000000

*4

y9

*2

-1

0

*2

-1

o

-1

*2 -Y7-1

*4

*2

*4

-y7

*4

-y7

*2

-1

*4

2y7+&4

0

-1

-1

0

CA

-y7

*2

*4

*2

*4

-y7

-1

*2

*4

*2

*2

o

o

o 217+&4

*4

o

o

-1

0

o

2

o

-1

o

o y9

2

2

0

-1

-1

0

BA

2000000

y7-&4

*2

.4

o

o

o

21 BA

000000

0

o

-1

o

o -1

2

.4

y7-&4

*2

2

*2

.4

y7-&4

-1

-1

o

o

o -1

-1

o -1

-1

21 AA

AA

21 CA 28A

18 AA

CA

AA

CA

18 CA

@

C*3 laA B*7 C*S 21A B*2 C*4

18 BA

56 AA

@@@@@@@

-1

56 CA BA B*5

@

-1

56 BA AA 14A

@

"-"

N

",-......

o ..j:::;.

VJ

L3 (11)andA 12 Linear group L (11) ;; A2(11) 3 Order = 212.427.600 = 2 4 .3.52 .7.11 3 .19

Out = 2

Mult =

Constructions

Linear

GL 3 (1l)

= 10

x G : all non-singular 3 x 3 matrices over F 11;

PGL 3 (11) ;; 3L (11) ;; P3L (11) ;; G 3 3

Maximal subgroups Order

Index

1597200

Specifications

Structure

G.2

Character

Ab stract

Linear

133

11 2 : (5x2L 2 (11) .2)

l1l+2 : 10 2 ,

1a+132a

N(11 2 )

point

1597200

133

11 2 : (5x2L 2 (11) .2)

[

1a+132a

N(11 2 )

line

1320

160930

L2 (11):2

600

354046

10 2 :3

: L2 (11):2 x 2 : 102 :D 12

N(2A 2 ). N(5 2 )

base

399

532400

133:3

: 133: 6

N(7AB). N(19A-F)

L 1 (1331)

168

1264450

L2 (7 )

: L2 (7):2

3

2L 2 (11).2.2

0 (11 ) 3

The characters of L 3(ll) are not printed

Al ternating group A 12

Order

= 239.500.800 = 2 9 .35 .52 .7.11

Mult = 2

Out = 2

Constructions Al ternating

3 12 =- G.2 : all permutations on 12 letters; A12 =- G : all even permutations; 2.G and 2.G.2 : the Schur double covers

G =- <x,' ••• 'X1olxr=(XiXj)2:1); G.2 =-

Presentations

Remark: A12 is the largest alternating group involved in the l-bnster group F,. It arises in several large groups involved in F" notably the Fischer group Fi

24

and the Harada-Norton group.

Maximal subgroups

Specifications

Crder

Index

19958400

12

All

: 3 11

1a+ 11a

3628800

66

S10

: S10 x 2

1a+ 11a+54a

C(2D)

duad

1088640

220

(A9x3):2

: S9 x S3

1a+ 11 a+54a+ 154a

NOA)

triad

518400

462

(A6xA ):22 6

: (3 6 x36):2

1a+54a+132a+275a

N(2A. 3A. 3B. 4A. 5A)2

bisection

483840

495

(A8 xA 4 ):2

: S8 x S4

1a+11a+54a+154a+275a

N(2A 2)

tetrad

302400

792

(A 7 xA 5 ):2

: 37 x 35

1a+11a+54a+154a+275a+297a

N(2A.3A.5A)

pentad

95040

2520

M12

95040

2520

M12 2 6 :3 3 : 3 4

: S4wrS 3

N(2 6 )

structure

G.2

I

L2 (11):2

Character

Abstract

AI ternating point

1a+132a+462a+1925b

S(5.6.12)

1a+ 132a+462a+ 1925b

3(5.6.12)

41472

5775

23040

10395

25 :S 6

: 26 :3 6

N(2B)

15552

15400

34 :23 '3 4

: S3wr3 4

4 N(3 )

= N(2AgB27C27)

trisection hexasection

= N(3A4B12C8D16)

quadrisection

~EJ43

BEJn 43 37

[91]

Al2

@

@ 161

@ 23

@

@

@

@

@

45446

@

@

@

@

@

@

12

2

@

@

@

@

@

239500800 280 040 608 320 460 972 486 880 38' 192 128 600 50 440 576 1411 144 144 P power A A A A A A A A C A C A A AA AC BA BB BC pi part A A A A A A A A A A A A A AA AC BA BB BC ind lA 2A 2B 2C 3A 3B 3C 3D !lA .C 40 SA 58 6A 6B 6C 6D 6E

.

x,

54

22

6

6

Zl

9

55

19

-5

-1

28

10

x,

..

132

20

20

11

6

6

6

x.

+

154

42

-6

10

119

7

4

x,

+

165

21

5 -11

57

12

3

3

5

x,

+

215

55

15

11

50

5

-4

5

5

x,

+

291

49 -15

9

21

9

0

0

x"

+

320

6ll

0

0 104

14

..Ji

..Ji

16

0

0

0

x"

+

330

-6

10

15

-3

6

-6

-2

2

-2

x, X.

[92]

@

@

@

@

@

@

@

@

@

@

@

@

@

@

36 CB CB 6F

18 420 16 DC A B DC A A 50 7A BA

16

27

D

D

A

A

8B

9A

27

27 120

10 AA BB

11

11

2'

@

@

@

@

@

@

@

28 180

45 20 21 60 35 AB AA AA AAA AA A A AA BA AB BC AA A A AA BB AB AA AA AAA AA 98 C.. lOA lOB l1A Bn 12A 120 12C 12D 14A 15A 158 20A 21A 30A 35A 0

D

A

72

72

@

'8

A AA CA BB CC AA

AA AA

@ 35 AA AA

en

x,

• • • •

x,

@

1

11

7

-1

3

-6

8

18

5

-1 0

2 0

-3

5

3

-1

6

•

-1

0

3

_1

10

2

2

2

14

-1

7

3

9

3

-3

-1

15

0

4

-4

-2

-2

2

0

4

0

4

-8

2

2

-2

2

2

-2

10

-2

2

-2

111

-1

9

3

-3

20

0

-3

o

2

-2

-1

0

0

10

2

3

_1

0

-3 -13

2

1

3

-3

-3

35

0

4

0

-2

0

0

15

0

-6

6

3

-3 -1

3

5

330

0

0 -1

4

5

-1

000

-1

-1

0

0

0

0

2

-1

0

0

_4

_1

-1

0

0

0

_1

0

0

0

0

0

5

0

0

-1

-1

-1

-1

0

o

0

0

0

0

-1

0

2

0

_4

-3

-1

x"

2

-2

-1

-3

x..

+

616

56

211

8 112

-5

4

-5

4

0

4

0

21

-4

8

-1

3

x"

+

891 111

-9

3 162

0

0

0

9

-5

-3

-1

21

x"

+

945

49 -15 -15 189

9

0

0

x"

o

1050 -10

10

10 105

15

-3

-3 _10

-2

-2

-2

0

0

5

-1

o

0

x"

o

1050 -10

10

10 105

15

-3

-3 -10 -2

-2

-2

0

0

5

-1

o

0

x..

+

1155

91

-5

21

21

-6

-6

x"

+

1320

56 -40

8 -48

15

6

x"

o

1320

8

40

8 -84

6

-3

-3

0

0

0

0

10

x"

o

1320

8

110

8 -84

6

-3

-3

0

0

0

0

10

x"

+

1408 128

0

0 112

11

4

4

0

0

0

0 -22

X~

+

1485

29

45

-3 -21

18

0

0

5

-3

x"

+

1650

50 -30

2 -15 -15

12

3 -10

-2

-2

x"

+

1128 -6ll

18

0

0 -16

0

0

x"

+

1925 105 -15

13 140 -25

..Ji

-1

-5

-3

-1

Xu

+

1925 -35

45

21

35 -10

8

8

5

5

-3

-9

15 189 -18

0

0

-5

1

0

0 216

x"

+

2019

X~

+

2112

64

x"

+

2376

56

x"

+

2613 111

x»

+

2910

34 -30

X~

+

3080

56 -40 -24

x»

+

3520 -64

X~

+

3564-132 -36

1

0

0-120

45

0

o 0

_1

0

0

0

0

0

3

0

0

0

-2

0

0

0

-1

0

0

0

0

0

0

0

_1

0

0

-2

o

0

0

-2

0

0

-1

2

_1

4

0

-1

-3

0

-1

-1

0

0

-2

0

0

5

-7

5

3

-1

o

-1

5

0

0

0

0

13

-2

_4

0

2

0

0

0

0

-1

0

0

o

0

0

4

3

-3

-3

0

0

-5

3

-2

0

0

0

0

0

3

-1

14

-1 _11

-3

-2

0

0

0

0

0

-1

1

4

0

-2

0

0

0

0

5

0

_4

0

-1

3

3

0

0

-4

0

0 0

0

0

0

0

-9

-3

3

-6-216

9

0

0 -111

6

2

56

11

2

2

4

0

-8

10

-8

10

16

12 162

0

0

0

6

6

2

-21-200100006 _2

35

0

4

0

4

0 -35

0

_4

0

0

0

0 -15

0

-4

0

4

0

4

_1

-3

0

0

0

0 -14

-2

10

-2

2

-2

0

-2

10

-2

2

-2

0

2

-2

2

28

-6

0

0

0

0

-2

-1

_1

-1

0

0

0

0

_1

-1

0

0

0

2

o

0

0

0

0

0

-1

0

0

-5

-2

o

_1

_1

_1

-1

o

0

0

-2

o

o

0

0

-2

-2

0

0

-1

0

0

0

0

-2

0

000

0

3

-1

-3

-3

0

0

0

0

0

0

0

-1

0

0

0

0

-2

8

0

ub210000

-1

o

0

0

_1

_1

_1

0

0

0

0

0

2

-1

-1

X20

b35

**

X21

**

b35

X22

_1

-1

X23

o

X~

0

o

-3

0

-3

0

_1

-1

-1

0

0

X30

o

",

0 _1

0

x" 0

0

X33

o

0

X34

-1

X3s

-1

-1

X~

0

00_200X37

-1

0

0

0

0

0

0

0

0

X38

-1

0

0

0

0

0

0

0

0

X39

o

0

3

0

-1

o

o 0

0

0 -1

-1

-1

o

0

0

0

0

0

0

2

-1

-2

8 8

9 18

9 18

9 18

20

20 20

11 22

11 22

211

24

12

211

28

15 30

15 30

40 40

21 42

60

35 70

35 70

2

-1

-1

-1

_1

0

0

0

0

0

4

_2

0

_2

0

-1

_1

u

0

0

0

0

0

4

0

0

0

0

0

4

-

32

0

0

0 -16

8

2

_4

0

0

0

0

-8

2

0

0

0

0

0

0

0

4

0

0

0

0

X~~

o

160

0

0

0 -56

16

-2

-2

0

0

0

0 -20

0

0

0

0

0

0

0

0

6

0

0

o

0-bl1

X%

o

160

0

0

0 -56

16

-2

-2

0

0

0

0 -20

0

0

0

0

0

0

0

0

6

0

0

o

0

x" x..

o

104

0

0

0-160

20

2

2

0

0

0

0 -36

_1

0

0

0

0

0

0

0

4

0

0

-1

-1

-1

0

15

0

0

0

0

0

0

0

0

0

0

o

104

0

0

0-160

20

2

2

0

0

0

0 -36

-1

0

0

0

0

0

0

0

4

0

0

-1

-1

-1

0 -15

0

0

0

0

0

0

0

0

0

0

x,"

+

1408

0

0

0 112

4

4

4

0

0

0

0 -22

-2

0

0

0

0

0

0

0

-6

0

0

o

0

0

0

0

0

0

0

0

2

_1 rl0

0

X~o

+

1408

0

0

0 112

4

4

4

0

0

0

0 -22

-2

0

0

0

0

0

0

0

-6

0

0

o

0

0

0

0

0

0

0

0

2

-1-rl0

x"

+

1760

0

0

0 128

8 -16

-4

0

0

0

0 -20

0

0

0

0

0

0

0

0

-11

0

0

-1

0

0

0

0

0

0

0

0

0

-2

-2

x"

o 1760

0

0

0-232

8

2

-4

0

0

0

0 -20

0

0

0

0

0

0

0

0

_11

0

0

-l-b27

u

0

0

0

0

0

0

0

0

0

-2

x"

o

1160

0

0

0-232

8

2

_11

0

0

0

0 -20

0

0

0

0

0

0

0

0

_1I

0

0

-1

u-b21

0

0

0

0

0

0

0

0

0

-2

X~

+

1160

0

0

0 104

-4

..J.i

5

0

0

0

0 -20

0

0

0

0

0

0

0

3

_11

0

0

-1

-1

-1

0

0

0

0

0

0

0

0

0

-1

-1

-1

0

0

0

0

0

0

0

0

-1

X42

00000000X4J

X~

2

X40

x..

-1

o

0000000004_20

u_b11

X2S\

X29

-1

0000000_2

b21uOOOO

X19

-1

o _1

-2

0

0

2

0

o

0

-1

0

11

7 14

XI'"

_1

0

0

-2

6 6

0

0

o

0

0

12

0

0

0

-1

0

6

XIS

0 X28

0

0

0

12

0

0

0

0

-1

12

o

0

0

0

-1

6

X14

0

0

_1

-2

-3

12

0

X27

0

0

-3

5 10

o

0

0

-1

5

5 10

XI3

0

0

0

5

4

0

0

0

0

0

8

0

Xu;

0

0

0

4

x"

-1

-1

-1

-2

8

X10

_1

o

2

0

3 6

-1

0

0

3 6

-1

0

0

3 6

-2

-1

0

0

-1

0

-2

0

0

0

o

-1

0

0

-2

0

0

-1

_1

0

-1

-2

-2

o

-3

2

0 -10

0

0

-2

0

-2

0

2

-1

-2

0

8

-5

2

0

0

-2

2

-1

0

8

5

-1

0

2

0

-2

-3

2

0

0

0

6

o

-3

0

0

15

0

2

2

-1

_1

0

0

3 6

_1

0

4

2

_1

o

0

4

2

0

0

4

2

0

0

1 2

2

o

0

0

ind

0

0

0

1-210

0

0

0

-5

0

-6

0

o

-1

0

-1

6

2

-1

0

OOOOX12

0

~

0

-8

_1

x,

-1

0

0

0

0

-1

o

0

-1

0

0

o

0

0

0

3

0

0

0

0

-1

0

0

0

-2

-1

0

0

0

-8

_1

-1

2

2

64 -32

_1

0

-1

0

-4

2

0

0

-1

0

-1

-1

0

-1

0

_1

0-1

0

-1

0

2

-3

0

4

0

0

X8

0

0

0

0

0

_1

0

0

6

0

0

0

0

-6

o

0

-3

0

2

0

0

o

0

3

15

0

0

0

_1

3

X7

-1

0

_1

-3

-1

-2

0

0

_1

-1

-2

0

0

9

2

0

0

0

0

220

_1

0

0

0

X6

2

0

0

0

0

4

0

-5

0

0

0

0

0

-3

-1

_1

0

0

o

0

2

_1

-2

3

0 -10

35

-2

3

-3

0

_1

o

-2

_1

0

5115

0

0

-9

+

0

0

30 -18-189

x"

0

0

0

0

0

0

1

0

0

2

x~

00000000XI8

0

-3

-1

-1

0

-7

_1

_1

0

0

-3

bll

u

0

-20

-1

0

-1

0

o

-2

_2

x, X.

0

2

0

o

0

00000000X17

0

0

-1

0

0-1

~

0

0

0

2

X2.

-1

o

0

0

_1

-1

.*

0

0

-1

-1

0 bl1

0

0

0

0

_1

0

0

-3

-1

3

o

0

-6

2 -2

0

_1

-1

0

0

_1

-1

-1

-4

0

0

0

2

o

0

0

0

-3

o

0

o

0

-3

000

-4

-5

0

0

0

-6 -35

5632

0

8

10

+

0

-2

-5

x"

0

2

0

15-162

0

2

-6 -35

11455 -21 -45

0

-2

10

+

0

-2

16 -16 168

x..

0

2

3696-112

14

-1

2

3850 -10

4158

0

-4

o

+

0

-4

+

x"

o

_4

x"

X~

0

0

X~

o 3850 -10

0

-4

-4

0

0

0

0

0

0

-3

o

0

0

0

3

-2-200

0

-1

o _1

0

6-600000-5

o

0-1 -1

o

0

6 -16

0

-1

8

0

9

6 -12

-1

-4

-9

24 -24 216

0

-1

2

-1

0

-1

1

-1

-1

-5

1

0

2

-1

2

0 -15

-2

2

2

_4

0

-1

2

0

0

0

2

-3..Ji

0

0

2

-3

0

0

-6

0

0

_1l

-6

-1 -35

0

0

3

-1

0

0

3

1

0

0

12

-1

0

-1

3

3

0

_1

21

-3

-1

0

11

84

0

-1

_2

14

-1

0

0

-2..Ji2

0 _1

-1

0

30

-3

-1

o

14

-3

2

o

462

0

0

0

o

0

2

0

0

462..Ji2 -10

0 -11

0

-1

0

+

35

0

2

_1

+

-1

-1

2

-3 -1

-1

6-1

x"

3 _14

2

X~

000

x.,

000

X%

o

-1

-1

X~7

o

_1

-1

X~8

0

-1

_1

X49

0

0

-1

-1

Xso

0

2

0

x"

-2

0

-1

0

x"

-2

0

-1

0

x"

1I

o

_1

0

X~

0

4

o

-1

0

x"

0

0

x"

+

1160

0

0

0 104

-4

-4

5

0

0

0

0 -20

0

0

0

0

0

0

0

-3

_11

0

0

X~

+

1192

0

0

0-2211

-8

-8

10

0

0

0

0 -28

2

0

0

0

0

0

0

0

0

0

0

o

0

-1

-1

0

0

0

0

0

_4

2

0

0

0

XS6

x"

o

2112

0

0

0 -48

24

6

6

0

0

0

0

32

2

0

0

0

0

0

0

0

-2

0

0

0

0

0

0

0

0

0

0

0

0

0

0

2

_1

0

O...b35

..

XS7

x"

o

2112

0

0

0..Ji8

211

6

6

0

0

0

0

32

2

0

0

0

0

0

0

0

-2

0

0

0

0

0

0

0

0

0

0

0

0

0

0

2

_1

0

o

u-b35

X58

XS9

0

2640

0

0

0-168

12

-6

-6

0

0

0

0

20

0

0

0

0

0

0

0

0

-6

0 212

0

0

0

0

0

0

0

0

0

0

0

0

2

2

0

0

0

_1

-1

XS9

Xw

0

26110

0

0

0-168

12

-6

-6

0

0

0

0

20

0

0

0

0

0

0

0

0

-6

0-212

0

0

0

0

0

0

0

0

0

0

0

0

2

2

0

0

0

_1

-1

XW

x"

+

38110

0

0

0

24 -12

6

0

0

0

0

20

0

0

0

0

0

0

0

0

4

0

0

0

0

0

0

0

o

0

0

0

0

-4

-1

0

-2

0

_1

-1

X61

8

2

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

2

-1

-1

0

0

OOOOOOOOOOOOX62

-8

-2

0

0

0

0

-8

2

0

0

0

0

0

0

0

4

0

0-2

12

12

-6

0

0

0

0 -28

0

0

0

0

0

0

0

96

x"

+

5600

0

0

0-280 -40

X~

+

5632

0

0

0

XM

+

1392

0

0

0 336

XM

+

1176

0

0

0

64 -32

0

36

0000000004-20

2

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

-4

0

0

0

0

0

0

0

6

0

0

0

0

0

0

0

-1

-1

0

0

0

o

-1

0

0

0

0

0

-11

2

0

000

0

0

0

0

000

_1

X<'>J

0 X64

x.,

@ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ 3628 17 3 80 15 2 800 280 840 640 768 384 192 64 120 160 432 432 432 162 54 54 48 96 32 600 120 A A A A A A B C AD BE AE BD BE DD CE DE SF B B AD AE A A A A A A A A AD BE AE BD BE DD CE DE BF A A AD AE fus ind 20 2E 2F 4E !IF 4G 'H 41 6H 61 6J 6K 6L 6M 6N 60 6P BC 80 lOC 100

@

@

@

50 10 720 BD BF AE BD BF AE

++

x,

++

9

5

x,

++

36

12

X.

++

x,

7

_1

ij

20

11

35

7 -5

21

-3

++

48

8

0

-8

8

0

x,

++

84

20

11

28

_4

x,

++

75

-5

-5

35

x,

++ 125

25

5

15

@

@

@

'8

2'

6

'2

@ 9

@ 60

@ 12

@ 14

@ 30

@ 15

@

21

@

30

AG GF FH AD AM AE CC AE ACH BDI ASH ABE BE AG BF CH AD AD AE AC AE ACH BDI ASH ABE 10E 10F 12E 12F 12G 12H 121 14B leA 20B 24A 28A 30B 30C 42A 50A

3-1

CE

•

o

-1

-1

2

0

2

0

0

o

0

-1

-1

6

5

2

3

-1

0

-1

2

3

-1

4

0

0

15

9

3

3

3

0

0

0

2

2

6

2

-1

14

10

-2

2

-2

-1

-2

3

_t

5

-3

0

0

6

0

-2

0

0

0

6

_4

2

0

_4

3

2

_1

0

0

0

-2

-2

-2

0

1.1

-2

0

2

0

-1

0

2

11

0

0

21

5

5

3

-1

3

2

_,

-2

-2

4

0

-1

-1

-1

0

0

0

-2

3

-5

_1

_1

15

10

-5

0

11

3

o

0

0

0

5

2

o

-1

-2

0

0

7

3

3

o

0

0

0

0

3

-1

-1

0

1.1

0

-1

20-545

-2 -2

-1

-3

-1

-1

x,

++ 117

21

5

-7

-7

15

-9

3

3

3

0

0

0

_1

_1

-1

-3

2

x"

++ 160

16

0

611

0

0

0

0

34

16

_2

-2

-2

-2

-2

-2

0

0

0

5

o

Xu

++

90 -22

10

34

2

-6

-2

_2

6

5

2

3

-1

0

-1

2

x"

++

42 -111

10

14

_2

-2

2

-2

0

4

3

-5

-3

x"

++

42

2

10 _14

2

2

6

-2

0

5

_4

3

-1

-3

x..

++ 224

8

16

56

8

0

0

0

14

5

2

-7

5

_1

x"

++ 351

27

-9

69

-3

-3

_1

36

0

0

0

0

0

0

0

x"

++ 315 -21

-5

91

-5

-5

3

3

21

9

-3

-3

-3

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

o

-2

-2

-3

0 0

5 4

-2

0

-3

2_1

o

0

-3

2-1

o

0

-1

0

0

-3

2

0

2

-3

2

0

-2

3

-1

1

_4

-1

-1

o

o

0 -1

-1

00-53004

o

o

5-1

0

0

o

_1

-2

0 0

-2

_1

-1

-5

-1

0

0

0

0

o

0

0

0

-1

0

-1

x"

++ 315

19

-5 -49

_1

_1

3

_1

-5

-1

0

0

11

x"

++ 240

8

0 -56

-8

0

0

0

6

5

2

-3

5

-3

2

-1

-3

0

0

-5

3

0

0

4

0

0

0

000

0

0

0

0

0

0

0

0

0

0

0

0

o

0

0

0

0

0

0

O'

0

-1

_1

-1

0

0 -1

o o

3

-1 0

0

0

o

0-1

-1

0

0

-1

0

0

0

-1

_1

_1

XII

o

2

X12

-2

XI)

0

0

o

0

0

0

0

0

_1

0

0

-1

o

0

-1

0

0

0

0

0

0

0

0

-1

0

0

0

o

0

x.. _1

-1

0

0

0

0

0

+

0

21 -11

-3

-1

_1

-1

o 0

0

0

0

0

0

o

_1

o

2

0

-1

0

0

o

-1

-1

X20

0

0

0

0

0

0

0

0

0

X21

0

x"

x"

x" x"

++ 1148

X~

++ 225 -15 -15 -55

x"

++ 300

32

0

0

20 -20 -20

0

0

9 -4

0

0

-3

-3

-4

0

28 -16

_4

4

_4

_2

2

2

0

0

0

-2

2

-2

0

0

0

0

-1

-1

o

0

0

0

5

2

2

2

0

0

-7

-3

-2

3-1

o

0

0

15

0

3

-6

-6

0

0

0

0 -15

5

5

3

-1

3

2

-1

6

0

0

0

0000-5

x"

++ 288 -48

0

64

0

0

0

0

-6

0

6

6

x"

++ 525

25

5

35

-5

-1

3

-3

0

-5

4

-3

x"

++ 175

15

15

35

3

3

3

_1 -35

0

-3

-2

6

4

0

0

0

x"

++ 441 -15

9

77

-3

5-3

-21

0

3

-6

-6

0

0

0

0

x"

++ 288

0 -64

0

0

0

0

-6

16

-2

6

-2

0

-2

-2

0

0

0

x"

++ 576 -24 -16

56

8

0

0

0

6

-9

_6

3

3

0

0

0

_1

0

0

0

3

-1

-3

-3

o

0

-5

-1

5

:

16

3 -2

0 _1

++ 567

27

15 -63

9

-3

-3

3

0

0

0

0

0

0

0

0

90

-1

-1

0

-7

:-6

10 -34

-2

6

-6

_2

6

9

_6

3

3

0

0

0

X~

++ 560 -24

0 -56

-8

0

0

0

14

-9

_6

-1

3

2

0

0

3

0

0

)6,

++ 160 -48

0 -64

0

0

0

0

34

0

6

-2

6

-2

0

0

0

0

0

5

-3

X~

++ 216

48

0

8

0

0 -36

0

0

0

0

0

0

0

0

0

0

-4

0

x"

++ 336 -48

16

0

0

0

0

0

0

-6

0

6

-6

3

0

-3

-2

0

0

6

2

0

0

0

0

0

0

0

0

0

0

0

0

0

000

0

0

o

0

+

0

0

0

o

0

-1

-1

0

0

0

4

-2

0

0 -10

-1

2

o

0

o

0

0

o

0

-1

_1

-1

2

-1

0

-2

_1

_1

o

0

0

-1

o

0

-1

0

-1

0

0

0

o 0

0 X2S -1

0

0

0

o

-1

0

0

-1

_1

o

0

-1

0

2

0

o

0

2

0

0

0

0

0

0

0

0

0

0

0

0

-4

-1

0

o

-1

0

o

o

0

-1

0

XM

0

-1

0

o

0

-4

0

X2)

0

_1

0

-1

0

-1

2

o

00.

0

xu;

0000000000X27

-20-42000

++

0 -24

0

0

000052-3000 -'I

x"

-4

2

0

0

0

-1

6

0

2

0

0

-1

0

-3

-1

o -1

-2

X29

x"

-1

0

X211

h,

0

0

-1

-1

0

X32

x"

0

0

-1

o

o

-1

-1

0-1

x"

0

-1

-1

0-1

X~

-1

X34

o

0

0

0

0

0

0

0

0

0

0

-1

0

0

X37

0

0

0

0

0

0

0

0

0

0

0

0

0

0

x~

h,

x" Xw

++ 252

12 -20,-23

x..

++ 135

27

15

x.,

++ 768

0

0

x.,

~7

x"

x"

x"

XIS

x"

x"

X32

Xs

X10

0

0-1

0

-1

_1

-1

x,

-1

2

o

Xs

o x,

-1

-1

_1

o x,

0

0

X2

X.

0

000 0

-1

o x,

0-1

o

-1

000

0

o

o

_1

-60-200

-3

0

+

72

@

x,

x,

x"

@

A 12

++ 525 -15

4

-4

0

0 -21

9

3

3

3

0

0

0

21

-3

-7

-3

_1 -36

0

0

0

0

0

0

0

0

-3

0

0

0

0

0 -48

0

0

0

0

-6

0

0

0

0

5-105

-2

-2

0

2

2

2

0

_1

-1

_1

5

-3

0

0

6

0

2

0

o

0

0

0

2

0

2 o

o

-1

0

_1

-1

0

_1

x..

0_1

80-2000000-2000020 0

X40

o

X~2

0

X~3

-1

3

3

o

15

0

-3

-3

3

0

3

-1

-3

o

0

0

0

0

-3

0

-1

0

0

0

0

0

0

0

0

fus ind

2

4

2

8

8

8

8

4

6

12

12

6

12

6

12

12

6

8

8

10

20

10

10

24

24

24

24 24

211 24

14

18 18

40

24 24

56 56

30

60 60

X~

00

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

o

0

0

0

0

0

0

0

i6

0

0

0

0

0

0

0

0

0

x"

+

0

0

0

0

0

0

000

0

0

0

0

0

0

0

0

0

0

0000000000000000

0

OX~s

42 120 42 120

X~

X~

x"

+

0

0

0

0

0

000

0

0

0

0

0

0

0

0

0

0

0

0000000000000000

0

OX~7

x..

x..

x"

OX~\I

+

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0000000000000000

0

x"

++

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

o

Or30

x"

+

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

000

x"

x" 0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

XSl

000000000000000000XS2

x"

x" X~

X~~

+

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

o

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

x~

x"

x"

x"

++

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

o

0

0

0

0

0

0

0

0

0

3

0

0

0

0

0

0

0

XS6

x"

+

0

0

0

0

0

0

0

0

0

0

0

0

0

0

000

0

0

o

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

XS7

0

x" x"

X~

+

0

0

0

0

000

000

000

0

0

0

0

0

0

000000000000000000X59

x" x"

++

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

o

0

0

0

0

0

0

0

0

0

0

0

0

0

0 r15

0

0

X61

x"

++

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

o

0

0

0

0

0

0

0

0

0

0

0 2r3

0

0

0

0

0

X62

x"

++

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

o

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0 r21

0

X63

XM

++

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

000000

02r6

0

0

0

0

0

0

0

0

0

0

XM

x"

++

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

o

0

0

0

0

0

0 rt 4

0

0

0

0

X6S

x"

0

0

0

0

0

0

[93]

M 24 .

Sporadic Ma thieu group M

24

Order

= 244.823,040 = 2'°.33.5.7.11.23

Out

Mult

Constructions Golay (2) code

G : the automorphism group of the (extended) binary GoIsy code, the unique dimension 12 length 24 code over F 2 of minimal weight 8, which is defined in several ways below. The code and cocode (words modulo the code) have

weight distributions 0'8759,22576,675924' and 0',24 2276 32024 4 '771 respectively. The support of a code word is called a C-set, and C-sets of size 8 and 12 are called Qctads and dodecads respectively. A pair of complementary dodecads 1s a duum. and a triple of disjoint octads is a trio.

Elements of the cocode are called C*-sets. Their minimal representatives are unique except that each tetrad is one of a set of six disjoint mutually congruent tetrads (forming a sextet). We usually arrange the coordinates in the 4 x 6 MOG array, which is particularly valuable because it simultaneously exhibits the elements of several distinct maximal subgroups. Steiner system

G: the automorphism group of the Steiner system S(5,S,24) of 159 special octads f'o-om a 24-set such that each

pentad is in a unique special octad. The octads are the supports of the weight S words of the binary GolaY code (see above). If {apa 2 , •••• aS} is any octad, then the number of octads (resp. dodecads) that meet {ap •• etai} in precisely {a1 .... aj} is the (j+1) l th entry in the (i+1)fth roW of the left (resp. right) table below.

2576

759 506 330 210

46

30

Hexacode

16

12

48

o

4

4

120

5 4

8

o

280

21

16

28

16

16

56

o

4

o

o

o

4

16

o

o

o

o

336

120

16

32

40

o

o

16

16

24 24

4e

72

88

24

o

16

160

48

40

16

280

336

88

32

616

176

160

72

16

1288

672

616

77

40

20

32

o

30

120

52

78

176

80

130

1288

253

16

o

o

In this construction it is easy to recognise codewords and to complete octads from 5 given points. The 64 hexacode words (see A6 ) (ab cd eO are obtained from yz yz yz

Ox Ox yz

00 xx xx

xx yy zz

00 00 00

(in which (x,y,z) is any cyclic permutation of (1,w,w)) by freely permuting the three couples ab, cd, ef and reversing the order in any even number of them. The Golay code words are obtained from hexacode words either by giving all their digits

~

interpretations in any way for which the top row receives an

~

number of

entries, or by giving the digits odd interpretations in any way for which the top row receives an odd number of entries. even interpretations

odd interpretations

80r880r8Bor8 80r8

Bor8 80r 8Bor8 80ra o

o

w

w

w

w

We give one even and one odd interpretation of (01 01 wW):

• • • ••• •• • •• •

•••• • • •• • • ••

o

0101ww

1 0 1

wW

To complete an octad from 5 given points: In the example (a) below, if the parity of the answer were even, we must change the three columns? of (b). So the remaining columns are correct, and the digits below them determine the hexacodeword and trial octad of (c), whose top row has the wrong parity. So the parity is odd, and the digits in (d) determine the hexacodeword and octad as in

(e).

EEBEEB O?w?O?

( a)

I!D CLCJ ~

01ww01

(b)

· EEB EEE °•°

••

?w?1?w (d)

(c)

•

~Oww10w

(e)

Using the fact that a hexacodeword is uniquely determined by any 3 correct digits, or by 5 digits of which at most one may be mistaken, the method more generally finds the unique Golay codeword nearest to any given set of odd cardinal. (The method also works for sets of even cardinal, although the answer need not be unique.) Sextet

The group fixing the sextet of MOG columns has structure 26 :3.36' the stabilizer of the top row being a subgroup 3°S6, and the code, as constructed above, is invariant under this group. The elements of the subgroup 26 correspond to hexacodewords (a), and there are other correspondences between hexacodewords and the remaining elements of 26 :3 (b) and the involutions of 3.36 (c), (d):

[94]

r:~:~~1 0101ww

mm

I;; id

(a)

(b)

(c)

0101ww

o

1 0 1 .or W

8J .~~ . .~

.

~

0101ww (d)

Triad

In terms of the 21 point projeotive plane over JF 4 = {0.1,w,w} the ootads of the Steiner system are the points of a line

+

3 extra points I.

a hyperconic

+

2 of I.

an IF'2-subPlane

+

one of I,

n,

]I.

M24

ill

ill

] I , ill

or the points of two lines without their intersection. Every maximal set of points with no three in line is a hyperconic, consisting of the five points of a conic and the unique point not on any chord. Hyperconics whose equations have coefficients in IF'2 mate with {IT ,ill}, while the IF'2-subplane of all points with coordinates in lF 2 mates with I. A~' element of GL (4) with determinant w 3 effects the permutation (I

ill). In fact the construction is invariant under PrL (4), with the field

]I

3

automorphism effecting (IT ill). In the MCXi array below points (x,y.O) in the left-hand octad are denoted ylx, while the points (x,y)=(x,y,1) form the right-hand square. We also show the elements of M24 obtained from some elements of prL (4): 3 y/x

00

o

,

x'=y, yf=X

n

o points (x,y)

w

II

y

w

= (x.y,1)

m w

xf=>:, yf=y

x'=x, y'=wy

w

w

0

I

Trio

x

W

1111 111

11111 X

We select two disjoint extended Hamming codes on V = {oo,0 .... ,6}, for example (i) 0, {co, i+3, 1+5, i+6}, and complements (11) 0, {co, i+1, i+2, i+4}, and complements

The binary Oolay code consists of all words of the form (X+t,y+t,Z+t) for which X, y, Z are elements of the

°

first Hamming code with X+Y+Z =

or V, while t is in the second Hamming code. Using the indicated numbering

for each brick of the MCXi, this construction exhibits a maximal sUbgroup 26:(S3xL2(7)) of M24 in which the typical element of

z5

consists of three translations (see below) with trivial product, S3 permutes the three

bricks, and L2 (7) acts on each brick in the same way. The brick numbering and the translations I

I

00

0

3

2

1

4

I I

5 6

I

t

!

-

i

\\ H

~ X

X X

-

The eight 3-sets permuted by the L2 (7) in this construction give rise to 14 dodecads by taking unions. There is one other type of family (octern) of eight 3-sets with this property, whose stabilizer is a maximal subgroup L2 (7), acting transitively. Octad

This construction uses the isomorphism between AS and L4(2), and is only briefly sketched here. The 24 points are taken as the points of an affine 4-space V over JF 2 (whose automorphism group is 24 : L4 (2) =- 24 : AS)' together with those of the S-point set S

{',2, ••• ,S} on which AS acts. They can be written in the MCXi array thus:

5

o

a

b

a+b

2

6

c

a+<:

b+c

a+b+c

3

7

d

a+d

b+d

a+b+d

4

8

c+d

a+c+d

b+c+d

a+b+c+d

s

v

The stabilizer of a system of four parallel 2-spaces in V is also the stabilizer of a bisection of S into two complementary tetrads. The six 4-sets so obtained form one of a familY of 35 sextets which suffice to define the code. The 759 octads of S(5,S,24) are the 1 octad S, 280 = 35x2x4 meeting Sin 4 points (obtainable from the above sextets). 448 meeting Sin 2 points, and the 30 affine 3-spaces in V. Modulo 23

The binary Golay code is a quadratic residue code, modulo 23. In the projective line PL(23), let Q={O.quadratic residues},. N={co,non-residues}. The code is spanned by the words Woo = !e i (all i), Wt = !en_t (n iri N). We have

~ci wi

= 0

~~

(ci) is in the code. G is generated by the elements A : t ---7 t+ 1, B : t ---7 2t. C :

t -'?-1/t and D: t ---7t 3 /g (t in Q) or gt 3 (t in N), in which =- PSL 2 (23) and =- M23 • Mathieu '2

The Mathieu group M12 acts on two 12-sets L={n+}, R={n-}, interchanged by its outer automorphism (duality). The stabilizer of a tetrad in one stabilizes a tetrad in the other while the stabiliZer of a duad in one stabilizes a pair of complementary hexads in the other. The resulting octads splitting 4+4, 2+6, 6+2 form the Steiner system S(S.S,24). Thus (using the shuffle numbering, see M12 ) the binary Oolay code is spanned by {1+2+3+4+5+6+0-1-}, {0+1+2+3+7+a+1-2-}, {0+1+2+4+5+9+2-3-}, {0+1+3+4+6+7+3-4-}, {O+1+2+3+S+10+4-5-}, {O+'+2+4+6+a+S-6-}. {O+2+3+4+S+7+6-7-}. {O+1+2+3+6+9+Y-8-}, {0+1+3+4+5+a+S-g-}, {0+1+2+S+6+7+9-'0-}, {0+'+2+3+4+'1+10-'1-} and {0+1+2+3+4+5+6+7+a+g+10+11+}. The shuffle and modulo 23 numberings are related by: n+ = whichever of n, -n is in Q

= whichever

n-

of a/n, -8/n is in N.

These and the lexicographic numbering (see below) appear in the Mex:; array as follows:

o

Lexioographio

11

00

2

22

0+

0-

1+

7-

8-

dO

d4

dS

d'2 d'6 d 20

3+

5- 4+ 10- 5+

d1

d5

dg

d 13 d 17 d 21

d2

d6

d 10 d 14 d'8 d 22

2+

19

3

20

4

,0

18

2-

'5

6

,4

'6

'7

8

1-

6+

6-

5

9

2'

'3

7

'2

r

g+

4- '0+ 11- 11+

7+

g-

8+

d 3 d7 d"

d'5 d '9 d23

Each code word can be defined in succession as the lexicographically earliest word {dO •••• ,d23' dln'ering in at least S places from all its predecessors: (0 24 )

(0'6,8)

(0 '2,4 04,4)

( ,24)

[95J

M 24 Presentation

G ;;


I a=(Cd)5:(Cde)5,f=(cdg)5,e=(bcf)3,1=(abf)3 )

g

f

Maximal sUbgroups Order

Index

10200960

24

887040

276

Specifications Structure

Character

. Abstract

"23

1a+23a

1a+23a+252a

322560

759

M22 :2 2 4 :A

190080

1288

M'2:2

1a+252a+1035a

138240

1771

26 :3'3 6

1a+252a+483a+1035a

120960

2024

L (4):S3 3

1a+23a+252a+483a+1265a

64512

3795

6072

40320

168

1457280

N(2A 4 )

1a+23a+252a+483a

e

N(2 6 ) : N(2A 45 B,8 )

Mathieu

Golay

point

weight 1 cocode word

duad

weight 2 cocode word

octad

weight 8 code word

doom

weight 12 code words

sextet

weight !I cocode word

triad

weight 3 cocode word

N(2 6 ) : N(2A 21 B42 )

trio

"2(23)

N(2B,3B,4C,6B,llA,12B,23AB)

projective line

"2(7)

N(2B,3A.,4C,7AB)

octern

26:(L3(2)xS3)

1a+252a+483a+1035a+2024a

G26 26 @

@

@

@

@

@

@

21 7 1 244823040 504 680 080 504 384 128

p power p' part ind lA

[96]

A

A

A

A

A

3A

3B

A 4A

A 4B

A A

A A

A

2A

28

@

@

@

@

@

@

96 60 24 24 42 42 BAAABBA A AAAABBA A 4C 5A 6A 6B 7A Bn

x,

+

x,

+

23

7

-1

5

x,

o

45

-3

5

0

3-3

o

X.

o

45

-3

5

0

3-3

o

x,

o

231

7

-9

-3

0

-1

-1

x,

o

231

7

-9

-3

0

-1

x,

+

252

28

12

9

0

4

x,

+

253

13 -11

10

x,

+

483

35

3

6

0

3

3

3

-2

Xw

o

770 -14

10

5

-7

2

-2

-2

x"

o

770 -14

10

5

-7

2

-2

x"

o

990 -18 -10

0

3

6

x"

o

990 -18 -10

0

3

x"

+

1035

27

35

0

x"

o

1035 -21

-5

x"

o

1035 -21

x"

+

1265

x..

+

1771 -21

x"

+

2024

x"

+

x"

@

16

@

@

@

11

12

A A

AA

@

@

@

@

@

@

@

@

@

12 14 14 15 15 21 21 23 23 BC AA BA AA AA BB AB A A AA BC AA BA AA AA AB BB A A BA lOA 11A 12A 128 14A Bn 15A Bn 21A Bn 23A Bn B A

20 AB AB

x, -1

-1

_1

-1

0

0

0

-1

-1

0

0

X2

n

0

0

b7

n

-1

-1

X3

-b7

0

0

**

b7

-1

-1

X4

0

0 b15

**

0

0

x,

0

0

0

n

b15

0

0

x,

o

0

0

-1

-1

0

0

-1

-1

0

0

0

0

-1

2

2

0

-1

b7

**

-1

0

o

-b7

0

-1

**

b7

_1

0

o

**

3

o

0

0

-1

o

-1

0

-1

3

o

0

0

-1

o

-1

11

0

o

0

0

0

2

-1

-1

-1

0

0

3

-1

3

2

3 -2

-3

0

_1

-1

X,

o

0

X8

0000X9

0

-1

-2

-1

0

0

o

0

0

0

0

-1

o

0

0

0

0

0 b23

**

XIO

-2

0

o

0

0

0

0

-1

o

0

0

0

0

0

**

b23

XII

2

-2

0

0

-1

b7

**

0

0

0

0

b7

**

0

0

b7

n

x"

6

2

-2

0

0

-1

**

b7

0

0

0

0

**

b7

0

0

n

b7

x"

6

3

-1

3

0

0

2

-1

-1

0

-3

3

3

-1

0

0

2b7

**

-5

0

-3

3

3

-1

0

0

**

2b7

49 -15

5

8

-7

-3

0

-2

-2

11

16

7

3

-5

-1

24

-1

8

8

0

0

2277

21 -19

0

6

-3

+

3312

48

16

0

-6

0

x"

+

3520

64

0

10

-8

0

x"

+

5313

49

9 -15

0

XM

+

5544 -56

24

9

0

x"

+

5796 -28

36

-9

Xu

+ 10395 -21 -45

0

o o

-1

-1

-1

0

-3

-3

0

2

0

0

-3

0-2

0

0

0

-3

-3

-8

0

0

0

-4

4

0

0

3

-1

3

o

o

0

-1

-1

0

0

_1

_1

0

0

Xl4

-1

0

o

- 1

0

0

0

0 -b7

**

0

0

X15

-1

0

o

- 1

0

0

0

0

-b7

0

0

XI6

0

-1

0

0

0

0

0

o

0

X17

o

0

-1

0

0

0

0

XI8

0

-1

0

o

0

X19

o

0

0

0

0

0

X20

o

0

-1

o

0

X21

0

0

o

0-1

o 2

2

-1

-1

o

n

o -1

-1

0

0

0

-1

0

0

o

0

-1

-1

0

-1

x"

-1

0

-1

-1

0

0

0

3

o

0

0

-1

-1

0

000000000X23

-1

o

0

0

0

-1

0

o

0

0

-1

0

0

0

0

0

0

0

0

0

-2

0

0

0

0

8

2

-1

o

_1

-1

0

0

0

0

0

0

0

0

-1

_1

0

0

XM

0000X25 0

0

0

0

-1

-1

X26

G2 (4) Chevalley group G2 (4) Order

= 2 12 .3 3 .5 2 .7.13

= 251,596,800

Mult

2

Out

=2

Constructions Chevalley

G2 (4)

=G

: the adjoint Chevalley gt'OUp of type G2 over IF 4;

G.2: the automorphism group of a gene:'alized hexagon of or'del' (4,4). consisting of 1355 vertices and 1355 edges, each object being incident with 5 of the other type. Cayley

G : the automorphism gr'oup of the lF 4 Cayley algebra. the tensor product of the field lF li = {O.l.w,w} with the non-associative ring of Cayley integers. Calculations are complicated by the fact that there is no very symmetric base for the Cayley integers. They can be generated (in :R 8 ) by the even sums of Xoo ' Xo ' •••• X6 together' with

= ~!Xn'

1

with multiplication defined by the equations

2X~ = Xn -1, 2X OXoo = 1-X 1-X 2 -X 4 , 2X ooXO

= 1-X3-X5-X6

their images under L2 (7). Additively. this is a scaled copy of the E8 root lattice under the norm

and

N{~anXn) = ~~a~.

Each lF li Cayley number' has the form A+wB where A and Bare Cayley integers modulo doubles of Cayley integers. (Note: since Xn is not a Cayley integer. 2X n is not 0 in this algebra.) There are 1 + 4095 + 2080x2 + 2015x4 elements of determinant 1. with respective order's 1. 2, 3 and 5. The elements of order 2 fall into 1355 triples c,f the form {l+X,l+wX,l+wX}. The map X ~ A- 1XA is an automorphism of the algebra just if A has order 3 (or 1). There is a subgroup isomot'phic to the Hall-Janko gl'OUp J Quaternionic 2G

2

containing 280x2 of these maps.

: the automorphism group of the quaternionic Leech lattice. whose typical element is written (qoo;Q0 ••••• q4). The quaternions ::1, .:ti, ±j • .:tk. w =

~(-1+i+J+k)

multiplicatively generate a group 2A li and additively generate the

Hurwitz ring of integral quaternions. The lattice is closed Undel" the scalar left multiplications by these numbers. There is a monomial group 2 5+6 .OXA ) (acting on the l'ight) generated by diag(i;i.J.k,k,j). 5 diag(w;w.w,w,w.w), (qoo)(qo ql ••• q4) and (qoo qO)(ql q4). There are 195560

= 24

x 8190 minimal vectors, which

generate 3190 1-spaces. and span the lattice. They are the images under the monomial group of (2+2i;05) (6), (2;2,0 4 ) (120). (O;O,l+k.l+j.l+j.l+k) (1920). (i+j+k;1 5 ) (6144), and fall naturally into 1365 congruence bases mod <1+i>. The corresponding basic decompositions. such as <e

oo

)m ••• e<e4) each refine 5 trio decompositions such as

<e oo ,eO)@<el,e4)@<e2,e ). which in turn each refine to 5 basic decompositions. 3 The quaternionic form of a real Leech lattice vector is obtained by interpreting the columns of the MOO as the quat ern ion coordinates of a vector (qoo;qO ••••• qij) =

~qtet,

according to the scheme:

qoo qo

ql

q2

qij

q3

k

j

k

j

k

j

i

k

i

k

i

k

j

i

j

i

j

i

and then diViding by 2. The tmit gt'OUp 2Aij extends in 2Co

to 2A • which addiUvely generates the fcosian ring of integt'al quaternions, and 5

1

converts the lattice into a 3-dimensional module over this ring, whose automorphisll gt'OUP is 2J 2 (q.v.). The subgroup 3L (4) can be written over the complex subring Z[w]. 3 Graph

G.2 : the automorphism group of a rank 3 graph of valence 100 on 416 points, in which the point stabilizer is J 2 .2.

Presentations

G.2

=< a

G "

<e

b d 8 c

c 8 d

e

f

g

I

a=(cd)4,{bcde)8=1 >

I~ 1 a.(cd)4, h(abf)5.(abcf)5.(abcdecbdc)7[.(abc)5] > a

Remark

G2 (4) is a maximal subgt·oup of the Suzuki group. 'and contains the Hall-Janko group J?"

Specifications

Max imal subgroups Order

Index

604800

416

Structure

G.2

Character

J2

J 2 :2

la+658+350a

Abstract

Chevalley

Cayley

Quaternionic

Graph

icosian ring

point non-edge

184320

1365

22+8: OxA ) 5

~+8:(3XA5):2 la+350a+364a+550a

N(2A 2 )

184320

1365

2 4 +6: ( A x3) 5

24+6: (ASx3):2

1a+350a+3641>+650a

N(2A 4 )

124800

2016

U (4):2 3

U (4):4

la+6508+1365a

120960

2080

3""3(4):2

3'L3(4):22

1a+3508+36ijI>+1365a

12096

20800

U3 0):2

"30):2 x 2

C(2C)

G2 (2)

subfield

edge

3600

69888

A5

(A xA ):2 5 5

N(2A.3B,5AB). N{2B,3A.5CD)

"2(4 )

quaternion suoalgebra

6-point coclique

1092

230400

"2(13):2

N(2B,3B.6B.1A.13AB)

x

A5

"2(13)

3

3

vertex

null sUbalgebra

trio decomposition

edge

order 2 triple

base

2A2 (4 )

order 5 element

NOA)

A (4 ) 2

order 3 element

complex sUbring

[97]

G 2 (4)

@

251596800 p power

pi part ind lA X.

+

x,

+

65

x,

+

78

X.

o

x,

@

@

@

@

@

@

@

61 3 60 1 ijijO 8ijO ij80 180 536 768 512 A A A A A A A A A A A A A A 2A 28 3A 38 ijA ij8 ijC

@

300

A A 5A

@

@

300

300 A A 5C

A A

B.

@

@

300 192 A AA A AA

a.

6A

@ 12 BB BB 6B

@ 21 A A 7A

@ 32 A A 8A

@

@

32 2a C DA A CA 8B lOA

@

20 CA DA B*

@

20 BB BB

lac

@ 20 AB AB

@ ij8 AA AA

D* 12A

@

@

ij8 AB AB 128

ij8 AB AB

@ 13

A A

C.. 13A

@ 13

@ 15

@

@

15

15 DB CB B. 15C

A BA AA A AA BA B. 15A

@

@

15 21 CB AA DB AA D* 21A

@ 21 AA AA B*

X.

o

o

5

5

ij

-1

-2

3

3

3

3

-1

0

8

-ij

o

o

a

o

o

ij

8

-ij

o

o

o

o

o

6

2-2

o

o

o

o

5

20

-1

-6

15

0

6

2

300 -20

0 -15

0

ij

o

300 -20

0 -15

0

x,

+

350

30

10

35

5

x,

+

36ij -20

ij

ij9

x,

+

36ij

ijij

0 -lij

x,

+

378

-6

-6

x"

+

650

10

x..

+

x"

lij

9-3

3

2

000 3

o

o

0

0

0

0

-1

-1

0

0

0

a

o

0

-1

-1

-1

-1

x, x,

-2

2

-1

-1

-1

-1

-1

0

0

0

0

0

0

1 2i3-1-2i3-1

o

0

0

0

-1

-1

x.

-1

0

0

0

0

0

0

1-2i3-1 2i3-1

a

0

0

0

-1

-1

x,

o

2

-2

0

0

0

0

3

-1

0

0

o

0

0

0

X6

o

0

a

0

0

-1

-1

-3

0

-1

-1

o

0

X7

-1

0

0

Xs

-1

-1

-1

12

ij

-ij

-1

-1

ij

ij

ij

ij

8

-ij

ij

ij

-1

-1

2

0

0

0

0

-1

-1

0

0

-2

2

200

0

0

-6

-6

10

3

3

3

3

0

0

0

2

-2

-1

-1

-1

-1

0

o

o

10

20

5

-6

-6

10

o

o

o

o

ij

-1

-2

2

0

0

0

0

0

o

o

819 -13

-9

63

0

11

-1

3 2r5-1-2r5-1

-3b5

*

-1

0

0

-1

-1

b5

*

-1

-1

+

819 -13

-9

63

0

11

-1

3-2r5-1 2r5-1

•

-3b5

-1

0

0

-1

•

b5

-1

-1

Xu

+

819

51 -13

0

3

3

3

3

-3b5

*-2r5-1 2r5-1

0

-1

0

-1

-1

b5

*

0

o

x"

+

819

51 -13

0

3

3

3

3

•

-3b5 2r5-1-2r5-1

0

-1

0

_1

-1

•

b5

0

o

x"

+

1300

20

20

85

-5

ij

ij

ij

o

o

o

o

5

-1

-2

0

0

0

0

x"

+

1365

85

21 -21

0

5

5

5

5

5

o

o

-5

0

x"

+

2925

ij5 -15 -ij5

0

5

-7

-3

o

o

o

o

3

x"

+

2925

ij5 -15 -ij5

0

5

-7

-3

o

o

o

o

x"

+

2925

ij5 -15

90

0

5

-7

-3

o

o

o

x'"

+

3276 -52

12

63

0

-ij

-ij

_ij

*

b5+4

x"

+

3276 -52

12

63

0

-ij

-ij

-ij

b5+ij

•

x"

+

3276

12

8

0

3 -12

0

-ij

3b5

x"

+

3276

12

8

0

3 -12

0

-ij

x"

+

ij095 -65

3 -63

0

7

-5

X25

+

ij095 -65

3 -63

0

7

-5

b

+

ij095

63

-5

0

-3

-9

3-1

x"

+

ij095

63

-5

0

-3

-9

3-1

x"

+

ij096

0

0

6ij

ij

0

0

x"

+

ij 160

6ij

0

20

-ij

0

Xoo

+

ij725 -75 -15

0

0

x"

+

ij725 -75 -15

0

Xn

+

5ij60

20

2ij -8ij

ind

1

2

2

2

x"

12

-ij

x"

10ij

x"

10ij

0

0

0

o

0

0

-1

o

0

0

3

0

-1

o

0

o

-6

0

-1

o

•

3b5

-1

0

0

0

0

3b5

*

-1

0

0

0

0 -b5

•

*

b5+4

0

-1

0

0

0

*

•

3b5

b5+ij

*

0

-1

0

0

0

b5

-1

-5b5

•

o

o

o

0

-1

-1

-1

•

-5b5

o

o

o

0

-1

-1

o

o

-5b5

o

o

0

-ij

0

0

-3

9

0

-3

0

3

3

6

o

-1

o

0

o

0

0

0

X9

0

0

0

0

o

0

-1

-1

XIO

-1

0

0

• -b5

o

0

0

0

XII

-1

0

0 -b5

•

o

0

0

0

X12

0

0

0 -b5

*

0

0

XIJ

0

000

*

-b5

0

0

XI4

0

o

0

o o

0

000

x"

-1

-1

-1

0

0

-1

-1

o

0

0

0

XI6

0

-1

-1

-1

0

0

0

0

o

O-b21

*

XI7

0

0

-1

-1

-1

0

0

0

0

o

0

*-b21

XI8

0

0

0

2

2

2

0

000

o

0

• -b5

•

b5

-1

-1

-1

0

0 -b5

•

OOOQX20

*

b5

*

-1

-1

-1

0

0

• -b5

OOOOX21

b5 -b5

*

0

o

0

0

0

0 -b5

*

0

0

X22

*

-b5

0

o

0

0

a

a

*

-b5

0

0

X23

0

0 -b5

•

a a

b5

•

OOOOX24

0

0

• -b5

*

b5

OOOOX2S

*

0

0

0

a

OaQOOb5*OQX26

• -b5

0

0

0

a

a

*

o o

o

0

-1

-1

Xl9

•

0

o

-1

-1 -b5

•

-5b5

0

o

-1

-1

-ij

-ij

-ij

0

0

o

0

0

0

0

0

0

o

a

o

o

-5

-5

ij

0

0

0

-1

-1

0

0

0

a

000

a a

5

o

o

o

o

000

o

0

0

0

0

a

a b13

0

0

OOOOX30

9

5

o

o

o

o

0

0

0

a

0

0

0

0

a

a

a

o

12

0

ij

-5

-5

o

o

-ij

0

0

0

0

0

0

-1

-1

0

a

a • b13 o a a

ij

ij

6

ij ij

7 lij

8

8 8

10 10

10 10

20

20

12 12

12

-6

0

-ij

0

002

a

8

0 -22

2

-8

0

0

8

0 -22

2

•

0

0-2

a o

x"

36ij -36

0 -lij

ij

x"

560 -16

0 -70

x"

1260 -36

0

0

6

ij

0

2

5

5

5

10

10

5 10

6 6

12

10 0

2

2

-3

-3

2

0

-2

0

2

0

0

2b5

•

b5-3

•

2

0

-1

0

0

-8

0

0

•

2b5

•

b5-3

2

0

-1

0

0 -b5

-ij

0

0

ij

ij

_1

-1

-6

0

0

0

2

-1

-1

0

0

2

-ij -16

0

0

o

o

-5

-5

2

0

0

0

0

-1

-1

0

0

2

12

a

0

o

o

-5

-5

0

0

0

0

-2

-1

-1

0

0

0

0

0

0

0

0

-2

• -b5

-2

x" x..,

-

1800

ijO

0 -90

0

-8

0

0

o

o

o

o

-2

0

o

-

1800

ijO

0 -90

0

-8

0

0

o

o

o

o

-2

0

000000-2

x••

-

1820 -52

0 -70

2

-ij

0

0

o

o

-5b5

•

2

0

0

0

-2 -b5

x.,

-

1820 -52

0 -70

2

-ij

0

0

o

o

•

-5b5

2

0

0

0

-2

x.,

-

2016

96

0

0

6

0

0

0

6

6

X#

-

218ij -88

0

ij2

0

-8

0

0 2r5+ij-2r5+ij

-3b5

•

2

0

0

0

0

b5

•

0

0

-2

x"

-

218ij -88

0

ij2

0

-8

0

0-2r5+ij 2r5+ij

•

-3b5

2

0

0

0

0

•

b5

0

0

-2

x...

-

3276

60

0

0

-6

12

0

0

6

6

-ij

-ij

0

0

0

0

-2

0

0

0

0

0

x.,

-

3600

80

0

90

0 -16

0

0

o

o

o

o

202

0

0

0

0

0

0

2

x"

-

37ijij

32

0 -36

0

0

0

0-3r5-1 3r5-1

-3b5

•

-ij

0

-1

0

0

b5

•

0

0

0

x"

-

37ijij

32

0 -36

0

0

0

0 3r5-1-3r5-1

•

-3b5

-ij

0

-1

0

0

•

b5

0

0

0

x"

-

3900 -20

0 -60

0

12

0

0

o

o

o

o

ij

0

020

0

0

0

x"

-

3900 -20

0 -60

0

12

0

0

o

o

o

o

ij

0

o

2

0

0

0

x"

-

ij096

0

0

6ij

ij

0

0

0

-ij

-ij

-ij

-ij

0

0

o

0

0

0

0

x"

-

582ij -6ij

0

28

-2

0

0

0

2b5

*-2r5-1 2r5-1

-4

0

0

0

x,.

-

582ij -6ij

0

28

-2

0

0

0

*

2b5 2r5-1-2r5-1

-4

0

0

x"

-

7ij88

0

36

0

0

0

0

-2

ij

0

-2

•

0

0

2

• -b5

0

0

2

00000

000

-2

3

3

a

a -1

13

•

x" -1

0

0

-1

0

15

15

15

15

21

21

3a

~

~

~

~

a

-1

-1

-1

-1

o

0

0 -b5

o

a a

a a

a • a

a a a

a a -1 a b13

a a a

a • b13 a 0 o 0 0 0 0 a a a a a

a a a o o o

a

-1 •

X29

X31

a

0

*

b5

*

-1

-1

X34

-b5

*

b5

-1

-1

X3S

-1

-1

0

0

X36

o

0

X37

o

0

X38

a 0 a 0 a a

o

a a

b5

*

o

*b5

x"

x"

x.., 0

0

X41

OOX42

o

a a

0

X43

a a o 0 a o 0 a o -1 -1 o 0 0

b5

•

OOOOX44

•

b5

OOOOX4S

0

0

0

0

-1

a a

-1

o

-1

0

0

x...

0

-1

-1

X47

-1

o a

-1

-1

x"

-1

o

0

-1

-1

X49

a

o

0 b21

000

o

0

o

0

0

0

o

a a

0

0

o

o

0

000

00

O-b5

0

0

000

a o

o

a

a

0

0

0

o

a a

0

0

-1

3a

0

0

-1

OOXZ7

13 26

0

-1

-1

*b5

26

o

-1

a

oaOOX32

a o

[98] 6ij

12

a

0

-1

•

-1

-1

o

*

* b21

Xso XSl

-1

-1

*-b5

*

0

QXS3

*-b5

0

0

-b5

o

0

x"

XS4

x"

@

@

12 096 192 A A fus 1nd

A

A

2C

4D

@

@

48 216 B AC A AC 4E 6C

@ 18 BC BC 6D

@

@

96 A A 8C

48 B A

8D

@

@

@

@

@

@

@

++

x,

++

7 -1

x,

++

6

-2

0

-3

x.

00

6

-2

0

x,

00

6

-2

0

x,

++

28

4

2

x,

++

14

-2

2

5

x,

++

14

6

0

-4

x,

000000000000

x"

++

20

4

4

2-1

o

0

0

-2

f

+

0

0

0

0

0

0

0

0

0

t

+

0

0

0

0

0

0

0

0

0

x,

o

_1

0

2

4

-2

o

-1

o

0

-3

0

-2

0

2

o

-1

o

0

-3

0

-2

0

2

o

-1

o

0

-13

13

x,

-1

-1

x,

2

-1

-1

420

-1

000

-1

2

-1

o

2

-2

-2

2

0

0

000

x,

-1 13 -13

x.

x,

-1

0200000-20axs

o

21 -21

o

0

0

X9

-1

o

0

0

0

0

Xw

0

0

0

0

0

0

0

Xll

0

0

0

0

0

0

0

x"

x"

Xl)

x"

x"

++

14

x"

++

21

x"

+

0

-2

-2

5-1

5 -3

3

0

0

0

0

0

2

0

2

0

o

2

0

o

0

-1

o

0

-1

-1

0

0

0

0

0

-1

-1

-1

0

0

0

XIS

x"

x"

Xl7

x"

x"

++

x"

+

27

3

-3

0

0

3

-3

-1

0

0

-1

1

1

0

0

000000000000000

0

x"

+

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

x"

Gho

x" 0

x" X24

x,

3 -3

-2

x" XI)

G z(4)

@

32 24 6 8 8 12 12 12 7 A AD BE AC B B AC BD CD A AD BE AC A A AC AD AD 8E 12D 12E 14A 16A Bn 24A 248 Cn

x,

XII

@

x" x"

~EJ32

BEJ23 32 16

x" f

+

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

x"

X24

x"

X26'+OOOOOOOOOOOOOOOOXU

x"

x"

x"

++

64

0

0

-8

-2

0

0

0

0

0

++

56

-8

0

2

2

0

0

0

-2

0

aOOOQX28

0

0

0

0

0

0

X29

:

X29

Xm

f+OOOOOOOOOOOOOOOOX30

x"

x" ++

42

2

0

6

0

-6

0

-2

2

0

0

0

0

0

0

0

rus ind

2

4

8 8

6

6

8

8 8

8

12

24 24

14 14

16 16

16 16

24 24

24 24

24 24

x"

o

0

0

0

0

0 2r2

0

0

0

0

r2

r2

o -r2

-r2

X33

x"

o

0

0

0

0

0

0

0

0

0

0

0

0

x"

X32

:

0

0

0

x"

X32

x"

x"

o

0

0

0

0

0 2r2

0

0

0

0 -r2 -r2

0 -r2 -r2

X36

x"

o

0

0

0

0

0 4r2

0

0

0

0

0

0

r2

r2

X37

0

0 212

0

0

0

0

0 -12

0

i 2 -12

0

0

0

X38

X38

:

00

0

0

x" x@

000000000000000

x..

o

Ob

x@ 0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

x"

x" X43

:

00

x«

0

0 412

0

0

0

0

0

0

12

0

0

0

0

0

0

X43

o

0

0

0

0

0

0

0

0

0

0

0

0

0

0

x«

0

x.,

x.,

X46

:

00

0

0 212

0

0

0

0

0

0 -12

0 -12

X47

:

00

0

0

0

0

0

0

0

0

0

0

0

o

0

0

0

0

0

0

0

0

0

0

x"

12

0

0

0

0

0

0

0

0

0

X46

16 -16

X47

0

0

o

x"

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

x"

x"

x" 00

x"

0

o

0

0

0

0

0

0

0

0

17

o

0

0

0

0

0

0

0

0

0

0

o

0

0

0

0

Xs2

0

0

0

0

0

x"

x"

x" XS5

x" x"

x"

x"

X.,

00

0

0

0

0

0

0

0

0

0

0

0

0

0 213

0

0

x"

[99J

MCL Sporadic M"Laughlin group MCL Order = 898,128,000 = 27 .3 6 .5 3 .7.11

Out = 2

Mult = 3

Constructions Graph

G.2 : the automorphism group of the M"Laughlin graph, a rank 3 graph of valence 112 on 275 points in which the point stabilizer is U4 (3).2. Under the simple group G, the 2-point stabilizers are L3 (4) and 34:A6' and there are two families of 22-point cocliques stabilized by M22 , and interchanged by the outer automorphism. G : the pointwise stabilizer of a triangle ABC of type 322 in the Leech lattice (see C0 1 ). G.2 : the setwise

Leech

stabilizer. The 275 vertices V of the M"Laughlin graph are represented by the lattice points for which VA, VB, VC are all of type 2, and V is joined to V' in the graph just when VV' has type 3. Taking A = (0,0,0 22 ), B = (4,4,0 22 ), C = (5,1,1 22 ) these points are (4,0,4,0 21 ) (22), (2,2,2 60 16 ) (77) and (3,1,_1 7 115) (176), and the containment of M22 is visible. In the Leech lattice modulo 2, the 3-spaces containing the fixed 2-space (of type 223) become points in the quotient 22-dimensional representation over F 2 • They are described below by the types of their non-zero vectors. The remaining maximal sUbgroups arise as intersections of G with maximal sUbgroups of C0 1 (q.v.). Unitary (5)

3. G has a 45-dimensional unitary representation written over F 25 • Explicit matrices have been computed.

Presentation

3.G " < a

b 5 c

d I a=(cf)2,b=(ef)3,1=(eab)3=(bce)5=(aecd)4=(cef)21

\I;/~7 e f

> ; for G adjoin (cef)7=1

6

Remark

3.G.2 is a maximal subgroup of the Lyons group.

Specifications

Max imal sUbgroups Structure

Leech

1a+22a+252a

vertex

223-2223-point

1a+22a+252a+1750a

coclique

223-2234-point

1a+22a+252a+1750a

coclique

223-2234-point

Index

3265920

275

443520

2025

M22

443520

2025

M22

126000

7128

58320

15400

U (5) 3 4 : 2S

58320

15400

34 : M10

: 34 :(M lO x2)

N(3 4 ) = N(3A 10 B30 )

edge

A2 12-hole

40320

22275

L (4):2 2 3

: ~(4):22

N(2A,3B,4A,4A,4A,5B,7AB)

non-edge

223-2333-point

40320

22275

2'A 8

: 2'S 8

N(2A)

octad space

40320

22275

2 4 :A 7

[ 22+4: (S3 xS 3)

N(2A 4 )

223-2344-point

40320

22275

24 :A 7

N(2A 4 )

223-2344-point

7920

113400

Ml1

: M11 x 2

C(2B)

223-3334-point

3000

299376

51+2 :3:8

: H.2

N(5A)

2 53 1O _S_lattice

U4 (3)

31+

G.2

Graph

Order

: U4 (3):2 3

[ 5

: U (5):2 3 1 4 •• 3++ •'4S 5

Character

Abstract

1a+22a+252a+1750a+5103a

N(2A,3B,4A,5A,5B,5B,5B,7AB)

223-2333-point

N(3A)

227 336 _s_lattice

~B24

EJF21 24 14 [100]

A l3 andRe Alternating group A 13

Order: 3,113,510,400 : 29.35.52.7.11.13

Mul t :; 2

Out:; 2

Constructions Al ternating

3 13 :: G.2 : all permutations cf 13 letters; A13 :: G : the even permutations; 2.G and 2.G.2 : the Schur double covers

G:: <X,t ••• ,Xl1Ixr=(XiXj)2:1>; G.2::

Presentations

Max imal 5ubgroups

Specifications

Order

Index

239500800

Structure

G.2

Character

13

A12

S12

1a+12a

39916800

78

Sll

3

10886400

286

(A 10 x3):2

4354560

715

2419200

x 2

Abstract

Alternating point

la+ 12a+65a

C(2D)

duad

SlO x S3

la+ 12a+65a+208a

N(3A)

triad

(A 9xA 4 ):2

S9 x S4

la+12a+65a+208a+429b

N(2A 2)

tetrad

1287

(A8 xA 5 ): 2

S8 x S5

la+12a+65a+20Ba+429b+572b

N(2A,3A,5A)

pentad

1814400

1716

(A7 xA 6 ):2

S7 x S6

la+ 12a+65a+208a+429bb+572b

N(2A, 3A, 38, 4A, 5A)

hexad

5616

554400

L (3) 3

N(2B,3C,3D, .•. )

S(2, 4, 13)

5616

554400

L (3) 3

N(2B, 3e, 3D, •.. )

3(2,4,13)

78

39916800

13:6

11

13: 12

N(13AB)

~EJ55

66 5 -49

27

Spor ad ic Held group He :: F7

Order

= 4,030,387,200 = 2 1°.3 3 .5 2 .7 3 .17

~fult

=

Out =

~

Constructions

Held

G : the automorphism group of a directed graph of rank 5 and out-valence 136 on 2058 points; the point stabilizer is 3 4 (4) .2, and the suborbit lengths are 1, 136, 136, 425, 1360. The outer automorphism rever ses the directi.'ons of the edges, thereby fusing the two orbits of size 136. The eentralizer of a 2B-involution is isomorphie to the centralizers of suitable involutions in L (2) and t1 . 24 5

Monster

7 x G : the centralizer in the t10nster group F 1 (q.v.) ofan element of class 7A. The normalizer is <7.3 x GL2. An element of order 3 in the normal subgroup ? 3 of this is centralized by 3.Fi

24

, and this leads to an embedding

of G.2 in Fi 24 . The group er has been explicitly constructed as the subgroup of Fi stabilizing a set of 2058 24 3-transpositions in F'i 24 . Presentations

G.2;;

< a 4-

b

C 10

1

e 8 f

I g:(ef)4:

Max imal subgroups

3pecifications

Order

Index

1958400

2058

S4(4):2

483840

8330

138240

3tructure

G.2

Character

Abstract, Held

: 34 (4):4, (S5 xS5):2

1a+51 ab+680 a+ 1275 a

point

22 'L 3 (4) ,S3

: 22· L3 (4).D

1a+51 ab+680a+ 1275a+5272a

N(2A 2 )

29155

26 :3'36

1 24+4 .(3 3x3 3 )·2

138240

29155

6 2 :3" S6

21504

187425

16464

244800

15120

12

N(2 6 ) : N(2A 18 B45 ) N(2 6 ) : N(2A 18 B45 )

21+6 •L (2) 3 72 :2L 2 (7)

: 21+6 .L (2).2 3 : 72 :2L 2 (7l.2

266560

3'S7

6174

652800

: 3'37 x 2 : 71+2 :(S3x6 )

NC3A), C(2C)

71+2 : (3 x3) 3

4032

999600

34 x L3 (2)

: 34 x L3 (2):2

N(2A 2 ), N(2B,3A,7AB)

3528

1142400 3358656

: 7:6 x L (2) 3 : 52 :43 4

N(7AB), N(2A,3B,7C)

1200

7:3 x L3 (2) 52 :4A 4

N(2B) NC7C 2 )

N(7C)

N(5A 2 )

~B33 33

[104]

19

0 7 (3)
"3, 3

u

~

~

38

~

x"" 02

Zl

x.., 02

351

x.,

02

351

x~,

02

Jl'8

-126

34

x~_,

02

1728

3$4

M

90

x...

02

1728

-384

64

90

77 -15

"

111

% U

~

31

lS

0

35

0

-6

o o

'15

0

2457

-315

02

2457

-399

xo,

02

2457

441

x..

02

2808

-216 -56

2~

0

~5

x""

02

2808

456 104

24

0

45

X".,

02

~914

714

18

0

99

o

x""

02

4914

546..ll6 -30

0

99

o

x",,,

02

5265

-855 105 -15

x,,,,

02

7020

74

300 140

x.."

02

7371

..ll41 111

x,,,_,

02

7371

-189

x"..

0,2

7560

840

zr

51

40 -24

x,,,,

02

7560

840

~0-24

02

9477

729

81

45

x..'"

02

9828

420 -92

36

x,...

02 12285

-735 -55

21

x".

02 12285

-315-155

-3

x",

02 12285 -1155 125

-3

x'l.'

0212285

525-115-3

X",

02 12285

945

x,,,

02 14742 _1386

x,,"

02 16848

x",

02 19656

25

90

o

90

21

15

6

6

"

6

6

6

_1

o "

o

o

x".

02 19683

-729 -81

Xu..

02 22464

384

54

x." x,,,

02 22464

-384

64

56 -24

., 1)..128

-3

_1

1

-3

0

-2

14

o

0

0

02 12095

x.,_

02 12095

x",.

02

1~0~0

o

1)..180

x",

02 15120

o

0 180

x,,"

02 19655

o

0

x".

02 20736

X,W

02 22464

o

0

Xl."

02 22464

36

h"

02 24192

o -35

x"",

02 24192

x""

02 29160

o

-5

10

21

0

_4

_1

o

~,

1

9A

8

9 9 9

_1

54

o

30 0 -13

o

o

-5 _15

-3

_1

-6

-6

_2

o o

o

O..ll

0

1_1

o _,

o

_1

0 -15

o "

,

_4

, , 3

8 24 24

3

0 -10

_1

o

-3

o

-3

o -3 o

-,

o

_5

_4

o

_5

_1

0

-7

() _15

0 -3

, 12 12

o -16

o _1 o _,

_1

o

-3

_1

o

_1 -24 -12

24

12

0

, ""

• 12 12

8 24 24

5 ~

15

•

12 12 12

12 12 12

12

o

-5

0

_1

-8

o _,

-8

-,

•"

12 12 \2

"""

_1

0

_1

, -,

~

o

2

_1

_1

_1

_1

1

_1

-3

_1

-3

,

o

o _,

o

2

o

_1

Complex

o _,

o -<

, -,

o

o -3

-2

1

Xo'

,

x",

_1

x.,

_1

_1

0

_1

_1 Xu"

0

_1

_1

-13

13

13 -13 13

o 0

13 -13

0

o 13

o

o _1 o _,

678899 6 42 24 24 18 18 2124249 14 8 18 21 24 9 42 24 18

o

o

1 -13

o

-13

1.3

13-13

o

0 -13

0

o

o _,

1 -13

_1

13 0

12 12

'"

2~

24

24 24

2~ 2~

12 12 12

12 12

12 12

_1 -13

0

13

o

\3

28

P

Out

x G.2 : all 7 x 7 matrices over lF

3

15 ~

84

~

_1

_1

36

36

36

36

o

».

(P (H)

W

,

o

x",

OX".. OX",

OX,,"

o

ox",

o

OX,," OX,,,,,

_1

_1

0_1

o

_1

_1

-,

Q-rl0 XL','

o

Ox""

= w-w = i3:

6 on ab, cd, and 1 on ac, bd, ad. bc, ef, gh, eg.fh, eh. fg

antilinear in each coordinate, and 3.G.2 is generated by the antilinear maps ("reflections") r as r ranges over the root vectors. The square of a coordinate vector is (2-11w)/27 times

2 10 itself, and the product of two distinct coordinate vectors has the form (A ,8 ,016), where A = (7w-4)/216, B = w/8, and the coordinates are distributed as in the corresponding root vector. The reduction modulo 2 of this

(3).

0

o

6 on. ac, bc, and 1 on ad. bd, ae. be. af, bf, ag, bg, ah,bh

°7

40

OX,,,,

~IXijI2/4 - I~XijI2/112. The root

The Fischer group Fi 22 contains two conjugacy classes of maximal subgroups isomorphic to

x,,,

Ox","

in which there is an orbit of 3 x 378

representation extends to 3Fi 22 and 3·(2 E6 (2)).3.

1

OX,,,

o -r7

The subgroup 3 (2) permutes the coordinates (see 3 (2), "Hesse"). There is an invariant multiplication x*y 6 6

= x*r

x,,,,

o c,

preserving a non-singular quadratic form;

16 2 1 vectors are the multiples by powers of w of the following vectors of shape (6 .1 °,0 ), where 6

x".

_1

120 120

-,

2

The norm of the vector (x 12 •••• ,x 7 8) is

36

15

o _,

o

= z3),

x,,"

_1

36120

o

_5

PG0 (3) • 507(3) • P50 7 (3) • G.2; 07(3) • G 7 3.G has a 27-dimensional complex unitary representation over Z(w] (w

x",

1

0-2

1536

U U 28 ~ D P

• b13

o

6

_1

1

-,

Mult

1

_1

13

o _,

= 29 .3 9 .5.7.13

0 X'"

~

0-20-2_5

o

0

nnp~36363636120

"

x",

_1

~

o b13

-5

0 Xu"

~

o

o

x,,'" Xl<..

~

n n

-5

13

-,

_1

1

o

_1

1_113_13 _1

x,,~

Ox,,,

, -, o

x,..,

o

1 x"'

, -, _2

Xl<...

1

Ox".

o

_2

x.."

ox",

o

_1

0 _1

o -3 -3

o

o

x,..,

ox""

o -3 -3

_1

99202012 18 18 60 60 12

60

_1

013-130

o

1

x""

Ox,..,

o o

0

Xo.'

, ,.

o

o _,

x._, x...

_1

13 _13 _1

1

_1

o -<

-3

_1

o

o

and work modulo the l-space «1 28

[108]

o 13

1

root vectors. It is simplest to take 28 coordinates indexed by unordered pairs from the 8-set {1,2.3,4,5,6,7,81

Remark

x.,

,

= 8 3 (3)

r x """7 x

_1

o

, -<

=2

X",

_1

o -13 13

.-b13

o _1 o _,

-2

o -<

, -,

_12

G0 (3) 7

0 -13

1

0

_1

Constructions Orthogonal

-3

1

0

o

-3

o o

-3

o 13 -13 o _,

I)..b13

_1

o

1

60 60

o '"

o

o o

" '" 18 18

o

o

"

13-13

o _,

, -,

"""

_1

o

-3

_1

18

o

o -<

, -,

_1

o

18e 180 20A

o -<

_1

o

1

o _1

-2

0 -36

Order; 4,585,351.680

_1

o

-,

,

Orthogonal group 0 (3) 7

o

_1

, -,

-6

o

18 18

_1 -13

_1

36

o

_1

-3

o "

o

18

18 18

-<

-3

o o ~

18

15 1S

o -<

,

o -.

o

1S

42 42

o

,

o -.

o " o _12

o

-3

o -.

o

14

39 39

o _,

o -<

0

an

13

o _, o

-3

o

o

B. 14A 1SA 18A

o -3

o o

o _4

39 39

1-1-1313

-2

o -< -<

_1

o -3

1

0

1

0160-8

o "

0

_1

-3

-9

12 12

o

o _1 o _,

, -,

o

,

0-7

o -12

1..ll8

~

o

0_9016 -3'-36

12F 12G 1211 13A 12 12 12 13

o _,

, _4

'"" " , -, "

_1

-3 0-1

o _,

-3

o

o

1-6

-3

_5

0

12C 12D 12 12 12 12 12 12

0-2-2-2

o

o_~

12

o

, -<

0-30-2-6

o

0

0

0_30_2_5

15

12

, -,

o _,

-3

-9

•• ••

o -,

-23513012

~

10

_1

0_4

24 -15 43

10

_1

o

')

9

_1

o

o "

-,

W1M1OO1~1~

_1

1

1$ -18

ge 9

o _,

o _1 -3

'" 0-6

98 9

o

o

-3

8B

S

-,

o -.

36

-3

0-215

o

_4

0

24 -15

, -,

1~4

72

()

12

_1

o o

'". o

-9

-2

-3

1)..108

x",

36

_6

-3

3

o

2

-3 -30

_1

-3

3

02

0-3

_2

-3

o

o

3

x""

3 _24 -12

1 -35

, , ,

3

-,

8A

7

, -,

o -.

o "

_4

-5

1)..126

,

-9

1 -3

o

o -15

1)..126

12 \2

3 -18

7A

21 ·2~ 24 212424

012_40-26-240_20-3

0-9

0 -36

0

o -18 -9

o

45

135

18

o "

"'

o

o

o _,

~

6

o

324120-$

-5

o -9

zr

0

1

3 -12

"'

0-9

1175

-3

_11

-3

6P

o -<

, "

-,

36

504- 104 -24

02 19655

~536

12

o -9 -3

0-5~

12 12

12

90

~5

6K 6L 6M 6N 60 660$&66

12

12

"'

61 6J

12

o

0

~

6

0-90

0

~

6

-3

90

M 6

O_H

o o

0

~

6

-3

0 -54

~

6

o -15

0

~

6

'35

48..ll8

x".

,, ,

o

0

9

15

o -5'

x,,'"

,.d

o

0

12

U

~

12

o -16

02

41

~D

12

"

x",

02 24192

~

12

-,

x....

-720

~B

~~4~5666666S66

12

9

9911-9036

3

is

3D 3E

~

2223333333 6 6 6 3 6 6 6 3

o

0 rl0 Xl."

0 7 (3) , , ", "• "• "'• , "',

2D

fUll lnd

2E

OH

6Q

6

"

6

65

61

61J

6V

6

6

6

6

" " 6

6Z 6Al 681 6 6 6

6Y 6

6

8

s* 2eA

2~A

S* 30A 36A

" " " " " " " " " " " " " " " " '" '" '"" " " "

8D loe 100 10E 121 12J 121( 121. 12M 12" 120 12f' 120 1211 16E 18F 18G 20B

BC

8

30

)6

)0

72 72

,-

,., ,.

x~,

x._,

,. ,. ,. ,. ,. ,. XI<..

x"" x""

X,,,_, XI<.. x,,~

x"..

x,,,, x".. XI("

x,,,, x,,, x"'

x,,_, x". x", x".

I

·• ·· ·• · · • · ·· ·• ·· • · ·• ·· ·• · · • · ·· · ·· ··· ·· •· · ·· ··• · ·

page

I I I I I I

106

page

107

G1Bss

[~]E}o

-----------~-----------

BiF34 I

BiF'2 S8

:

46

I

I I

page

108

I I

page

109

x", x". x"..

". x",

x", fus 1nd

, ,

2

8

8 8

8

8

,

" " "

6

" "

6

6

" "

6

6

8

!

"" '" '" "" "" "" "" "'" " " "

12

'" " " " "" '"'" "

24

52

52 52

56 56

56 55

"

x"-'

x".

x", x,,"

x", X'l,

". XlJ"

Presentation

X,."

3.G

~
b

c

1: .

f

I (abCdefg)9~1 >

replacing the last relation by

Xl.'>

x'" x" ...

(bcdefdghd)10=1 gives 2 x G; both relations give G. See page 232.

Maximal subgroups

Specifications

Order

Index

Structure

13063680

351

2U 1/3);2 2

: 2U q (3).(2 2 )122

12597120

364

3 5 :"4 (2): 2

: 35 :(U q (Z):2x2)

12130550

378

L4 (3):22

: L4 (3):22 x 2

4245696

1080

G (J) 2

4245696

1080

G2 (3)

I

4094064

1120

33+3: L3 (J)

: 33+3:(L (3)x2) 3

1451520

3159

S6 (2)

1451520

3159

S6(2)

1259712

3640

3~+6:(2A4xA4)·2

362880

12636

S9

362880

12636

S9

207360

22113

151280

28431

17280 13824

Character

Abstract

Orthogonal

1a+168a+182a

N(2A)

minus point

1a+158a+195a

N(3 5 )

1a+182a+195a

C(2D)

plus point

N(JA 3 )

isotropic plane

: 3~+6:(2S4XS4)

N(JA)

isotropic line

(2 2 x U (2)):2 4 26 :A 7

: D x U4 (2):2 8

N(2B)

non-isotropic line

: 26~S7

N(2 6 ) ~ N(2A7B21C35)

base

265356

S4 x S6

: 2 x S4 x S6

N( 2S 2). C(2E)

°3(3) xOii(J)

331695

S4x2(A4xA4) .2

: S4 x2 (A 4 xA 4 )·4

N(2C). N( 2S 2)

°3(3)x01j(3)

G.2

I

~ N( 3A 40 B36 C45)

isotropic point

la+260a+819a la+260b+819a 18+105a+195a+819a 18+ 1688+250a+2730a la+168a+260b+2730a

[109]

G2 (5)

Chevalley group G2 (S)

= 5.859,000.000 = 26 .3 3 .56 .7.31

Order

Mult

Out

Constructions 02(5) =: G : adjoint ChevalleY group of type G2 over F S :

Chevalley

the automorphism group of a generalized hexagon of order (S,S), consisting of 3906 "vertices" and 3906 "edges". each

object being incident with 6 of the other type; Cayley

(subscr~pts

G : the automorphism group of the Cayley algebra modulo 5, consisting of vectors Iani n • with an in F S

modulo 7). where n runs over {<:x>.O ••••• 6} and 100 =1, 1 0 + 1 ~ i, 1n+ 2 '"""'7 j, 1n+ 1I ~ k form a quaternion subalgebra. Remark: G2 (S) is a maximal subgroup of the Lyons group.

Maximal subgroups

Order

@

5 859000000 p power p' part ind lA

x,

x,

Specifications

Index

Structure

Character

Abstract

Chevalley

Cayley

~44

1500000

3906

Sl+4: GL2 (S)

la.930a.l085a.1890a

N(5A)

vertex

null subalgebra

1500000

3906

S2+3:GL2 (5)

la.930a.l085b+1890a

N( 5A 2)

edge

isotropic point

756000

7750

3'U (5):2 3

N(3A)

2 A2 (5)

minus point

744000

7875

L (5):2 3

A2 (5 )

plus point

14400

406875

2'(AsxAs).2

D2 (5)

quaternion subalgebra

12096

484375

U (3):2

1344

4359375

2 3 'L (2) 3

@

@

@

@

@

N(2A)

G2 (Z)

3

@

@

44

N( 2A 3)

@

@

@

@

@

14 378 3 1 1 375 15 400 000 720 480 480 000 000 750 875 250 720 720 A A A A A A A A A AA BA A A A A A A A A A A A AA BA 2A 3A 38 4A 4B SA 58 5e 50 5E 6A 68

@

36

@

21 A

BA

BA 6C

@

@

base

@

@

24 600 600 B AA BA

2' A

A

50

@

@

@

50

@

@

@

75 AA CA AA CA

2' 750

2' EA AA EA AA

CA CA

BA

AA

@

BB BB

A A 7A 8A

8B lOA lOB 10e 100 12A 128 15A 15B

-2

0

_2

-1

4

-1

_1

o

2

-3

0

@

75

75

OA OA 15e

OA OA

@

@

30

20

20

BB BA BB BA

AB AB

Ou 15E

@

@

@

@

21 211 AA AA AA AA 8* 211A

21 AA AA 21A

20A 20B

@

@

@

@

@

@

@

@

@

x,

+

•

124

o

4 -20

x,

•

280

-8

28

X.

•

651

11

21

x,

+

930

18

4

9

-6

-1

30 -10

0

-5

5

-1

o

0

-3

-5

-5

26

26

48

0

6

6

55

15

4

4

-2

0

0

0

2

2

5

5

_1

0

-1

_1

6

6

-1

0

-5

5

0

0

0

0

5

0

o

o

3

3

-2

-2

-, -2

0033-2300-23

X.

o

960

0 -48

0

0

0 -40

0 _10

-5

10

0

0

0

o

0

0

0

0

0

0

0

2

2 523.2

x,

o

960

0 -48

0

0

0 -40

0 _10

-5

10

0

0

0

o

0

0

0

0

0

0

0

2

2

x.

•

1085

5

77

-1

5 -40

5

0 -10

10

5

-1

-1

0

_1

0

5

0

0

x,

•

1085

5 -49

5

5

85

10

-1

5

_1

0-1

5

0

0

0

-1

x"

+

1240

-8 -20

4

0

0 -10 -10

15 -10

11

4

-2

2

2

-3

2

0

0 -10 15

-5

Xu

+

1890

0

0

6

6

15

15

x"

•

2480 -16 104

-4

0

0 -20 -20

5

5

Xu

•

2604

20

84

-3

0

4 -21 -16

14

-1

x,.

+

2604 -12

21

0

4

0 104 -21

18

15

15 -10

o

0

-1

n

5

_1

_1

_1

0

0

_1

o

-1

0

0

o

o

_1

o

o

0

0

0

5

8

-4

2

2

0

0

4

4

_1

_1

0

0

4

_1

_1

_1

, , ,

4

_4

5-1

002-50000

9

_1

-1

-1

-3

0

0

0

o

0

0

0

0

-1

_1

o

0

0

0

0

0

0

0

0

_1

0

0

0

0

0-2

0

0

0

-1

_1

_1

_1

-1

0

o

0

2

0

_1

o

-1

o

-1

0

o

0

5820200-500

o

-1

o

o

0

-1

o

o

0

o

0

10

15

0

5

x..

+

3906 -30 126

0

-2

6

31

36

16

x"

+

3906 -30

0

6

6

-2 156

31

6

6

6

0

6

0

0

0

0

0

-5

0

0

0

-2

0

o

o

x,.

+ 10416 -16

84

0

0

0 -84

36 -19

-4

-9

_4

8

2

0

0

0

4

4

-1

_1

0

0

9-1

-1

-1

x"

+ 12096

0

0

0

0

0

96 -24

6

-9

_4

0

0

0

0

0

0

0

0

0

0

0

0

0

0

o

x"

+ 12096

0

0

0

0

0

96 -24

6

-9

_4

0

0

0

0

0

0

0

0

0

0

0

0

0

0

x"

+ 12096

0

0

0

0

0

96 -24

6

-9

-4

0

0

0

0

0

0

0

0

0

0

0

0

0

Xu

+ 12096

0

0

0

0

0

96 -24

6

-9

_4

0

0

0

0

0

0

0

0

0

0

0

0

0

x"

+ 12096

0

0

0

0

0

96 -24

6

-9

-4

0

0

0

0

0

0

0

0

0

0

0

0

0

x,.

• 13020

4 -84

9

0

4-105

20

-5

-5

-5

4

o

0

2

-1

4

_1

_1

o

x"

+ 13020

36 105

0

4

0

15 -15

0

-5

9

0

-2

0

-4

-9

o

-5

_1

6

-1

_1

5

0

-1

-1

o

_1

o

0

0

-8

-2

0

2

0

4

-1

_1

_1

0

_4

5

_1

2

0

-2

-5

0

0

0

0 124

-6 _11

4

-1

12

0

0

0

0

0

-11

6

0 124

-6 _11

,

_1

6

0

0

0

0

0

11

-6

0-5

-1

_1

5

0-1

o

_1

XI3

-2

0

0

0

0

0

Xl5

00000X16

000000

0

000000

o

0

0

0

o

0

o

0 e31

*2

*11

*8 *16

XZ1

o

o

0

0

0

000000

o

0 *16 e31

*2

*4

*8

XZ2

0

o

o

0

0

0

000000

o

0

*8 *16 e31

*2

*4

X23

0

o

o

0

0

0

000000

o

0

*11

*8 *16 e31

*2

Xu

0

o

o

0

0

0

000000

o

0

*2

*11

*8 *16 e31

X2~

_1

0

o

o

0

o

0

0

o

0

o

o

0-1 -1

0

0

o

_1

0

0

_1

0

0

200000X28

o

OOOOOOX29

0

0

_1

_1

_1

0

0

0

0

0

0

0

0

0

0-1

r6 -r6

0

0

_1

-1

000000X'32

r6

0

0

_1

-1

0

0

0

0

0

0

X33

2

0

0

0

0

0

X~

o

o

0-1

0

o

o

0

0

-1

00-2-2

-2

-2

0

0

0

o

0

o

2

0

o

0 0

0

0

-1

_1

o 0

_1

2

0

-11

6

o

0

o

0

0

x"

+ 156211 _24 126

0

0

0 1211

-6 _11

11

_1

-6

0

0

0

0

0

-4

6

o

0

o

0

0

o

0 -r6

X~

+ 156211 -211

0 -12

0

0-126

11

9

-6

_1

o

12

0

0

0

0

6

_11

o

0

0

0

o

o

-2

0

0

o

0

0

0

0

0-1

o

X~

+ 15624

211

0

6

0

-8-126

11

9

-6

_1

o

6

0

0

0

0

-6

11

o

-2

0

0

o

o

o

2

o

0

0

0

0

0-1

o

X~

+ 156211 -211

0

6

0

0-126

11

9

-6

-1

0-600006-11

o

0

0

0

o

o

o

0

o

0

0

0

0-.600006-4

o

0

o

0

0

0 -r6

0

0

-1

_1

o

0

0

0

o

o

0

0

0

-1

_1

o

0

o

o

5

5

0

0

-1

_1

-5

0

o

o

0

_1

_1

r6 -r6

-1

OOOOOX26

0

0

0

0

0

0

0

0

0

0

0

0

Xv

X30

000000X31

0

o

00000X19 _200000X20

0

0

_1

0

o

0

_1

0

-2

0

-1

0

0

0

-1

0

000000X1S

-6

-6

0

000000X17

_1

9

0

0

4

11

00000X12

0

-6 _11

0-126

XIl

_1

0 124

0

-1

2000000X1"

0

6

-1

0

0

0

-1

-1

+ 15624 -24 126

+ 156211 -211

_1

0

x"

_1

X9

0-1

o

_1

0

0

5

-2

0

o

000000

-5

o 0

0

OOOOOX",

_1

o

_1

0

o -1 -1

0

-2

-9

o

0

0

_2

_1

OOOOOXS

o

6600000-5000-20

0

-8

o

o -,

5

0

0

0

o -,

-1

24 126

0

o

_2

-5

+ 15624

o

o

,

X>

2

0

-1

0

X>

0

6

0

x,

_2

-1

.0

-1

3

o

0

-1

3

-5

0

-1

3

-2

0

-1

0

3

0

-1

0

8

0

-1

0

0

X.

_1

0

-2

C

-1

0

0

0

_1

3

x"

0

0

_1

31 105

0

0

0

0

3255

0

0

0

0

+

• 15624 -24-252

0

0

0

x"

x,.

_1

-1

o 0

5

0

_1

0

0

15 _15

-1

0

5

20

-1

x,

0

5

4-125

_1

0

0

5

0

0

0

0

-5 130

4

o

0

0

-1

0

0

0

_1

0

9

0

0

_1

0

31 -21

5

2

o

0

o

3255

20

0

o

•

4 105

0

0

x"

20

0

0

4

+ 13020

0

0

0

+ 15500

0

0

21

x,.

o

x,

0

20

x"

0

_1

0

2604

20

0

0

_1

**

•

0 104 -21

o

_1

0

-, 0

.1

523+2 2

2-3

o

-2

Xu

x"

@

@

211 25 30 30 31 31 31 31 "31 BB A AAA EBB A A A A A BB A AAA EBB A A A A A 0* 25A 30A 30B 31A B*2 C*4 D*8E*16

211 211 AA BB AA BB B* 211C

00000X3S

-1

0

r6

_1

0-1

-1

_1

0

0

0

o

_1

_1

0

0

0

0

0

X3f>

00000X37 X~

X~

+ 15625

25-125

-5

5

5

0

0

0

0

0

-5

-5

x,.

+ 16275

35-105

-9

_1

_1

25

25

0

0

0

_1

_1

_1

o

x~

+ 17856

0-288

0

0

0-11111 -211

-11

11

6

0

0

0

-1

0

0

0

0

0

0

0

0

12

2

2

2

0

0

0

_1

_1

+ 17856

0 11111

0

0

0_11111 -211

_11

11

6

0

0

0

-1

0

0

0

0

0

0

0

0

-6

_1

-1

-1

0

0

0 -b21

*

0

0

0

0

o

0

0

0

*-b21

o

0

0

0

O O O O O O O X.. 2

0

0

o

0

0

0

0

0

0

_1

0

0

0

0

0

0

X4'

o

0

0

0

0

0

0

0

_1

0

0

0

0

0

Xu

x.,

x.,

+ 17856

0 11111

x.,

+ 19530

-6-126

[114] xu

+ 19530

-6

0

0

0

0 _10 -6

_11

11

6

0

0

0

-1

0

0

0

0

0

0

0

0

-6

_1

-1

-1

30

5

5

5

-6

0

0

0

0

0

_1

-6

-1

_1

2

0

-1

_1

_1

-1

35

15

0

5

0

-6

0

0

0

0

-6

-1

-1

_1

o

2

0

0

0_11111 -24 6 155

6 -10

30

o

o

_1

o

o

0 0

0

0

0

0

0

0

0

0

X39

OOOOOOOX4<'

0

0

0

0

0

0

X.. 1

M Sporadic Fischer-Griess llMonster" or "Friendly Giant" Broup H :; FG :; F 1

Order = 808.017.424,794,512,875,886,459,904,961,710,757,005,754,368,000,000,000 246.320.59.76.112.133.17.19.23.29.31.41.47.59.71

Mult

=

Out

Constructions Griess

G : the group of orthogonal autOl1'lorphisms of a certain 195834-dimensional algebra. There is a fixed vector It and

G acts irreducibly on the 195833-space orthogonal

1. A base for the space can be defined in terms of the

'\"..0

action of the maxi'llal subgroup 2'+24Co ,. The algebra and an element of G outside this sUbgroup have been computed explicitly in this base. The mUltiplication used below differs slightly from that of Griess, so care

is necessary in comparing results. Algebra

The inner product (a,b) and algebra product a ~nd

( a

*

*

a )

lli

*

a , c

C

b

defi~ed

*

= a

lli

a

(

c ) > ( a

c ) = (a,b,c), say;

*

c ), then ( a

lli

*

b

)

.'3

*

{ x I x alternates with all s in S

~lternat~

If C

a ,

b

*

c ) for all b (and we say that a and c

c , a *" c ) , h'ith equality just when a and c alter'nate.

I f a set S is closed under taking algebra squares

and

later have the following properties:

(a,h) is positive definite;

( a , b

if ( a

*

*

s = 0 for all s in 3 }

are subalgebras. If elements v1' v2'

all alternate with

S , then its annihilator { x

I

x

their own squares and Hith eac:' other, then the subalgebra they generate is associative. The algebra has a unit 1, and the projection 1s of 1 onto any subalgebra S is a unit for 3. For an idempotent i, the eigenvalues of the map x --7 x are suoalgebras wi th Vo (i)

*

lli

i lie in [0,1], and the 0- and 1-eigenspaces Va(i) and V 1 (i)

V1 (i) = 1. tloreover, i f i and j are irje'llpotents with (i ,j) = 0, then

To each element of class 2A there is a corresponding lattice containing

~_~s..p9sition

vector t with t

*t

= 64t, and there is a

and these vectors that is closed under algebra 'Ilultiplication. When equipped with the

inner product 2(a,b) this lattice is integral (and probably unimodular). For further details see the section lIVectors" below. In the following sections, we shall describe a simplified construction for G and the algebra using the loop P described under Fi 24 , section "Loop". The several sections construct: The group N, which is a !.I-fold cover of the normalizer NO of a certain four_group {1,x,y,z} in the Monster. The Leech link, which is a homomorphism from a certain group Qx onto the Leech lattice modulo 2. Representations of various groups, which can be assembled into representations of degree 19688q for Nx ' Ny ' Nz' the centralizers of x, y, z in N. A dic.!-~onary which identifies the 195884-spaces for Nx ' Ny ' Nz compatibly with the action of their intersection Nxyz • A

multiplic~tion

on the resulting 19688 11-space, invariant under G.

Enlargements of certain quotients NXI)' NyO ' NzO of Nx ' My' Nz to groups GxO ' GyO' GzO which preserve the algebra and generate the Monster, in which they are the centralizers of x, y, z. Vectors corresponding to various elements of G. The action of a transposition can be defined in terms of the corresponding vector and the algebra structure. The group N

This is defined using the loop P described under Fi 24 , section "Loop". It is the group of permutations of the triples (A,a,C) of loop elements with ABC = 1 generated by the particular maps (specified by the irnage of CA,B,C)):

(DAD, DB, CD)

YD X

s Ys

(AD, DBD, DC)

ZD : (DA,BD,DcD)

(AS,BS,C S )

(~S = U, Le. 3 even)

(CS,SS,XS)

zS: (SS,XS,CS )

(D

(US

in P)

= -U,

Le. S o-dd)

where S is a standard automorphisrn of P and X = .0.- 1 is the loop inverse of A. Important particular cases of xs' yS' Zs are xd' Yd' zd for d a C*-set. The composition factors in the structure (22X22L211.222.(S3xM24) are defined modulo earlier ones by K

v• 2 11

<

2 22

< xD'YD,zD I

D in P

S3

< xi'Yi'zi >

for some i in (U], regarded as an (odd) C*-set

xd

< Xs

I d an even C*-set >

or ys or Zs

>,

in which we have xnYnzo=1

all 3

>.

in which xS,yS'zS all act as (S], the permutation corresponding to S.

By considering the subgroups S2 and S1 in 3 , we see that N has subgroups Nx ' Ny '

3

Nz of index 3 whose

intersection N has index 5 in N. The group Nx = (22x22).211.222.(S2xM24) has the alternative structure xyz 2.21+24.C212:M24) in which the normal subgroup 2.2 1+24 is called ~ and is generated by the Xo and xd' while the quotient Nx/Q x is isomorphic to the group of monomial automorphisms of the Leech lattice, and is generated by yO (or zD)' acting as the sign-change on (D], and xs' acting as the permutation (S].

[228]

M The Leech link

The typical element XAXgXC'"

}Idl

xDxE = z~

of Ox is written xA.8.C ••••• and we write x_R = x_1.R' The formula

xDExd' where d is the intersection of ~D] and ~J::], regarded as a C*-set, shows that Ox contains z,

and so contains k,=xU?.' There is a natural homomorphisrn taking xR

~

vR' from Ox onto the Leech lattice modulo

2, defined by

where i is one of the 24 points of (U], regarded as a C*-set. The preimages of the 98280 type 2 vectors of the Leech lattice (modulo 2) are called the shor_t_ el.e;nents of Ox' We USe x'R and x'R to mean xU.R and x_U.R when these are short, and otherwise xR' Then typical short elements, and their images in the Leech lattice, are Xi:. ij

~ (4 on i' _4 on j .Oelsewhere)

X;ij ~

.Oelsewhere)

(,I· on i.j

xo.i ~ (-3 0n i"elsewhere)sign-Changed on [0] xt.d ~ (:?:on (D],Oelsewhere)sign-changed on d where in the last line [0] is an octad and d an even subset of it. Representations

The group Nx has the followi":'lg representations (named by their degrees sUbscripted by x): 24 x : kernel (xO.xd> of order 2-26· , l'nage 2'2.u "·'24' 1 8'

This acts on the space generated by 24 orthogonal vectors i 1 (i in (U]) of norm

The image is the group of

monomial autollorphisms of a copy of the Leech lattice L (with (""xi"") interpreted as

~xii1)

in which yO

and zD act as the sign-change on [0]. and Xs as the per"llutation (S] in M24 that corresponds to S. 4096 x : kernel (YU,z_U> of order 2 2 , i.mage 21+24.(2 11 :M 24 ). This is contained in the matrix normal i zer of the faithful representation of the extraspecial group 2' +24 • There are 2 x 4096 norm' basis vectors D1" and Di. permuted like the triples (?,O,?) and (7,7,D) under conjugation (7 indicates an immaterial entry) with the identifications (-UO)1

= 0"

(UD)1

= 01'

(-O)f

= -(D~).

98280 x : kernel (k"k 2 ,k ,x> of order 2 3 • 3 This is a monomial representation on norm 1 vectors XR with XR

= -X R

permuted in the way that the short

elements xR of Qx are permuted under conjugation by the quotient group NX1

= Nx/(k 1>

of Nx '

300 x : the symmetric tensor square of 24 x ' This acts on symmetric 24 x 24 matrices A.A.s basis elements, we take the matrices (ij)1 (for i 'I j) and (H)" whose only :'lon-zero entries are 1 in places (i,j) and U,i) (for (ij),) and (i,t) (for (11)1)' In tensor

•

98304 x : the tensor product of 4096 x and 24 x ' We take the basis elements 0, Dictionary

= 300 x

We define 196884 x NxO

= Nx/K.

~

i, to have norm 1.

+ 98280 x + 98304x' Since K is in the kernel, this can be regarded as a representation of

We define similar representations· 196884 y of Nyor Nyo and 196884 z of Nz or NzO by cyclicallY

permuting the letters (x ,y, z), (X, Y, Z), (A, 8, C), and the subscripts (',2,3) in all our notations. Thus there is a representation 300 y of Ny with basis elements (ij)2 and (ii)2' and typical element B. The dictionary reads: (A. X, , )-name

(8, Y, 2)-namp.

(C,Z,3)-name

(ij) ,

Zij - ZIj

Xij - Xlj

(iJ)2

Zij + ztj

Xij + Xlj

Yij - ytj

(iJ)3

(ii) ,

(ii)2

(11) 3

20

O

XO• i 0

10

i,

03 03

i2

YD• i

2~

O

D1" 0 i 1

i2

0 i3 ~ i3

x;).d

8~-e-deYD.e

~~-d-deZ;'e

+ 8~-d-deXD.e

YO•d

1 + 8~-e-deZD.e

~~-e-deXD.€

8 4 -d-de Yo.e

1,

(11). say

ZO.i

1

1

=

+

ZO.d

The sums in the last three rows are over all C*-sets e which can be represented by even subsets of (0], and the

l ldl

symbol -d means (_1)2 MUltiplication

.. In the three previous rows it is supposed that i is not in (0] ..

There is a symmetric invariant multiplication * defined on 196884-space .. We specify it below in the notation of

A.

*

At

2(AA' + AlA) (A, vR 81 vR) ,XR (X R• S

= X±1)

(X R• S short) (otherwise)

W '&I vA X • R

(W €I v)

=

(W '&I v) • X R

(W 8l v) • (W' 0 Vi)

1

+

81 tr

A • W~ v

R

8 W 81 (v - 2(v,vR)·vR]

~ ~(WR.wt)[(v,vt) - 2(v,vR)(v l ,vR)],XR + + (W,WI).( (v,v')I + 4.. v €I v' + 4.v' 0 v]

Vectors of 300 x are represented by 24 x 24 symmetric matrices A, A', ..... The general vector of 24 x is called v, and vR denotes either of the two type 2 Leech lattice vectors corresponding to the short element xR of Ox' W is the typical vector in 4096 x-space. Its image under xR is written WR.. The sum in the last line is over all short XR (the factor ~ may be omitted if we sum instead over just one of each pair XR, X_R). The quadratic form on '96884-space is defined by the condition that (ij), has norm 2, and all our other basis vectors have norm '. Our basis vectors are orthogonal except when opposite or equal.

[229]

M Enlargements

= 2 12 :M 24

The images of Nx in the representations 211 x and l.1096 x are matrix groups Nx (211) 21+24.(211:M24) 'xhich have a common quotient 2 11 :M 21t • and can be extended

to

and Nx (!W96) =

matrix groups Gx (24)

=

2°Co, and

Gx (4096) = 21+21~.Col having a common quotient Co, (extending 211:~24)' Since the intersection of the kernels of Nx for 24 x and 4096:< is K, Nx (24) ~

2

11

:M 24

= ,

i t follows that NX1 = Nxf!<:, is isomorphic to the diagonal product

Nx (l./096) which can be extended to a larger group Gx1

= Gx (2li).t::.

Co

1

Gx (4096) of structure

2'+24. 2co ,. (The diagonal product A!::J. c 8 is the set of (a,h) in the Cartesian product whose coordinates have

the same image in C.) Plainly the rapresentations 24 x and 4096 x extend to Gx1 ' as does 98280 x since

~1

= Qx /!<'

is still normal and GX1 still permutes the short glements of Ox, by conjugation. So the representation '96884 x extends to Gx '. Since k 2 ;::md k are represented trivially, the result is actually a representation of GxO 3 Gx ,/K 2 , where :'<2

= .

The three such groups GxO ' GyO ' GzO whi-::lh can be supposed (using the dictionary) to

act on a single '963811-space, generate the Monster group G. Vectors

There are certain vectors in the algebra that are permuted by G in the way that i t transforms the elements of certain conjugacy classes. Thus the

~ntire

300 y or 300 z ' -",hlch S:::lt isf ies Cl, 1) = 24, I

group fixes the unit. 24x24 matrix I, regarded as a vector of 300 x '

*

x = 4x, and so ~I is a unit element 1 for the algebra,

corresponding to the unit element of G. Again, the centraliozer (a double cover of the Baby Monster) of an element of class 2.t1.. (a transposition) fixes a further vector t (which we call a transposition vector) satisfying (t,t) :: '28, t

*

t:: 64t. The lattice generated by 1, t, t • t', where t and t' range over all

transposition vectors, is closed under algebra multiplication, and integral (and probably unimodular) with respect to the inner product 2(a,b). There are similar vect·?rs corresponding to (written

~)

elements of certain other conjursacy classes. We describe

the structure of the algebra generated by transposition nctors to and t, corresponding to two transpositions a and b. The conju:sacy classes of ab <md its powers are given in the table, followed by certain relations in the algebra. These involve the vectors t n correspQndi~g to transpositions a(ab)n, ~nd t, u, v, r~ corresponding to any powers of ab that lie in t'1.e respective conjugacy classes 2!1., 3A, 4A, SA..

*

lA

to

2A, 'A

to. t,

2B, ,A.

to • t 1 :: 0 = (to,t,)

3A, 1.0.

to to

* *

to :: 64t O' (to,t O) :: '28, (t o ,1)

= !3(t O+t 1-t),

(to,t,)

= 16,

t

= 2, ~

ab

t, :: 4tO+4t,+2t2-3u, (to,t,) = '3/2, u :: 'O(2t O-t,-t 2 +U), (to'u) :: IlS, u

= 3/2

(1,1)

U ~

912

ab, (u,1)

* u = 90u,

(u,u)

= 1l0S

3C, 1.0.

to. t, :: to+t,-t;:::, (to,t,) :: 2

!lA., 28, '1\..

to. t, = 3tO+3t1+t2+t3-v, (to,t,) :: 4, v ~ ab, (v,1) :: '2 144, v

4B, 2A, 1A.

t o+t,-t 2-t 3+t, (to,t,)

SA, 1A

1

*

v :: 192v, (v,v) :: 2304

'2<3tO+3t,-t2-trt4+N) , 1

2(3to+3t2-t4-t,-t3-w) , to

* 1,4::

'4(t,-t 2-t 3+t 4+w), (to'w) :: 0, (1,4,1) :: 0

W * w:: 350(tO+t1+t2+t3+t4)' (1,4,1,4) :: 3500 6A, 3A., 21\.., 'A

to

*

to+t1-t2-t3-t4-ts+t+u, (to,t,) = 5/2, t ~ (ab)3, u ~ (ab)2

t,

(t ,u)

The complete structure of the algebra generated by to and t, can in fact be read off from the table. Thus if ab is in class 6A., then (ab)2 is in class 3A, and so to

*

t 2 :: !ltO+4t2+2t4-3u.

In terms of the algebra, the action of a transposition can be recovered from the corresponding vector t by the formula x -? x -

141 [

24(x,t)t + ,6x

*

t -

(x

*

t)

*

t )

The vector corresponding to a short element xR of Qx is 2A R - 8X R, where the matrix AR = v R

~

vR

XR • XR•

In particular, for Xij :: Yij :: Zij the vector is 4(ii)+(jj)-(ij)1-(ij)2-(ij)3). The MOnster is generated by NO together with any transposition outside NO' We exhibit the vector t for such a transposition, from which the action of that transposition can be recovered from the above formula. Take an involution of 2Co, outside 212M24 which fixes an ~-dimensional sublattice L8 • For each of the type 2 vectors vR in L8 (modulo 2) select one of the preimages xR in Ox' in such a way that the chosen preimages generate an elementary abelian group 28 • Then t :: A. - 1 XR (summed over the corresponding XR), where A is the matrix of the projection onto L8 •

"Presentation"

(G x G).2 ':

<

I more relations >

The three outermost dots are redundant. By omitting the lowest two dots we obtain generators for G. Further redundant generators may be added so as to complete the diagram to the 26-point graph whose nodes are the 13 points and 13 lines of the order 3 projective plane, joined by incidence. See page 232.

[230]

M Moonshine

There is an infinite sequence of characters I\t (the head characters) of G such that for each element g of G the is fixed up to scalar factors by a

function Tg{z) :: q-l + !il(g).Q + H2 (g).q2 + ••• (where q:: eKp{21Tiz»

certain group ECg) of transformations A

B

C

D

Az + 8

):z--)

Cz + D

In particular for g :: 1 we find that EO) is the group PSL 2 (Z) and T,(z) :: j(z) - 744, where j(z) is the

classical 'elliptic :nodular function q-l + 144 + 196884q + ?1493760q2 + 864299970q3 + 20245856256q4 + 333202640600q5 +

The group ECg) always has the form r o(nlh + el'e 2 •••• ), where the parameters are integers with h dividing n and each et a divisor of nlh prime to nIChe i)' This group is deflned to consist of all matrices

ae

b/h

cn

de

of determinant e

for which a, 0, et d are integers and e is one of1, e1' e2' '.' • He abbreviate its name by omitting !Ilhll when

h = 1, and by writing r')(nlh) or ro(nlh-) when no ei (save 1) is present, and ro(nlh+) when all possible e1 are present. The scalar factors for elements of S(g) are : for elements ()f the normal subgroup rO(nh + e1.e2".') ex p( -2rri/h)

for the ele:nent (

exp(±217i/h)

for the element (

1 1/h

o 1 1

0

n

1

)

).

where the sign is + or - according as nlh is or is not among the e i • These rules completely deter:nine the subgroup F(g) of index h in

ra(nlh+el"~2"••

) that fixes Tg(z) exactly. In fact Tg(Z) is characterized as the

canonioal Hauptmodul for this group F(g) - that is to say, it generates the field of all modular functions invariant under F( g) and has a series of the form 1 Iq + a1 q + a2q2 + •••• The symbols n\h+e1,e 2 , ... serve as names for the conjugacy classes of G, with the slight ambiguities that g and g-1 have the same symbol, and that classes 27A and 27B both have symbol 27+. In this notation the "class" of gk

= n/(n,\(),

is n' Ih'+e',e} •••• where n'

h'

= h/(h,\(),

and e,.

e2' ...

are the divisors of n'lh l allong

e1' e2' •••• Many properties of the elements of G and their centralizers are illU'llinated by this parametrization. Using Fn1 h+ e 1'e2.... for the non-.!\..belian composition factor (when this exists) of the appropriate centralizer, we have

Suz

B

The na'lles F1' F2 , f'3' and FS often used for

SI2 _

F 4 j2+

F!.I_

F

F'1.I(2)

S6(2)

G2 (4)

FS+ HN

He

8, 'rh, :md HN may be regarded as abbreviations of these F_narnes.

~·l.

The full forms can be used to exhihit the analogies between these groups. There are

repli~~ion

H2o (g)

formulae relating the various characters

= Hn+1 (g)

+!' ;ii(g);.!n_i(g) (1

where:£.1 indicates that when n

= 2m

~

Hn,

for instance the

~~~~i:~tion

formula

i": n/2)

the term l\nCg)2 must be replaced by ~2-(S)

= ~(Hm(g)2

- Hm(g2)). The

duplication formula can be used to express all the Rn in terms of Hp H2 , 11 3 and HS ' The head characters H_ 1 , H9 are the linear combinations of the first few irreducible characters with the coefficients

o 4 5

3

2

4 7

5

3

7 11

7 6 3 4 2 2

a 1S 12

8

12 23 16 14

Groups involved

3

4

8

4

4

8 12

7

7

0 3

o

2

0

Which simple groups are involved in M ? We list all simple groups whose order divides that of M, in an abbreviated notation. The relevant parameter is followed by x when the group is not involved in M, by'? when the involvement is undecided, and by , when the simple group intended is the derived group of the group named,

C2 ,3,S,7,11,13,17,19,23,29,31,1.I1,47,S9,71

AS- 12 ,13x-32x

L2 (4,S,7,8,9,11,13,16,17,19?,23,2S,27?,29?,31?,32x,41?,47x,49?,S9?,64x,71?,a1,12Sx,169x,102I.1x) L3 C2, 3, 4. S, 7x, 9x, 16x ,2Sx) U (3,I.I,S,a) 3

L4 (2,3, 4x, Sx, 7x, 9x)

UI.I(2,3,4x,Sx,8x)

S4(21,3,I.I,Sx,7x,ax,9x)

US (2,4x)

S6(2,3x,l.Ix,5x)

~7(2,3,Sx)

08:C2,3)

Sz(8?,32x)

G2 C2',3,4,Sx)

0'8(2,3)

1. 6 C2x, 3x ,4x)

U6 (2,l.Ix) SaC2,3x)

09(2,3x)

3 04 (2)

LS (2, 3x,4x)

F4 (2)

S10(2x)

0iO(2,3x)

01'O(2)

2FI.I(21)

2 E6 (2)

S12(2x) 011C2x)

[231J

The Y-groups We amplify the remarks in the section "presentation", and in so doing provide presentations for

a large number of groups. The wreath square of the Monster is a homomorphic image of the Coxeter group defined by Figure 1. Redundant generators can be added to form a graph of 26 nodes a, zi' ai' b i , ci' d it ei' fit 8i' f (1 = 1,2,3) in which the joins are from a

to bit bj • bk , f

zi to ai' Cj' ok' ei

ai to zi' b i , f j • f k

bi to a , ai' ci' 8i

cl to Zj' zk' bit di

di to ci' ei' .gj' gk

ei to zi' d it fit f

f 1 to aj' sk' ei' gi

gi to b i , dj' dk , f i

f

to a , ei' ej' e k

where {i,j,k} = {1,2,3}. Abstractly, this is the incidence graph of the 13 points (zi,bi,di,fi,f) and 13 lines (a,ai,ci,ei,gi) of the projective plane

~f

order 3.

We define Ypqr to be the subgroup of Mwr 2 generated by a and the first p terms of the sequence b 1, cl' d 1 , el' f 1 the first q terms of the sequence b 2 , c2' d 2 , e2' f 2 the first r terms of the sequence b 3 , c 3 ' d 3 , e3' f 3 (see Figure 1). Similarly, we define Xpqr to be the group generated by a and the first p terms of the sequence b 1 , cl' d 1 , el (p

~

4)

the first q terms of the sequence b2 , c2' d2 , e2 (q

~

4)

the first r terms of the sequence b3 , a3 (r and if p, q

~

~

2) (see Figure 2a),

3 obtain Xpqrs by adjoining the first s terms of f, e3 (see Figure 2b). Similarly,

Qpqr is generated by f 1 , f 2 , f and 3 the first p terms of the sequence al' b 1 , cl' d 1 the first q terms of the sequence a2' b 2 , c2' d 2 the first r terms of the sequence a3' b 3 , c3' d 3 (see Figure 3). The structures of these groups, and multiple covers of certain of them, are tabulated on the next page. The intersection of Ypqr ' Xpqr(s)' Qpqr with M x M is a subgroup of index 2 which projects isomorphic ally into M in all cases of the table except Y444 (image

= M)

X2222 (image

= 210+16.09(2»

X3222 (image

= 2 10+16 .0;0(2»

We have Y5qr = Y4qr if q > 1, r > 0

= Q3qr = Y3 ,q+l,r+l

Xpqr = Ypqr i f r < 2

if q > 0

·X 33rs = X32rs if r, s > 0

Defining relations. We define f ij

= (abibjbkCiCjdi)9,

whenever {i,j,k}

= {1,2,3}.

We conjecture

that Mwr 2 is the abstract group defined by the Coxeter relations of Y555 (Figure 1) together with f1

= f 12

or f 13 ,

f2

= f 23

or f 21 ,

f3

= f 31

or f 32 •

It is known that the eight alternatives here are mutually equivalent, and define the same group as the Coxeter relations of Y444 together with the relation S = 1 of the table, when the above relations are used to define the f i • With the further definitions zi

= fi(abibjbkCidi)5

f

= fi(cidibjCjdjZk)5

ai

= fjejdjCjbjabkCkdkekfk

gi

= ci(abjCjckeif)5

these relations imply all the Coxeter relations of the projective plane. The "relations" column gives relations that complete (or conjecturally complete) the Coxeter

relations to a presentation in the other cases. Using also the information in the "centre" column of the table one can derive presentations for many groups. Thus the Coxeter relations of Y332 together with 3=1;

are presentations for the respective groups

[232]

M==F 1 Relations

Group

Structure

Ql10

S6

none

Q210

25 :S 6

none

Q220

26 :2 5 :S 6

X=l

Q111

4 3 :S6

V=1

none

Q211

2.0;;(3):2

V=l

2 6S8

P=1

Q2 21

23 .U 6 (2)

V=l

Y221

0;;(2):2

none

Q222

24 .220. U6 (2)

V=l ?

Y321

°7(2) x 2

none

26 :0;;(2):2

W=l

Y421

08(2) :2

Q=l

X222

22 .2 6 .07(2)

W=l

R=l

f 12t

X322

08(2) :2

Y331

R=1

X422

W=l

°9(2) x 2

f 21

°9(2) x 2

Y431

W=l

°;0(2):2

R=l

X332

28 :°8(2) :2

Y441

35 :°5 (3):2

W=l

S=1

X432

°;0(2):2

Y222 Y322

° 7 (3) x 2

S=1

X1111

22.2 1+6 :S 4

Y422

° 8 (3):2

Q=S=l

X2111

2.2 4 .2 1+8 :s

Y332

22' Fi22

S=l

f 12 ,

X3111

2.2 4 .[2 11 ],S6

Y432

2 x Fi 23

S=l

f 21

X2211

2 8 .2 8 .2 6 .°-( 6 2):2

Y4 42

3Fi 24

S=1 ?

X3211

28 .2 8 .27 • ( 07(2 ) x2)

Y333

2 3 .2 E6 (2)

S=l ?

f 12 ,

X2221

1 16 2 °.2 '°8(2):2

Y433

22. 8

S=1 ?

f 21 , f 31

X3221

21O .2 16 .(09(2)X2)

Y443

2 x M

S=1 ?

f 31

X2222

(210+16x210+16).(09(2)X2)

Y444

M wr 2

S=l ?

X3222

(210+16x210+'6).0;0(2):2

2Y 511

27S8

none

P

P = (ab,b 2b3c,d 1"l f l)7

2Y 421

2·°8(2):2

none

Q

Q = (f'2",)3 or (ab 1b 2b C,C 2d,",)15 3

3Y 222

36 .° 5 (3):2

T=,

S

R = [f 12 ,d 2 ] or [f21 ,d 1]

3Y 322

3.° 7 (3) x 2

U,=l

f'2' S

S = (ab1clab2c2ab3c3)10

2Y 422

2.° 8 (3):2

S=l

Q

T = ( ablc1b2c2b3c3) 18

3Y332

(22 x 3) .Fi 22

U,=U 2 =,

f 12 , f , 21

3Y333

(2 3 x 3). 2E6 (2)

U1=U2 =U 3 =1 ?

f 12 , f 23 , f 31' S

Group

Structure

Relations

YmnO

Sm+n+2

none

Ylll

23S4

none

Y211

24S5

none

Y31 !1

25S6

none

Y41 h

26 S7

Y5 1,1

Centre

(ab 1b 2b )3 3

(ab1b2b3c1d1)

5

f 12

f 21

f 12

f 21

f 23 , f 31

5

Ui = f ijf ik or [fij,c k ] or [fik,c j V = (alf,a2f2a3f3) 4

S

]

W = ( ablb2b3c,c2a3) 12 X = (a,a2blb2f1f2f3)8

f1

f2 "1

"2 d1

d2 cl

c2 b1

b2 a "1 b

c

d1

cl

b1

a

3

Figure 2a

3

Figure 1

d

3

"3 f

d2

d1 cl

3

f b,

c2 3

b2

a1

a2

f2

f1

"3 f

d1

c,

b,

a3

a

b2

c2

b3

d2 Figure 3

b3

c3

a

d

Figure 2b

3

3

[233]

M Maximal p-local sUbgroups (complete for p #. 2) p

Structure

Specification

2

2°8

N(2A)

22.(2£6(2»:S3

N(2A. 2 }

2'+24· Co

N(28)

+

2

2.2

11

.2

1 22

.(S3xM2l\)

23.26.2'2.2'8.(L3(2)X3S6) 5

2.2

3

10

.2

20

,(S3xL5(2»

N( 2B 3) N(2B 5 ) N(2'O)

3°Fi 24

N(3A)

0

31+

:2 x

°8(3»

'S4

12 • 23uz:2

53 x Th

N(3A 2)

N(3B)

3 .3 5 .3'0.( M, 1x2Slj)

N(38 2 )

33.36.[38l.(L3(3)x[24l)

N(3B3)

38 '0"8(3)02

N(38)

(D

x HN)"2

lO

sl+6: 4J2 "2 2

(5 :(2

2

4]

x U3 (S» .S3

= NOA 221 4a'066)

N(5A) N(SB) N(5A 2 )

5 .5 .5 ,(S3XGL2(5»

4

N(5S 2 )

5 3+ 3 '(2 x L (5» 3

N(58 3 )

54: (Jx2L 2 (25) .2)

N(5B")

(7:3 x He):2

Ne?A)

11+4 :(3x2S 7 )

N(7B)

(72:(3x2A4) x L2 (7».2 7 2 .7.7 2 :GL 2 (7)

N(7A 2 ) N(78 2)

(11:5 x }1,2):2

N(11A)

11 2 : (5 x2A ) S

N(l1A2 )

(13:6 x L (3»'2 3

N(13A)

13~+2:(3x4S4)

N(13B)

13 2 :4L 2 (13)'2

N(13B 2 )

17

(17:8 x L (2»'2 3

N(17A)

19

(19:9 x A ):2 5

N(19A)

("3(8):3 x A5 ):2

7

11

13

2

= N(2A4968527)

N(3G)

2

5

N(2B 2 )

2 'O + 16 ·0tO(2)

2

Supergroups

(S4(4):2 x L (2))'2 3

23

23:11xSlj

N(23AB)

22. 2 1 1. 2 22. (M 24 x3 ) 3

29

(29:14x3)'2

N(29A)

3·Fi 24

31

31: 15 x 3

N (31 AB)

3

41

41:40

N(lnA)

L2 (41):2 ?

47

47:23 x 2

N(47AB)

2'B

59

59:29

N(59AB)

L2 (S9) ?

71

71:35

N(71AB)

L2 (71) ?

3

3

x Th

Some non-local subgroups (AS x A'2):2

N(2A,3A,5A)

(L 2 (11) x H,2 ):2

N(2A, 3A, SA, 11 A), N(2A, 26, 38, 3A, SA)

A °2 2 6

N(2A,3A,5A,11A), N(2B,3B,3B,4A,5A)

M '1

x

(A 6 x A6 x A6 ) .(2 x 54)

N(2A,3A,3A,4B.5A)3

('7

N(2A,3A,3A.5A,7A), N(2A,3A,SA)2

x (AS x A5 )·4).2

(L (2) x S4(4):2)'2 3

N(2A.3A,4B,7A)

(AS x "3(8):3):2

N(2A,3C.SA)

(SS x S5 x 3 5 ):5 3

N(2A.3A,5A)3

(L 2 (11) x L2 (11»:4

N(2A,3A,5A,11A)2

~194 194

[234]

Chevalley group E8 (2) Order = 337,804,753,143,634,806,261,388,190,614,085,595,079,991,692,242,467,651,576,160,959,909,068,800,000

= 212°.313.55.74.112.132.172.19.312.41.43.73.127. 151.241.331 t1ul t = 1

Out

=

1

Constructions

Chevalley

G :

adjoint Chevalley group of type E8 over the field lF 2

Max imal 2-10cal sUbgroups (278 J:014(2) (2 98 J: (5 xL (2» 3 7

(2106J:(53xL3(2)XL5(2» (2104J:(A8xL5(2» (297J:(L3(2)x010(2» (2 83 J:(5 xE 6 (2) 3 57 (2 J:E 7 (2) (2 92 J :L8 (2) Some other sUbgroups S3 x E7 (2)

(L (2) x E6 (2»:2 3 3'(32 :Q8 x 2E6 (2»:S3

(L 5 (2) x L5 (2».4 (U 5 (2) x U5 (2».4

L9 (2):2 3'U 9 (2):S3 0;6 (2)

5·U5 (4).4 U5 (4).5.4 (08(2) x 08(2».012 °8(4)·°12 (304(2) x 304(2».6 3°4(4).6 32+8.24+8.32.2S4 (L (2) x L (2) x L (2) x L3 (2».2S 4 3 3 3 (U (4) x U (4».8 3 3 U (16).8 3 (S3) 8 .2 3L3 (2) U (3):2 x F4(2) 3

[235J

Partitions and the classes and characters of Sn The table gives the usual correspondences between partitions and the classes and characters of Sn' In the table, N

is the largest part of the partition,

m+

refers to the m'th printed character of the ATLAS table,

m-

to its associated character, and

m,M to the sum of the m'th and M'th ATLAS characters of An' The characters of the Schur double covers are also indexed by partitions. More precisely, a partition of n into distinct parts refers either to two characters of 2An that fuse in 2Sn , or to two characters of 2Sn that agree on 2~.

The partition indicates the Sn image of the classes on which the two

characters differ. The following provides enough information to define the Schur double covers

2~, 2+Sn and 2-S n • The preimages in 2±Sn of the permutation (abc • ., )(ghi. ., ) (pqr., • ) • ., are ±e, where e

= [abc ••• l[ghi ••• l[pqr ••• l •••

and we have

= [a1a2l[a1a3l ••• [a1akl.

[a1 a 2a 3··· a k l

If 0 is either preimage of the permutation taking a "-?A, b "-?B, c "-?C,

then we have 0- 1e0

=±

[ABC ••• l[GHI ••• l[PQR ••• l •••

where the sign is - just when the images of

e

and 0 are both odd permutations.

We have [a1 a 2·.,ak lk according as

[236]

={ ..

k

+1

+1

+1

-1

-1

-1

-1

+1

in 2+Sn

+1

+1

-1

+1

-1

-1

+1

-1

in 2-S n

0

1

2

3

4

5

6

7

(mod 8)

form of n :: 12 n = 13 n = 11 partition (pp 102,103) (pp 92,93) (page 76)

N

13AB

1+

121

1+

l1A8

1+

Nl

12M

2+

llA8

2+

10D

2+

N2 N11

22A l1A

3+ 4+

lOB 10F

3+ 4+

18A 9A

3+ 4+

N3 N21 N111

30F 10C lOG

5+ 7+ 6+

98C 6+ 18A 10+ 9A 7+

24A 8A 8B

N4 N31 N22 N211 N1111

36A 98C 18A 18B 9A

8+ 17+ 15+ 18+ 12+

88 24A 8D 8A 8C

8+ 15+ 14+ 16+ 11+

N5 N41 N32 N311 N221 N2111 Nl1111

40A 88 24A 248 8D 8A 8C

13+ 19+ 23+ 29+ 27+ 22+ 14+

35A8 28A 42A 21A 14A 14B 7A

N6 N51 N42 N411 N33 N321 N3111 N222 N2211 N21111 Nl11111

428 35A8 28A 288 218 42A 21A 14C 14A 14B 7A

9+ 21+ 36+ 38+ 26+ 49+ 40+ 28+ 39+ 30+

N61 N52 N511 N43 N421 N4111 N331 N322 N3211 N31111 N2221 N22111 N211111 Nll11111

n = 10 49)

(page

n =8 22)

(page

n =7 10)

(page

n =6 5)

(page

n =5 (page

2)

1+

9A8

1+

8A

1+

7AB

1+

68

1+

5AB

1+

9A8

2+

8A

2+

7A8

2+

6C

2+

5AB

2+

4A

4+

8A 88

3+ 4+

14A 7A

5+ 6+

6B 6E

4+ 5+

lOA 5A

5+ 7+

4A 4B

6+ 7+

6A 3A

5+ 2,3

5+ 12+ 6+

21AB 6+ 14A 10+ 7A 7+

6G 6B 6F

10+ 13+ 11+

15A8 8+ lOA 13+ 5A 9+

12A 4A 4B

6+ 9+ 3,4

3B 6A 3A

34,5 7-

2A 2B

54-

28A 21A8 HA 148 7A

10+ 16+ 14+ 17+ 11+

12B 61 6H 6C 6E

8+ 18+ 14+ 19+ 9+

8+ 897-

2C 2A 2B

3+ 62-

9+ 23+ 27+ 31+ 29+ 26+ 12+

308 12C 6E 6L 6K 6D 6G

9+ 21+ 24+ 28+ 26+ 23+ 7,8

58 20A 30A 15A lOA 108 5A

5+ 16+ 22+ 24+

lA

1-

6F 30C 12H 12D 6N 6G 60 6D 6P 6E 10, 11 61

5+ 19+ 32+ 34+ 25+ 42+ 37+ 28+ 36+ 17,18 12-

5B 20A 20B 158 30A 15A 10C lOA 108 5A

13+ 25+ 27+ 20+ 31+ 30+

6H 308 30E 12F 12L 12E 68 6R 6G 6P 6F 6Q 6C 6J

16+ 34+ 35+ 37+ 53+ 46+ 47+ 50+ 54+ 45+ 44+ 453014-

20+ 24+ 30+ 43+ 35+ 40+ 41+ 38,39 3635372611-

12A 4F 48 12F 12E 128 12D 4C 4E 4A 4D

2931262728176-

N53 N521 N5111 N44 N431 N422 N4211 N41111 N332 N3311 N3221 N32111 N311111 N2222 N22211 N221111 N2111111 Nl1111111

15C 10F 58 20A 60A 20D 20B 20C 30D 158 30A 30C 15A lOA 10E 108 10D 5A

24+ 43+ 33+ 20+ 52+ 51+ 55+ 4448+

41283340434229243431167-

6H 3C 6C 6J 38 6A 61 6B 6F 3A

N441 N432 N4311 N4221 N42111 N411111 N333 N3321 N33111 N3222 N32211 N321111 N3111111 N22221 N222111 N2211111 N21111111 Nl11111111

41 12K 12C 4D 4H 4B 12H 12C 121 128 12J 12A 12G 4G 4C 4F 4A 4E

25+ 31,32 4851502825524743534927353829186-

3C 6M 3D 6L 6C 6K 3B 68 6J 6A 6H 3A

13302520322714192315104-

N3331 N3322 N33211 N331111 N32221 N322111 N321il11 N31111111 N22222 N222211 N2221111 N22111111 N211111111 Nl111111111

3D 6D 60 3C 6N 6E 6L 38 6K 6B 6M 6A 61 3A

20243726343623151621191774-

28 2F 2C 2E 2A 2D

598632-

N222221 N2222111 N22211111 N221111111 N2111111111 Nl1111111111

2C 28 2E 2A 2D

9138532-

lA

1-

Nl11111111111

lA

1-

2F

10E 58 60A 20A 208 15B 30A 30B 15A 10D lOA 10C 5A

4H 12C 4D 41 48 128 41,42 12F 12G 555412A 12E 394F 33464C 404G 224A 4E 12-

10C

n =9 37)

(page

4A 12A 4D 4B 4C

3+ 14+ 12+ 9-

3B 6A 6B 3A

12+ 16+ 18+ 181711-

6D 38 6A 6C 3A

6,7 1214135-

2C 2A 28

652-

231724197-

38 6E 3C 6D 6A 6C 3A

3,4 16141215136-

2A 2D 2B 2C

3842-

lA

1-

3C 68 6G 38 6F 6A 6D 3A

111522141618104-

28 2D 2A 2C

91052-

lA

1-

1520252414132116124_

2D 28 2E 2A 2C

58632-

lA

1-

2E 2B 2D 2A 2C

910532-

lA

1-

lA

1-

22+

302311-

4F 4C 12D 12A 12C 4A 4E 4B 4D

20A 9+ 15A8 15+ lOA 14+ 108 17+ 5A 7,8

4B 12A 128 23+ 4D 12,13 4A 4C 9-

15+ 17+ 11+ 20,21

10,11

, lA

1-

'"

15+ 29+ 2218,19

13+ 33+ 21,22

[237]

Involvement of sporadic groups in one another For each sporadic simple group G, we ask which smaller sporadic groups are involved in G (that is, are quotients of subgroups of G)? In the table the entry in the row for G and the column for S is if S is involved in G

+

if S is not involved in G if we don't know ( S

?

= J 1,

,

G

=M )

If J 1 is involved in M, then it must be as a maximal subgroup, with conjugacy classes lA, 28, 3C, 58, 6F, 78, 10E, llA, 150, 19A.

+,'

M11 M12

.¥ +

+,'t-

.¥

.¥

J1

':>'

.¥

M22

.¥

J2

M23 HS J

+'t-'t-

+

.¥

+

+

':>'t+'t-'?

.¥

+

~

.¥

3

M24

+

MCL

+

+

+

':>'?

.¥

+

+rt

.¥

+

~'v

.¥

He

~

.¥

Ru Suz

+

O'N

+

C0 3

+

CO 2

+

Fi 22

+

HN

+

Ly

.¥

+

+

,>::-V

+

+

+ +

,~

.¥

+ +

":l,;,'"

+

+ +

.¥

+

'

-

Cl

v°'?

.

+

.¥

+

+

+

+

v°'t-

¥

«-",'t-'t-

.

¥

+

+

.¥

+

23

COl J4

[238]

+ + +

+

+

+

+

+

+ +

+

+

+

+

+

+

8

+

+

+

+

+

+

?

+

+

+

+

+

+

+

+

«..""

¥«-'"'t-'?

.

+

+

Fi 24

M

'v'"

.¥

Th Fi

'X:-~

(,0'

.¥

+

.

¥

+ + + +

+

+ +

+

+

+ +

+

+

+

+ +

+

+ +

«-",'rt

.¥

+ +

':>~

.¥'9

+ +

+

+

+

.¥

+

Order of simple groups This list contains all the simple groups of order less than 10 25 , except that we have stopped the series

at orders

Name(s)

'5

="2(4) ="2(5)

Hult

Order

60

2 2 .3.5

2

'68

2 3 .3.7

2 6

="2(7) '6 ="2(9) =3 4 (2)' "2(8) =R<3l'

360

23 .32 .5

504

23 .3 2 .7

2

3

"2(11)

660

2 2 .3.5.11

2

2

"2(13)

1092

22. 3• 7 • 13

2

2

"2('7>

2448

2

2 .3 "7

2

2

'7

2520

23.32 .5.7

6

2

3420

22. 32. 5 • 19

2

2

4080

2 4 .3.5.17

4

5616

2 4 .3 3 .13

2

6048

2 5 .33 .7

2

6072

2 3 .3.11.23

2

7800

2 3 .3.52 .13

2

"11

7920

2 4 .32 .5.11

"2(27)

9828

2 2 .3 3 .7.13

2

6

"2(29)

12180

22. 3• 5 • 7 .29

2

2

"2<3 1 )

14880

2 5 .3.5.3'

2

2

20160

26 .32 .5.7

"3(4)

20160

26 .32 .5.7

"2<37>

25308

2 2 .3 2.19.37

25920

26 .34 .5

2

3z(8)

29120

26 .5.7,'3

3

"2(32)

32736

25 .3.11.3'

5

"2(41)

34440

2 3 .3.5.7.41

2

2

39732

2 2 .3.7.11.43

2

2

5'888

2 4 .3.23.47

2

58800

2 4 .3.52 .72

2

"3(4)

62400

26 .3.5 2 .13

"2(53)

74412

22. 33. 13 • 53

2

2

95040

2 6 .3 3 .5.11

2

2

102660

22. 3• 5 .29.59

2

2

113460

2 2 .3.5.31.61

2

2

126000

2'.32 .5 3 .7

3

"2(67)

150348

2 2 .3.11.17.67

2

J,

175560

2 3 .3.5.7.11. '9

"2(71)

'78920

2 3 .32 .5.7.7'

2

2

'9

181440

26 .3 4.5.7

2

2

"2(73)

194472

2 3 .32 .37.73

2

2

2 4 .3.5.13.79

2

2

"3(2)

"2(19) "2(16)

"3<3l U (3) =- G2 (2)' 3

"2(23) "2(25)

'8

="4(2)

"4(2)

=3 4 (3)

"2(43) "2(47) "2(49)

",2 "2(59) "2(61) "3(5)

"2(79)

4

2

D'2 2

2

262080

2 6.32.5.7. '3

"2(81)

265680

2 4 .3 4 .5.41

2

2 x 4

"2(83)

285852

22.3.7.4,.83

2

2

352440

2 3 .32 .5.11.89

2

2

372000

25 .3.53 .31

443520

2 7 .32 .5.7.11

'2

2

456288

2 5 .3.7 2 .97

2

2

515100

2 2 .3.52 .17.101

2

2

546312

23 .3.13.17. '03

2

2

J2

604800

2 7 .3 3 .52 .7

2

2

"2('07>

612468

22.33.53.107

2

2

"2(109)

647460

22. 33. 5• 11 • '09

2

2

"2('13)

721392

2 4 .3.7.19.113

2

2

"2(121)

885720

2 3 .3.5.n 2 .61

2

22

"2('25)

916500

22.32.53.7.31

2

6

979200

28 .32 .52 .17

"2(64 )

"2(89) "3(5) "22 "2(97) "2(101) "2(103)

3 4 (4) 3 (2) 6

'451520

29 .34 .5.7

',0

1814400

4 2 2 7 .3 .. 5 .7

6

2

4 2 2

2

[239]

Name(s)

Mult 3

Out 3 3

1876896

2 5 .3 2 .7 3 .19

".0)

3265920

2 7 .3 6 .5.7

32 x 14

D8

G2O)

4245696

2 6 .3 6 .7.13

3

2

3.(5)

4680000

2 6 .3 2 .5'.13

2

2

5515716

2 9 .3'.7.19

3

3 x 53

5663616

27 .3.7 3 • '3

6065280

2 7 .3 6 .5.13

9999360

2 10 .3 2 .5.1.31

"3(8) "3(7) L.O) L (2) 5

2 2

22 2

"23

10200960

2 1 .3 2 .5.1.11.23

"5(2)

13685760

2 10 .35 .5.11

2

L (8) 3 2F4 (2)1

16482816

2 9 .32 .12 .13

6

17911200

2 11 .3 3 .5 2 .13

2

All

19958400

21 .3 4 .52 .1.11

5z(32)

32537600

2 10 .5 2 .31.41

5

L (9) 3

42456960

27 .3 6 .5.7.13

22

"3(9)

42573600

25 .3 6.52 .73

M3

44352000

2 9 .32 .5 3 .7.11

2

2

J

50232960

2 7 .3 5 .5.17.19

3

2

10915680

2 5 .3 2 .5.11 3 .37

3

3

2

2

22

3

3

U (11) 3

2

2

• 3

3.(7)

138291600

2 8 .3 2 .5 2 .7'

0"8(2)

114182400

2 12 .35 .52 .1

0 (2)

197406120

2 12 .34 .5.1.11

2

3D.(2)

211341312

2 12 .3 4 .1 2 .13

3

212421600

2'.3.52 .7.11 3 .19

2

239500800

2 9 .3 5 .52 .1.11

"2.

244823040

2 10 .33 .5.1.11.23

G2 (·)

251596800

8

L (11) 3

A

3

2

2

2 12 .3 3 .52 .1.13

2

2

210118272

25 .32 .7.13 3 .61

3

3

"3(13)

811213008

2 4 .3.1 2 .13 3 .157

"cL

898128000

21 .36 .53 .1.11

L.(4)

987033600

2 12 .34 .52 .1.17

22

".(4)

1018368000

2 '2 .3 2 .53 .13.'7

3.(B)

2 12 .3 4 .5.12 .13

•

1056106560

L (16) 3

1425715200

2 12 .3 2 .52 .1.13.17

3

4 x 53

3.(9)

1121606400

28 .3 8 .52 .41

2

22

"3(17)

2311618212

2 6 .3 4 .7.13.11 3

3

3

An

3113510400

2 9 .3 5 .5 2 .1.11.13

2

2

He

4030381200

210.33.52.73.11

2

4219234560

2 12 .3.5.17 2 .241

8

'585351680

2 9 .3 9 .5.7.13

2

2

4585351680

29 .39 .5.7.13

6

2

5644682640

2\34 .5.19 3 .1 27

3

3

G2 (5)

5859000000

2 6 .3 3 .56 .7.31

L (17) 3

6950204928

2 9 .32 .173 .307

1254000000

27 .3 2 .5 6 .13.31

"6(2)

9196830120

2 15 .3 6 .5.7.11

R(27)

10073444412

2 3 .3 9 .7.13.19.37

3.(11)

12860654400

".(5)

'2

L (13) 3

"3(16) 3 6 0> 0 7 0> L (19) 3

L.(5)

3

2 3

2

6

3

3

2

•

D 8

22 x 3

3

2 6 .3 2 .52 .11 4 .61

2

2

14142000000

2 5 .3 4 .54 .7.13

2

22

"3"9)

16938986.00

25.3 2 .52 .73 .19 3

2

L (2) 6

20158709760

2'5 .3'.5.1.3'

2

"/23)

26056457856

2 7 .32 .11.132 .23 3

5z(128)

3.093383680

2 14 .5.29.113.127

A14

43589145600

210.35.52.72.11.13

3 (2) 8

47377612800

2'6.35.52.7.17

L (25) 3

50718000000

3

3

3

3

3

7 2

2

2 7 .3 2 .5 6 .7.13.31

3

685189814.0

2 6 .3 2 .5.7 2 .13 4 .11

2

D '2 2

78156525216

25 .3.7.11 2 .23 3 .19

145926144000

2".33.53.7.'3.29

"3(25)

152353500000

25 .3.5 6 .132 .601

"3(29)

166551358800

4 2 .32 .52 .7.29 3 .271

L (3) 5

237183237120

2 9 .3 10 .5.11 2 .13

2

"50)

258190571520

2 11 .3 10 .5.1.61

2

L (27) 3

282027186768

2 4 .39 .7.13 2 .751

6

"3(27)

2820564 4 5216

25 .3 9 .72 .13.19.37

6

L/31>

283991644800

2 7 .32 .52 .31 3 .331

3

3

"3 (32)

366151135872

2 15 .32 .11 2 .31.331

3

5 x 53

3.(13) L/23) Ru

[240J

Order

L (7) 3

2 2

• 3

3

3

3

Name(s)

Order

Su.

qq83q5q97600

O'N

460815505920

2 '3 .37 .52 .7""'3 q 29 .3 .5.73 .11.19.31

q95766656000

2 '0 .37 .5 3 .7.".23

q99631102880

25 .3.5.72 .293 .87'

65383718qOOO

2'0.36.53.72.11"3

C0 L

3

3(29)

',5 G2(7) U (31) 3

66q376'38q96

28 .33 .76 .817

852032133'20

Sq(7)

100qq97oqqqSO

.3.5.72 .19.3,3 " q q 2 'O .3 .5.17 .29

Sq(6)

10951999118800

216.32.52.172.257

Uq (7)

1165572172800

210.32.52.76.113

6

Out 2

3

2

'Mult

2 2

2

2

2 2

2 8

q

08

2

22

2

2

L q(7)

2317591180800

Sq(9)

30570' 7889600

S6(q)

4106059776000

q 29 .3 .52 .76 ,'9 q q 26 .3 .52 .19 .181 q 2 '8 .3 .53 .7.13.'7

G2 (8)

q329310519296

218.35.~. 19.73

°8(3)

4952179814400

212.312.52.7.13

22

Sq

2

22

2

2

° 80)

1015'968619520

2 'O .3 '2 .5.7.'3.q,

',6 3Dq O)

10461394944000

2,q.36.53.72.".'3

20560831566912

26.312.72.132.73

2 3

3

20674026236160

2 8 .32 .5.11 2 .23 4 .53

G2(9)

225911320403200

28 .3 '2 .52 .7.13.73

2

°;0(2)

23q992959q8800

220.35.52.7.17.31

2

°10(2)

250'5379558qOO

2

L q(8)

3q5585313382qO

220.36.52.7.1"'7 q 2 '8 .3 .5.73 .'3.73

6

Uq(8)

3q693789777920

2 '8 .3 7 .5.72 .13.19

6

5z(512)

35115786567680

2'8.5.7.13.37.73.109

9

CO

42305421312000

2

22

Sq(23)

2

2

2 Sq(25)

47607300000000

2 '8 .3 6.5 3 .7.".23 28.32 .5 8 .132 .313

L q (9)

507598Q3097600

210.312.52.7.13.111

Q

2 x D8

U (Q) 5

53443952640000

2 20 .32 .5 4 .13.17.41

5

5:Q

Fi 22

64561751654400

2 17 .3 9 .52 .7.11.13

6

2

UQ(9)

101798586Q32000

2 9.3 '2 .53 .Q1.73

2

2 x Q

SQ(27)

10280Q 15783Q560

26.3'2.5.72"32.73

2

6

7 (2)

1638Q9992929280

2 21 .34 .5.72 .31.127

1778437140118000

2,Q.36.53.72.11.13.17

2

2

210103196385600

2 6 .32 .52 ;72 .294 .421

2

2

227787'03272960

2 21 .38 .5.7.11.43

228501000000000

29 .3 Q.5 9 .7.13.31

2

2

228501000000000

Q 29 .3 .5 9 .7.13.3'

2

2

258492255436800

2 20 .3 5 .52 .7.11.17.31

22

273030912000000

2 14 .36 .5 6 .7.11.19

2

G2 (11)

376611192619200

26 .3 3.52 .7.11 6.19.37

SQ<3 1 )

1109387254681600

212.32.52.13.314.37

2

2

Q

D8

L

'17 SQ(29) U7 (2) S6(5) °7(5) L

5 HN

(Q)

2

2

U1I( 11)

1036388695Q78QOO

2 7 .3 4 .5 2 .11 6 .37.61

SQ(32)

11211799322521600

220.32.52.112.312.41

L Q(11

2069665112592000

27 .3 2 .53 .7.11 6.19.61

2

22

SQ<37>

2402534664555840

Q 26 .3 .5. 192 .37Q• 137

2

2

',8

3201186852864000

2'5.38.53.72.1,.13"7

2

2

FQ(2)

3311126603366QOO

22Q.36.52.72.13,'7

2

2

G2(13)

391Q077Q89672896

26.3 3.72 "3 6.61.157

SQ(Q1)

670733Q818822QOO

28.32.52.72.292.414

2

2

6(3)

21032402889738240

211.315.5.7.112.132

2

22

U6 0)

228374724320870110

2 13 .3 15 .5.72 .13.61

2

22

5 10 (2)

24815256521932800

2 25 .3 6 .52 .7.11.17.31

Sz(2048)

36011213418659840

222. 5.23.89.397.2113

11

Q9825657Q393Q0552

2 3 .3 15 .7.11 2 .31.61.271

5

51765179004000000

28.37 .5 6 .7.11.31.37.67

56653740000000000

211.32.510.11.13.31.71

2

U5 (5)

57604365000000000

2 9 .3 5 .5 10 .7.13.521

2

"9

60822550204416000

2'5.38.53.72.1,.13"7.19

2

2

S80)

6578Q75665QQ89600

2111.316.52.7.13.41

2

2

0 9 0)

6578Q75665QQ89600

214.316.52.7.13.41

2

2

°8(11)

670108955QQ320000

°a(Q)

67536Q711956Q8000

22Q.35 .5 Q.7.13.172 22Q.3Q.5 3 .7.13.17.257

D'2 Q

3DQ (Q)

67802350642790400

22Q.3 Q.52 .72 .132 .2Q1

6

G2(6)

7177611478302?ZQO

224.33.52.7.13.172.241

Q

Tb

90745943887872000

215.310.53.72.13.19.31

L

)

R(2Q3) Ly L

5 (5)

G2 (17)

167795197370551296

2 '0 .35 .7"3.'7 6 .307

10

[241]

Mult

Out

'6(7)

2734572'8604953600

2'2. 34. 52. 79"9.43

2

2

0 7 (7)

273457218604953600

2'2.34.52.79.19.43

2

2

G2(19)

796793353927300800

.26.35.52.73.196.127 2

2

Name(s)

A20

'216451004088320000

2'7.38.54.72.11.13"7"9

F1 23

4089470473293004800

2'8.3'3.52.7""'3"7.23

Co,

4157776806543360000

221.39.54.72.11.13.23

L (2) 8

5348063769211699200

228.35.52.~.'7.31.'27

"8(2)

7434971050829414400

228. 39.52 .7.11.17.43

°8(5)

8911539000000000000

2

2'2.35.512.7.132.31

2 2 22

'4

'6(8)

9077005607'76765440

G2 (23)

'157092'62'943780096

227 .37 .5.73 "3"9.73 28 .33 .7.1,2"32.236 .79

08(5 )

17880203250000000000

2'0.34.5'2.7.13.31.313

2

22

A21

25545471085854720000

2'7.39.54.73.1"'3.17"9

2

2

'z(8'92)

36888985097480437760

226 .5.53.157.1613.8191

13

G2(25)

37193298187500000000

28.33.5'2.7.132.31.601

2

3°4(5)

35817806390625000000

26.34.5'2.72.3,2.60'

0;2(2)

500275571482'6524800

230.38.52.72.11.'7.31

2

0'2(2)

51615733565620224000

230 .36.53 .7.1,.13.17.3'

2

'6(9)

54025731402499584000

2'2. 3'8. 53. 7• 13.4,.73

2

22

0 (9) 7

54021)73140;>1199'584000

2'2.3'8.53.7"3.4,.73

2

22

J4

86775571046077562880

221.33.5.7.1,3.23.29.3,.37.43

3

L5 (7)

187035'98320488089600

2".35.52.7'0.19.280'

2

"5(7)

188151359720376729600

2'5.32.52.7'0.'1.43.19'

2

L6 (4) A22

3613'0'34959341568000

230.36.53.72.11.13.17.31

3

562000363888803840000

2'8.39.54.73.1,2.13.17.19

2

°'0(3)

650084965259666227200

215.320.52.7.13.41.61

4

"6(4)

1120527288631296000000

230.34.56.7"32"7.41

°8 4

0;0(3)

1289512799941305'39200

215.320.52.7.112.13.41

2

22

'6(11)

3669292720793456064000

29 .34.53 .7.11 9 .19.37.61

2

2

0 7 (")

3669292720793456064000

29 .34.53 .7.11 9.19.37.6'

2

2

'8(4)

4408780839651901440000

232.35.54.7"3.'72.257

2

L5 (8)

463822600749'0'0887680

230 .34.5.74 ,'3.31.73.151

6

"5(8)

4656663745464977326080

230 .39 .5.72 .".13.19.331

6

°'2 2

A23

12926008369442488320000

218.39.54.73.112.13.11.19.23

2

2

"5(9)

15775810414207914240000

2".320.54.41.73.'18'

5

5:4

'z(32768)

37777778976635853209600

230 .52 .7.13.3,.41.61,'51.1321

15

L (3) 7

67034222101339041669120

2'3.32'.5.7.1,2.132.'093

2

"7(3)

72853912155490594652'60

2'6. 321. 5• 72"3.61.547

2

2E6 (2) L5 (9)

76532479683774853939200

236.39.52.72.11.13,'7,'9

18660280796419613491200

2'5.320.52.1.'12.13.41.61

°8(1)

112554991177198901160000

216.35.54.112.19.43

22

'4

'6(13)

1227969793359061'387'360

2

2

° 7 (13)

122796979335906113871360

29 -3 4.5.7 3.13 9.17.61.157 29.3 4.5.7 3.13 9.17.61.157

2

2

8 12 (2)

208114637736580743168000

236.38.53.72.11.13. '7. 31

E6 (2)

214841515522005575270400

236-36.52.73.13. '7. 31. 73

°8(7)

225297514001560801689600

213.34.52.712.19.43.1201

0(2'87>

239189910264352349332632

23 .3

.1 2 .43.541.1093.2269

7

2F4 (8)

264905352699586116614400

236.35.52.12.132.19.31.109

3

'24

310224200866619719680000

221.310.54.73,112.13.17.19.23

2

2

"9(2)

325473292721108444174400

2 36 .3 11 .5 2 .1.11.11.19.43

3

'3

304 (7)

450782974156649555296512

28,34.712.13.192.432,181

"6(5)

468755520181500000000000

2".36.5'5.72.13.3,.521

L (2) 9

6996'23'0033197642547200

236.35.52.73. 17.31. 73.127

F1 24

1255205109190661121292800

221.3'6.52.73.11.13.17.23.29

3

2

1383059421750000000000000

213.34.515.7.11.13.312.11

2

22

5

°10

21

22 x 3

'3

;,2

2 2

22

3 6

°12 2

L6 (5) L (11) 5 F4 (3)

1952052708565059186240000

29.32.54.7.1110.19.61.3221

5734420792816611844161600

215.324.52.72.132.41.73

'8(5)

6973279267500000000000000

214.35.516.7.132.31.313

2

2

°9(5)

6973219267500000000000000

214.35.5'6.7"32.3'.3'3

2

2

A25

1755605021665492992000000

22'.3'0.56.73.112.13.17"9.23

2

2

"5(11)

97750620209947436785'5200

211.35,52.1110.37.61.13421

2

8

-4154781 481226426191177580544000000

241.313.56.72.11.13.17.19.23.31.41

2

E (2) 7

7997476042075 799759100487262680802918400

2 63 .3" •52.7 3.11. 13. 17. 19-31. 43. 73. '27

,

M

8080174247945'2875886459904

246.320.59.76.112.133.17.19.23.29.31.

961710757005154368000000000

41.47.59.11

E8 (2)

[242]

Order

331804153143634806261

3881906'4085595079991692242 46765'576160959909068800000

2120.313.55.14.112.132.112.19.312.41. .43.13. 121. 151. 241. 331

Bibliography Our main bibliography refers only to the sporadic groups. We add here a few works chosen because they deal with most of the major families of simple groups. Many of them contain extensive bibliographies.

E.Artin, Geometric algebra, Wiley-Interscience, London and New York, 1957 R.W.Carter, Simple groups of Lie type, Wiley-Interscience, London and New York, 1972 R.W.Carter, Finite groups of Lie type, Wiley-Interscience, London and New York, 1985 C.Chevalley, Sur certains groupes simples, Tohoku Math. J. (2) 7 (1955) 14-66 H.S.M.Coxeter & W.O.J.Moser, Generators and relations for discrete groups, 2nd ed., Springer, Berlin, 1965 C.Davis, A bibliographical survey of simple groups of finite order, 1900-1965, Courant Inst. of Math. Sci., New York, 1969 L.E.Dickson, Linear groups with an exposition of the Galois field theory, 2nd ed., Dover, 1958 J.Dieudonne, La geometrie des groupes classiques, 2nd ed., Springer, Berlin, 1963 D.Gorenstein (ed.), Reviews on finite groups, Amer. Math. Soc., Providence, Rhode Island, 1974 D.Gorenstein, Finite simple groups: an introduction to their classification, Plenum Press, London and New York, 1982 B.Srinavasan, Representations of finite Chevalley groups, Springer, Berlin, 1979 R.Steinberg, Variations on a theme of Chevalley, Pacific J. Math. 9 (1959) 8'(5-891

[243]

Mathieu group M11

D.R.Hughes, A combinatorial construction of the small Mathieu designs and groups, Ann. Discrete Math. 15 (1982) 259-264 G.Keller, A characterization of A6 and M11 , J. Algebra 13 (1969) 409-421 H.Kimura, A characterization of A1 and M11 , I, 11, Ill, Hokkaido Math. J. 3 (1914) 213-217; 4 (1975) 39-44; 4 (1915) 213-211 W.O.J.Moser, Abstract definitions for M11 and M12 , Canad. Math. Bull. 2 (1959) 9-13 t.W.Norman, A characterization of the Mathieu group M11 , Math. Z. 106 (1968) 162-166 D.Parrott, On the Mathieu groups M22 and M11 , Bull. Austral. Math. Soc. 3 (1910) 141-142 D.Parrott, On the Mathieu groups M11 and M12 , J. Austral. Math. Soc. 11 (1910) 68-81 G.J.A.Schneider, The Mathieu group M11 , M.Sc. thesis, Oxford, 1919 K.C.Ts'eng and C.S.Li, On the commutators of the simple Mathieu groups, J. China Univ. ScL Tech. 1 (1965) 43-48 H.N.Ward, A form for M11 , J. Algebra 31 (1915) 340-351

Note See also M12 and M24

Mathieu group M12

K.Akiyama, A note on the Mathieu groups M12 and M23 , Bull. Centre Res. Inst. Fukuoka Univ. 66 (1983) 1-5 R.Brauer and P.Fong, A characterization of the Mathieu group M12 , Trans. Amer. Math. Soc. 122 (1966) 18-41 F.Buekenhout, Geometries for the Mathieu group M12 , in "Proceedings of the Conference on Combinatorics", (D.Jungnickel and K.Vedder, eds.), pp. 14-85, Sprj"ger Lecture Notes 969 (1982) J.H.Conway, Hexacode and 1 tracode

MOG and MINIMOG, in "Computational Group Theoryll, Academic Press, London, 1984

J.H.Conway, Three lectures on excel ional groups, in "Finite Simple Groups" (Powel! and Higman. eds.) pp. 215-247, Academic Press, 1971

H.S.M.Coxeter, Twelve points in PG(5,3) with

9~

40 self-transformations, Proc. Roy. Soc. London A 247 (1958) 279-293

R.T.Curtis, The Steiner system S(5.6,12). the Mathieu group M12 and the IIKitten", in "Computational Group Theory" (M.Atkinson. ed.), Academic Press, 1984 G.Frobenius, Uber die Charaktere der mehrfach transitiven Gruppen, Berliner Berichte (1904) 558-571 M.Hall, Note on the Mathieu group M12 , Arch. Math. 13 (1962) 334-340 D.R.Hughes, A combinatorial construction of the small Mathieu designs and groups, Ann. Discrete Math. 15 (1982) 259-264 J.F.Humphreys. The projective characters of the Mathieu group M12 and of its automorphism group, Math. Proc. Cambridge Philos. Soc. 87 (1980) 401-412 B.H.Matzat, Konstruktion von Zahlkorpern mit der Galoisgruppe M12 uber Q(/-5), Arch. Math. 40 (1983) 245-254 W.O.J.Moser, Abstract definitions for M11 and M12 , Canad. Math. Bull. 2 (1959) 9-13 D.Parrott, On toe Mathieu groups M1 I and M12 , J. Austral. Math. Soc. 11 (1970) 68-81 G.J.A.Schneider, On the 2-modular r"present,·' ions of M12 , in "Representations of Algebras ll (Auslander and Lluis, eds.), pp. 302-314, Springer Lecture Notes 90 G.J.A.Schneider. The vertices

(1981) the simple modules of M12 over a field of characteristic 2, J. Algebra 83 (1983) 189-200

R.G.Stanton, The Mathieu groUpJ, Canad. J. Math. 3 (1951) 164-174 G.N.Thwaites, A characterization of M12 by centralizer of involution, Quart. J. Math. Oxford (2) 24 (1973) 537-551 J.A.Todd. On representations of the Mathieu groups as collineation groups, J. London Math. Soc. 34 (1959) 406-416 J.A.Todd, Abstract definitions for the Mathieu groups, Quart. J. Math. Oxford (2) 21 (1910) 421-424 T.A.Whitelaw, On the Mathieu group of degree twelve, Proc. Cambridge Philos. Soc. 62 (1966) 351-364 E.Witt, Die 5-fach transitiven Gruppen von Mathieu, Abh. Math. Sem. Hamburg 12 (1938) 256-264 W.J.Wong. A characterization of the Mathieu group M12 , Math. Z. 84 (1964) 378-388

Note See also M24

Janko group J 1

P.J.Cameron, Another characterization of the small Janko group, J. Math. Soc. Japan 25 (1913) 591-595 G. R. Chapman, Generators and relations for the cohomology ring of Janko's first group in the first twenty-one dimensions, in "Groups - St. Andrews 1981" (Campbell and Robertson, eds.) Cambridge Univ. Press, 1982 J. H. Conway, Three lectures on exceptional groups, in "Finite Simple Groupsll (Powel! and Higman, eds.) pp. 205-214, Academic Press, 1971

P.Fong, On the decomposition numbers of J 1 and R(q), Symp. Math. 13 (1972) 415-422 T.M. Gagen, On groups with Abelian 2-Sylow subgroups, in liThe Theory of Groups" (Kovacs and Neumann, eds.), pp. 99-100, Gardon and Breach, London and New York, 1961 T.M.Gagen, A characterization of Janko's simple group, Proc. Amer. Math. Soc. 19 (1968)

1~93-1395

G. Higman. Construction of simple groups from character tables, in "Finite Simple Groups" (Powell and Higman, eds.) pp. 205-21-4, Academic Press, 1911

[244]

Z.Janko, A new finite simple group with Abelian Sylow 2-subgroups, Proc. Nat. Acad. Sei. USA 53 (1965) 657-658 Z.Janko, A new finite simple group with Abelian Sylow 2-subgroups, and its characterization, J. Algebra 3 (1966) 147-186 Z.Janko, A characterization of a new simple group, in liThe Theory of GrOUpSIl (Kovacs and Neumann, eds.), pp. 205-208, Q:>rdon and Breach, London and New York, 1967 P.Landrock and G.O.Michler, Block structure of the smallest Janko group, Math. Ann. 232 (1979) 205-238 C.S.Li, The commutators of the small Janko group J 1 , J. Math. (Wuhan) 1 (198) 175-179 D.Livingstone, On a permutation representation of the Janko group, J. Algebra 6 (1967) 43-55 R.P.Martineau, A characterization of Janko's simple group of order 175,560, Proc. London Math. Soc. 19 (1969) 709-729 M.Perkel, A characterization of J 1 in terms of its geometry, Geom. Ded. 9 (1980) 291-298 E.E.Shult, A note on Janko's group of order 175,560, Proc. Amer. Math. ·Soc. 35 (1972) 342-348 T.A.Whitelaw, Jankots group as a collineation group in PG(6,10), Proc. Cambrige Philos. Soc. 63 (1967) 663-677 H.Yamaki, On the Janko's simple group of order 175,560, Osaka J. Math. 9 (1972) 111-112

Mathieu group M22

S.M.Gagola and S.C.Garrison, Real characters, double covers and the multiplier, J. Algebra 74 (1982) 20-51 R.L.Griess, The covering group of M22 and associated component problems, Abstracts Amer. Math. Soc. 1 (1980) 213 D.Held, Eine Kennzeichnunug der Mathieu-Gruppe M22 and der alterniernden Gruppe A10 , J. Algebra 8 (1968) 436-439 J.F.Humphreys, The projective characters of the Mathieu group M22 , J. Algebra 76 (1982) 1-24 Z.Janko, A characterization of the Mathieu simple groups. I, J. Algebra 9 (1968) 1-19 W.Jonsson and J.McKay, More about the Mathieu group M22 , Canad. J. Math. 28 (1976) 929-937 P.Mazet, Sur le multiplicateur de Sehur du groupe de Mathieu M22 , C. R. Acad. Sei. Paris 289 (1979) 659-661 D.Parrott, On the Mathieu groups M22 and M11 , Bull. Austral. Math. Soc. 3 (1970) 141-142 J. Tits, Sur les systemes de Steiner associes aux trois llgrands" groupes de Mathieu, Rend. Math. e AppL (5) 23 (1964) 166-184 K.C.Ts'eng and C.S.Li, On the commutators of the simple Mathieu groups, J. China Univ. Sei. Tech. 1 (1965) 43-48

Note See also M24

Hall-Janko group J 2

A.M.Cohen, Finite quaternionic reflection groups, J. Algebra 64 (1980) 293-324 L.Finkelstein and A.Rudvalis, Maximal subgroups of the Hall-Janko-Wales group, J. Algebra 24 (1973) 486-493 S.M.Gagola and S.C.Garrison, Real characters, double covers and the multiplier, J. Algebra 74 (1982) 20-51 D.Q:>renstein and K.Harada, A characterization of Janko's two new simple groups, J. Fac. Sei. Univ. Tokyo 16 (1970) 331-406 M.Hall and D.B.Wales, The simple group of order 604,800, J. Algebra 9 (1968) 417-450, and in liThe Theory of Finite Groups" (Brauer and Sah, eds.), pp. 79-90, Benjamin, 1969 A.P.ll'inyh, A characterization of the Hall-Janko finite simple group, Mat. Zametki 14 (1973) 535-542 Z.Janko, Some new simple groups of finite order, in liThe Theory of Finite Groups" (Brauer and Sah, eds.), pp. 63-64, Benjamin, 1969 J.H.Lindsey, On a projective representation of the Hall-Janko group, Bull. Amer. Math. Soc. 74 (1968) 1094 J.H.Lindsey, Linear groups of degree 6 and the Hall-Janko group, in "The Theory of Finite Groupsll (Brauer and Sah, eds.), pp. 97-100, Benjamin, 1969 J.H.Lindsey, On a 6-dimensional projective representation of the Hall-Janko group, Pacific J. Math. 35 (1970) 175-186 J.McKay and D.B.Wales, The mUltipliers of the simple groups of order 604,800 and 50,232,960, J. Algebra 17 (1971) 262-272 F.L.Smith, A general characterization of the Janko simple group J 2 , Arch. Math. (Basel) 25 (1974) 17-22 J.Tits, Le groupe de Janko d'ordre 604,800, in "The Theory of Finite Groups" (Brauer and Sah, eds.), pp. 91-95, Benjamin, 1969 D.B.Wales, The uniqueness of the simple group of order 604,800 as a subgroup of SL6 (4), J. Algebra 11 (1969) 455-460 D.S.Wales, Generators of the Hall-Janko group as a subgroup of G2 (4), J. Algebra·13 (1969) 513-516 R.A.Wilson, The geometry of the Hall-Janko group as a quaternionic reflection group, Preprint, Cambridge, 1984

Mathieu group M23

K.Akiyama, A note on the Mathieu groups M12 and M23 , Bull. Centr. Res. Inst. Fukuoka Univ. 66 (1983) 1-5 N.8ryce, On the Mathieu group M23 , J. Austral. Math. Soc. 12 (1971) 385-392 U.Dempwolff, Eine Kennzeichnung der Gruppen A5 und M23 , J. Algebra 23 (1972) 590-601 Z.Janko, A characterization of the Mathieu simple groups. 11, J. Algebra 9 (1968) 20-41 L.J.Paige, A note on the Mathieu groups, Canad. J. Math. 9 (1957) 15-18 J. Tits, Sur les systemes de Steiner associes aux trois IIgrands ll groupes de Mathieu, Rend. Math. e AppL (5) 23 (1964) 166-184

Note See also M24

[245]

Higman-Sims group HS

A.E.8rouwer, Folarities of G.Higman's symmetric design and a strongly regular graph on 176 vertices, Aequationes Math. 25 (1982) 77-82

A.R.Calderbank and D.B.Wales, A global code invariant under the Higman-Sims group. J. Algebra 75 (1982) 233-260 J.S.Frame. Computation of the characters of the Higman-Sims group and its automorphism grou, J. Algebra 20 (1972) 320-349 D.Go,renstein and M.E.Harris. A characterization of he Higman-Sims simple group, J. Algebra 24 (1973) 565-590 D.G.Higman and C.C.Sims, A simpl" group of order 44,"::52,000, Math. Z. 105 (1968) 110-113 G.Higman, On the simple group of D.G.Higman and C.C.Sims, Ill. J. Math. 13 (1969) 74-80 J.F.Humphreys, The modular characters of the Higman-Sims simple group, Proc. Roy. Soc. Edinburgh A 92 (1982) 319-335 Z.Janko and S.K.Wong, A characterization of the Higman-3ims simple group, J. Algebra 13 (1969) 517-534 H.Kimura. On the Higman-Sims simple group of order 44,352.000. J. Algebra 52 (1978) 88-93 S.S.Magliveras. The maximal subgroups of the Higman-Sims group. Ph.D. thesis, Birmingham. 1970 S.S.Magliveras. The subgroup structure of the Higman-Sims simple group. Bull. Amer. Math. Soc. 77 (1971) 535-539 J.McKay and D.B.Wales, The multiplier of the Higman-3ims simple group. Bull. London Math. Soc. 3 (1971) 283-285 D.Parrott and S.K.Wong. On the Higman-3ims simple group of order 44,352.000. Pacific J. Math. 32 (1970) 501-516 A.Rudvalis. Characters of the covering group of the Higman-Sims simple group, J. Algebra 33'(1975) 135-143 C.C.3ims, One isomorphism between two groups of order 44.352.000, in liThe Theory of Finite Groups" (Brauer and 3ah. ed.), pp. 101-108. Benjamin, 1969 H.S.Smith, On rank 3 permutation groups, J. Algebra 33 (1975) 22-42 M.S.Smith. On the isomorphism of two simple groups of order 44,352,000, J. Algebra 41 (1976) 172-174 M.S.Smith, A combinatorial configuration associated with the Higman-Sims group. J. Algebra 41 (1976) 175-195 D.B.Wales. Uniqueness of the graph of a rank three group, Pacific J. Math. 30 (1969) 271-276 R.A.Wilson, On maximal subgroups of automorphism groups of simple groups. Preprint, Cambridge, 1984

Janko group J

3

J.H.Conway and D.B.Wales. Matrix generators for J , J. Algebra 29 (1974) 474-476 3 L.Finkelstein and ARudvalis, The maximal sUbgroups of Janko's simple group of order 50,232,960. J. Algebra 30 (1974) 122-143 D.Frohardt, A trilinear form for the third Janko group. J. Alg$bra 83 (1983) 349-379 D.Gorenstein and K.Harada. A characterization of Janko's two new simple groups, J. Fac. Sei. Univ. Tokyo 16 (1970) 331-406 G.Higman and J.McKay, On Janko's simple group of order 50,232.960, Bull. London Math. Soc. 1 (1969) 89-94. 219, and in lithe Theory of Finite Groups" (Brauer and 3ah. eds.), pp. 65-77, Benjamin, 1969 Z.Janko. Some new simple groups of finite order, Symp. Math. 1 (1968) 25-64, and in "The Theory of Finite Groups" (Brauer and 3ah, eds.), pp. 63-64. Benjamin, 1969 J.McKay and D.B.WaleG. The multipliers of the simple groups of order 604,800 and 50,232,960. J. Algebra 17 (1971) 262-212 R.M.Weiss. A geometric construction of Janko's group J , Math. z. 179 (1982) 91-95 3 R.M.Weiss. On the geometry of Janko·s group J 3 , Arch. Math. (Basel) 38 (1982) 410-419 S.K.Wong. On a new finite non-abelian simple group of Janko, Bull. Austral. Math. Soc. 1 (1969) 59-79

Mathieu group M24

E.F.Assmus and H.F.Mattson, Perfect codes and the Mathieu groups, Arch. Math. (Basel) 17 (1966) 121-13S N.8urgoyne and P.Fong, The Sehur mUltipiers of the Mathieu groups, Nagoya Math. J. 27 (1966)

733-74~,

and 31 (1968) 297-304

Chang Choi, On subgroups of M24 • I. Stabilizers of subsets, Trans. Amer. Math. Soc. 167 (1972) 1-27 Chang Choi, On subgroups of M24 • 11. The maximal subgroups of M24 • Trans. Amer. Math. Soc. 167 (1972) 29-47 J.H.Conway, The Miracle Oct<:ld Generator. in "Topics in group theory and computation, Proceedings of the Summer Sehool, Galway, 1973 11 , pp. 62-68, Academic Press, 1977 J.H.Conway, Hexacode and tetracode - MOG and MINIMOG, in "Computational group theory", Academic Press, 1984 J.H.Conway. Three lectures on exceptional groups, in "Finite Simple Groups" (Powell and Higman, eds.), pp. 215-247. Academic Press, 1971

R.T.Curtis. On the Mathieu group M24 and related topics, Ph.D. thesis. Cambridge 1972 R.T.Curtis, A new combinatorial approach to M24 • Math. Proc. Cambridge Philos. Soc. 79 (1976) 25-42 R.T.Curtis, The maximal subgroups of M24 , Math. Proc. Cambridge Philos. Soc. 81 (1977) 185-192 G.Frobenius. Uber die Charaktere der mehrfach transitiven Gruppen, Berliner Berichte (1904) 558-571 D.Garbe and J.Mennicke. Some remarks on the Mathieu groups, Canad. Math. Bull. 7 (1964) 201-211, and 15 (1912) 141 P.J.Greenberg. Mathieu groups. Courant Inst. of Math. SeL, New York. 1973 D.Held, The simple groups related to M24 , J. Algebra 13 (1969) 253-296 G.D.James, The modular characters of the Mathieu groups, J. Algebra 27 (1973) 57-111 M.Koike, Automorphic forms and Mathieu groups, in "Topics in Finite Group Theoryl1, pp. 47-56, Kyoto Univ., 1982 E.Kovac Striko, A characterization of the finite simple groups M24 , He and L5 (2), J. Algebra 43 (1976) 351-397 R.J.List. On the maximal subgroups of the Mathieu groups. I. M24 • Atti Accad. Naz. Lincei Rend. Cl. So1. Fis. Mat. Natur. (8) 62

[246]

(1977) 432-438

H.Luneburg, Uber die Gruppen von Mathieu, J. Algebra 10 (1968) 194-210 D.R.Mason, On the construction of the Steiner system S(5,8,24), J. Algebra 47 (1977) 77-79 E.Mathieu, Memoire sur 1 'etude des fonctions de plusieurs quantites, J. Math. Pures Appl. 6 (1861) 241-243 E.Mathieu, Sur les fonctions cinq fois transitives de 24 quantites, J. Math. Pures AppL 18 (1873) 25-46 P.Mazet, Sur les mUltiplicateurs de Schur des groupes de Mathieu, J. Algebra 77 (1982) 552-576 G.A.Miller, Sur plusieurs groupes simples, Bull. Soc. Math. de France 28 (1900) 266-267 V.Pless, On the uniqueness of the Golay code, J. Combinatorial Theory 5 (1968) 215-228 R.Rasala, Split codes and the Mathieu groups, J. Algebra 42 (1976) 422-471 M.Ronan, Locally truncated buildings and M24 , Math. Z. 180 (1982) 489-501 U.Schoenwaelder, Finite groups with a Sylow 2-subgroup of type M24 , J. Algebra 28 (1974) 20-45 and 46-56 J. de Seguier, Sur les equations de certains groupes, C. R. Acad. Sei. Paris 132 (1901) 1030-1033 J. de Seguier, Sur certains groupes de Mathieu, Bull. Soc. Math. de France 32 (1904) 116-124 R.G.Stanton, The Mathieu groups, Canad. J. Math. 3 (1951) 164-174 J.Tits, Sur les systemes de Steiner associes aux trois "grands" groupes de Mathieu, Rend. Math. e AppL (5) 23 (1964) 166-184 J.A.Todd, On representations of the Mathieu groups as collineation groups, J. London Math. Soc. 34 (1959) 406-416 J.A.Todd, Representation of the Mathieu group M24 as a collineation group, Ann. di Math. Pure ed Appl. (4) 71 (1966) 199-238, and Rend. Mat. e Appl. 25 (1967) 29-32 J.A.Todd, Abstract definitions for the Mathieu groups, Quart. J. Math. Oxford (2) 21 (1970) 421-424 E.Witt, Die 5-fach transitiven Gruppen von Mathieu, Abh. Math. Sem. Hamburg 12 (1938) 256-264 E.Witt, Uber Steinersche Systeme, Abh, Math. Sem. Hamburg 12 (1938) 265-274

McLaughlin group MCL

O.Diawara, Sur le groupe :Slmple de J.McLaughlin, Fh.D. thesis, Bruxelles, 1984 L.Finkelstein, The maximal subgroups of Conway's group C and McLaughlin's group, J. Algebra 25 (1973) 58-89 3 Z.Janko and S.K.Wong, A characterization of McLaughlin1s simple group, J. Algebra 20 (1972) 203-225 J.McLaughlin, A simple group of order 898,128,000, in "The Theory of Finite Groupsll (Brauer and Sah, eds.), pp. 109-111, Benjamin, 1969

H.S.Smith, On rank 3 permutation groups, J. Algebra 33 (1975) 22-42

He Id group He

A.Borovik, 3-local characterization of the Held group, Algebra i Logika 19 (1980) 387-404, 503 G. Butler , The maximal subgroups of the sporadic simple group of Held, J. Algebra 69 (1981) 67-81 J.J.Cannon and G.Havas, Defining relations for the Held-Higman-Thompson simple group, Bull. Austral. Math. Soc. 11 (1974) 43-46 M.Deckers, On groups related to Held's simple group, Arch. Math. (Basel) 25 (1974) 23-28 I.S.Guloglu, A-characterization of the simple group He, J. Algebra 60 (1979) 261-281 D.Held, Some simple groups related to M24 , in "The Theory of Finite Groupsl1 (Brauer and Sah, eds.), PP. 121-124, Benjamin, 1969 D.Held, The simple groups related to M24 , J. Algebra 13 (1969) 253-296, and J. Austral. Math. Soc. 16 (1973) 24-28 E.Kovac Striko, A characterization of the finite simple groups M24 , He and L5 (2), J. Algebra 43 (1976) 357-397 G.Mason and S.D.Smith, Minimal 2-local geometries for the Held and Rudvalis sporadic groups, J. Algebra 79 (1982) 286-3 06 J.McKay, Computing with finite simple groups, in uProceedings of the Second International Conference on the Theory of Groups" U.Schoenwaelder, Finite groups with a Sylow 2-subgroup of type M24 , J. Algebra 28 (1974) 20-45 and 46-56 R.A.Wilson, Maximal subgroups of automorphism groups of simple groups, Preprint, Cambridge, 1984

Rudvalis group Ru

S.B.Assa, A characterization of 2F4 (2)' and the Rudvalis group, J. Algebra 41 (1976) 473-495 J.Bierbrauer, A 2-local characterization of the Rudvalis simple group, J. Algebra 58 (1979) 563-571 J.H.Conway, A quaternionic construction for the Rudvalis group, in "Topics in groups theory and computation, Proceedings of the Summer School, Galway, 1973", pp. 69-81, Academic Press, 1977 J.H.Conway and D.B.Wales, The construction of the Rudvalis simple group of order 145,926,144,000, J. Algebra 27 (1973)

538-5~8

U.Dempwolff, A characterization of the Rudvalis simple group of order 2 14 .3 3 .5 3 .7.13.29 by the centralizers of non-central inVOlutions, J. Algebra 32 (1974) 53-88 M.Hall, A representation of the Rudvalis group, Notices Amer. Math. Soc. 20 (1973)

A~88

G.Mason and S.D.Smith, Minimal 2-local geometries for the Held and Rudvalis sporadic groups. J. Algebra 79 (1982) 286-306 V.D.Mazurov. Characterization of the Rudvalis group, Mat. Zametki 31 (1982) 321-J38, 473 T.Okuyama and T.Yoshida. A characterization of the Rudvalis group, Corom. Algebra 6 (1918) 463-474 M.E.O'Nan, A characterization of the Rudvalis group, Comm. Alg. 6 (1978)

101~141

D.Parrott, A characterization of the Rudvalis simple group, Proc. London Math. Soc. 32 (1976)

2~-51

A.Rudvalis, A new simple group of order 2 14 .3 3 .5 3 .7.13.29, Notices Amer. Math. Soc. 20 (1973) A-95

[247 J

A.Rudvalis, The graph for a new group of order 2 14 .33 .53 .7.13.29, Preprint. Michigan, 1972 A.Rudvalis. A rank 3 simple group of order 2 ,4 .33 .5 3.7.13.29. I, J. Algebra 86 (1984) 181-218 A.Rudvalis, A rank 3 simple group of order 2'4 .33 .5 3.7.13.29. 11. Characters of G and G, J. Algebra 86 (1984) 219-258 R.A.Wilson, The geometry amd maximal subgroups of the simple groups of A.Rudvalis and J.Tits. Proc. London Math. Soc. 48 (1984)

533-563 S.Yoshiara, The maximal subgroups of the sporadic simple group of Rudvalis, Preprint, Tokyo. 1984 K-e~Young.

Some simple subgroups of the Rudvalis simple group, Notices Amer. Math. Soc. 21 (1974) A-481

SUzuki group Suz

J.H.Lindsey, A correlation between PSU 4 (3), the Suzuki group. and the Conway group, Trans. Amer. Math. Soc. 157 (1971) 189-204 J.H.Llndsey. On the Suzuki and Conway groups, Bull. Amer. Math. Soc. 76 (1976) 1088-1090; and in "Representation Theory of Finite Groups and Related Topics ll (I.Reiner, ed.), Amer. Math. Soc., 1971 N.J.Patterson and S.K.Wong, A characterization of the Suzuki sporadic simple group of order 448,345,497,600, J. Algebra 39 (1976)

277-286 A.Reifart, A characterization of Sz by the Sylow 2-subgroup. J. Algebra 36 (1975) 348-363 A.Reifart, A characterization of the sporadic simple group of Suzuki, J. Algebra 33 (1975) 288-305 L.H.Soicher. A natural series of presentations for the Suzuki chain of groups, Preprint. Cambridge. 1984 M.Suzuki, A finite simple group of order 448,345,497.600. in liThe Theory of Finite Groups" (Brauer and Sah, eds.), pp. 113-119. Benjamin, 1969 R.A.Wilson, The complex Leech lattice and maximal subgroups of the Suzuki group, J. Algebra 84 (1983) 151-188 O.Wright, Irreducible characters of the Suzuki group. J. Algebra 29 (1974) 303-323 H.Yamaki. A characterization of the SUzuki simple group of order 448.345,497.600. J. Algebra 40 (1976) 229-244 H.Yamaki. Characterizing the sporadic simple group of Suzuki by a 2-10cal subgroup, Math. Z. 151 (1976) 239-242 S. Yoshiara, The complex Leech lattice and sporadic Suzuki group, in "Topics in Finite Group Theory". pp. 26-46. Kyoto Univ., 1982

O'Nan group O'N

S.Andrilli, On the uniqueness of O'Nanls sporadic simple group. Thesis. Rutgers. 1980 I.S.Guloglu, A characterization of the simple group ON. Osaka J. Math. 18 (1981) 25-31 A~P.Illinyh,

Characterization of the simple O'Nan-Sims group by the centralizer of an element of order 3, Mat. Zametki 24 (1978)

487-497, 589 M.E.O'Nan. Some evidence for the existence of a new simple group, Proc. London Math. Soc. 32 (1976) 421-479 A.J.E.Ryba, The existence of a 45-dimensional 7-modular representation of 3·0 ' N, Preprint, Cambridge, 1984 L.H.Soicher, Presentations of some finite groups with applications to the O'Nan simple group, Preprint, Cambridge. 1984 S.A.Syskin, 3-characterization of the O'Nan-Sims group, Mat. Sb. 114 (1981) 471-478. 480 R.A.Wilson, The maximal subgroups of the QINan group, to appear in J. Algebra S.Yoshiara, The maximal subgroups of the O'Nan group, Preprint, Tokyo, 1984

Conway group Co 3

D.Fendel, A characterization of Conway's group '3, Thesis, Yale, 1970, and Bull. Amer. Math. Soc. 76 (1970) 1024-1025. and J. Algebra 24 (1973) 159-196 L.Finkelstein, The maximal subgroups of Conway's group C3 and McLaughlin's group, J. Algebra 25 (1973) 58-89 B.Mortimer, The modular permutation representations of Conway's third group, Carleton Math. Ser. 172 (1981) M.F.WarbQYs, Generators for the sporadic group C0

3

as a (2,3,7)-group. Proc. Edinburgh Math. Soc. (2) 25 (1982) 65-68

T. Yoshida, A characterization of Conway's group 'C 3 • Hokkaido Math. J. 3 (974) 232-242

Note See also COl

Conway group Co 2

F.L.Smith. A characterization of the '2 Conway simple group, J. Algebra 31 (1974) 91-116 R.A.Wilson, The'maximal SUbgroups of Conwayts group '2. J. Algebra 84 (1983) 107-114 T.Yoshida. A characterization of the '2 Conway simple group, J. Algebra 46 (1977) 405-414

Note See also COl

[248]

Fischer group Fi 22

S.B.Assa, A characterization of M(22), J. Algebra 69 (1981) 455-466 J. H. Conway, A construction for the smallest Fischer group, in "Finite groups '12" (Gagen, Hale and Shult, eds.), pp. 27-35,

North-Holland, Amsterdam, 1913 G.M.Enright, The structure and subgroups of the Fischer groups F22 and F23 , Ph.D. thesis, Cambridge, 1916 G.M.Enright, A description of the Fischer group

F~2' J.

Algebra 46 (1911) 334-343

G.M.Enright, Subgroups generated by transpositions in F22 and F23 , Comm. Algebra 6 (1978) 823-831 B.Fischer, Finite groups generated by 3-transpositions, Preprint, Warwick D.Flaass, 2-10cal subgroups of Fischer's

group~

F22 and F23 , Mat. Zametki 35 (1984) 333-342

D.C.Hunt, A sporadic simple group of B.rischer of order 64,561,751,654,400, Ph.D. thesis, Warwick, 1970 D. C. Hunt , Character tables of certain finite simple groups, Bull. Austral. Math. Soc. 5 (1911) 1-42 D.C.Hunt, A characterization of the finite simple group M(22), J. Algebra 21 (1912) 103-112 J.Moori, On certain groups associated with the smallest Fischer group, J. London Math. Soc. 23 (1981) 61-67 D.Parrott, Characterization of the Fischer groups, !rans. Amer. Math. Soc. 265 (1981) 303-341 R.A.Wilson, On maximal sUbgroups of the Fischer group Fi 22 , Math. Proc. Cambridge Philos. Soc. 95 (1984) 191-222

Harada-Norton group HN

B.Beisiegel, A note on Harada's simple group F, J. Algebra 48 (1911) 142-149 K.Harada, On the simple group r of order 2 14 .3 6.5 6.7.11.19, in "Proceedings of the Conference on Finite Groups (Utah 1915)" (W.R.Scott and F.Gross,

".I:;.

>,

pp. 119-216, Academic Press, 1916

K.Harada, The automorphism group and the Schur multiplier of the simple group of order 2 14.3 6.5 6.7.11.19, Osaka J. Math. 15 (1918)

633-635 S.P.Norton, F and other simple groups, Ph.D. thesis, Cambridge, 1915 S.P.Norton and R.A.Wilson, Maximal subgroups of the Harada-Norton group, Preprint, Cambridge, 1984 P.E.Smith, On certain finite simple groups, Ph.D. thesis, Cambridge, 1915

Lyons group Ly

R.Lyons, Evidence for a new finite simple group, J. Algebra 20 (1972) 540-569, and 34 (1915) 188-189 W.Meyer and W.Neutsch, Uber 5-Darstellungen der Lyonsgruppe, Math. Ann., to appear W.Meyer, W.Neutsch and R.A.Parker, The minimal 5-representation of Lyons' sporadic group, Preprint C.C.Sims, The existence and uniqueness of Lyons' group, in "Finite groups '12" (Gagen, Hale and Shult, eds.), pp. 138-141, North-Holland, Amsterdam, 1913 R.A.Wilson, The subgroup structure of the Lyons group, Math. Proc. Cambridge Philos. Soc. 95 (1984) 403-409 R.A.Wilson, The maximal subgroups of the Lyons group, Math. Proc. Cambridge Philos. Soc., to appear

Thompson group Tb

R.Lyons, The Schur multiplier of F3 is triVial, Comm. Algebra 12 (1984) 1889-1898 R.Markot, A 2-10cal characterization of the simple group E, J. Algebra 40 (1916) 585-595 D.Parrott, On Thompson's simple group, J. Algebra 46 (1971) 389-404 A.Reifart, A characterization of Thompson's sporadic simple group, J. Algebra 38 (1916) 192-200 P.E.Smith, On certain finite simple groups, Ph.D. thesis, Cambridge, 1915 P.E.Smith, A simple sUbgroup of M? and E8 (3), Bull. London Math. Soc. 8 (1976) 161-165 J.G.Thompson, A simple subgroup of E8 (3), in "Finite Groups" (N.lwahori, ed.), pp. 113-116, Japan Soc. for the Promotion of Science, Tokyo, 1916

Fischer group Fi 23

F.Buekenhout, Diagrams for geometries and groups, J. Combinatorial Theory 21 (1979) 121-151 G.M.Enright, The structure and subgroups of the Fischer groups F22 and F23 , Ph.D. thesis, Cambridge, 1976 G.M.Enright, A description of the Fischer group F23 , .J. Algebra 46 (911) 344-354 G.M.Enright, SUbgroups generated by transpositions in F22 and F23 , Comm. Algebra 6 (1978) 823-831 B.Fischer, Finite groups generated by 3_transpositions, Preprint, Warwick D. C. Hunt , A characterization of the finite simple group M(23), J. Algebra 26 (1912) 431-439 D.C.Hunt, The character table of Fischer's simple group M(23), Math. Comp. 28 (1914) 660-661 D.Parrott, Characterization of the Fischer groups, Trans. Amer. Math. Soc. 265 (1981) 303-341 S.K.Wong, A characterization of the Fischer group M(23) bya 2-10ca1 subgroup, J. Algebra 4 (1917) 143-151

[249]

Conway group COl

J.H.Conway. A perfect group of order 8.315.553,613,086.720,000 and the sporadic simple groups, Proc. Nat. Acad. Sei. USA 61 (198) 398-400

J.H.Conway, A group of order 8,315,553,613,086,70,000, Bull. London Math. Soc. 1 (1969) 79-88 J.H.Conway, A characterization of the Leech lattice, Invent. Math. 7 (1969) 137-142 J.H.Conway. Three lectures on exceptional groups, in "Finite Simple Groups" (Powell and Higman, eds.), pp. 215-247, Academic Press, 1971

J.H.Conway, Groups, lattices and quadratic forms, in "Computers in algebra and number theory", pp. 135-139, Amer. Math. Soc. 1971 J.H.Conway, The automorphism group of the

26-dimensio~al

even unimodular Lorentzian lattice, J. Algebra 80 (1983) 159-163

J.H.Conway, R.A.Parker and N.J.A.Sloane, The covering radius of the Leech lattice, Proc. Roy. Soc. London A 380 (1982) 261-290 J.H.Conway and N.J.A.Sloane. 23 constructions for the Leech lattice, Proc. Roy. Soc. London A 381 (1982) 215-283 J.H.Conwayand N.J.A.Sloane, Lorentzian forms for the Leech lattice, Bull. AIDer. Math. Soc. 6 (1982) 215-211 R.T.Curtis, On the Mathieu group M24 and related topics. Ph.D. thesis, Cambridge, 1972 R.T.Curtis, On subgroups of '0. I. Lattice stabilizers. J. Algebra 21 (1973) 549-513 R.T.Curtis, On SUbgroups of '0. 11. Local structure, J. Algebra 63 (1980) 413-434 J.Lepowsky and A.Meurman, An E8-approach to the Leech lattice and the Conway group, J. Algebra 71 (1982) 484-504 S.P.Norton, A bound for the covering radius of the Leech lattice. Proc. Roy. Soc. London A 380 (1982) 259-260 N.J.Patterson, On Conway's group ·0 and some subgroups. Ph.D. thesis, Cambridge, 1972 A.Reifart, A remark on the Conway group '1, Arch. Math. 29 (1977) 389-391 A.Reifart, A 2-local characterization of the simple groups M(24)', ·1 and J 4 • J. Algebra 50 (1978) 213-227 J.G.Thompson, Finite groups and even lattices. J. Algebra 38 (1916) 1-1 J.Tits, Four presentations of Leech's lattice, in "Finite Simple Groups lP' (M.J.Collins, ed.). pp. 303-307, Academic Press, 1980 R.A.Wilson, The maximal subgroups of Conway's group Col' J. Algebra 85 (1983) 144-165

Janko group J 4

D.J.Benson, The simple group J 4 , Ph.D. thesis, Cambridge, 1980 1.S.Guloglu, A characterization of the simple group J 4• Osaka J. Math. 18 (1981) 13-24 Z.Janko, A new finite simple group of order 86,115,510.046,071,562.880, which possesses M24 and the full covering group of M22 as subgroups. J. Algebra 42 (1916) 564-596 W.Lempken, The Schur multiplier of J 4 is trivial, Arch. Math. 30 (1978) 267-270 W.Lempken, A 2-local characterization of Janko's simple group J 4 , J. Algebra 55 (1918) 403-445 G.Mason, Some remarks on groups of type J 4 , Arch. Math. 29 (1911) 514-582 S.P.Norton, The construction of J 4, in "The Santa Cruz Conference on Finite Groups" (Cooperstein and Mason, eds.), pp. 211-278, Amer. Math. Soc., 1980 A.Reifart. Some simple groups related to M24 , J. Algebra 45 (1977) 199-209 A.Reifart, Another characterization of Janko's new simple group J 4 • J. Algebra 49 (1971) 621-627 A.Reifart. A 2-10cal characterization of the simple groups M(24)', ·1 and J 4 • J. Algebra 50 (1918) 213-227 R.M.Stafford, A characterization of Janko's new simple %roup J 4, Notices Amer. Math. Soc. 25 (1918) A-423 R.M.Stafford, A characterization of Janko's new simple group J 4 by centralizer.s of elements of order three, J. Algebra 57 (1919) 555-566

G.Stroth, An odd characterization of J 4, Israel J. Math. 31 (1978) 189-192

Fischer group Fi 24

S.L.Davis and R.Solomon, Some sporadic characterizations. Comm. Algebra 9 (1981) 1725-1142 B.Fischer. Finite groups generated by 3-transpositions, Preprint, Warwick S.P.Norton, F and other simple groups, Ph.D. thesis, Cambridge, 1915 S.P.Norton, Transposition algebras and the group F24 , Preprint, Cambridge D.Parrott, Characterizations of the Fischer groups, Trans. AIDer. Math. Soc. 265 (1981) 303-341 A.Reifart, Some simple groups related to M24 , J. Algebra 45 (1911) 199-209 A.Reifart, A 2-10ca1 characterization of the simple groups M(24)', ·1 and J 4 , J. Algebra 50 (1978) 213-221

[250]

Baby monster group B

J.Bierbrauer, A characterization of the "Baby monster ll group F2 • including a note on 2E6 (2), J. Algebra 56 (1979) 384-395 D.G.Higman, A monom1al character of Fischer's Baby Monster. in lIProceeding of the Conference on Finite Groups" (W.R.Seott and F.Gross, eds.) pp. 277-284, Academic Press, 1976 O.Kroll and P.Landrock, The characters of some 2-blocks of the Baby Monster, its covering group. and the Monster, Comm. Algebra 6 (1978) 1893-1921

J.S.Leon, On the irreducible characters of a simple group of order 241.313.56.72.11.13.17.19.23.31.47, in lIProceedings of the Conference on Finite Groups" (W.R.SCott and F.Gross, eds.), pp. 285-299, Academic Press, 1976 J.S.Leon and C.C.Sims, The existence and uniqueness of a simple group generated by {3,4}-transpositions, Bull. Amer. Math. Soc. 83 (1977) 1039-1040

C.C.Sims, How to construct a Baby Monster, in "Finite Simple Groups II" (M.J.Collins, ed.), pp. 339-345. Academic Press, 1980 G.Stroth, A characterization of Fischer's sporadic group of order 241.313.56.72.11.13.17.19.23.31.47, J. Algebra 40 (1976) 499-531

Monster group M

J.H.Conway, Monsters and moonshine, Math. Intellingencer 2 (1979/80) 165-171 J.H.Conway, A simplified construction for the Fischer-Griess Monster group, Invent. Math., to appear J.H.Conway and S.P.Norton, Monstrous moonshine, Bull. London Math. Soc. 11 (1979) 308-339 S.L.Davis and R. Solomon , Some sporadic characterizations, Comm. Algebra 9 (1981) 1725-1742 P.Fong, Characters arising in the Monster-modular connection, in "The Santa Cruz conference on finite groups" (Cooperstein and Mason, eds.), pp. 557-559, Amer. Math. Soc., 1980 I.Frenkel, J.Lepowsky and A.Meurman, A natural representation of the Fischer-Griess Monster with the modular function J as character, Preprint, Berkeley, 1984 R. L. Griess, The structure of the "Monster" simple group, in "Proceedings of the Conference on Finite Groups" (W .R. Seott and F. Gross, eds.), pp. 113-118, Academic Press, 1976 R.L.Griess, A construction of F1 as automorphisms of a 196,883-dimensional algebra, Proc. Nat. Acad. Sei. USA 78 (1981) 689-691 R.L.Griess, The Friendly Giant, Invent. Math. 69 (1982) 1-102 R.L.Griess, The monster and its non-associative algebra, in "Proceedings of the Montreal Conference on Group Theory (1982)", (J.McKay, ed.) V.G.Kac, A remark on the Conway-Norton conjecture about the "Monster" simple group, Proc. Nat. Acad. Set. USA 77 (1980) 5048-5049 O.Kroll, The characters of a non-principal 2-block of the Monster F1 (?), Preprint Ser. Math. Inst. Aarhus Univ. 31 (1977) O.Kroll and P.Landrock, The characters of some 2-blocks of the Baby Monster, its covering group, and the Monster, Comm. Algebra 6 (1978) 1893-1921

S. P. Norton, The uniqueness of the Fischer-Griess Monster, in "Proceedings of the Montreal Conference on Finite Group Theory (1982 )11, (J.McKay, ed.) J.G.Tbompson, Uniqueness of the Fischer-Griess Monster, Bull. London Math. Soc. 11 (1979) 340-346 J.G.Thompson, Finite groups and modular functions, Bull. London Math. Soc. 11 (1979) 347-351 J.G.Thompson, Some numerology between the Fischer-Griess Monster and the elliptic modular function, Bull. London Math. Soc. 11 (1979) 352-353

J.Tits, Le Monstre. Seminaire Bourbaki (1983/84), No. 620 J. Tits, On R. Griess I "Friendly Giant", Invent. Math., to appear T. van Trung, On the "Monster" simple group, J. Algebra 60 (1979) 559-562

[251]

Index of groups The first column of page numbers refers to the text; the second to the character table 190

188-189

48

49

A"

75

76

Ly, LyS

174

174-175

A'2

91

92-93

L (11 ) 2

7

7

Rut Ruv, Rv

126

127

A

104

102-103

L2 (13)

8

8

R(27)

123

122

AS

2

2

L 2 (16)

12

12

R(2)'

74

75

A6

4

5

L 2 ( 17)

9

9

R(3)'

6

6

A7

10

10

L 2 (19)

11

11

A8

22

22

L 2 (23)

15

15

3~ (q), see ~ (q)

A9

37

37

L2 (25)

16

17

~,

L 2 (27)

18

19

SOn(q), see 0n(q)

2 (29)

20

20

SPn(q), see Sn(q)

L (31) 2

21

21

SUn(q), $ee Un(q)

L 2 (32)

29

29

Suz, Sz

131

128-131

L 2 (4), L 2 (5)

2

2

3z(32)

77

77

28

28

'3

216-217

B

208-219

Cn(q), see S2n(q)

PSPn(q), see Sn(q)

L

see

~

Co" Cl

180-183

184-187

L2 (7)

3

3

3z(8)

Co 2 , C2

154

154-155

L2 (8)

6

6

S2(Q), see L 2 (q)

Co , C 3 3

134

135

L2 (9)

4

5

S4(2)'

4

5

L (11 ) 3

91

3 4 (3)

26

27

L3 (2)

3

3

3 4 (4)

44

45

L (3) 3

13

13

3 4 (5)

61

62-63

L3 ("J

23

24-25

36 (2)

46

47

E (q). see G (q) 2 2

L ( 5) 3

38

38

3 6 (3)

112-113

110-113

E4 (q), see F4 (q)

L 3(1)

50

51

3 8 (2)

123

124-125

E

177

176

E6 (2)

191

L3 (8)

74

74

E7 (2)

219

L (9) 3

78

78

Th

177

176

E8 (2)

235

L (2) 4

22

22

Tits

74

75

L 4 (3)

68-69

68-69

L5 (2)

70

70

U3 (11) U3 (3)

84

84

14

14

F

166

164-165

FG

228-232

220-227

Fi 21 • F 21

115

116-121

Fi 22 , F22

162-163

156-163

Fi 23 , F23

177

178-179

M

Mlo

U2 (q), see L (q) 2

228-234

220-227

100

101

U (4) 3

30

30

4

5

34

35

18

U3 (5) U3 (7)

66

67

Fi 24 , F24

206-207

200-207

18

F,

228-232

220-227

31-33

33

U (8) 3

66

64-65

F2 , F2 +

216-217

208-219

M

21

23

24-25

U (9) 3

79

79

F3 , F 313

177

176

M22

39

40-41

U4 (2)

26

27

71

71

U4 (3)

52-53

54-59

U5 (2) U6 (2)

72-73

72-73

115

116-121

W(E 6 )

26

27

W(E 7 )

46

47

W(E 8 )

85

86-87

F3 +

206-207

200-207

M

F _ 3

131

128-131

M24

94-96

96

F 4 (2)

170

167-173

M(22)

162-163

156-163

164-165

M(23)

177

178-179

43

M(2Ji)'

206-207

200-207

23

105 ON, QIN, D'S G~(q),

132

133

m ffi 02n+1 (2 ), see S.zn(2 )

see Ln(q)

GUn(q), see Un(q)

[252J

POn(q),t PSOn(q), see 0n(q)

0;0(2)

146

142-145

147

148-153

W(~), see ~+1

2An(q), see U + (q) n 1

G (2)' 2

14

14

010 (2)

G2 (3)

60-61

60

03(Q). see L (q) 2

2B2 (2 2m + 1 ), 2 C2 (2 2m+ 1 ), see SZ(2 2m+ 1 )

G (4) 2

97

98-99

04(Q). see L (q2)

2Dn (q), see 02n (q)

G2 (5)

114

114

05(Q),

2

see S4(Q)

2E6 (2)

191

192-199

06"(q), see L4 (q)

2F4 (2)'

74

75

2G2 (27)

123

122

Ht He. HHM

104

105

06'(q), see U4 (q)

HaN, Ha No • HN

166

164-165

0 (3) 7

HiS, HS

80

81

HJ, HaJ

42

43

8(2) °8 m

HJM

82

83

0

°ii(2) °ii m

108-109

106-109

2G2 (3)'

6

6

85

86-87

3D4 (2)

89

90

140-141

136-139

"1

180-183

184-187

89

88

"2

154

154-155

"223

100

101

"233

80

81

"3

134

135

141

36

36

42

43

PGLn(q),

82

83

PGUn(q). psUn(q), see Un(q)

P3~(q),

see

~(q)