Conway group Co3 explained

In the area of modern algebra known as group theory, the Conway group

Co3

is a sporadic simple group of order

   495,766,656,000

= 210375371123

≈ 5.

History and properties

Co3

is one of the 26 sporadic groups and was discovered by as the group of automorphisms of the Leech lattice

Λ

fixing a lattice vector of type 3, thus length . It is thus a subgroup of

Co0

. It is isomorphic to a subgroup of

Co1

. The direct product

2 x Co3

is maximal in

Co0

.

The Schur multiplier and the outer automorphism group are both trivial.

Representations

Co3 acts on the unique 23-dimensional even lattice of determinant 4 with no roots, given by the orthogonal complement of a norm 4 vector of the Leech lattice. This gives 23-dimensional representations over any field; over fields of characteristic 2 or 3 this can be reduced to a 22-dimensional faithful representation.

Co3 has a doubly transitive permutation representation on 276 points.

showed that if a finite group has an absolutely irreducible faithful rational representation of dimension 23 and has no subgroups of index 23 or 24 then it is contained in either

\Z/2\Z x Co2

or

\Z/2\Z x Co3

.

Maximal subgroups

Some maximal subgroups fix or reflect 2-dimensional sublattices of the Leech lattice. It is usual to define these planes by h-k-l triangles: triangles including the origin as a vertex, with edges (differences of vertices) being vectors of types h, k, and l.

found the 14 conjugacy classes of maximal subgroups of

Co3

as follows:
Maximal subgroups of Co3
No. Structure Order Index Comments
1McL

2

1,796,256,000
= 28·36·53·7·11		276
= 22·3·23 		McL fixes a 2-2-3 triangle. The maximal subgroup also includes reflections of the triangle.

Co3

has a doubly transitive permutation representation on 276 type 2-2-3 triangles having as an edge a type 3 vector fixed by

Co3

.
2HS 44,352,000
= 29·32·53·7·11		11,178
= 2·35·23 		fixes a 2-3-3 triangle
3U4(3).22 13,063,680
= 29·36·5·7 		37,950
= 2·3·52·11·23
4M23 10,200,960
= 27·32·5·7·11·23 		48,600
= 23·35·52 		fixes a 2-3-4 triangle
535:(2 × M11)3,849,120
= 25·37·5·11 		128,800
= 25·52·7·23 		fixes or reflects a 3-3-3 triangle
62·Sp6(2) 2,903,040
= 210·34·5·7 		170,775
= 33·52·11·23 		centralizer of an involution of class 2A (trace 8), which moves 240 of the 276 type 2-2-3 triangles
7U3(5):S3 756,000
= 25·33·53·7 		655,776
= 25·34·11·23
83:4S6 699,840
= 26·37·5 		708,400
= 24·52·7·11·23 		normalizer of a subgroup of order 3 (class 3A)
924·A8 322,560
= 210·32·5·7 		1,536,975
= 35·52·11·23
10PSL(3,4):(2 × S3) 241,920
= 28·33·5·7 		2,049,300
= 22·34·52·11·23
112 × M12 190,080
= 27·33·5·11 		2,608,200
= 23·34·52·7·23 		centralizer of an involution of class 2B (trace 0), which moves 264 of the 276 type 2-2-3 triangles
12[2<sup>10</sup>.3<sup>3</sup>] 27,648
= 210·33 		17,931,375
= 34·53·7·11·23
13S3 × PSL(2,8):3 9,072
= 24·34·7 		54,648,000
= 26·33·53·11·23 		normalizer of a subgroup of order 3 (class 3C, trace 0)
14A4 × S5 1,440
= 25·32·5 		344,282,400
= 25·35·52·7·11·23

Conjugacy classes

Traces of matrices in a standard 24-dimensional representation of Co3 are shown.[1] The names of conjugacy classes are taken from the Atlas of Finite Group Representations.[2] [3] The cycle structures listed act on the 276 2-2-3 triangles that share the fixed type 3 side.[4]

ClassOrder of centralizerSize of classTraceCycle type
1A all Co3 124
2A 2,903,040 33·52·11·238 136,2120
2B 190,080 23·34·52·7·230 112,2132
3A 349,920 25·52·7·11·23-3 16,390
3B 29,160 27·3·52·7·11·236 115,387
3C 4,536 27·33·53·11·230 392
4A 23,040 2·35·52·7·11·23-4 116,210,460
4B 1,536 2·36·53·7·11·234 18,214,460
5A 150028·36·7·11·23-1 1,555
5B 300 28·36·5·7·11·234 16,554
6A 4,320 25·34·52·7·11·235 16,310,640
6B 1,296 26·33·53·7·11·23-1 23,312,639
6C 216 27·34·53·7·11·232 13,26,311,638
6D 108 28·34·53·7·11·230 13,26,33,642
6E 72 27·35·53·7·11·230 34,644
7A 4229·36·53·11·233 13,739
8A 192 24·36·53·7·11·232 12,23,47,830
8B 192 24·36·53·7·11·23-2 16,2,47,830
8C 32 25·37·53·7·11·232 12,23,47,830
9A 162 29·33·53·7·11·230 32,930
9B 81 210·33·53·7·11·233 13,3,930
10A 60 28·36·52·7·11·233 1,57,1024
10B 20 28·37·52·7·11·230 12,22,52,1026
11A 22 29·37·53·7·232 1,1125power equivalent
11B 22 29·37·53·7·232 1,1125
12A 144 26·35·53·7·11·23-1 14,2,34,63,1220
12B 48 26·36·53·7·11·231 12,22,32,64,1220
12C 36 28·35·53·7·11·232 1,2,35,43,63,1219
14A 14 29·37·53·11·231 1,2,751417
15A 15 210·36·52·7·11·232 1,5,1518
15B 30 29·36·52·7·11·231 32,53,1517
18A 18 29·35·53·7·11·232 6,94,1813
20A 20 28·37·52·7·11·231 1,53,102,2012power equivalent
20B 20 28·37·52·7·11·231 1,53,102,2012
21A 21 210·36·53·11·230 3,2113
22A 22 29·37·53·7·230 1,11,2212power equivalent
22B 22 29·37·53·7·230 1,11,2212
23A 23 210·37·53·7·111 2312 power equivalent
23B 23 210·37·53·7·111 2312
24A 24 27·36·53·7·11·23-1 124,6,1222410
24B 24 27·36·53·7·11·231 2,32,4,122,2410
30A 30 29·36·52·7·11·230 1,5,152,308

Generalized Monstrous Moonshine

In analogy to monstrous moonshine for the monster M, for Co3, the relevant McKay-Thompson series is

T4A(\tau)

where one can set the constant term a(0) = 24,

\begin{align}j4A(\tau) &=T4A(\tau)+24\\ &=(\tfrac{η2(2\tau)}{η(\tau)η(4\tau)})24\\ &=((\tfrac{η(\tau)}{η(4\tau)})4+42(\tfrac{η(4\tau)}{η(\tau)})4

2\\ &=1
q
)

+24+276q+2048q2+11202q3+49152q4+... \end{align}

and η(τ) is the Dedekind eta function.

References

External links

Notes and References

  1. Conway et al. (1985)
  2. Web site: ATLAS: Conway group Co3.
  3. Web site: ATLAS: Conway group Co1.
  4. Web site: ATLAS: Co3 — Permutation representation on 276 points.

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