Jump to content

Conway group Co3

From Wikipedia, the free encyclopedia

In the area of modern algebra known as group theory, the Conway group is a sporadic simple group of order

   495,766,656,000
= 210 · 37 · 53 ·· 11 · 23
≈ 5×1011.

History and properties

[edit]

is one of the 26 sporadic groups and was discovered by John Horton Conway (1968, 1969) as the group of automorphisms of the Leech lattice fixing a lattice vector of type 3, thus length 6. It is thus a subgroup of . It is isomorphic to a subgroup of . The direct product is maximal in .

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

Representations

[edit]

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.

Walter Feit (1974) 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 or .

Maximal subgroups

[edit]

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.

Larry Finkelstein (1973) found the 14 conjugacy classes of maximal subgroups of as follows:

Maximal subgroups of Co3
No. Structure Order Index Comments
1 McL: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. has a doubly transitive permutation representation on 276 type 2-2-3 triangles having as an edge a type 3 vector fixed by .
2 HS 44,352,000
= 29·32·53·7·11
11,178
= 2·35·23
fixes a 2-3-3 triangle
3 U4(3).22 13,063,680
= 29·36·5·7
37,950
= 2·3·52·11·23
4 M23 10,200,960
= 27·32·5·7·11·23
48,600
= 23·35·52
fixes a 2-3-4 triangle
5 35:(2 × M11) 3,849,120
= 25·37·5·11
128,800
= 25·52·7·23
fixes or reflects a 3-3-3 triangle
6 2·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
7 U3(5):S3 756,000
= 25·33·53·7
655,776
= 25·34·11·23
8 31+4
+
:4S6
699,840
= 26·37·5
708,400
= 24·52·7·11·23
normalizer of a subgroup of order 3 (class 3A)
9 2A8 322,560
= 210·32·5·7
1,536,975
= 35·52·11·23
10 PSL(3,4):(2 × S3) 241,920
= 28·33·5·7
2,049,300
= 22·34·52·11·23
11 2 × 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 [210.33] 27,648
= 210·33
17,931,375
= 34·53·7·11·23
13 S3 × 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)
14 A4 × S5 1,440
= 25·32·5
344,282,400
= 25·35·52·7·11·23

Conjugacy classes

[edit]

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]

Class Order of centralizer Size of class Trace Cycle type
1A all Co3 1 24
2A 2,903,040 33·52·11·23 8 136,2120
2B 190,080 23·34·52·7·23 0 112,2132
3A 349,920 25·52·7·11·23 -3 16,390
3B 29,160 27·3·52·7·11·23 6 115,387
3C 4,536 27·33·53·11·23 0 392
4A 23,040 2·35·52·7·11·23 -4 116,210,460
4B 1,536 2·36·53·7·11·23 4 18,214,460
5A 1500 28·36·7·11·23 -1 1,555
5B 300 28·36·5·7·11·23 4 16,554
6A 4,320 25·34·52·7·11·23 5 16,310,640
6B 1,296 26·33·53·7·11·23 -1 23,312,639
6C 216 27·34·53·7·11·23 2 13,26,311,638
6D 108 28·34·53·7·11·23 0 13,26,33,642
6E 72 27·35·53·7·11·23 0 34,644
7A 42 29·36·53·11·23 3 13,739
8A 192 24·36·53·7·11·23 2 12,23,47,830
8B 192 24·36·53·7·11·23 -2 16,2,47,830
8C 32 25·37·53·7·11·23 2 12,23,47,830
9A 162 29·33·53·7·11·23 0 32,930
9B 81 210·33·53·7·11·23 3 13,3,930
10A 60 28·36·52·7·11·23 3 1,57,1024
10B 20 28·37·52·7·11·23 0 12,22,52,1026
11A 22 29·37·53·7·23 2 1,1125 power equivalent
11B 22 29·37·53·7·23 2 1,1125
12A 144 26·35·53·7·11·23 -1 14,2,34,63,1220
12B 48 26·36·53·7·11·23 1 12,22,32,64,1220
12C 36 28·35·53·7·11·23 2 1,2,35,43,63,1219
14A 14 29·37·53·11·23 1 1,2,751417
15A 15 210·36·52·7·11·23 2 1,5,1518
15B 30 29·36·52·7·11·23 1 32,53,1517
18A 18 29·35·53·7·11·23 2 6,94,1813
20A 20 28·37·52·7·11·23 1 1,53,102,2012 power equivalent
20B 20 28·37·52·7·11·23 1 1,53,102,2012
21A 21 210·36·53·11·23 0 3,2113
22A 22 29·37·53·7·23 0 1,11,2212 power equivalent
22B 22 29·37·53·7·23 0 1,11,2212
23A 23 210·37·53·7·11 1 2312 power equivalent
23B 23 210·37·53·7·11 1 2312
24A 24 27·36·53·7·11·23 -1 124,6,1222410
24B 24 27·36·53·7·11·23 1 2,32,4,122,2410
30A 30 29·36·52·7·11·23 0 1,5,152,308

Generalized Monstrous Moonshine

[edit]

In analogy to monstrous moonshine for the monster M, for Co3, the relevant McKay-Thompson series is where one can set the constant term a(0) = 24 (OEISA097340),

and η(τ) is the Dedekind eta function.

References

[edit]
  1. ^ Conway et al. (1985)
  2. ^ "ATLAS: Conway group Co3".
  3. ^ "ATLAS: Conway group Co1".
  4. ^ "ATLAS: Co3 — Permutation representation on 276 points".
[edit]