Conway group Co3


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

History and properties

is one of the 26 sporadic groups and was discovered by as the group of automorphisms of the Leech lattice fixing a lattice vector of type 3, thus length. It is thus a subgroup of [Conway group|]. 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

Co3 acts on a 23-dimensional even lattice with no roots, given by the orthogonal complement of a norm 6 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 or.

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 being vectors of types h, k, and l.
found the 14 conjugacy classes of maximal subgroups of as follows:
No.StructureOrderIndexComments
1McL:21,796,256,000
= 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.
2HS44,352,000
= 29·32·53·7·11		11,178
= 2·35·23		fixes a 2-3-3 triangle
3U4.2213,063,680
= 29·36·5·7		37,950
= 2·3·52·11·23
4M2310,200,960
= 27·32·5·7·11·23		48,600
= 23·35·52		fixes a 2-3-4 triangle
535:3,849,120
= 25·37·5·11		128,800
= 25·52·7·23		fixes or reflects a 3-3-3 triangle
62·Sp62,903,040
= 210·34·5·7		170,775
= 33·52·11·23		centralizer of an involution of class 2A, which moves 240 of the 276 type 2-2-3 triangles
7U3:S3756,000
= 25·33·53·7		655,776
= 25·34·11·23
83:4S6699,840
= 26·37·5		708,400
= 24·52·7·11·23		normalizer of a subgroup of order 3
924·A8322,560
= 210·32·5·7		1,536,975
= 35·52·11·23
10PSL:241,920
= 28·33·5·7		2,049,300
= 22·34·52·11·23
112 × M12190,080
= 27·33·5·11		2,608,200
= 23·34·52·7·23		centralizer of an involution of class 2B, which moves 264 of the 276 type 2-2-3 triangles
1227,648
= 210·33		17,931,375
= 34·53·7·11·23
13S3 × PSL:39,072
= 24·34·7		54,648,000
= 26·33·53·11·23		normalizer of a subgroup of order 3
14A4 × S51,440
= 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. The names of conjugacy classes are taken from the Atlas of Finite Group Representations.
The cycle structures listed act on the 276 2-2-3 triangles that share the fixed type 3 side.
ClassOrder of centralizerSize of classTraceCycle type
1Aall Co3124-
2A2,903,04033·52·11·238136,2120-
2B190,08023·34·52·7·230112,2132-
3A349,92025·52·7·11·23-316,390-
3B29,16027·3·52·7·11·236115,387-
3C4,53627·33·53·11·230392-
4A23,0402·35·52·7·11·23-4116,210,460-
4B1,5362·36·53·7·11·23418,214,460-
5A150028·36·7·11·23-11,555-
5B30028·36·5·7·11·23416,554-
6A4,32025·34·52·7·11·23516,310,640-
6B1,29626·33·53·7·11·23-123,312,639-
6C21627·34·53·7·11·23213,26,311,638-
6D10828·34·53·7·11·23013,26,33,642-
6E7227·35·53·7·11·23034,644-
7A4229·36·53·11·23313,739-
8A19224·36·53·7·11·23212,23,47,830-
8B19224·36·53·7·11·23-216,2,47,830-
8C3225·37·53·7·11·23212,23,47,830-
9A16229·33·53·7·11·23032,930-
9B81210·33·53·7·11·23313,3,930-
10A6028·36·52·7·11·2331,57,1024-
10B2028·37·52·7·11·23012,22,52,1026-
11A2229·37·53·7·2321,1125power equivalent
11B2229·37·53·7·2321,1125power equivalent
12A14426·35·53·7·11·23-114,2,34,63,1220-
12B4826·36·53·7·11·23112,22,32,64,1220-
12C3628·35·53·7·11·2321,2,35,43,63,1219-
14A1429·37·53·11·2311,2,751417-
15A15210·36·52·7·11·2321,5,1518-
15B3029·36·52·7·11·23132,53,1517-
18A1829·35·53·7·11·2326,94,1813-
20A2028·37·52·7·11·2311,53,102,2012power equivalent
20B2028·37·52·7·11·2311,53,102,2012power equivalent
21A21210·36·53·11·2303,2113-
22A2229·37·53·7·2301,11,2212power equivalent
22B2229·37·53·7·2301,11,2212power equivalent
23A23210·37·53·7·1112312power equivalent
23B23210·37·53·7·1112312power equivalent
24A2427·36·53·7·11·23-1124,6,1222410-
24B2427·36·53·7·11·2312,32,4,122,2410-
30A3029·36·52·7·11·2301,5,152,308-

Generalized Monstrous Moonshine

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 = 24,
and η is the Dedekind eta function.