Algebraic structure → Group theory Group theory |
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In the area of modern algebra known as group theory, the Conway group is a sporadic simple group of order
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.
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 .
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:
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 |
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 (OEIS: A097340),
and η(τ) is the Dedekind eta function.