In the area of modern algebra known as group theory, the Conway group \( {\displaystyle \mathrm {Co} _{3}} \( is a sporadic simple group of order
210 · 37 · 53 · 7 · 11 · 23
= 495766656000
≈ 5×1011.
History and properties
\( {\displaystyle \mathrm {Co} _{3}} \) 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 \( \Lambda \) fixing a lattice vector of type 3, thus length √6. It is thus a subgroup of \( {\displaystyle \mathrm {Co} _{0}} \). It is isomorphic to a subgroup of \( {\displaystyle \mathrm {Co} _{1}} \). The direct product \( {\displaystyle 2\times \mathrm {Co} _{3}} \) is maximal in \( {\displaystyle \mathrm {Co} _{0}} \).
The Schur multiplier and the outer automorphism group are both trivial.
Representations
\( {\displaystyle \mathrm {Co} _{3}} \) 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.
\( {\displaystyle \mathrm {Co} _{3}} \) has a doubly transitive permutation representation on 276 points.
(txt) 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 \( {\displaystyle \mathbb {Z} /2\mathbb {Z} \times \mathrm {Co} _{2}} \) or \({\displaystyle \mathbb {Z} /2\mathbb {Z} \times \mathrm {Co} _{3}} \).
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.
Larry Finkelstein (1973) found the 14 conjugacy classes of maximal subgroups of C o 3 {\displaystyle \mathrm {Co} _{3}} {\displaystyle \mathrm {Co} _{3}} as follows:
- McL:2 – McL fixes a 2-2-3 triangle. The maximal subgroup also includes reflections of the triangle. \( {\displaystyle \mathrm {Co} _{3}} \) has a doubly transitive permutation representation on 276 type 2-2-3 triangles having as an edge a type 3 vector fixed by \( {\displaystyle \mathrm {Co} _{3}} \) .
- HS – fixes a 2-3-3 triangle.
- U4(3).22
- M23 – fixes a 2-3-4 triangle.
- 35:(2 × M11) - fixes or reflects a 3-3-3 triangle.
- 2.Sp6(2) – centralizer of involution class 2A (trace 8), which moves 240 of the 276 type 2-2-3 triangles
- U3(5):S3
- 31+4:4S6
- 24.A8
- PSL(3,4):(2 × S3)
- 2 × M12 – centralizer of involution class 2B (trace 0), which moves 264 of the 276 type 2-2-3 triangles
- [210.33]
- S3 × PSL(2,8):3 - normalizer of 3-subgroup generated by class 3C (trace 0) element
- A4 × S5
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]
| 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
In analogy to monstrous moonshine for the monster M, for Co3, the relevant McKay-Thompson series is T\( T_{{4A}}(\tau ) \) where one can set the constant term a(0) = 24 (OEIS: A097340),
\( {\displaystyle {\begin{aligned}j_{4A}(\tau )&=T_{4A}(\tau )+24\\&={\Big (}{\tfrac {\eta ^{2}(2\tau )}{\eta (\tau )\,\eta (4\tau )}}{\Big )}^{24}\\&={\Big (}{\big (}{\tfrac {\eta (\tau )}{\eta (4\tau )}}{\big )}^{4}+4^{2}{\big (}{\tfrac {\eta (4\tau )}{\eta (\tau )}}{\big )}^{4}{\Big )}^{2}\\&={\frac {1}{q}}+24+276q+2048q^{2}+11202q^{3}+49152q^{4}+\dots \end{aligned}}} \)
and η(τ) is the Dedekind eta function.
References
Conway et al. (1985)
http://brauer.maths.qmul.ac.uk/Atlas/v3/spor/Co3/#ccls
http://brauer.maths.qmul.ac.uk/Atlas/v3/spor/Co1/#ccls
http://brauer.maths.qmul.ac.uk/Atlas/v3/permrep/Co3G1-p276B0
Conway, John Horton (1968), "A perfect group of order 8,315,553,613,086,720,000 and the sporadic simple groups", Proceedings of the National Academy of Sciences of the United States of America, 61 (2): 398–400, doi:10.1073/pnas.61.2.398, MR 0237634, PMC 225171, PMID 16591697
Conway, John Horton (1969), "A group of order 8,315,553,613,086,720,000", The Bulletin of the London Mathematical Society, 1: 79–88, doi:10.1112/blms/1.1.79, ISSN 0024-6093, MR 0248216
Conway, John Horton (1971), "Three lectures on exceptional groups", in Powell, M. B.; Higman, Graham (eds.), Finite simple groups, Proceedings of an Instructional Conference organized by the London Mathematical Society (a NATO Advanced Study Institute), Oxford, September 1969., Boston, MA: Academic Press, pp. 215–247, ISBN 978-0-12-563850-0, MR 0338152 Reprinted in Conway & Sloane (1999, 267–298)
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