Conway group Co3

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In the area of modern algebra known as group theory, the Conway group [math]\displaystyle{ \mathrm{Co}_3 }[/math] is a sporadic simple group of order

   210 · 37 · 53 ·· 11 · 23
= 495766656000
≈ 5×1011.

History and properties

[math]\displaystyle{ \mathrm{Co}_3 }[/math] 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 [math]\displaystyle{ \Lambda }[/math] fixing a lattice vector of type 3, thus length 6. It is thus a subgroup of [math]\displaystyle{ \mathrm{Co} 0 }[/math]. It is isomorphic to a subgroup of [math]\displaystyle{ \mathrm{Co}_1 }[/math]. The direct product [math]\displaystyle{ 2\times \mathrm{Co}_3 }[/math] is maximal in [math]\displaystyle{ \mathrm{Co}_0 }[/math].

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.

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 [math]\displaystyle{ \Z/2\Z \times \mathrm{Co}_2 }[/math] or [math]\displaystyle{ \Z/2\Z \times \mathrm{Co}_3 }[/math].

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 [math]\displaystyle{ \mathrm{Co}_3 }[/math] as follows:

  • McL:2 – McL fixes a 2-2-3 triangle. The maximal subgroup also includes reflections of the triangle. [math]\displaystyle{ \mathrm{Co}_3 }[/math] has a doubly transitive permutation representation on 276 type 2-2-3 triangles having as an edge a type 3 vector fixed by [math]\displaystyle{ \mathrm{Co}_3 }[/math].
  • 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 [math]\displaystyle{ T_{4A}(\tau) }[/math] where one can set the constant term a(0) = 24 (OEISA097340),

[math]\displaystyle{ \begin{align}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{align} }[/math]

and η(τ) is the Dedekind eta function.

References

External links