# Conway group Co3

In the area of modern algebra known as group theory, the Conway group Co3 is a sporadic simple group of order

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

## History and properties

Co3 is one of the 26 sporadic groups and was discovered by (Conway 1968, 1969) as the group of automorphisms of the Leech lattice Λ fixing a lattice vector of type 3, thus length √ 6. It is thus a subgroup of Co0. It is isomorphic to a subgroup of Co1. The direct product 2xCo3 is maximal in Co0.

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.

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 Z/2Z × Co2 or Z/2Z × Co3.

## Maximal subgroups

Finkelstein (1973) found the 14 conjugacy classes of maximal subgroups of Co3 as follows:

• McL:2 – can transpose type 2 points of conserved 2-2-3 triangle. Co3 has a doubly transitive permutation representation on 276 type 2-2-3 triangles containing a fixed type 3 point.
• HS – fixes 2-3-3 triangle.
• U4(3).22
• M23
• 35:(2 × M11)
• 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
• A4 × S5

## Generalized Monstrous Moonshine

In analogy to monstrous moonshine for the monster M, for Co3, the relevant McKay-Thompson series is ${\displaystyle T_{4A}(\tau )}$ where one can set the constant term a(0) = 24 (),

{\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.