# Conway group Co3

In mathematics, the Conway group Co3 is a sporadic group of order

210 · 37 · 53 ·· 11 · 23 (= 495,766,656,000)

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.

## 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.

## Generalized Monstrous Moonshine

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

\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}

and η(τ) is the Dedekind eta function.

## Maximal subgroups of Co3

Finkelstein (1973) showed that there are 14 conjugacy classes of maximal subgroups, 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, 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, which moves 264 of the 276 type 2-2-3 triangles
• [210.33]
• S3 × PSL(2,8):3
• A4 × S5