# Conway group Co3

For general background and history of the Conway sporadic groups, see Conway group.

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.

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.

## 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

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, 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