Conway group Co3

In the area of modern algebra known as group theory, the Conway group $$\mathrm{Co}_3$$ is a sporadic simple group of order
 * 210·37·53·7·11·23
 * = 495766656000
 * ≈ 5.

History and properties
$$\mathrm{Co}_3$$ is one of the 26 sporadic groups and was discovered by as the group of automorphisms of the Leech lattice $$\Lambda$$ fixing a lattice vector of type 3, thus length $\sqrt{6}$. It is thus a subgroup of $\mathrm{Co}_0$. It is isomorphic to a subgroup of $$\mathrm{Co}_1$$. The direct product $$2\times \mathrm{Co}_3$$ is maximal in $$\mathrm{Co}_0$$.

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.

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 $$\Z/2\Z \times \mathrm{Co}_2$$ or $$\Z/2\Z \times \mathrm{Co}_3$$.

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.

found the 14 conjugacy classes of maximal subgroups of $$\mathrm{Co}_3$$ as follows:


 * McL:2 – McL fixes a 2-2-3 triangle. The maximal subgroup also includes reflections of the triangle. $$\mathrm{Co}_3$$ has a doubly transitive permutation representation on 276 type 2-2-3 triangles having as an edge a type 3 vector fixed by $$\mathrm{Co}_3$$.
 * 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. The names of conjugacy classes are taken from the Atlas of Finite Group Representations. The cycle structures listed act on the 276 2-2-3 triangles that share the fixed type 3 side.

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.