Conway group Co2

Representations

Co₂ acts as a rank 3 permutation group on 2300 points. These points can be identified with planar hexagons in the Leech lattice having 6 type 2 vertices.

Co₂ acts on the 23-dimensional even integral lattice with no roots of determinant 4, given as a sublattice of the Leech lattice orthogonal to a norm 4 vector. Over the field with 2 elements it has a 22-dimensional faithful representation; this is the smallest faithful representation over any field.

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.

The Mathieu group M₂₃ is isomorphic to a maximal subgroup of Co₂ and one representation, in permutation matrices, fixes the type 2 vector u = (-3,1²³). A block sum ζ of the involution η =

${\displaystyle \mathbf{1}/2\left(\begin{array}{cccc}1& -1& -1& -1\\ -1& 1& -1& -1\\ -1& -1& 1& -1\\ -1& -1& -1& 1\end{array}\right)}$

and 5 copies of -η also fixes the same vector. Hence Co₂ has a convenient matrix representation inside the standard representation of Co₀. The trace of ζ is -8, while the involutions in M₂₃ have trace 8.

A 24-dimensional block sum of η and -η is in Co₀ if and only if the number of copies of η is odd.

Another representation fixes the vector v = (4,-4,0²²). A monomial and maximal subgroup includes a representation of M₂₂:2, where any α interchanging the first 2 co-ordinates restores v by then negating the vector. Also included are diagonal involutions corresponding to octads (trace 8), 16-sets (trace -8), and dodecads (trace 0). It can be shown that Co₂ has just 3 conjugacy classes of involutions. η leaves (4,-4,0,0) unchanged; the block sum ζ provides a non-monomial generator completing this representation of Co₂.

There is an alternate way to construct the stabilizer of v. Now u and u+v = (1,-3,1²²) are vertices of a 2-2-2 triangle (vide infra). Then u, u+v, v, and their negatives form a coplanar hexagon fixed by ζ and M₂₂; these generate a group Fi₂₁ ≈ U₆(2). α (vide supra) extends this to Fi₂₁:2, which is maximal in Co₂. Lastly, Co₀ is transitive on type 2 points, so that a 23-cycle fixing u has a conjugate fixing v, and the generation is completed.

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.

Wilson (2009) found the 11 conjugacy classes of maximal subgroups of Co₂ as follows:

Conjugacy classes

Traces of matrices in a standard 24-dimensional representation of Co₂ are shown. The names of conjugacy classes are taken from the Atlas of Finite Group Representations.

Centralizers of unknown structure are indicated with brackets.