Conway group Co2

In the area of modern algebra known as group theory, the Conway group Co2 is a sporadic simple group of order
 * 218·36·53·7·11·23
 * = 42305421312000
 * ≈ 4.

History and properties
Co2 is one of the 26 sporadic groups and was discovered by  as the group of automorphisms of the Leech lattice Λ fixing a lattice vector of type 2. It is thus a subgroup of Co0. It is isomorphic to a subgroup of Co1. The direct product 2×Co2 is maximal in Co0.

The Schur multiplier and the outer automorphism group are both trivial.

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.

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 =

{\mathbf 1/2} \left ( \begin{matrix} 1 &  -1 & -1 & -1 \\ -1 &  1 & -1 & -1 \\ -1 & -1 &  1 & -1 \\ -1 & -1 & -1 & 1 \end{matrix} \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.

found the 11 conjugacy classes of maximal subgroups of Co2 as follows:


 * Fi21:2 ≈ U6(2):2 - symmetry/reflection group of coplanar hexagon of 6 type 2 points. Fixes one hexagon in a rank 3 permutation representation of Co2 on 2300 such hexagons. Under this subgroup the hexagons are split into orbits of 1, 891, and 1408. Fi21 fixes a 2-2-2 triangle defining the plane.
 * 210:M22:2 has monomial representation described above; 210:M22 fixes a 2-2-4 triangle.
 * McL fixes a 2-2-3 triangle.
 * 21+8:Sp6(2) - centralizer of involution class 2A (trace -8)
 * HS:2 fixes a 2-3-3 triangle or exchanges its type 3 vertices with sign change.
 * (24 × 21+6).A8
 * U4(3):D8
 * 24+10.(S5 × S3)
 * M23 fixes a 2-3-4 triangle.
 * 31+4.21+4.S5
 * 51+2:4S4

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.