Conway group Co2

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 Co2 is a sporadic simple group of order

218 · 36 · 53 ·· 11 · 23
= 42305421312000
≈ 4×1013.

History and properties

Co2 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 2. It is thus a subgroup of Co0. It is isomorphic to a subgroup of Co1. The direct product 2xCo2 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.

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 (-3,123). A block sum of the involution η =

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

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 the block sum involution 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 (4,4,022). This includes a permutation matrix representation of M22:2. It also includes 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. It should be noted that -η sends (4,4,0,0) to (0,0,4,4); there is permutation matrix in M24 that restores (4,4,0,0). There follows a non-monomial generator for this representation of Co2.

Maximal subgroups

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

• U6(2):2 Fixes a point of the rank 3 permutation representation on 2300 points.
• 210:M22:2
• McL (fixing 2-2-3 triangle)
• 21+8:Sp6(2)
• HS:2 (can transpose type 3 vertices of conserved 2-3-3 triangle)
• (24 × 21+6).A8
• U4(3):D8
• 24+10.(S5 × S3)
• M23
• 31+4.21+4.S5
• 51+2:4S4