Conway group Co2

In the area of modern algebra known as group theory, the Conway group Co₂ is a sporadic simple group of order

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.

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 =

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.

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