Conway group Co2
|Algebraic structure → Group theory
- 218 · 36 · 53 · 7 · 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.
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.
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.
- McL (fixing 2-2-3 triangle)
- HS:2 (can transpose type 3 vertices of conserved 2-3-3 triangle)
- (24 × 21+6).A8
- 24+10.(S5 × S3)
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