Conway group Co3

In the area of modern algebra known as group theory, the Conway group Co₃ is a sporadic simple group of order

Co₃ 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 3, thus length √ 6. It is thus a subgroup of Co₀. It is isomorphic to a subgroup of Co₁. The direct product 2xCo₃ is maximal in Co₀.

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

Co₃ acts on the unique 23-dimensional even lattice of determinant 4 with no roots, given by the orthogonal complement of a norm 4 vector of the Leech lattice. This gives 23-dimensional representations over any field; over fields of characteristic 2 or 3 this can be reduced to a 22-dimensional faithful representation.

Co₃ has a doubly transitive permutation representation on 276 points.

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.

Finkelstein (1973) found the 14 conjugacy classes of maximal subgroups of Co₃ as follows:

In analogy to monstrous moonshine for the monster M, for Co₃, the relevant McKay-Thompson series is ${\displaystyle T_{4A}(\tau )}$ where one can set the constant term a(0) = 24 ( ),

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