In the area of modern algebra known as group theory, the Conway group ' is a sporadic simple group of order
' is one of the 26 sporadic groups and was discovered by as the group of automorphisms of the Leech lattice fixing a lattice vector of type 3, thus length . It is thus a subgroup of . It is isomorphic to a subgroup of . The direct product is maximal in .
The Schur multiplier and the outer automorphism group are both trivial.
Co<sub>3</sub> acts on a 23-dimensional even lattice with no roots, given by the orthogonal complement of a norm 6 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.
Co<sub>3</sub> has a doubly transitive permutation representation on 276 points.
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 or .
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 14 conjugacy classes of maximal subgroups of as follows:
Traces of matrices in a standard 24-dimensional representation of Co<sub>3</sub> are shown. The names of conjugacy classes are taken from the Atlas of Finite Group Representations.
The cycle structures listed act on the 276 2-2-3 triangles that share the fixed type 3 side.
In analogy to monstrous moonshine for the monster M, for Co<sub>3</sub>, the relevant McKay-Thompson series is where one can set the constant term a(0) = 24 (),
and ÷(ÃÂ) is the Dedekind eta function.