Conway group Co3

History and properties

\mathrm{Co}_3 is one of the 26 sporadic groups and was discovered by as the group of automorphisms of the Leech lattice \Lambda fixing a lattice vector of type 3, thus length . It is thus a subgroup of \mathrm{Co}_0. It is isomorphic to a subgroup of \mathrm{Co}_1. The direct product 2\times \mathrm{Co}_3 is maximal in \mathrm{Co}_0.

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

Representations

Co3 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.

Co3 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 \Z/2\Z \times \mathrm{Co}_2 or \Z/2\Z \times \mathrm{Co}_3.

Maximal subgroups

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 \mathrm{Co}_3 as follows:

Conjugacy classes

Traces of matrices in a standard 24-dimensional representation of Co3 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.

Generalized Monstrous Moonshine

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

:\begin{align}j_{4A}(\tau) &=T_{4A}(\tau)+24\ &=\Big(\tfrac{\eta^2(2\tau)}{\eta(\tau),\eta(4\tau)} \Big)^{24} \ &=\Big(\big(\tfrac{\eta(\tau)}{\eta(4\tau)}\big)^{4}+4^2 \big(\tfrac{\eta(4\tau)}{\eta(\tau)}\big)^{4}\Big)^2\ &=\frac{1}{q} + 24+ 276q + 2048q^2 +11202q^3+49152q^4+\dots \end{align}

and η(τ) is the Dedekind eta function.

References

• Reprinted in

References

1. Conway et al. (1985)
2. "ATLAS: Conway group Co3".
3. "ATLAS: Conway group Co1".
4. "ATLAS: Co3 — Permutation representation on 276 points".