Conway group Co3

Representations

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.

Walter 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 ${\displaystyle \mathbb{Z}/2\mathbb{Z}\times {\mathrm{C}\mathrm{o}}_2}$ or ${\displaystyle \mathbb{Z}/2\mathbb{Z}\times {\mathrm{C}\mathrm{o}}_3}$.

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.

Larry Finkelstein (1973) found the 14 conjugacy classes of maximal subgroups of ${\displaystyle {\mathrm{C}\mathrm{o}}_3}$ 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 ${\displaystyle T_{4A}(\tau )}$ where one can set the constant term a(0) = 24 (OEIS: A097340),

${\displaystyle \begin{array}{cc}{j}_{4A}(\tau )& =T_{4A}(\tau )+24\\ & =(\frac{{\eta }^2(2\tau )}{\eta (\tau )\eta (4\tau )}{)}^{24}\\ & =((\frac{\eta (\tau )}{\eta (4\tau )}{)}^4+4^2(\frac{\eta (4\tau )}{\eta (\tau )}{)}^4{)}^2\\ & =\frac{1}{q}+24+276q+2048q^2+11202q^3+49152q^4+\dots \end{array}}$

and η(τ) is the Dedekind eta function.