Conway group Co3 - Knowlpedia

The Conway group $\mathrm{Co}_3$ is the stabilizer

\mathrm{Co}_3=\{g\in O(\Lambda):g(v)=v\}

of a vector $v$ of squared length $\langle v,v\rangle=6$ in the Leech lattice $\Lambda$ . Here $O(\Lambda)$ is the group of real linear isometries preserving the lattice. These vectors form one orbit, so all choices give conjugate stabilizers.

Order and simplicity

This is a nonabelian simple group, with

|\mathrm{Co}_3|=495\,766\,656\,000 =2^{10}\cdot3^7\cdot5^3\cdot7\cdot11\cdot23.

A finite matrix model

The two matrices in Co3G1-f2r22B0.m1 and Co3G1-f2r22B0.m2 generate a faithful copy in $\mathrm{GL}_{22}(\mathbb F_2)$ , the group of invertible matrices over the field with two elements.

Subgroup relationships

Projection to $\mathrm{Co}_1$ is injective on this vector stabilizer, since $-I$ does not fix $v$ . Its image is maximal of index $8\,386\,560$ .

The Higman–Sims group occurs as a maximal subgroup of index $11\,178$ . A maximal subgroup $\mathrm{McL}:2$ , containing the McLaughlin group with index two, has index $276$ . Consequently the embedded $\mathrm{McL}$ has index $552$ and is not maximal.

References

Robert A. Wilson, Finite Simple Groups, Chapter 5 notes, §5.3.4 and equation (5.17), for vector transitivity and the stabilizer.
ATLAS: Co3, order and maximal-subgroup tables; ATLAS: Co1, Co3 maximal-subgroup row.
Representation Co3G1-f2r22B0, “About this representation”; exact MeatAxe data generator a and generator b.