Conway group Co2 - Knowlpedia

The Conway group $\mathrm{Co}_2$ is the stabilizer

\mathrm{Co}_2=\{g\in O(\Lambda):g(v)=v\}

of a vector $v$ of squared length $\langle v,v\rangle=4$ 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}_2|=42\,305\,421\,312\,000 =2^{18}\cdot3^6\cdot5^3\cdot7\cdot11\cdot23.

A finite matrix model

The two matrices in Co2G1-f2r22B0.m1 and Co2G1-f2r22B0.m2 generate a faithful copy of this group in $\mathrm{GL}_{22}(\mathbb F_2)$ , the group of invertible matrices over the field with two elements. The representation dimension $22$ is not the dimension $24$ of the original lattice.

Subgroup relationships

Projection to $\mathrm{Co}_1$ is injective on this vector stabilizer, since $-I$ does not fix $v$ . Its image is a maximal subgroup of index $98\,280$ .

The McLaughlin group is isomorphic to a maximal subgroup of $\mathrm{Co}_2$ , of index $47\,104$ .

References

Robert A. Wilson, Finite Simple Groups, Chapter 5 notes, §5.3.4 and equation (5.16), for vector transitivity and the stabilizer.
ATLAS: Co2, order and maximal-subgroup tables; ATLAS: Co1, Co2 maximal-subgroup row.
Representation Co2G1-f2r22B0, “About this representation” and “Checks applied”; exact MeatAxe data generator a and generator b.