Conway group Co1 - Knowlpedia

The Conway group $\mathrm{Co}_1$ is the quotient

\mathrm{Co}_1=O(\Lambda)/\{I,-I\},

where $\Lambda$ is the Leech lattice and $O(\Lambda)$ consists of the real linear isometries of $\mathbb R^{24}$ that preserve $\Lambda$ . The subgroup $\{I,-I\}$ is central, so the quotient is a group.

Order and simplicity

The group is nonabelian and simple, with

|\mathrm{Co}_1|=4\,157\,776\,806\,543\,360\,000 =2^{21}\cdot3^9\cdot5^4\cdot7^2\cdot11\cdot13\cdot23.

The full lattice isometry group is often denoted $\mathrm{Co}_0=2.\mathrm{Co}_1$ . It has twice this order and is not the simple group $\mathrm{Co}_1$ .

A finite matrix model

The two binary matrices in Co1G1-f2r24B0.m1 and Co1G1-f2r24B0.m2 generate a faithful copy of $\mathrm{Co}_1$ in $\mathrm{GL}_{24}(\mathbb F_2)$ , the group of invertible matrices over the field with two elements. The files specify all entries row by row after their format header.

Neighboring Conway groups

There are maximal subgroups isomorphic to $\mathrm{Co}_2$ and $\mathrm{Co}_3$ , of indices $98\,280$ and $8\,386\,560$ , respectively.

References

Robert A. Wilson, Finite Simple Groups, Chapter 5 notes, §5.3.2 for the central quotient and §5.3.3 for simplicity.
ATLAS: Co1, order and maximal-subgroup tables.
Representation Co1G1-f2r24B0, “About this representation”; exact MeatAxe data generator a and generator b.