Conway groups

Description

The Conway groups $\mathrm{Co}_1$, $\mathrm{Co}_2$ and $\mathrm{Co}_3$ are three sporadic groups, found by John Conway from the symmetries of the Leech lattice. Let $\mathrm{Co}_0$ be the automorphism group of the Leech lattice, that is, the group of linear isometries of $\R^{24}$ that map the lattice onto itself. Its centre is $\set{\pm 1}$, and its order is $8315553613086720000$.

1. $\mathrm{Co}_1 = \mathrm{Co}_0 / \set{\pm 1}$, of order $4157776806543360000 = 2^{21} \cdot 3^9 \cdot 5^4 \cdot 7^2 \cdot 11 \cdot 13 \cdot 23$.
2. $\mathrm{Co}_2$ is the stabiliser in $\mathrm{Co}_0$ of a lattice vector of norm $4$, of order $42305421312000 = 2^{18} \cdot 3^6 \cdot 5^3 \cdot 7 \cdot 11 \cdot 23$.
3. $\mathrm{Co}_3$ is the stabiliser in $\mathrm{Co}_0$ of a lattice vector of norm $6$, of order $495766656000 = 2^{10} \cdot 3^7 \cdot 5^3 \cdot 7 \cdot 11 \cdot 23$.

All three are simple groups. Here the lattice is scaled so that its shortest nonzero vectors have norm $4$.