A computer-friendly construction of the monster

Abstract

Let $\mathbb{M}$ be the monster group which is the largest sporadic finite simple group, and has first been constructed in 1982 by Griess. In 1985, Conway has constructed a 196884-dimensional representation $\rho$ of $\mathbb{M}$ with matrix coefficients in $\mathbb{Z}[\frac{1}{2}]$. So these matrices may be reduced modulo any (not necessarily prime) odd number $p$, leading to representations of $\mathbb{M}$ in odd characteristic. The representation $\rho$ is based on representations of two maximal subgroups $G_{x0}$ and $N_0$ of $\mathbb{M}$. In ATLAS notation, $G_{x0}$ has structure $2_+^{1+24}.\mbox{Co}_1$ and $N_0$ has structure $2^{2+11+22}.( M_{24} \times S_3)$. Conway has constructed an explicit set of generators of $N_0$, but not of $G_{x0}$. This paper is essentially a rewrite of Conway's construction augmented by an explicit construction of an element of $G_{x0} \setminus N_0$. This gives us a complete set of generators of $\mathbb{M}$. It turns out that the matrices of all generators of $\mathbb{M}$ consist of monomial blocks, and of blocks which are essentially Hadamard matrices scaled by a negative power of two. Multiplication with such a generator can be programmed very efficiently if the modulus $p$ is of shape $2^k-1$. So this paper may be considered a as programmer's reference for Conway's construction of the monster group $\mathbb{M}$. We have implemented representations of $\mathbb{M}$ modulo 3, 7, 15, 31, 127, and 255.