New computations in the Monster

Summary

Abstract

We survey recent computational results concerning the Monster sporadic simple group. The main results are: progress towards a complete classification of the maximal subgroups, including showing that L₂(27) is not a subgroup; showing that the 196882-dimensional module over GF(2) supports a quadratic form; a complete set of explicit conjugacy class representatives; small representations of most of the maximal subgroups; and a partial classification of the ‘nets’ (in the sense of Norton).

Introduction

Our aim in this paper is to update the survey by describing the various explicit computations which have been performed in the Monster group, and the new information about the Monster which has resulted from these calculations. We begin by summarising for the benefit of readers who do not have that paper to hand.

The smallest matrix representations of the Monster have dimension 196882 in characteristics 2 and 3, and dimension 196883 in all other characteristics. Three of these representations (over the fields of orders 2, 3, and 7) are now available explicitly. It is hoped that the data and programs to manipulate them will be made available in the next release of Magma. The generating matrices are stored in a compact way, and never written out in full. The basic operation of the system is to calculate the action of a generator on a vector of the underlying module.

Our first construction was carried out over the field GF(2) of two elements in the interests of speed, and proceeded by amalgamating various 3-local subgroups.