Explicit construction of the maximal subgroups of the Monster

TL;DR

This paper presents an explicit computational construction and verification of the maximal subgroups of the Monster group, providing a new database and correcting previous claims about its subgroup structure.

Contribution

It introduces a comprehensive database of maximal subgroups of the Monster group using the mmgroup package, including explicit generators and verification methods, and corrects prior misconceptions.

Findings

Constructed explicit generators for all maximal subgroups

Verified the subgroup classifications independently

Identified a new maximal subgroup 59{:}29

Abstract

Seysen's Python package mmgroup provides functionality for fast computations within the sporadic simple group $\mathbb{M}$, the Monster. The aim of this work is to present an mmgroup database of maximal subgroups of $\mathbb{M}$: for each conjugacy class $C$ of maximal subgroups in $\mathbb{M}$, we construct explicit group elements in mmgroup and prove that these elements generate a group in $C$. Our generators and the computations verifying correctness are available in accompanying code. The maximal subgroups of $\mathbb{M}$ have been classified in a number of papers spanning several decades; our work constitutes an independent verification of these constructions. We also correct the claim that $\mathbb{M}$ has a maximal subgroup $\mathrm{PSL}_2({59})$, and hence identify a new maximal subgroup $59{:}29$.