Maximal subgroups of ${}^2E_6(2)$ and its automorphism groups

TL;DR

This paper presents a new computer-assisted proof classifying maximal subgroups of the simple group ${}^2E_6(2)$ and its automorphisms, combining computational methods with statistical validation.

Contribution

It provides the first computational proof of the classification of maximal subgroups for ${}^2E_6(2)$ and its automorphism extensions, enhancing verification methods.

Findings

Successful computational classification of maximal subgroups

Development of automated subgroup analysis techniques

Validation of results through statistical analysis

Abstract

We give a new computer-assisted proof of the classification of maximal subgroups of the simple group ${}^2E_6(2)$ and its extensions by any subgroup of the outer automorphism group $S_3$. This is not a new result, but no earlier proof exists in the literature. A large part of the proof consists of a computational analysis of subgroups generated by an element of order 2 and an element of order 3. This method can be effectively automated, and via statistical analysis also provides a sanity check on results that may have been obtained by delicate theoretical arguments.