Computing the Character Table of a 2-local Maximal Subgroup of the Monster

TL;DR

This work computes the character table of the Monster's maximal 2-local subgroup $2^{5+10+20} \ (S_{3} \times PSL_{5}(2))$ by leveraging mmgroup's rapid Monster computations in tandem with a hybrid group representation that encodes the solvable radical. The authors implement Brauer's induction framework to construct irreducible characters from induced characters of carefully chosen subgroups, addressing the computational bottleneck posed by the large Sylow 2-subgroup with parallelization and memory-efficient techniques. They report the successful derivation of all irreducible characters (718 in total) and provide a reproducible workflow, including code and data, with the resulting character table slated for inclusion in the GAP Character Table Library. This approach completes the Monster's maximal-subgroup character tables and enables streamlined analysis of the Monster via its subgroups.

Abstract

We employ the recently developed hybrid and mmgroup computational models for groups to calculate the character table of $N(\rm{2B}^5) \cong 2^{5+10+20}.( \rm{S}_3 \times \rm{L}_5 {2} )$, a maximal subgroup of the Monster sporadic simple group. This completes the list of the character tables of maximal subgroups of the Monster. Our approach illustrates how the aforementioned computational models can be used to calculate relatively straightforwardly in the Monster.

Theorems & Definitions (10)

• Lemma 2.1
• proof
• Remark 2.2
• Proposition 3.1
• proof
• Remark 3.2
• Lemma 3.3
• proof
• Proposition 3.4: Adapted from Thümmel thummel, Lemma 2
• Remark 3.5