Extraspecial towers and Weil representations

TL;DR

This paper explores Weil representations of extraspecial groups, their extensions, and form-preserving properties, motivated by a subgroup of the sporadic simple group Fi23, revealing complex group structures and representations.

Contribution

It introduces new Weil representations of extraspecial groups that extend to larger groups and preserve specific quadratic forms, connecting to a subgroup of Fi23.

Findings

Construction of Weil representations that extend to larger groups

Identification of form-preserving properties of these representations

Description of a solvable group with derived length 10 and composition length 24

Abstract

This paper was motivated by a remarkable group, the maximal subgroup $M=S_3\ltimes 2^{2+1}_{-}\ltimes3^{2+1}\ltimes2^{6+1}_{-}$ of the sporadic simple group ${\rm Fi}_{23}$, where $S_3$ is the symmetric group of degree 3, and $2^{2+1}_{-}$, $3^{2+1}$ and $2^{6+1}_{-}$ denote extraspecial groups. The representation $3^{2+1}\to{\rm GL}(3,\mathbb{F}_4)\to{\rm GL}(6,\mathbb{F}_2)$ extends (remarkably) to $S_3\ltimes 2^{2+1}_{-}\ltimes3^{2+1}$ and preserves a quadratic form (of minus type) which allows the construction of $M$. The paper describes certain (Weil) representations of extraspecial groups which extend, and preserve various forms. Incidentally, $M$ is a remarkable solvable group with derived length 10, and composition length 24.