Extraspecial towers and Weil representations

Abstract

This paper was motivated by a remarkable group, the maximal subgroup $M=S_3⋉2_{-}^{2+1}⋉3^{2+1}⋉2_{-}^{6+1}$ of the sporadic simple group ${\mathrm{Fi}}_{23}$, 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 $3^{2+1}\to \mathrm{GL}(3,{\mathbb{F}}_4)\to \mathrm{GL}(6,{\mathbb{F}}_2)$ extends (remarkably) to $S_3⋉2_{-}^{2+1}⋉3^{2+1}$ 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.

arXiv comment26 pages