A class of 2-groups admitting an action of the symmetric group of degree 3

TL;DR

This paper classifies biextraspecial 2-groups with an action of the symmetric group of degree 3, showing that such groups exist only for even ranks and are uniquely determined by their rank and type.

Contribution

It provides a complete classification of biextraspecial groups, establishing their existence conditions, uniqueness, and automorphism group structure.

Findings

Rank m must be even for biextraspecial groups to exist.

Exactly two biextraspecial groups exist for each even rank m, distinguished by a sign.

Automorphism groups are extensions of orthogonal spaces by orthogonal groups.

Abstract

A biextraspecial group of rank $m$ is an extension of a special 2-group $Q$ of the form $2^{2 + 2m}$ by $L\cong L_2(2)$, such that the 3-element from $L$ acts on $Q$ fixed-point-freely. Subgroups of this type appear in at least the sporadic groups $J_2$, $J_3$, $McL$, $Suz$, and $Co_1$. In this paper we completely classify biextraspecial groups, namely, we show that the rank $m$ must be even and for each such $m$ there exist exactly two biextraspecial groups $B^\varepsilon(m)$ up to isomorphism where $\varepsilon\in{+,-}$. We also prove that $\Out(B^\varepsilon(m))$ is an extension of the $m$-dimensional orthogonal GF(2)-space of type $\varepsilon$ by the corresponding orthogonal group. The extension is non-split except in a few small cases.