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

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.