Tri-extraspecial groups

Abstract

We classify the groups which are certain split extensions of special 2-groups of the form 23+3m, m ≥ 1, by the group L3(2). These groups behave very much like extraspecial 2-groups and we call them tri-extraspecial groups. A tri-extraspecial group of this form exists if and only if m is a positive even integer, and for every n ≥ 1 there are exactly two tri-extraspecial groups of the form 23+6n : L3(2). We denote these groups by T+(2n) and T−(2n). Let ε be + or −. Then the isomorphism type of Q (2n) ≔ O2(T ε(2n)) is independent of ε. The automorphism group Aε(2n) of T ε(2n) is a non-split extension of Q (2n) by the direct product L3 (2) × S2n (2) × 2. The group Aε(2n) permutes transitively the conjugacy classes of L3 (2)-complements to Q (2n) in T ε(2n). If S ε(2n) is the stabilizer in Aε(2n) of one of these classes of complements, then S ε(2n) is a split extension of Q (2n) by L3 (2) ×