Tri-extraspecial groups

Abstract

We classify the groups which are certain split extensions of special 2-groups of the form 2^3+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 A^ε(2n) of T ^ε(2n) is a non-split extension of Q (2n) by the direct product L₃ (2) × S_2n (2) × 2. The group A^ε(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 A^ε(2n) of one of these classes of complements, then S ^ε(2n) is a split extension of Q (2n) by L₃ (2) × (2). The group S ^ε(2n) is isomorphic to the stabilizer in the orthogonal group (2) of a 3-dimensional totally singular subspace in the natural module. Even more remarkably, a subgroup of A⁺(4) which is a non-split extension of Q (4) by L₃ (2) × S₅ is the so-called pentad subgroup in the fourth sporadic simple group of Janko J₄, while a subgroup of index 2 in A⁻(4) which is a non-split extension of Q (4) by L₃ (2) × S₆ ≅ L₃ (2) × S₄(2) is a maximal 2-local subgroup in the largest Fischer 3-transposition group Fi₂₄. The two sporadic examples were the primary motivation for our interest in tri-extraspecial groups.