A. A. Ivanov, S. Shpectorov
November 18, 2005
We classify the groups which are certain split extensions of special 2-groups of the form 2 3+3 m , m ≥ 1, by the group L 3 (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 2 3+6 n : L 3 (2). We denote these groups by T + (2 n ) and T − (2 n ). Let ε be + or −. Then the isomorphism type of Q (2 n ) ≔ O 2 ( T ε (2 n )) is independent of ε . The automorphism group A ε (2 n ) of T ε (2 n ) is a non-split extension of Q (2 n ) by the direct product L 3 (2) × S 2 n (2) × 2. The group A ε (2 n ) permutes transitively the conjugacy classes of L 3 (2)-complements to Q (2 n ) in T ε (2 n ). If S ε (2 n ) is the stabilizer in A ε (2 n ) of one of these classes of complements, then S ε (2 n ) is a split extension of Q (2 n ) by L 3 (2) × (2). The group S ε (2 n ) 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 L 3 (2) × S 5 is the so-called pentad subgroup in the fourth sporadic simple group of Janko J 4 , while a subgroup of index 2 in A − (4) which is a non-split extension of Q (4) by L 3 (2) × S 6 ≅ L 3 (2) × S 4 (2) is a maximal 2-local subgroup in the largest Fischer 3-transposition group Fi 24 . The two sporadic examples were the primary motivation for our interest in tri-extraspecial groups.