ON THE FAITHFUL REPRESENTATIONS, OF DEGREE 2(N), OF CERTAIN EXTENSIONS OF 2-GROUPS BY ORTHOGONAL AND SYMPLECTIC GROUPS

If R is a 2-group of symplectic type with exponent 4, then R is isomor phic to the extraspecial group 2(epsilon)(1+2n), or to the central pro duct 4 o 2(1+2n) of a cyclic group of order 4 and an extraspecial grou p, with central subgroups of order 2 amalgamated. This paper gives an explicit description of a projective representation of the group A of automorphisms of R centralizing Z(R), obtained from a faithful represe ntation of R of degree 2(n). The 2-cocycle associated with this projec tive representation takes values which are powers of -1 if R is isomor phic to 2(epsilon)(1+2n) and powers of root-1 otherwise. This explicit description of a projective representation is useful for computing ch aracter values or computing with central extensions of A. Such central extensions arise naturally in Aschbacher's classification of the subg roups of classical groups.