J Austral Math Soc Ser A 58 pp232--247, 1995.

On the Faithful Representations, of Degree 2ⁿ, of Certain Extensions of 2-Groups by Orthogonal and Sympletic Groups

S. P. Glasby

Abstract

If R is a 2-group of sympletic type with exponent 4, then R is isomorphic to the extra special group 2_e ^{1 + 2n}, or to the central product 4 °2^{1 + 2n} of a cyclic group of order 4 and an extraspecial group, 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 representation of R of degree 2ⁿ. The 2-cocycle associated with this projective representation takes values which are powers of -1 if R is isomorphic to 2_e ^{1 + 2n} and powers of Ö(-1) otherwise. This explicit description of a projective representation is useful for computing character values or computing with central extensions of A. Such central extensions arise naturally in Aschbacher's classification of the subgroups of classical groups.

1991 AMS Subject Classification: 20C15, 20G05

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Authors

S. P. Glasby

School of Mathematics and Statistics, The University of Sydney, NSW 2006, Australia.