Nontrivial difference sets in groups of order a power of 2 are part of the family of difference sets called Menon difference sets (or Hadamard), and they have parameters (22d+2, 22d+1 ±2d, 22d±2d). In the abelian case, the group has a difference set if and only if the exponent of the group is less than or equal to 2d+2. In [14], the authors construct a difference set in a nonabelian group of order 64 and exponent 32. This paper generalizes that result to show that there is a difference set in a nonabelian group of order 22d+2 with exponent 2d+3 . We use representation theory to prove that the group has a difference set, and this shows that representation theory can be used to verify a construction similar to the use of character theory in the abelian case.

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Copyright © 1994, Kluwer Academic Publishers. This article first appeared in Journal of Algebraic Combinatorics: 3:2 (1994), 137-151.

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