A note on certain 2-groups with Hadamard difference sets (Q5939678)

Let \(G\) be a group of order \(2^{4t+m}\) (\(m=2\) or 4) and \(K\) a normal cyclic subgroup of \(G\) generated by \(x\) having order equal to the exponent \(e\) of \(G\). If \(G\) has a Hadamard difference set and \(x\) is not conjugate to \(x^{-1}\) then for \(m=4\) we have \(e\leq 2^{3t+3}\) and for \(m=2\) we have \(e\leq 2^{3t+2}\) (this last bound is sharp). Proofs are obtained by a description of \(G\).