Non-abelian hadamard difference sets

Difference sets with parameters (ν, k, λ) may exist even if there are no abelian (ν, k, λ) difference sets; we give the first known example of this situation. This example gives rise to an infinite family of non-abelian difference sets with parameters (4t2, 2t2 − t, t2 − t), where t = 2q · 3r · 5 · 10s, q, r, s ⩾ 0, and r > 0 ⇒ q > 0. No abelian difference sets with these parameters are known.