Nonexistence of Hadamard difference sets in groups of order 4p² with p≡ 3 4, p≥ 7

We show that the Frobenius group x,y,z xᵖ=yᵖ=z⁴=1,\ xy=yx,\ zxz⁻¹=y,\ zyz⁻¹=x⁻¹ of order 4p² contains no (4p²,2p²-p,p²-p) difference set if p≡ 3 4 is a prime with p ≥ 7. Together with results of McFarland, Iiams and Liebler this shows that no group of order 4p² with p ≡ 3 4 and p ≥ 7 contains a Hadamard difference set.