Non-abelian skew Hadamard difference sets fixed by a prescribed automorphism

Let p be a prime larger than 3 and congruent to 3 modulo 4, and let G be the non-abelian group of order p3 and exponent p. We study the structure of a putative difference set with parameters in G which is fixed by a certain element of order p in Aut(G). We then give a construction of skew Hadamard difference set in the group G for each prime p>3 that is congruent to 3 modulo 4. This is the first infinite family of non-abelian skew Hadamard difference sets. Finally, we show that the symmetric designs derived from these new difference sets are not isomorphic to the Paley designs.