Paley type partial difference sets in non p-groups

By modifying a construction for Hadamard (Menon) difference sets we construct two infinite families of negative Latin square type partial difference sets in groups of the form where is any odd prime. One of these families has the well-known Paley parameters, which had previously only been constructed in -groups. This provides new constructions of Hadamard matrices and implies the existence of many new strongly regular graphs including some that are conference graphs. As a corollary, we are able to construct Paley–Hadamard difference sets of the Stanton-Sprott family in groups of the form when is a prime power. These are new parameters for such difference sets.