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 Z32 × Zp4t where p is any odd prime. One of these families has the well-known Paley parameters, which had previously only been constructed in p-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 Z32 × Zp4t × Z9p4t±2 when 9p4t ± 2 is a prime power. These are new parameters for such difference sets.