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 $${\mathbb {Z}_3^2 \times \mathbb {Z}_p^{4t}}$$ 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 $${\mathbb {Z}_3^2 \times \mathbb {Z}_p^{4t} \times {\it EA} (9p^{4t} \pm 2)}$$ when $${9p^{4t} \pm 2}$$ is a prime power. These are new parameters for such difference sets.