Free actions of finite groups on S n × S n S^n \times S^n

Let pp be an odd prime. We construct a non-abelian extension Γ\Gamma of S1S^1 by Z/p×Z/p\mathbf {Z}/p \times \mathbf {Z}/p, and prove that any finite subgroup of Γ\Gamma acts freely and smoothly on S2p−1×S2p−1S^{2p-1} \times S^{2p-1}. In particular, for each odd prime pp we obtain free smooth actions of infinitely many non-metacyclic rank two pp-groups on S2p−1×S2p−1S^{2p-1} \times S^{2p-1}. These results arise from a general approach to the existence problem for finite group actions on products of equidimensional spheres.