Infinitely many nonsolvable groups whose Cayley graphs are hamiltonian

We show there are infinitely many finite groups G, such that every connected Cayley graph on G has a hamiltonian cycle, and G is not solvable. Specifically, we show that if A_5 is the alternating group on five letters, and p is any prime, such that p ≡ 1 (mod 30), then every connected Cayley graph on the direct product A_5 * Z_p has a hamiltonian cycle.