Given a group (G, ·) and positive integers r, s ≤ |G|, we denote with μG (r,s) the least possible size of the sumsets AB = {a · b : a ∈ A and b ∈ B}, where A, B run over all subsets of G, such that |A| = r and |B| = s. Let H = Q × (Z/2Z)k × C be a Hamiltonian group, where k is a non-negative integer, Q is the quaternion group of 8 elements and C is a cyclic group of odd order. We present an explicit formula for μH(r,s).