For a finite group G, a positive integer λ, and subsets X, Y of G, write λG = XY if the products xy (x ∈ X, y ∈ Y ), cover G precisely λ times. Such a subset Y is called a code with respect to X, and when λ = 1 it is a perfect code in the Cayley graph Cay(G,X). In this paper we present various families of examples of such codes, with X closed under conjugation and Y a subgroup, in symmetric groups, and also in special linear groups SL2(q). We also propose conjectures about the existence of some much wider families.