On minimal coverings and pairwise generation of some primitive groups of wreath product type

The covering number of a finite group G, denoted σ(G), is the smallest positive integer k such that G is a union of k proper subgroups. We calculate σ(G) for a family of primitive groups G with a unique minimal normal subgroup N, isomorphic to Anm with n divisible by 6 and G/N cyclic. This is a generalization of a result of Swartz concerning the symmetric groups. We also prove an asymptotic result concerning pairwise generation.