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