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The covering number of a finite group $G$, denoted $\sigma(G)$, is thesmallest 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 uniqueminimal 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 concerningthe symmetric groups. We also prove an asymptotic result concerning pairwisegeneration.

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