Covering certain monolithic groups with proper subgroups

Given a finite non-cyclic group $G$, call $σ(G)$ the least number of proper subgroups of $G$ needed to cover $G$. In this paper we give lower and upper bounds for $σ(G)$ for $G$ a group with a unique minimal normal subgroup $N$ isomorphic to $A_n^m$ where $n \geq 5$ and $G/N$ is cyclic. We also show that $σ(A_5 \wr C_2)=57$.

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