Covering certain Wreath Products with Proper Subgroups

For a non-cyclic finite group $X$ let $σ(X)$ be the least number of proper subgroups of $X$ whose union is $X$. Precise formulas or estimates are given for $σ(S \wr C_{m})$ for certain nonabelian finite simple groups $S$ where $C_m$ is a cyclic group of order $m$.

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