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Non-cyclic graphs of (non)orientable genus one

Let $G$ be a finite non-cyclic group. The non-cyclic graph $Γ_G$ of $G$ is the graph whose vertex set is $G\setminus Cyc(G)$, two distinct vertices being adjacent if they do not generate a cyclic subgroup, where $Cyc(G)=\{a\in G: \langle a,b\rangle\ \text{is cyclic for each } b\in G\}$. In this paper, we classify all finite non-cyclic groups $G$ such that $Γ_G$ has (non)orientable genus one.