Non-cyclic graphs of (non)orientable genus one

Let $G$ be a finite non-cyclic group. The non-cyclic graph ${\mathrm{\Gamma }}_G$ 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 ${\mathrm{\Gamma }}_G$ has (non)orientable genus one.