Non-cyclic graphs of (non)orientable genus one

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