Crosscap of the non-cyclic graph of groups

The non-cyclic graph CG to a non locally cyclic group G is as follows: take G∖Cyc(G) as vertex set, where Cyc(G)={x∈G|〈x,y〉  is cyclic for all  y∈G} is called the cyclicizer of G, and join two vertices if they do not generate a cyclic subgroup. In this paper, we classify the finite groups whose non-cyclic graphs are outerplanar. Also all finite groups whose non-cyclic graphs can be embedded on the torus or projective plane are classified.