Genus of commuting conjugacy class graph of groups

For a non-abelian group $G$, its commuting conjugacy class graph $\mathcal{C}\mathcal{C}\mathcal{C}(G)$ is a simple undirected graph whose vertex set is the set of conjugacy classes of the non-central elements of $G$ and two distinct vertices $x^G$ and $y^G$ are adjacent if there exists some elements $x^{\mathrm{\prime }}\in x^G$ and $y’ \in y^G$suchthat$x’y’ = y’x’$\mathrm{.}Inthispaperwecomputethegenusof$$\mathcal{C}\mathcal{C}\mathcal{C}$(G)$ for six well-known classes of non-abelian two-generated groups (viz. $D_{2n},S{D}_{8n},Q_{4m},V_{8n},U_{(n,m)}$ and $G(p,m,n)$) and determine whether $\mathcal{C}\mathcal{C}\mathcal{C}(G)$ for these groups are planar, toroidal, double-toroidal or triple-toroidal.