genus of commuting conjugacy class graph of certain finite groups

For a non-abelian group g, its commuting conjugacy class graph ccc(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 xg and yg are adjacent if there exists some elements x′ ∈ xg and y′ ∈ yg such that x′y′ = y′x′. in this paper we compute the genus of ccc(g) for six well-known classes of non-abelian two-generated groups (viz. d₂n, sd₈n, q₄m, v₈n, u₍n,m₎ and g(p, m, n)) and determine whether ccc(g) for these groups are planar, toroidal, double-toroidal or triple-toroidal.