Domination polynomial of clique cover product of graphs

Let $G$ be a simple graph of order $n$. We prove that the domination polynomial of the clique cover product $G^\mathcal{C} \star H^{V(H)}$ is
$$D(G^\mathcal{C} \star H,x) =\prod_{i=1}^k\Big[\big((1+x)^{n_i}-1\big)(1+x)^{|V(H)|}+D(H,x)\Big],$$
where each clique $C_i \in \mathcal{C}$ has $n_i$ vertices. As an application, we study the $\mathcal{D}$-equivalence classes of some families of graphs and, in particular, describe completely the $\mathcal{D}$-equivalence classes of friendship graphs constructed by coalescing $n$ copies of a cycle graph of length $3$ with a common vertex.