Combina)on of the concepts of power graph and Cayley graph associated to groups has led to the introduc)on to two new varia)ons of Cayley graph known as the union power Cayley graph and the intersec)on power Cayley graph. The set of ver)ces for both graphs consist of the elements of a finite group ????. Consider any inverse-closed subset ???? of ????, two ver)ces ???? and ???? are adjacent in the union power Cayley graph if ????????!\\\\\\\" ∈ ???? or if either one is an integral power of the other. Furthermore, ???? and ???? are adjacent in the intersec)on power Cayley graph if ????????!\\\\\\\" ∈ ???? and if either one is an integral power of the other. In this paper, the generaliza)on of the union power Cayley graphs and the intersec)on power Cayley graphs of the dihedral groups with order 2????, for ???? ≥ 3 and ???? = ????#; ???? is prime and ???? is a natural number, rela)ve to a specific subset containing rota)on elements in the groups is found. In addi)on, proper)es of these graphs including the clique numbers, vertex chroma)c numbers, girths and diameters are computed. Finally, the characteris)cs of the graphs, whether they are connected, regular, complete, and planar are also determined.

Keywords