As a vital link between group theory and graph theory, Cayley graphs provide a geometric framework for encoding algebraic structures. This study explores the properties of Cayley graphs derived from cyclic groups whose order is the square of the product of three distinct prime numbers. We specifically examine cases where the connecting set is defined by the collection of all elements with an order equal to the square of a prime. A comprehensive analysis of these graphs is presented, focusing on structural characteristics such as connectivity, Eulerian properties, and Hamiltonicity. Furthermore, we determine several key graph parameters, including the clique number, chromatic number, independence number, and diameter.