It is well known that the Petersen graph does not contain a Hamilton cycle. In \(1983\), Alspach completely determined which Generalized Petersen graphs are Hamiltonian \([1]\). In this paper, we define a larger class of graphs which includes the Generalized Petersen graphs as a special case, and determine which graphs in this larger class are Hamiltonian, and which are \(1\)-factorable. We call this larger class spoked Cayley graphs.