Order Of 2 Times Prime Power Of Five Degrees Symmetrical Map

Let Γ be a graph, and let AutΓ denote its full automorphism group. A graph Γ is called X—arc transitive (X-symmetric) if X is transitive on the set of arcs of Γ, and a graph Γ is called locally primitive if for each vertex a, Xa:={x X|αx=α} acts primitive on Γ(α). Since a transitive permutation group of prime degree is primitive, a pentavalent symmetric graph is a locally primitive graph.Symmetric graphs with prime valency have many nice properties and interesting examples, and have received much attention of many reseachers. For example, cubic symmetric graphs on up to768vertices are presented in [6]. Let p be a prime, for pentavalent graphs, the symmetric cases of order2pq with q≠p a prime are classified in [17]; the symmetric cases of order2p,2p2or2pq with q≠p a prime are classified in [5,30,17], respectively. For more results, see [1,21,31] and references therein.In this paper, we study pentavalent symmetric graphs of order twice a prime power. It is proved that such graph is a normal cover of one of the following Cayley graphs:K6, K5,5, G(16), G(32), G(2p,5), or the standard double cover of a directed normal symmetric pentavalent Cayley graphs of Zdp with p an odd prime and2≤d≤4. Moreover, a classification of pentavalent symmetric graphs of order twice a prime cube is given.