On symmetric graphs of order four times an odd square-free integer and valency seven

A graph is called symmetric if its automorphism group is transitive on its arcs. In this paper, we classify symmetric graphs of order four times an odd square-free integer and valency seven. It is shown that, either the graph is isomorphic to one of 9 specific graphs or its full automorphism group is isomorphic to PSL(2,p), PGL(2,p), PSL(2,p)×Z2 or PGL(2,p)×Z2 with p≡±1(mod7) a prime.