Pentavalent symmetric graphs of order four times an odd square-free integer

A graph is said to be symmetric if its automorphism group is transitive on its arcs. Guo et al. in 2011 and Pan et al. in 2013 determined all pentavalent symmetric graphs of order 4pq. In this paper, we shall generalize this result by determining all connected pentavalent symmetric graphs of order four times an odd square-free integer. It is shown in this paper that, for each such graph Gamma, either the full automorphism group Aut Gamma is isomorphic to PSL(2, p), PGL(2, p), PSL(2, p) x Z(2) or PGL(2, p) x Z(2), or Gamma is isomorphic to one of 9 graphs.