Classification of half-arc-transitive graphs of order 4p4p

A vertex-transitive graph XX is said to be half-arc-transitive   if its automorphism group acts transitively on the set of edges of XX but does not act transitively on the set of arcs of XX. A classification of half-arc-transitive graphs on 4p4p vertices, where pp is a prime, is given. Apart from an obvious infinite family of metacirculants, which exist for p≡1(mod4) and have been known before, there is an additional somewhat unique family of half-arc-transitive graphs of order 4p4p and valency 1212; the latter exists only when p≡1(mod6) is of the form 22k+2k+122k+2k+1, k>1k>1.