Tetravalent half-arc-transitive graphs of order a product of three primes

A graph is half-arc-transitive   if its automorphism group acts transitively on its vertex set, edge set, but not arc set. Let nn be a product of three primes. The problem on the classification of the tetravalent half-arc-transitive graphs of order nn has been considered by Xu (1992), Feng et al. (2007) and Wang and Feng (2010), and it was solved for the cases where nn is a prime cube or twice a product of two primes. In this paper, we solve this problem for the remaining cases. In particular, there exist some families of these graphs which have a solvable automorphism group but are not metacirculants.