PENTAVALENT 1-TRANSITIVE DIGRAPHS WITH NON-SOLVABLE AUTOMORPHISM GROUPS

A digraph (Gamma) over right arrow is said to be 1-transitive if its automorphism group acts transitively on the 1-arcs but not on the 2-arcs of (Gamma) over right arrow. We give a tentatively complete classification of pentavalent strongly connected 1-transitive digraphs of order 2(a)p(b)q, where p and q are two distinct odd primes, a is an element of {3, ... , 8},b is an element of {1, ... , 4}, whose automorphism groups are non-solvable. It is shown that such digraphs exist if and only if q = 3 or 13 and p is an element of {7, 11, 17, 19, 31, 41}.