Symmetric cubic graphs with solvable automorphism groups

A cubic graph ΓΓ is GG-arc-transitive if G≤Aut(Γ) acts transitively on the set of arcs of ΓΓ, and GG-basic   if ΓΓ is GG-arc-transitive and GG has no non-trivial normal subgroup with more than two orbits. Let GG be a solvable group. In this paper, we first classify all connected GG-basic cubic graphs and determine the group structure for every GG. Then, combining covering techniques, we prove that a connected cubic GG-arc-transitive graph is either a Cayley graph, or its full automorphism group is of type 2222, that is, a 22-regular group with no involution reversing an edge, and has a non-abelian normal subgroup such that the corresponding quotient graph is the complete bipartite graph of order 66.