Four-valent oriented graphs of biquasiprimitive type

Let OG(4) denote the family of all graph-group pairs (Γ, G) where Γ is 4-valent, connected and G-oriented (G-half-arc-transitive). Using a novel application of the structure theorem for biquasiprimitive permutation groups of the second author, we produce a description of all pairs (Γ, G) ∈ OG(4) for which every nontrivial normal subgroup of G has at most two orbits on the vertices of Γ, and at least one normal subgroup has two orbits. In particular we show that G has a unique minimal normal subgroup N and that N ^∼= T^k for a simple group T and k ∈ {1, 2, 4, 8}. This provides a crucial step towards a general description of the long-studied family OG(4) in terms of a normal quotient reduction. We also give several methods for constructing pairs (Γ, G) of this type and provide many new infinite families of examples, covering each of the possible structures of the normal subgroup N.