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Using a novel application of the structure theorem for biquasiprimitive permutation groups of the second author, we produce a description of all pairs $(\\Gamma, G) \\in\\mathcal{OG}(4)$ for which every nontrivial normal subgroup of $G$ has at most two orbits on the vertices of $\\Gamma$. In particular we show that $G$ has a unique minimal normal subgroup $N$ and that $N \\cong T^k$ for a simple group $T$ and $k\\in \\{1,2,4,8\\}$. 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Praeger, Nemanja Poznanovi\\'c","submitted_at":"2019-02-28T00:51:59Z","abstract_excerpt":"Let $\\mathcal{OG}(4)$ denote the family of all graph-group pairs $(\\Gamma,G)$ where $\\Gamma$ 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 $(\\Gamma, G) \\in\\mathcal{OG}(4)$ for which every nontrivial normal subgroup of $G$ has at most two orbits on the vertices of $\\Gamma$. In particular we show that $G$ has a unique minimal normal subgroup $N$ and that $N \\cong T^k$ for a simple group $T$ and $k\\in \\{1,2,4,8\\}$. 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