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Pentavalent 1-transitive digraphs with non-solvable automorphism groups. (English) Zbl 07344163
Summary: A digraph $$\overrightarrow{\Gamma}$$ is said to be 1-transitive if its automorphism group acts transitively on the 1-arcs but not on the 2-arcs of $$\overrightarrow{\Gamma}$$. We give a tentatively complete classification of pentavalent strongly connected 1-transitive digraphs of order $$2^ap^bq$$, where $$p$$ and $$q$$ are two distinct odd primes, $$a\in\{3,\dots,8\}$$, $$b\in\{1,\dots,4\}$$, whose automorphism groups are non-solvable. It is shown that such digraphs exist if and only if $$q=3$$ or 13 and $$p\in\{7,11, 17,19,31,41\}$$.
##### MSC:
 05C25 Graphs and abstract algebra (groups, rings, fields, etc.) 05C20 Directed graphs (digraphs), tournaments 20B25 Finite automorphism groups of algebraic, geometric, or combinatorial structures
Magma
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