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Smallest tetravalent half-arc-transitive graphs with the vertex-stabiliser isomorphic to the dihedral group of order $8$

A connected graph whose automorphism group acts transitively on the edges and vertices, but not on the set of ordered pairs of adjacent vertices of the graph is called half-arc-transitive. It is well known that the valence of a half-arc-transitive graph is even and at least four. Several infinite families of half-arc-transitive graphs of valence four are known, however, in all except four of the known specimens, the vertex-stabiliser in the automorphism group is abelian. The first example of a half-arc-transitive graph of valence four and with a non-abelian vertex-stabiliser was described in [Conder and Marušič, A tetravalent half-arc-transitive graph with non-abelian vertex stabilizer, J. Combin. Theory Ser. B 88 (2003) 67--76]. This example has $10752$ vertices and vertex-stabiliser isomorphic to the dihedral group of order $8$. In this paper, we show that no such graphs of smaller order exist, thus answering a frequently asked question.