  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
Maru\v{s}i\v{c}, 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.
