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A graph $X$, with a group $G$ of automorphisms of $X$, is said to be $(G,s)$-transitive, for some $s\ge 1$, if $G$ is transitive on $s$-arcs but not on $(s+1)$-arcs. Let $X$ be a connected $(G,s)$-transitive graph of prime valency $p\ge 5$, and $G_v$ the vertex stabilizer of a vertex $v\in V\left(X\right)$. Suppose that $G_v$ is solvable. Weiss (1974) proved that $|G_v{|\mid p(p-1)}^2$. In this paper, we prove that $G_v\cong ({\mathbb{Z}}_p⋊{\mathbb{Z}}_m)\times {\mathbb{Z}}_n$ for some positive integers $m$ and $n$ such that $ndivm$ and $m\mid p-1$.

Let $X$ be a finite simple undirected graph with a subgroup $G$ of the full automorphism group $\mathrm{Aut}\left(X\right)$. Then $X$ is said to be $(G,s)$-transitive for a positive integer $s$, if $G$ is transitive on $s$-arcs but not on $(s+1)$-arcs, and $s$-transitive if it is $\left(\mathrm{Aut}\right(X),s)$-transitive. Let $G_v$ be a stabilizer of a vertex $v\in V\left(X\right)$ in $G$. Up to now, the structures of vertex stabilizers $G_v$ of cubic, tetravalent or pentavalent $(G,s)$-transitive graphs are known. Thus, in this paper, we give the structure of the vertex stabilizers $G_v$ of connected hexavalent $(G,s)$-transitive...