On basic graphs of symmetric graphs of valency five

A graph $\text{G}$ is {\em symmetric} or {\em arc-transitive} if its automorphism group $\text{Aut}(\text{G})$ is transitive on the arc set of the graph, and $\text{G}$ is {\em basic} if $\text{Aut}(\text{G})$ has no non-trivial normal subgroup $N$ such that the quotient graph ${\text{G}}_N$ has the same valency with $\text{G}$. In this paper, we classify symmetric basic graphs of order $2q{p}^n$ and valency 5, where $q<p$ are two primes and $n$ is a positive integer. It is shown that such a graph is isomorphic to a family of Cayley graphs on dihedral groups of order $2q$ with $5\text{di}(q-1)$, the complete graph $K_6$ of order $6$, the complete bipartite graph $K_{5,5}$ of order 10, or one of the nine sporadic coset graphs associated with non-abelian simple groups. As an application, connected pentavalent symmetric graphs of order $k{p}^n$ for some small integers $k$ and $n$ are classified.