  A graph $\G$ is {\em symmetric} or {\em arc-transitive} if its automorphism
group $\Aut(\G)$ is transitive on the arc set of the graph, and $\G$ is {\em
basic} if $\Aut(\G)$ has no non-trivial normal subgroup $N$ such that the
quotient graph $\G_N$ has the same valency with $\G$. In this paper, we
classify symmetric basic graphs of order $2qp^n$ 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
$5\di (q-1)$, 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 $kp^n$ for some small integers $k$ and
$n$ are classified.
