  For a graph $G$ and a positive integer $k$, a vertex labelling
$f:V(G)\to\{1,2\ldots,k\}$ is said to be $k$-distinguishing if no non-trivial
automorphism of $G$ preserves the sets $f^{-1}(i)$ for each
$i\in\{1,\ldots,k\}$. The distinguishing chromatic number of a graph $G$,
denoted $\chi_D(G)$, is defined as the minimum $k$ such that there is a
$k$-distinguishing labelling of $V(G)$ which is also a proper coloring of the
vertices of $G$. In this paper, we prove the following theorem: Given
$k\in\mathbb{N}$, there exists an infinite sequence of vertex-transitive graphs
$G_{i}=(V_i,E_i)$ such that $\chi_D(G_i)>\chi(G_i)>k$ and
$|\mathrm{Aut}(G_i)|=O_k(|V_i|)$, where $\mathrm{Aut}(G_i)$ denotes the full
automorphism group of $G_i$. In particular, this answers a problem raised in
the paper $\chi_D(G)$, $|\mathrm{Aut}(G)|$ and a variant of the Motion lemma.
