  This paper deals with the Cayley graph $\mathrm{Cay}(\mathrm{Sym}_n,T_n),$
where the generating set consists of all block transpositions. A motivation for
the study of these particular Cayley graphs comes from current research in
Bioinformatics. As the main result, we prove that
Aut$(\mathrm{Cay}(\mathrm{Sym}_n,T_n))$ is the product of the left translation
group by a dihedral group $\mathsf{D}_{n+1}$ of order $2(n+1)$. The proof uses
several properties of the subgraph $\Gamma$ of
$\mathrm{Cay}(\mathrm{Sym}_n,T_n)$ induced by the set $T_n$. In particular,
$\Gamma$ is a $2(n-2)$-regular graph whose automorphism group is
$\mathsf{D}_{n+1},$ $\Gamma$ has as many as $n+1$ maximal cliques of size $2,$
and its subgraph $\Gamma(V)$ whose vertices are those in these cliques is a
$3$-regular, Hamiltonian, and vertex-transitive graph. A relation of the unique
cyclic subgroup of $\mathsf{D}_{n+1}$ of order $n+1$ with regular Cayley maps
on $\mathrm{Sym}_n$ is also discussed. It is shown that the product of the left
translation group by the latter group can be obtained as the automorphism group
of a non-$t$-balanced regular Cayley map on $\mathrm{Sym}_n$.
