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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 forthe study of these particular Cayley graphs comes from current research inBioinformatics. As the main result, we prove thatAut$(\mathrm{Cay}(\mathrm{Sym}_n,T_n))$ is the product of the left translationgroup by a dihedral group $\mathsf{D}_{n+1}$ of order $2(n+1)$. The proof usesseveral 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 uniquecyclic subgroup of $\mathsf{D}_{n+1}$ of order $n+1$ with regular Cayley mapson $\mathrm{Sym}_n$ is also discussed. It is shown that the product of the lefttranslation group by the latter group can be obtained as the automorphism groupof a non-$t$-balanced regular Cayley map on $\mathrm{Sym}_n$.

Automorphism groups of Cayley graphs generated by block transpositions and regular Cayley maps