This paper deals with the Cayley graph Cay ( 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 ( Cay ( Sym n , T n ) ) is the product of the left translation group and a dihedral group D n + 1 of order 2 ( n + 1 ) . The proof uses several properties of the subgraph Γ of Cay ( Sym n , T n ) induced by the set T n . In particular, Γ is a 2 ( n − 2 ) -regular graph whose automorphism group is D n + 1 , Γ has as many as n + 1 maximal cliques of size 2 , and its subgraph Γ ( 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 D n + 1 of order n + 1 with regular Cayley maps on Sym n is also discussed. It is shown that the product of the left translation group and the latter group can be obtained as the automorphism group of a non- t -balanced regular Cayley map on Sym n .