  This paper deals with the Cayley graph $\Cay,$ 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. We
prove that ${\rm{Aut}}(\Cay)$ is the product of the right translation group by
$\textsf{N}\rtimes \textsf{D}_{n+1},$ where $\textsf{N}$ is the subgroup fixing
$S_n$ element-wise and $\textsf{D}_{n+1}$ is a dihedral group of order
$2(n+1)$. We conjecture that $\textsf{N}$ is trivial. We also prove that the
subgraph $\Gamma$ with vertex-set $S_n$ is a $2(n-2)$-regular graph whose
automorphism group is $\textsf{D}_{n+1}$. Furthermore, $\Gamma$ has as many as
$n+1$ maximum cliques of size $2.$ Also, its subgraph $\Gamma(V)$ whose
vertices are those in these cliques is a $3$-regular, Hamiltonian, and
vertex-transitive graph.
