Sets of Operations on Set of 3 Elements/Automorphism Group of D

Theorem
Let $S = \set {a, b, c}$ be a set with $3$ elements.

Let $\BB$ be the set of all operations $\circ$ on $S$ such that the groups of automorphisms of $\struct {S, \circ}$ form the set $\set {I_S, \tuple {a, b, c}, \tuple {a, c, b} }$, where $I_S$ is the identity mapping on $S$.

Then:
 * $\DD$ has $19 \, 422$ elements.