Sets of Operations on Set of 3 Elements/Automorphism Group of C n

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

Let $\CC_1$, $\CC_2$ and $\CC_3$ be respectively the set of all operations $\circ$ on $S$ such that the groups of automorphisms of $\struct {S, \circ}$ are as follows:

Then:
 * Each of $\CC_1$, $\CC_2$ and $\CC_3$ has $3^4 - 3$ elements.