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

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:

where $I_S$ is the identity mapping on $S$.

Let $\CC = \CC_1 \cup \CC_2 \cup \CC_3$.

Let $\oplus \in \CC$.

Then the isomorphism class of $\oplus$ consists of $3$ elements: exactly one operation from each of $\CC_1$, $\CC_2$ and $\CC_3$.