Symmetric Group on 3 Letters is Isomorphic to Dihedral Group D3

Theorem
Let $S_3$ denote the Symmetric Group on 3 Letters.

Let $D_3$ denote the dihedral group $D_3$.

Then $S_3$ is isomorphic to $D_3$.

Proof
Consider $S_3$ as presented by its Cayley table:

Consider $D_3$ as presented by its group presentation:

and its Cayley table:

Let $\phi: S_3 \to D_3$ be specified as:

Then by inspection, we see:

and the result follows.