Symmetric Group on 3 Letters/Generators

Generators of the Symmetric Group on 3 Letters
Let $S_3$ denote the Symmetric Group on 3 Letters, whose Cayley table is given as:

Let:

Then:

where $\gen G$ denotes the group generated by a subset $G$ of $S_3$.

Proof
For $G_1$:

For $G_2$: