Show that $\operatorname{Aff}(3)$ is isomorphic to $S_3$, the symmetric group of all permutations of 3 objects. Show that $\operatorname{Aff}(3)$ is isomorphic to $S_3$, the symmetric group of all permutations of 3 objects.
Where 
$$\operatorname{Aff}(3):={\{( \begin{array}{cc}
a & b \\ 
0 & 1
\end{array}): a,b\in\mathbb{Z}_3}, a\neq0\}$$,  $\mathbb{Z}_3$ are the integers module 3.
Idea: I know that $ S_3 $ has 6 elements, and the $\operatorname{Aff}(3)$ matrices are: $( \begin{array}{cc}
1 & 0 \\ 
0 & 1
\end{array}) $, $( \begin{array}{cc}
1 & 1 \\ 
0 & 1
\end{array})$ $( \begin{array}{cc}
1 & 2 \\ 
0 & 1
\end{array}) $, $( \begin{array}{cc}
2 & 0 \\ 
0 & 1
\end{array})$ $( \begin{array}{cc}
2 & 1 \\ 
0 & 1
\end{array}) $, $( \begin{array}{cc}
2 & 2\\ 
0 & 1
\end{array})$
But how do I find an isomorphism? Can you help me please

If $a\mapsto \begin{pmatrix} 1 & 1 \\ 0 & 1\end{pmatrix}$ and $b\mapsto \begin{pmatrix} 2 & 0 \\ 0 & 1\end{pmatrix}$, then $\operatorname{Aff}(3)$ has as a presentation $$\langle a, b\mid a^3, b^2, bab=a^{-1}\rangle,$$ which is, in turn, a presentation for $S_3$. Hence $\operatorname{Aff}(3)\cong S_3.$
