Presentation complex for the symmetric group on three letters
= Presentation complex for the symmetric group on three letters
Attach three 2-cells to $S^1\vee S^1$ along loops representing $a^2$, $b^3$, and $(ab)^2$. The resulting group is
$$
\langle a,b\mid a^2,b^3,(ab)^2\rangle\cong S_3.
$$
Indeed, $a\mapsto(12)$ and $b\mapsto(123)$ gives a surjection to $S_3$, while $aba=b^{-1}$ reduces every word to one of $b^i$ or $ab^i$, so the presented group has at most six elements.