Two Möbius bands attached to a torus
= Two Möbius bands attached to a torus
Attach Möbius bands to a torus along the classes $ab$ and $a^2b^3$. If their core generators are $x,y$, the resulting presentation is
$$
\langle a,b,x,y\mid[a,b],\ x^2=ab,\ y^2=a^2b^3\rangle.
$$
Because the exponent vectors $(1,1)$ and $(2,3)$ form a unimodular basis of $\mathbb Z^2$, eliminating $a,b$ gives
$$
\langle x,y\mid[x^2,y^2]=1\rangle.
$$
This group maps onto $S_3$ by $x\mapsto(12)$ and $y\mapsto(23)$, so it is nonabelian.