Klein's bottle:

Let $Y = \mathbb{R}^2$ and $G$ be the group generated by the affine maps $T$ and $S$ given by

$$T(x, y) = (x + 1, y),\quad S(x, y) = (1-x, y+1).$$

Note that $T$ and $S$ are isometies and the group generated by these acts properly discontinuously on $\mathbb{R}^2$. The orbit space is the Klein's bottle $K$. Thus $\pi_1(K) = G$. Now

$$TS(x, y) = (2-x, y+1), \quad ST(x, y) = (-x, y+1)$$

The fundamental group of the Klein's bottle is non-abelian. Note that $TST = S.$ There are no other independent relations and the fundamental group of the Klein's bottle is the group on two generators $T$ and $S$ with one relation $TST = S$. Summarizing we have