Let $\ell_{1}$ be the line in $\mathbb{R}^{2}$ joining $(0,0)$ and $\left(\frac{1}{2}, \frac{\sqrt{3}}{2}\right)$, and $\ell_{2}$ the line in $\mathbb{R}^{2}$ joining $(0,0)$ and $\left(\frac{\sqrt{3}}{2}, \frac{1}{2}\right)$. Consider the group of bijections $\mathbb{R}^{2} \rightarrow \mathbb{R}^{2}$ under composition, and its subgroup $G$ generated by the reflections about $\ell_{1}$ and $\ell_{2}$. Then $G$ has exactly $12$ elements.
