Let $G$ be the group generated by the two transformations $f_1,f_2:\mathbb{R}^2\to \mathbb{R}^2$ given by $f_1(x,y)=(x+1,y)$ and $f_2(x,y)=(x+1,-y)$.
What is the orbit space generated by the action $G\times \mathbb{R}^2\to\mathbb{R}^2$.
I would like to know if there is a way to see the orbit space geometrically.