Orbitopes of finite groups are symmetric convex polytopes

Question

I am reading the following paper:

• Raman Sanyal, Frank Sottile, Bernd Sturmfels, Orbitopes, arXiv:0911.5436.

In the first page, the paper says "Orbitopes of finite groups are symmetric convex polytopes".

Consider an orbit $\mathcal{O}$ of $x_0$ under the finite group $G$

1. Convexity:
   Because of an orbitope is the convex hull of an orbit.
2. Symmetry:
   If this is the group with operator "addition", $g\in G$ guarantees $-g\in G$, then it is not hard to think about "symmetry" since we have $-gx_0, gx_0$. However it the operator is "multiplication", how to explain this? Or even more, the operator is composition.
3. I also have no idea how to view it as a polytopes. Could anyone give me a strict way to view it?