Given an $n$-dimensional convex polytope $P$, one may set into motion a point-mass, starting on one of the facets of $P$, which travels along a straight trajectory inside $P$ except on collision with the walls, when it is subjected to an elastic response (i.e., its direction vector undergoes a reflection in the facet the point-mass has collided with).
The total trajectory covered by the point-mass is its orbit, and such an orbit is periodic if the point-mass eventually returns to its starting spot and its starting velocity. This billiards flow defines a fairly well-studied dynamical system.
I'm wondering if there are conditions on the symmetry group of $P$ which have consequences for the existence of periodic orbits inside $P$. More precisely, are there any conditions on a finite group $G$ which forbid the existence of a periodic orbit in any convex polytope having $G$ as its symmetry group?