Hopefully an easy question: the icosahedral group of order 60 (orientation preserving symmetries of a regular icosahedron) is isomorphic to the alternating group on 5 points. In terms of the icosahedron, what are the 5 "points"?
It would be ideal if the "points" were actually points. The wikipedia article mentions some compounds of inscribed solids, and I think I'd need a physical demonstration to see that. However, according to this question and this question, we should be able to give just a few points, so that the stabilizer of those points has order 12.
However, I am not sure what the points would be. Maybe they are the vertices of an associated tetrahedron. It would be nice if it was easy to describe those vertices, as I certainly don't see any tetrahedra myself, but I can imagine a collection of 4 vertices easily enough.