Showing that the cube and the octahedron have the same symmetry group I am aware that the octahedron and the cube have the same symmetry group but I was wondering how we show this concretely. I have looked/been thinking about for an answer to this and I have got that as they are the dual of each other then they have the same symmetry group. However whilst I can intuitively understand why this is true I can't find/write a proof of the following statement which the argument seems to rely on: Every polyhedra has the same symmetry group as its dual Thanks very much for any help

How about: put the octahedron in a cube in the natural way (each vertex of the octahedron is at the center of one of the cube's faces). Now prove that any symmetry of the octahedron corresponds to a symmetry of the cube, and vice-versa.