I know this is a very basic question, but I couldn't find an answer with Google:
What is the symmetry group of the usual sphere packing?
By the usual one, I mean the FCC lattice or HCP lattice. I am interested in both the rotation group and the group of all orthogonal symmetries. I can tell it has $48$ (or $96$ respectively) elements.
My reasoning for it having 48 elements is that - focusing only on the spheres touching the origin sphere - a sphere can be mapped to any of the twelve others. Say we map A to A′. Then a sphere adjacent to A can be mapped to any sphere adjacent to A′. I think there are 4 such spheres. That gives 12×4 possible choices. When reflections are allowed, there is an extra choice between the two spheres adjacent to both of those. Is this incorrect?
I'd be happy with just a link, as I'm sure its structure has been proven countless times elsewhere.
You can see on wikipedia that the centers of the spheres around a single sphere form either (the vertices of) a cuboctahedron (For FCC lattice), or a triangular orthobicupola (For HCP lattice). So the symmetry is either octahedral symmetry or triangular-prism-symmetry.
The order of this symmetry group is thus 48, not 96, because some spheres form triangles and other for squares, and you cannot put a triangle on a square, that is not a symmetry.