Rubik's Cube group
The following uses the notation described in How to solve the Rubik's Cube. The orientation of the six centre facets is fixed.
Putting all the pieces together we get that the cube group is isomorphic to
When the centre facet symmetries are taken into account, the symmetry group is a subgroup of
The symmetry group of the Rubik's Cube obtained by disassembling and reassembling it is slightly larger: namely it is the direct product