I would propose initially modeling the "larger group", where
invisible changes in location and orientation are visible. Number
all 16 facelets of each face on the 4x4x4, for example. [...]. For
example, to make invisible orientation changes visible, you have to
give a facelet four numbers rather than just one.

The only facelet for which invisible orientation changes are even
possible is the center facelet on an odd-order cube. Other facelets
always have a fixed orientation with respect to the center of the face
they're on at the moment. (On the 4x4x4, for example, if you mark
every facelet for orientation, you will find that each center facelets
always has the same corner to the face center, regardless of which face
it's on.)

der Mouse

mouse@collatz.mcrcim.mcgill.edu