4x4 Rubik's cube centers permutations count About a 4x4 Rubik's cube. We know the count of permutations.
But how about the permutation of the centers only ?
X X X X
X O O X
X O O X
X X X X

There are 6 colored faces containing 4 stickers at centers (marked with "O").
How many permutations do we have ?
This is 24 facelets but the answser is not 24! because when we have placed 5 faces, the last face is obviously made with the 4 last facelets of the same color.
How could we calculate the number of the centers position ?
 A: Posting this to get something started.
Using the counting convention described in Jaap's comment it can be shown that we can get all the even permutations of the 24 facelets, giving a total of
$$
\dfrac12\,24!=310224200866619719680000
$$
ways of positioning them.
It is well known that all the 3-cycles generate the group of all even permutations. It is easy to milk more out of the idea and show that a handful of 3-cycles will suffice (we can use conjugation to get more of them). The key is thus to show that enough 3-cycles can be realized by sequences of moves. The animation below shows one
such.

The visual appearance is a bit deceiving in that it may look as if only a single white sticker and a single blue sticker trade places. If you look at it more closely you will see that actually they move in a 3-cycle White->White->Blue (->White). In other words: the net effect of this sequence is that white facelet A moves to the place of the white facelet B that moves to the blue area, and a blue facelet moves to the place where the white facelet A was before the beginning of the sequence.
