Notes on the various Super-Groups
---------------------------------

I have calculated the size of the super-groups for various subgroups
of the cube. I have suffixed the standard group names with the
letter c to show that the centre orientations are significant.

The groups are (ranked smallest to largest):

Size (slice)       =         768
Size (slicec)      =      24,576     Size (slicec) / Size (slice)   = 32

The following reference confirms this calculation and expounds further
on the nature of the slice group...

The Slice Group in Rubik's Cube,
by David Hecker, Ranan Banerji
Mathematics Magazine, Vol. 58 No. 4 Sept 1985

Size (antisl)      =       6,144
Size (antislc)     =      49,152     Size (antislc) / Size (antisl) =  8

Size (sq)          =     663,552
Size (sqc)         =   5,308,416     Size (sqc) / Size (sq)         =  8

Size (ur)          =  73,483,200
Size (urc)         = 587,865,600     Size (urc) / Size (ur)         =  8

Size (domino) =      406,425,600
Size (dominoc)=    3,251,404,800     Size (dominoc) / Size (domino) =  8

Size (cube)  =     43,252,003,274,489,856,000
Size (cubec) = 88,580,102,706,155,225,088,000

Size (cubec) / Size (cube) = 2,048

The case of the super squares group (sqc) is interesting. It is only
possible to rotate opposite centres 180 degrees. There are actually
8 centres in the super square's group:

(1 way)    Identity
(1 way)    All 6 centres rotated 180 degrees
(3 ways)   2 opposite centres rotated 180 degrees
(3 ways)   2 pairs of opposite centres rotated 180 degrees