Orientation of Rubik's cube?

Question

Properly assembled Rubik's cube 2x2x2
(note: ordinary cubes are 3x3x3)
floats in space and does not spin.

On each step weightless ant is allowed to rotate
any layer in any direction, provided that each individual
rotation ends in properly assmebled cube again.

How many spacial orientations of the cube are possible?


The cube is made of 8 eight identical (except colors)
cubes, held together by weightless suspension.
'Assmebled' cube means 'solved', ie each face has
uniform color.

Answer 1

the point is that if the ant is rotating one layer, the other layer rotates in the other direction.

so the ant can rotate 4 faces at the same time, while keeping 2 intact.

these 4 faces have 2 possible orientations.

lets call the original orientation
(1)TBLRFK (Top Bottom Left Right Front bacK)
keeping LR in place, goes to BTLRKF(2)
keeping TB in place, goes to TBRLKF(3)
keeping FK in place, goes to BTRLFK(4)

(2)BTLRKF
keeping BT in place, goes to BTRLFK(4)
keeping LR in place, goes to TBLRFK(1)
keeping KF in place, goes to TBRLKF(3)

no new positions :(

by symmetry argument, neither will (3) or (4) produce anything new

thus, 4 poisitons in all.

Answer 2

(8!*3^7)/24 = 3,674,160

Because --
The eight little-cubes can be assorted into any of the eight slots.
Seven of the cubes can be rotated within the slot, (one can't for reason of parity).
Divide by 24 to eliminate duplicate views.

Answer 3

3*3*3*3*3*3*3**3 = 19,683