Representing the cube as a subgroup of $S_{48}$
Label the facets of the Rubik's Cube as shown in the diagram.
![](images/rubikscube3-labeling.png)
It may help to view this labeling on the flattened cube as follows.
![](images/rubikscube3-labeling-flat.png)
The permutations corresponding to each of the basic moves of the Rubik's Cube are: $$\begin{array}{l} R = (25,27,32,30)(26,29,31,28)(3,38,43,19)(5,36,45,21)(8,33,48,24) \\ L = (9,11,16,14)(10,13,15,12)(1,17,41,40)(4,20,44,37)(6,22,46,35)\\ U = (1,3,8,6)(2,5,7,4)(9,33,25,17)(10,34,26,18)(11,35,27,19)\\ D = (41,43,48,46)(42,45,47,44)(14,22,30,38)(15,23,31,39)(16,24,32,40) \\ F = (17,19,24,22)(18,21,23,20)(6,25,43,16)(7,28,42,13)(8,30,41,11)\\ B = (33,35,40,38)(34,37,39,36)(3,9,46,32)(2,12,47,29)(1,14,48,27) \end{array}$$ $R^{-1}, L^{-1}, U^{-1}, D^{-1}, F^{-1}, B^{-1}$ correspond to the inverses of these permutations.
Since the centre's of the cube are fixed by these moves then any two of these moves are inequivalent. This means that $RC_3$ can be represented by the subgroup of the Symmetric group $S_{48}$ generated by these permutations: $$RC_3 = \langle \mathrm{R}, \mathrm{L}, \mathrm{U}, \mathrm{D}, \mathrm{F}, \mathrm{B} \rangle.$$ We can define $RC_3$ in SageMath as follows.