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(1) Let c be a coloring of the cube. The identity id is a rotation, and id(c)=c, so (c,c)∈S. S is reflexive.
Let (c,d)∈S, and r the rotation such as r(c)=d. r−1(d)=c, (d,c)∈S and S is symmetric.
If (c,d)∈S and (d,e)∈S, r(c)=d and r′(d)=e, r′(r(c))=e, then (c,e)∈S and S is transitive.
(2) Suppose that the colors are a, b, c, d, e and f. Suppose that a is up.
If b is down, suppose that c is front. There are 3!=6 possibilities for the three other colors.
If b is front, there are 4!=24 possibilities for the four other colours.
|s|=30.
This can be proved with a quotient: there are 6!=720 possibilities to colorize a cube, and the rotation group that preserve the cube is isomorphic to 𝔖4, whose cardinal is 4!=24, so |S|=6!/4!=6×5=30. Note that this proof is a bit incomplete.