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**Deveno** ...so if G acts transitively on the vertices, the orbit of one (and thus any) vertex v has order 8. which means that the index of the stabilizer [G:Gv] = 8, so there are 8 cosets of Gv in G.

if one knows that the rotational symmetry group of the cube is S4, this tells you that Gv has order 3.

G also acts transitively on the faces, so Gf (for any face f) has order 4.

finally, G also acts transitively on the edges, so Ge (for any edge e) has order 2.

(the class equation for these subsets of {faces, vertices, edges} is particularly simple).

it occurs to me, that it is possible you are being asked to compute the orbits of V,F,E under the respective actions induced by each group:

Gv,Gf, and Ge, for some particular stabilizer in each set. this is somewhat of a different matter.

for example, say V = {v1,v2,v3,v4,v5,v6,v7,v8}. Gv1 fixes v1, so it's orbit is: {v1}. if the opposite vertex is v7, Gv1 also fixes v7,

so its orbit is: {v7}. the other two orbits have to have order 3, since there are no other points fixed by Gv1, except v1 and v7,

and the size of the orbits has to divide |Gv1| (it helps to think about WHICH rotations Gv1 must be: rotations about the axis between v1 and v7).

you can carry out a similar analysis for E and F.