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How many 3-colorings of cube's faces exist if we consider two colorings the same iff it's possible to rotate and/or mirror the cube such that one coloring goes to another?

I do know a similar question considering only the rotations was asked here. I do understand where the solution comes from: Burnside's lemma and the geometrical classification of cube rotations (by face, by edge, by vertex, etc). But I can't seem to construct a convenient geometrical classification of all the symmetries (is there really a sensible geometrical approach to this problem???)

I sense that the full symmetry group of a cube is $S_4 \times \mathbb Z_2$ since we can fix an arbitrary mirroring and combine it with all the rotations of the cube (which form a group isomorphic to $S_4$). I have heard of a solution which uses an embedding $S_4 \times \mathbb Z_2 \rightarrow S_6$ which "realizes all cube symmetries in permutations of cube's faces". I though have no idea how to build such an embedding and how can it help?

Basically I am looking for any approach to this problem which doesn't involve calculating $\text{Fix}(g)$ for all of the 48 cube symmetries $g$ separately and plugging it in Burnside's lemma (this is going to be on a test with limited time, so that is not acceptable).

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The rotation group permutes the four diagonals, whence $S_4$. The rotations and index two in the full subgroup, since reflection x reflection = rotation, and the central involution $-I_3$ commutes with everything so we find the full symmetry group is $S_4\times\mathbb{Z}_2$. Anyway, the conjugacy classes of rotations in $S_4$ are:

  • $1~$ - $~0^{\circ}$ rotation
  • $6~$ - $~90^{\circ}$ rotations around face axes
  • $8~$ - $120^{\circ}$ rotations around vertex axes
  • $3~$ - $180^{\circ}$ rotations around face axes
  • $6~$ - $180^{\circ}$ rotations around edge axes

It's also a good exercise to figure out the cycle types in $S_4$ corresponding to each item above.

The other possibilities are plane reflections and rotation-reflections (which are reflections across planes combined with rotation in said planes). Here's a list:

  • $3$ coordinate plane reflections
  • $3$ coordinate plane reflections + $180^{\circ}$ rotation
  • $6$ coordinate plane reflections + $90^{\circ}$ rotation
  • $6$ antipodal-edge plane reflections
  • $6$ antipodal-edge plane reflections + $180^{\circ}$ rotation

Finding $\mathrm{Fix}(g)$ for these $g$s should be doable.

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