# Formal counting of cube faces colorings wrt full symmetry

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).

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.