colouring a non-cubic prism

Question

How many distinct ways are there to colour the faces of a rectangular non-cubic prism, if you have 3 colours available to use?

attempt of solution:

we have done this same question with a cube but this question will have a different rotational axis

I wanted help in figuring out the size of the group acting on the prism since it is supposed to be smaller than the group acting on the cube

Any help on this will be highly appreciated

Thanks

...I just found what a non-cubic prism is. You just have to take account on the lateral symmetries, actually the cube case was harder!(Or I am interpreting the problem terribly wrong) — chubakueno, Mar 27 '14 at 03:34
There is a discussion of this question by various users at this MSE link. — Marko Riedel, Mar 27 '14 at 20:47

score 0 · Answer 1 · answered Mar 27 '14 at 11:58

Use the theorem giving the order of the group in terms of sizes of orbits and stabilizers. Consider vertices, they are easier to visualize.

It is a cube "stretched upwards": There is just 1 operation that keeps one of the vertices fixed, and any of the 8 vertices can take its place. The order of the group is $1 \cdot 8 = 8$. Operations are 4 rotations along the long axis, and rotations followed by flipping over.
The ends are also non-square rectangles (a brick-shape): Again, 1 operation that keeps a vertex fixed, but now there are only 4 that can take its place. The order is $1 \cdot 4 = 4$. Operations are 2 rotations along the long axis, followed by flipping over.

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