# Face Permutations of a Square Based Prism

I am trying to count the face permutations of a square based prism.

The goal of this question is to establish the cycles of the face permutations because that is needed to find the cycle index.

I am considering that the there are five planes of symmetry (Am I right?) May you please tell me if my solution is correct?

I will use T for the face of the square top of the square based prism, B for the face of the square bottom, and 1, 2, 3, 4 for the other rectangular faces in clockwise direction. For the rotations I will also use clockwise direction.

Here it is my solution:

1) Rotation 1

I (Identity) (Also Rotation 0 or 360 degrees)

(B)(T)(1)(2)(3)(4)

2) Rotation 2

R90 (Rotation 90 degrees)

(B)(T)(1234)

3) Rotation 3

R180 (Rotation 180 degrees)

(B)(T)(13)(24)

4) Rotation 4

R270 (Rotation 270 degrees)

(B)(T)(1432)

5) Flip 1-3

(Flip along the axis perpendicular to faces 1 and 3)

(BT)(1)(3)(24)

6) Flip 2-4

(Flip along the axis perpendicular to faces 2 and 4)

(BT)(2)(4)(13)

7) Flip Diagonal A

(BT)(14)(23)

8) Flip Diagonal B

(BT)(12)(34)

Also, if all is correct, can we conclude that this group is isomorphic to the Dihedral Group of order 8 because we have 4 rotations and 4 flips?

In my perspective the answer is no. For example "our" identity element has six 1-cycles while the identity of the Dihedral Group of order 8 has four 1-cycles. My perspective is that our group of permutations is created by the action of the dihedral group of order 8 over a square based prism.

I looked with all the search engines information about this kind of problem, but I only found articles that talk about the face permutations of the cube. Do you know websites, articles, about the face permutation of the cuboid?