Homomorphism

1. List all the symmetries of the triangular prism in cycle form as permutations of the set $\{1,2,3,4,5\}$.

First, notice that the triangular faces are labeled as $1$ and $5$, while the rectangular faces are labeled as $2$, $3$ and $4$.

Looking first at rotational symmetries, we can rotate the triange faces by $120^{\circ}$, which corresponds to the cycle $(2,3,4)$, or by $240^{\circ}$, which is the cycle $(2,4,3)$. If we rotate the rectangle face marked as $2$ by $180^{\circ}$, we get the cycle $(1,5)(3,4)$. Rotating the other rectangular faces, we get $(1,5)(2,4)$ and $(1,5)(2,3)$, in much the same way.

Now we look at the reflection symmetries. There are three reflection symmetries of a triangle, and looking at them in terms of the prism, those reflections are the same ones that reflect one of the rectangles vertically. These reflections give $(2,3)$, $(3,4)$ and $(2,4)$. Lastly, we can reflect about the plane that cuts through the middle of all 3 rectangles, and get $(1,5)$.

Then, there are the symmetries that are combinations of those already mentioned, combinations of reflections and rotations. Namely, if we reflect through the plane parallel to the triangles and then rotate the triangles, we get $(1,5)(2,3,4)$ and $(1,5)(2,4,3)$.

Of course, there is also the symmetry that does nothing, the identity element of the symmetric group. Thus, we have a total of 12 symmetries.