If I have a natural number $o=2n$ or $4n$ I can create a polyhedron whose group of symmetries has order $o$ by making a polygon like $C_n$ and then dragging it out to make a prism (I believe this is dihedral symmetry in the case $o=4n$; you need colors in the $o=2n$ case, not sure what this is called).
What if $o=3n$? Is the group $C_n \times C_3$ isomorphic to the group of symmetries for any polyhedron? It seems like there should be some clever way to make a pyramid-type structure, but I can't quite figure it out.