Asked this question in Physics Stack Exchange, but may be readers of Maths Stack Exchange can answer it too.
When considering improper rotations (roto-reflections), we can derive that if $n$ is odd, then $C_n$ and $\sigma_h$ (reflection plane normal to $C_n$ axis) must exist. Similarly, we can also derive that if $n$ is even, then $C_n/2$ and $\sigma_h$ (reflection plane must exist).
Further, when considering crystallographic point groups, we can easily derive that there are no rotation operations other than $1$-, $2$-, $3$-, $4$- and $6$- fold axes. Now, considering the above-mentioned fact that for improper rotations and for $n$ = even, $C_n/2$ must exist, we can say that $n/2 = 4 $ and $6$ must be allowed and therefore, improper rotations of the kind $S_8$ and $S_{12}$ must be allowed as well. However, that is not the case. Is there any algebraic/geometrical proof for this? I tried looking everywhere including textbooks and various online pages, but no luck. Any help would be much appreciated. Thank you.