I want to find the number of rotational symmetries of the 120-cell but I am not very familiar with polytopes nor counting symmetries. So, I don't know if someone can give me an idea or an example with some other polytope or a reference to read about it. Any help is welcome.
Thank you!
According to Coxeter's "Regular Polytopes" (p. 130) the order of the reflectional symmetry group of an $n$ dimensional regular polytope $\{p,q,...,v,w\}$ is given by
$$g_{\ p,q,...,v,w}=N_{n-1}N_{n-1,n-2}\cdots N_{2,1}N_{1,0}$$
simply by counting the according fundamental regions of the mirror hyperplane dissections.
Now consider the according incidence matrix of the 120-cell $\{5,3,3\}$.
$$\begin{array}{cccc} 600 & 4 & 6 & 4\\ 2 & 1200 & 3 & 3\\ 5 & 5 & 720 & 2\\ 20 & 30 & 12 & 120 \end{array}$$
we get accordingly
$$g_{\ 5,3,3}=120\cdot 12\cdot 5\cdot 2=14400$$
As 2 consecutive reflections make up a rotation (even within 4D) you would have simply divide the above value by 2 to get the order of the rotational subgroup, i.e. 7200.
--- rk