# Any finite group can be realized as a subgroup of the group of symmetries of a regular polytope?

Follow up of my previous doubt (Finite groups as subgroups of dihedral groups) , can anyone tell me is that fact correct or not, that any finite group can be realized as a subgroup of the group of symmetries of a regular polytope?

• A more interesting and harder question is whether any finite group can be realized as exactly the group of symmetries of a regular polytope. – Qiaochu Yuan Sep 9 '16 at 19:38
• @QiaochuYuan I don't consider myself an exert in group theory. So it will be very helpful if you give me some reference. Actually this problem came to my mind while I was trying to solve a problem in mapping class group. – Anubhav Mukherjee Sep 9 '16 at 20:09
• @QiaochuYuan you probably didn't mean to include the word "regular", as that's not a very interesting question as stated... – Nate Sep 10 '16 at 14:38
• @Nate: why not? – Qiaochu Yuan Sep 10 '16 at 16:33
• There aren't that many regular polytopes, and their symmetry groups are known. Above dimension $4$ there are only $3$ regular polytopes of each dimension, and their symmetry groups are symmetric and hyperoctahedral groups. – Nate Sep 10 '16 at 17:49

• @Anubhav One can realize a simplex as the basis coordinate vectors in $\mathbb{R}^n$. One uses them to create a polytope with $n$ vertices. Any symmetry is determined by the permutation it induces on the set of vertices, and every permutation is realizable as a symmetry (just use the associated permutation matrix as a transformation of space). – arctic tern Sep 10 '16 at 2:13