# How would one classify point groups?

By point group I mean a finit subgroup of $$\mathrm O(\Bbb R^n)$$.

Lists of point groups for some small dimensions are found on Wikipedia, but I am not certain about their completeness. As there seem to exist lists, I wonder the following:

Question: How would one go about classifying the point groups in some fixed dimension $$n$$?

I know that all (finite) reflection groups can be listed systematically via Coxeter diagrams, but not all point groups are reflection groups $$-$$ in fact, they are not even subgroups of reflection groups as explained in this answer.

Can the classification of point groups be obtained from the classification of reflection groups in some simple way, e.g. by only taking products and subgroups?

• It is unclear what you mean by "classify". Every finite group embeds in some permutation group, hence, in some $O(n)$, and we do not even have names for all finite groups. In contrast, finite simple groups are classified. If you are asking about these, then it becomes a rather difficult question of representation theory of these groups which is a vast area of research. – Moishe Kohan Apr 17 at 3:16
• @MoisheKohan I know that every group is isomorphic to a point group. This is why I asked about classification for a fixed dimension $n$. I am also less interested in the actual list of point groups, but more in the methods and tools used to find it: how to know that I found all, or how to construct more. I am specifically interested in the mentioned connection between point groups and reflection groups. E.g. explaining to me that every point group can be obtained from a reflection group in some simple way would be a helpful answer. In fact, this was told to me. But is it true? – M. Winter Apr 17 at 6:36
• I do not know what a "simple way" would mean. Take a look here for references etc: mathoverflow.net/questions/37136/… One more thing to know is Jordan-Schur theorem en.wikipedia.org/wiki/Jordan%E2%80%93Schur_theorem – Moishe Kohan Apr 17 at 21:12