I always tried to understand how *finite reflection groups* from $ Bbb R ^ d $ (of some fixed dimension $ d $) relate to the *point groups* of the same space $ smash { Bbb R ^ d} $ (finite subgroup of the orthogonal group $ smash { mathrm O ( Bbb R ^ d)} $)

Initially, I had the impression that each group of points is a subgroup of a finite reflection group. This turned out to be incorrect, which is obvious in retrospect. Many reflection groups have symmetries in the placement of their mirrors that can be used to enlarge the group.

So let's take these expanded groups instead. From my geometric understanding, by that I mean the symmetry groups of the uniform polytopes. Then i will call you **uniform dot groups**. Most (or all?) Of uniform polytopes can be generated from a reflection group, and then have all the symmetries of this group, but could have more.

Question:Is each group of points a subgroup of a uniform group of points?

With regard to the answer to that question, I am open to any statement that sheds light on the location of reflection groups (or groups easily derived from them) within the family of general point groups.