I wonder where I can find something written on symmetric point arrangements (see definition below). I am interested in general references, preferably books that introduce (or papers that use) some notation/terminology which is accepted as standard. I would also be happy if you could provide helpful terminology, as my search terms have not brought up much. Terms I used: "point arrangements/constellations/configurations".

A point arrangement is a (finite) family of points $p_i\in\Bbb R^d,i\in N:=\{1,...,n\}$. It is said to be symmetric w.r.t. a group $\Gamma\subseteq\mathrm{Sym}(N)$, if there is a representation $\rho :\Gamma\to\mathrm{O}(\Bbb R^d)$ so that

$$\rho(\phi)p_i=p_{\phi(i)},\qquad\text{for all $\phi\in \Gamma$ and $i\in N$}.$$

One prototypical question that I might ask is the following:

Given $n\in\Bbb N$ and a group $\Gamma\subseteq\mathrm{Sym}(N)$ acting as permutation group on $N:=\{1,...,n\}$. What are the symmetric point arrangements of $n$ points w.r.t. group $\Gamma$?

I have an answer to this question, basically using (real) representation theory of finite groups. It is not really deep, and this is why I think someone should have come up with this before. I am currently writing a paper, and struggle with myself whether to include these "basics" when I can possible reference them somewhere. I am also worried about using a non-standard language, which then has to be corrected as soon as a referee hints me to suitable literature.

Note, that my interests are different from e.g. orbit polytopes (= orbit of a point under some group $\Gamma\subseteq\mathrm O(\Bbb R^d)$). I start with a permutation group on a fixed number of points instead of a given representation.