Let $V := \mathbb{R}^2$, $q \in \mathbb{N}$ and let a set $P := \{p_1,\dotsc,p_q\} \subseteq V$ be given. Define a set $\mathcal{S}$ where its members fullfill following properties:
i.) S is an invertible, linear map from $V$ to $V$.
ii.) $\exists \,\sigma \in S_q \; \forall \, i \in \{1,\dotsc,q\} \colon S(p_i) = \pm
p_{\sigma(i)}$
In words: Every member $S$ of the group $\mathcal{S}$ is an invertible, linear map of $V$ to $V$ and it permutes up to a sign the "point cloud" $P \subseteq V$.
One can show that the set $\mathcal{S}$ becomes a group under the composition of maps.
Question: What kinds of subgroups might occur, depending on properties on the set $P$? For example the group $\mathcal{S}$ might be any subgroup of the dihedral group depending on the parameter $q \in \mathbb{N}$, what else?
This question seems very classic to me. Maybe someone has a reference or an idea on how to prove which subgroups of $\mathcal{S}$ might occur.
There is a concept of root system in Lie algebras. They are a finite set of vectors in a vector space endowed with an inner product. It has an associated group called the Weyl group which permutes these root vectors. This Weyl group is the group generated by reflections corresponding to these vectors.
They can be far different from dihedral groups. When the Lie algebra comes from $SL(n)$ it is the whole of $S_n$.
The reference is the textbook by J.E.Humphreys Lie Algebras and Representation Theory.
EDIT: In dimension 2: Let $P$ be the set of vectors $\displaystyle{m\choose 1}$ for all integers $m$. Let $S$ be the infinite group of matrices (horizontal shears) $\displaystyle {1\quad n\choose0\quad 1}$. This group $S$ acts on this infinite set $S$, by translation. This can be imitated for uncountable case also.
In general one can take the group $SL(2,\mathbf{Z})$ as $S$ and the set $P$ as the set of lattice points of the plane.