Consider a square. We can rotate it by multiples of 90 degs or reflect it in 4 ways, and we get $D_4$. This is the typical basic approach to constructing symmetry groups.
However, these are not all the transformations that preserve the square. For example, we can take two points on the square and replace them, or in general "shuffle around" points in an arbitrary way. These are not considered part of what defines the symmetry group of the square, because they do not preserve distances (e.g. the two points that were replaced are no longer of the same distance to some third point).
Nevertheless, if we include all such transformations, they seem to also form groups (is this true?). Is there any name and/or use to such "enlarged symmetries"?
Considering all possible ways to reshuffle points seems uninteresting, because every geometric object is trivially invariant to relabeling of its points. Thus preservation of distances gives a meaningful constraint that limits possible permutations to more "rigid" transformations. But how about slightly less stringent constraints, or replacing distance preservation with some other desirable property?
Are there examples of such constructions?
You can construct the "full transformation" of $n+1$ points as points of same distance to each other in $\mathbb R^n$. Then the transformation always preserves distance.
For example we have equilateral triangle in $\mathbb R^2$, regular tetrahedron in $\mathbb R^3$, and so on.
The of the "full transformation" of $n$ points is usually called the Symmetric Group of order $n$, denoted as $S_n$. And usually $D_{n}$ is called the Dihedral Group, which means $n$ points in a plane under reflection and rotation. It is sometimes also written as $D_{2n}$ to denote the group having $2n$ elements.