Choose one of the Cyclic Groups, and try clicking the generator "a" button at different speeds. It keeps track of how many times you've clicked and directs each disc straight toward its ultimate destination, causing a pleasing tightening of the disc circle...
Something is weird: for example with the dihedral group D2, it shows two identical generators (1 2) and (1 2), which is wrong, as the generators must be different permutations.
D1 and D2 are a little weird. The way they’re usually defined, they can’t be defined as subgraphs of S1 and S2 respectively. See this relevant stackexchange answer [0].
By the symmetry of the n-gon you should, in this case, think of the group of all the isometries of the plane that fix the outline of the n-gon. For n=1 there are two such isometries, namely the identity and the reflection through the midline.
What’s being shown in the linked visualizer is actually the action of D2 on the labeled vertices of a 2-gon. I suspect two identical generators are shown because this is how each of the two generators of D2 (180 degree rotation and reflection about midpoint of 2-gon) act on the vertices.
https://nathancarter.github.io/group-explorer/index.html
The book is also very nice.
edit: Oh right, that's linked from this page ...