When studying groups of order 12, I learned about dicyclic groups, of which the quaternion group is the first example, and there is one of every order $4n$ for $n > 1$. I can do the basic calculations involving them, but I was wondering if there is some mechanical/geometric model I could use as an aid to intuition. I know every finite group is a subset of the symmetric group on $k$ elements for some $k$, but I was hoping for something more specific. For example, the alternating group on four elements can be visualized as the rotations of a tetrahedron. I haven't been able to find one on my own, and I was wondering whether that is because the "simple visualization" is four dimensional, like the quaternions?
I tried drawing the elements in two superimposed rings, with a variety of orderings of elements, but I wasn't able to represent the actions of the group elements, the way the symmetries of an equilateral triangle can be thought of as positions or as actions (which I think of as the "noun and verb problem".)
Is there a common model in use for dicyclic groups? Or should I just get used to them as an abstraction with no picture?
The smallest dicyclic group is just the quaternion group $Q_8$. Realizing $Q_8$ as the group of symmetries of some "visualizable" object is already a bit of a challenge, and it is treated here. It is a pretty accessible read; the discussion there hints at how difficult it might be for larger groups to be realized in this way.