Is the group of symmetries of the rectangle-not-square related to the Klein bottle mathematically?


The reason I am asking is because I want to put a Klein bottle coffee cup in a joke about V\_4 and wonder if this would be a good idea.



Is the Klein group related to the Klein bottle?
No, according to wikipedia ([bottle](https://en.wikipedia.org/wiki/Klein_bottle), [group](https://en.wikipedia.org/wiki/Klein_four-group)) they're both named after Felix Klein, but appeared in different papers on completely different topics. The bottle comes from his [notes](https://www.kleinbottle.com/The%20First%20Klein%20Bottle.html) from 1882 and the group in [Lectures on the icosahedron and the solution of equations of the fifth degree](https://archive.org/details/cu31924059413439/mode/2up).


Klein group is also a bit of a pun, since Klein means small in German.


