§ Line of investigation to build physical intuition for semidirect products
To quote wikipedia:
In crystallography, the space group of a crystal splits as the semidirect product of the point group and the translation group if and only if the space group is symmorphic
The if and only if is interesting: The geometry ofthe crystal lattice truly
appears to capture the structure of the semidirect product. It's a discrete
object as well, which makes it way easier to visualize. I'm going to hunt down
the definitions involved so I can finally feel like I truly understand semidirect
products from the "action" perspective.