I'd like to request a hint
So far, I've enumerated all 16 symmetries. I know that $D_8$ has order $16$, so I've got them all. I know that I could in theory just check them all, which would be tedious but doable. I'm sure there is a trick to this though. Unfortunately, I do not have any other ideas. I know that in general, a rotation and a reflection do not commute, but I'm not sure if I can extend that to every rotation and reflection.
Obviously I have that the identity commutes.
If someone could give a hint that would be much appreciated.