Well, the idea of using commutators in the Rubik cube is what leads to a nice, elementary method of solving it, but it's not really a theorem or result from group theory that's telling us something we didn't already know. Plus I suspect many people would regard this as just a toy, a form of "play for play's sake" and hence still irrelevant.

The fact that the quaternions form a field (hence two kinds of group, one of which is interesting) where the multiplication doesn't commute, and then that can be used to model rotations is again interesting, and can be asserted to be useful (and it is) but it's still not really what I'm looking for.

But keep the ideas coming. There's got to be a good one out there somewhere - isn't there?