Rubik Cube finite non-abelian group
From a theoretical perspective, solving Rubik's cube is a nice example of the Schreier-Sims algorithm in action, although applying the algorithm will not give in general the quickest solution.
The book by Alexander Frey and David Singmaster, Handbook of Cubik Math, solves the cube using such group-theoretic concepts as cycles, equivalences, identities, inverses, order, commutativity, and conjugates.
There's also a book by David Joyner, Adventures in Group Theory, which has a lot of stuff about the Rubik cube, but I don't know whether it does what you want.