Orders of moves in the Rubik's cube

Take a solved cube and do $RU$; then trace the cycle structure of the permutation it realizes.

The combination moves 5 corner cubies in a cycle where a cubie is twisted by a third of a turn when it gets back to its original position, so that's a factor of 15.

It also twists a the FRU corner by one third of a turn; that's taken care of by the factor of 15 too.

Then it permutes 7 edges cyclically, but this time every edge has the correct orientation when it gets back.

So the order is the least common multiple of 7 and 15, namely 105.

(For a subercube we need another factor of 4 to get the centers back into the original orientation).

For some time ago I wrote my bachelor thesis about the element of greatest order:

http://www.math.kth.se/~boij/kandexjobbVT11/Material/rubikscube.pdf

Actually, I came up with a new theorem for the generalized symmetric group that confines, as a special case, the orders inside the Rubik's cube and the result is quite interesting. The greatest order is 1260.