What are some interesting coding projects (doable in Java) that relates to group theory? I would like some ideas of possible programs I can write in Java that involves some computational aspects of group theory.  My only ideas so far is to write a program that computes the product of two elements of $S_n$, but this is too easy.  Any suggestions/ideas will be greatly appreciated.  

One option (off the top of my head) would be to determine the order of an element in a cyclic group.
I suggest looking at this book (free): http://abstract.pugetsound.edu/
It has some programming problems (designed for a different language, but should be close enough) that relate to whatever topic the chapter discusses.  Obviously, group theory is included.
