TITLE: An efficient method for discovering finite groups
QUESTION [5 upvotes]: I wrote a program to "discover", by means of constructing Cayley tables, finite groups.
Due to the method used (the one describe on the wikipedia article for Cayley tables, starting with a identity skeleton) the runtime is on the order of $O(n!)$
This allows me to find small groups with order at most 12 before the time taken to generate the tables becomes impractical.
Is there a faster method for finding finite simple groups? This clearly rules out composing them as a product of smaller groups. Is there a procedure that could discover the Monster group, for example?

REPLY [2 votes]: There is a vast literature on computational group theory: you could start here http://en.wikipedia.org/wiki/Computational_group_theory.