Computing size of permutation group from generators
The most efficient (and also simplest) algorithms for this are based on the notion of a strong generating set, introduced by Sims. Strong generating sets can be computed efficiently using the Schreier-Sims algorithm. Essentially any introductory book that talks about computational group theory should have a decent introduction to this (and if it contains an introduction to this that isn't decent, I'd think it's not a very good book). While the standard deterministic algorithm for this takes quadratic time, there is a randomized nearly-linear time algorithm. See Seress's book "Permutation Group Algorithms," Chapter 4 (the nearly-linear time Monte Carlo algorithm is Theorem 4.5.5).