The algorithm is an efficient method of computing a base and strong generating set (BSGS) of a permutation group.
The Monte Carlo variations of the Schreier–Sims algorithm have the estimated complexity: Modern computer algebra systems, such as GAP and Magma, typically use an optimized Monte Carlo algorithm.
It is meant to leave out all finer details, such as memory management or any kind of low-level optimization, so as not to obfuscate the most important ideas of the algorithm.
Notable details left out here include the growing of the orbit tree and the calculation of each new Schreier generator.
The tree is rooted at the identity element, which fixes the point stabilized by the subgroup.
By the orbit-stabilizer theorem, these form a transversal of the subgroup of our group that stabilizes the point whose entire orbit is maintained by the tree.