I am interested in algorithms for finite groups as implemented in the GAP package. It seems that all known algorithms in this field deal with permutation groups/matrix groups; two fundamental ones are Schreier-Sims  and Butler , see e.g. 'Algorithms for Permutation groups' by Alice Niemeyer as a possible reference (?)
Hence, I was wondering whether there had been significant progress in the field in the last 50 years. I've seen that user NisaiVloot asked some questions about braid groups that may constitute an interesting extension of known results about permutation groups, though it's unclear to me what is the current state of research in this field as the mathematics/algorithmics communities seem somewhat out-of-sync nowadays.