While working on the pattern for incrementing the S-group order I realized that S-groups can be modeled as a series of layers, where a layer is a (count, inc) pair in seqN_inc. Group[N] will consist of layers[1...N] and the only difference in a layer's use by G[N] and G[N+1] is that its invocation arguments will be shifted. This is why seq4_inc >>67 starts by computing incS1 and countS1 in a way that is equivalent to how seq3_inc >>36 computes incS and countS. This means that with G[N] in hand, G[N+1] need not be derived from scratch. The first N layers will match modulo a shift in arguments, and only the new layer[N+1] has to be derived.