Thanks for the explanation about N=5 and SmallGeneratingSet.

I understand the name now ... i.e. why it isn't called MinimalGeneratingSet.

Ok, but the real upshot is that this demonstrates what I said earlier about rotate being more "powerful" than flip - the rotate permutation still completed the group, without having a complete subgroup of row/band permutations.

This explains why my x and y operations using (N-1) + (N-2) shifts still work, even though the pair {x, y} does not generate the full band/row subgroup! It doesn't actually have to, because of the power of rotate to fill in the gaps (pun intended)!

Yes, I witnessed the "power of rotate", in my earlier scheme.

PS: Yes, my shifts are done left to right (+ means "followed by"). I haven't tested whether this order is critical or not, but will do this.

[EDIT] The order of the shifts is apparently not critical at all, the "+" actually commutes in this particular case!

The order of the operations will make a difference in the actual generators, though, since row operations in band 1, don't commute with band operations involving band 1.

I don't know how detailed your testing was, but unless I have a bug, for N=3, the ordering did matter, as to whether {x,y} generates the "bands & rows" group. It's possible that even though switching the order (apparently) doesn't work for that part, adding "z = rotate" to the set, still works for generating the full group.

You will have gathered that all my verifications simply involve examining the group size when I plug a set of permutations into GAP ... very fast.

Nothing wrong with that. I should really learn GAP

We have done well, I think! We have produced truly elegant, generally applicable minimal generators ... good job all round!

I would be much happier if we could come up with an actual proof (or a clear vision that one exists), for general N, and some particular specification of generators.

I think I could do that for my earlier setup for the full group, but I'ld be much happier if we could do it for a pair of generators for the "bands & rows" subgroup. I'm close to having something, I think. (Edit: ... not as close as I hoped ) If it works, it would involve an "odd/even" difference, again, in the way the generators are specified.