Solvable Groups of High Derived Length

[David Harden]

I have thought of a sequence I would like to submit to the OEoIS, but I need some help from someone with a system like GAP or Magma to get more terms.

If G is a solvable group, the derived length of G is defined recursively to be 0 if G is trivial and
1 + the derived length of the commutator subgroup [G,G] otherwise.
The derived length of a solvable group is, in some sense, a measure of its 'complexity', so it is natural to ask about what the smallest group with a given derived length is.

The sequence I want to submit is the one whose nth term is the order of the smallest group with derived length n. I have the first 5 terms of this sequence (since the starting index is 0, these are a_0 through a_4), and they are 1,2,6,24 and 48. The groups giving rise to 6, 24 and 48 are, respectively, S_3, S_4 and SL(2,3) (at this order we have a tie!), and GL(2,3) (I don't know that this is the only group of derived length 4 of order 48, but I know it is one).
I put 1,2,6,24,48 into the OEoIS and the only thing which came up is A099144.

A few things about this sequence:

i. If G is solvable, then the commutator subgroup of G has index >=2 in G and it has derived length equal to the derived length of G minus 1. Therefore each term is at least twice the previous term. In particular, this means a_5 in this sequence is at least 96 and no search through the groups of order 64, 72 or 80 needs to be conducted to look for a solvable group of derived length 5.

ii. A p-group of order p^k has derived length <=  ceiling(k/2). The reason is that unless the group is cyclic (in which case the derived length is 1, in keeping with this statement), its Frattini subgroup has index >= p^2 and contains the commutator subgroup.

iii. I am pretty sure that the general affine group 3^2:GL(2,3) has derived length 5. In this case, the next term is bounded from above by 432 and from below by 96. With further work, I have reduced the possible candidates for this number to one of 144, 192, 288, 320, 336, 360, 384, 400, or 432. In any 
of these cases, it is not a factorial or a double of a factorial. Therefore this sequence is different from A099144!

---- David Harden