Is there a class of solvable groups $G$ having a derived length $O(\log|G|)$? 

See [here][1] for the definition of Big-Oh$(O)$ and here for the definition of [derived series][2] of a group][2].

Any help would be appreciated. Thank you in advance!


  [1]: https://en.wikipedia.org/wiki/Big_O_notation
  [2]: https://groupprops.subwiki.org/wiki/Derived_series