These questions are by Moshe Newman
Let $G$ be a finite solvable group of derived length $d$, with the property that every proper subgroup and every proper quotient of $G$ has derived length less than $d$.
1) Can one give an upper bound on the number of different prime divisors of $|G|$, the order of $G$, a bound that depends only on $d$?
2) Can one give an upper bound on the number of elements required to generate $G$, a bound that depends only on $d$?
For $d=2$ the answer to both questions is yes, and the bound is $2$ in both questions. (See Miller and Moreno, for example.) What happens in general? Comment : one cannot bound the composition length, even for the case d = 2. Thus one can only hope to bound the number of distinct prime divisors of $|G|$, and not the total number of prime divisors counting multiplicity. Comment : assuming the answer to 2) is yes, it is the case that the bound must grow with $d$. Is this true for 1) as well?