Much is known about the derived length of a solvable group given the degrees and cardinality of the set of degrees of the irreducible characters. Martin Isaacs and Donald Passman pretty much started this area of study in 1960.
Say you are only given the set of degrees of the monomial irreducible characters of a solvable group. Then how much is known about its derived length? Let's denote this set by m.c.d(G). It is not too difficult to show that if |m.c.d(G)|=1 then G must be abelian. Is it known that if |m.c.d(G)|=2 then G is metabelian for example? Or perhaps something similar?