I am currently trying to bound the derived length of certain solvable groups assuming that they have only two irreducible monomial complex character degrees. Using induction, it often suffices to consider groups $G$ satisfying that its set of irreducible character degrees, $\textrm{cd}(G)$, satisfies

$\textrm{cd}(G)=\lbrace 1,pq,pr,qr\rbrace$

where $p,q$ and $r$ are distinct primes. I would in particular like to know whether these (solvable) groups have derived length at most 3, but I have been unable to prove this. The paper

M.L.Lewis. Determining Group Structure from Sets of Irreducible Character Degrees - Journal of Algebra (1997)

studies similar groups, where Mark Lewis is able to prove strong statements about the groups he studies. This might suggest that it is reasonable to think that one can also put similar restraints on the groups above (and in particular bound its derived length by 3).


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