Can we find finite metabelian (ie with derived length 2) groups with arbitrarily many distinct degrees of irreducible complex characters?


If we cannot, can we somehow find a bound of the form $|cd(G)|\leq f(dl(G))$ for some "interesting" function $f$ (linear would be very cool for instance).


The motivation is that we of course have the bound $dl(G)\leq 2|cd(G)|$ for solvable groups (and conjectured to actually be $dl(G)\leq |cd(G)|$), so I was wondering if a bound in the other direction also existed.



Are there finite metabelian groups with arbitrarily many character degrees?
Sure, you just take direct products of metabelian groups with different character degrees: cd(G×H) = cd(G)×cd(H) and (G×H)′ = G′×H′.


I suggest taking G(p) = AGL(1,p) = Hol(p) to be the normalizer of a Sylow p-subgroup in the symmetric group of degree p, for each prime p, but there are lots of examples. For instance:


* G(3) = Sym(3) has character degrees {1,2},
* G(5) = F20 has character degrees {1,4},
* G(7) = F42 has character degrees {1,6},
* and G(3) × G(5) × G(7) has character degrees { 1, 2, 4, 6, 8, 12, 24, 48 }.


