  Let $F(G)$ and $b(G)$ respectively denote the Fitting subgroup and the
largest degree of an irreducible complex character of a finite group $G$. A
well-known conjecture of D. Gluck claims that if $G$ is solvable then
$|G:F(G)|\leq b(G)^{2}$. We confirm this conjecture in the case where $|F(G)|$
is coprime to 6. We also extend the problem to arbitrary finite groups and
prove several results showing that the largest irreducible character degree of
a finite group strongly controls the group structure.
