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It may be relevant to note a 1965 Theorem of Isaacs and Passman ( Pacific Journal of Mathematics), which may be interpreted as follows: for any positive integer $m$, there are only finitely many possibilities for $G/F(G)$ if all complex irreducible characters of the finite group $G$ have degree at most $m$. Here, $F(G)$ denotes the unique largest nilpotent normal subgroup of $G$.