Feit numbers and $p'$-degree characters
Suppose that $χ$ is an irreducible complex character of $G$ and let $f_χ$ be the smallest integer $n$ such that the cyclotomic field $\mathbb Q_n$ contains the values of $χ$. Let $p$ be a prime, and assume that $χ\in \textrm{Irr}(G)$ has degree not divisible by $p$. If $G$ is solvable and $χ(1)$ is odd, then there exists $g \in {\bf N}_G(P) /P'$ with $o(g)=f_χ$. In particular $f_χ$ divides $|{\bf N}_G(P) /P'|$.
math.GR↗