= Coprime-degree determinant extension theorem
Suppose $N\triangleleft G$, $G/N$ is solvable, $\theta\in\operatorname{Irr}(N)$ is invariant under $G$, and $\theta(1)$ is coprime to $[G:N]$. If a <linear character> $\mu$ of $G$ restricts to the <determinant character> $\det\theta$, then there is a unique <character extension> $\chi\in\operatorname{Irr}(G)$ with <determinant> $\mu$. The coprime degree makes determinant normalization remove the ambiguity from twisting an extension by <linear characters> of the <quotient group>.
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