For a group $G$ there is a well-defined map $\operatorname{Irr}(G) \to \operatorname{Lin}(G)$ which sends $\chi \mapsto \det \chi$, where $\det \chi$ the linear character of $G$ given by taking the determinant of the representation affording $\chi$.

In general, is there a good way to go about computing $\det \chi$ without having to construct a representation affording $\chi$, or are there some conditions on $G$ or $\chi$ under which this can be done easily?