projective image as $G$. (Inside $\operatorname{GL}_n({\mathbb C})$, this central product just corresponds to adjoining more scalar matrices to $G$.) Conversely, if $G \times_{Z(G)} C_m$ is an irreducible subgroup of $\operatorname{GL}_n({\mathbb C})$, so is $G$. We say $G$ is a minimal lift of $H$ to $\operatorname{GL}_n({\mathbb C})$ if $G$ is an irreducible subgroup of $\operatorname{GL}_n({\mathbb C})$ with $\bar G \simeq H$ such that $G$ is not isomorphic to $G_0 \times_{Z(G_0)} C_m$ for any proper subgroup $G_0$ of $G$.

Serre’s construction {#sec:serre}
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Here we explain the proof of Proposition \[prop1\] and describe Serre’s construction.

Suppose $\chi_1$ and $\chi_2$ are two distinct irreducible degree $n$ characters of a finite group $G$. Let $Y$ be the set of elements of $G$ such that $\chi_1(g) \ne \chi_2(g)$. Then we have $$|G| ( (\chi_1, \chi_1) - (\chi_1, \chi_2) ) =
\sum_Y \chi_1(g) (\overline{\chi_1(g)} - \overline{\chi_2(g)}).$$ Using the bound $|\chi_i(g)| \le n$ for $i=1,2$ and orthogonality relations, we see $$|G| = |G| ( (\chi_1, \chi_1) - (\chi_1, \chi_2) )
\le 2n^2 |Y|.$$ This proves Proposition \[prop1\].

We now recall Serre’s construction proving Theorem \[thm:serre\], which is briefly described in [@ramakrishnan:motives] using observations from [@serre Sec 6.5].

Let $H$ be an irreducible subgroup of $\operatorname{GL}_n({\mathbb C})$, containing 