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The minimal dimension of a nontrivial representation of a central extension of a group $G$ can be smaller than the minimal dimension of a nontrivial representation of $G$. What about non-central extensions? More precisely:

Suppose a finite group $G$ has no nontrivial central extensions, i.e. the Schur multiplier of $G$ is trivial. Let $\tilde{G} \to G$ be a finite nontrivial extension of $G$, but of course not a central one.

Let $d$ be the smallest dimension of a nontrivial representation of $G$. Is the dimension of a nontrivial representation of $\tilde{G}$ necessarily greater than or equal to $d$?

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  • $\begingroup$ What is meant by a "nontrivial extension"? Presumably this excludes direct products, but how about semidirect products? The regular wreath product of an abelian group by G is an extension that has a nontrivial representation of degree 1. $\endgroup$
    – Marty Isaacs
    Commented Jun 9, 2015 at 17:46

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