  For a finite group $G$, let $\text{rdim}(G)$ denote the smallest dimension of
a faithful, complex linear representation of $G$. It is clear that
$\text{rdim}(H)\leq \text{rdim}(G)$ for any subgroup $H$ of $G$. We consider
$G$ with the property that $\text{rdim}(H)<\text{rdim}(G)$ whenever $H$ is a
proper subgroup of $G$, in particular proving a classification of such groups
when $G$ is abelian or $\text{rdim}(G)\leq 3$.
