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I want to construct an irreducible representation $V$ of any group $G$ such that there exist a mapping of form $\sum_{i=1}^{100} a_i g_i, a_i \in C, g_i \in G$ with kernel of dimension $dim V-1$. In general it is not hard to construct such representation. Does it possible to construct such representation with group $G$ small and $\dim V$ large? For example does it possible that $\frac{\log|G|}{\log (dim V)}<c$ when $\dim V \to \infty$?

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