I am interested in constructing small (as possible) group $G$ with large dimensional irreducible representation $\rho,V$ such that exist three elements of $g_1,g_2,g_3\in G$ such that for some $c_1,c_2,c_3\in C$ the matrix $c_1\rho(g_1)+c_2\rho(g_2)+c_3\rho(g_3)$ has rank one.($3$ elements here is arbitrary it may be any constant number)

In fact I know that if there is only two elements then $|G|\geq 2^{n}$, where $n=dim V$. It is easily floows from the post: Irreducible representation flipping two elements

My question is how to construct such irreducible representation of dimension $n>>log|G|$?

Example when $|G|>2^n$ is symmetric group $S_n$ acting on $n$ elements induces reps on $F^n$. If $\rho$ is $n-1$ dimensional irreducible sub-representation then $id-(1,2)$ acts as rank one matrix.