I am interested in intransitive irreducible linear subgroups $G\subseteq\mathrm{GL}_n(\mathbb{F}_p)$ acting on $V-\{0\}=\mathbb{F}_p^n-\{0\}$ in the natural way, such that all of the orbits are very large, with constant density in $V$.
Actually I am wondering if there exists an infinite family of groups satisfying this property. Here $p$ is supposed to be a fixed prime.
The question can be formalized as follows:
Do there exist a prime $p>0$, a constant $c\in (0,1)$, and infinitely many integers $n\in\mathbb{N}^+$ and irreducibe linear groups $G_n\subseteq \mathrm{GL}_n(\mathbb{F}_p)$, such that each $G_n$ acts intransitively on $V_n-\{0\}:=\mathbb{F}_p^n-\{0\}$ in the natural way, and the size of every $G_n$-orbit is at least $cp^n$?
Transitive groups e.g. $G=\mathrm{GL}_n(\mathbb{F}_p)$ certainly have this property but I wonder if there exist infinitely many intransitive examples. I think one can easily reduce to the case that $G$ is a primitive linear group but then I don't know how to proceed.
