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.



Intransitive finite irreducible linear groups whose orbits are all large
Take an odd prime $p$, let $q=p^n$ and consider $V=\mathbb{F}\_q$ as a vector space over $\mathbb{F}\_p$. For $G\_n=(\mathbb{F}\_q^\*)^2$ you get $c=1/2$.


