I have a finite-dimensional vector space $E$ over the finite prime field $\mathbf{F}_p$ and a solvable subgroup $G\subset\mathbf{GL}(E)$ for which $E$ is an irreducible representation. Do the groups $H^1(G,E)$ and $H^2(G,E)$ both vanish ?

I have a finite-dimensional vector space $E$ over the finite prime field $\mathbf{F}_p$ and a solvable subgroup $G\subset\mathbf{GL}(E)$ for which $E$ is an irreducible representation. Do the groups $H^1(G,E)$ and $H^2(G,E)$ both vanish ?

Addendum (2016/08/17) For those curious as to why I needed this result, see Solvable primitive $p$-extensions (http://arxiv.org/abs/1608.04673).

I have a finite-dimensional vector space $E$ over $\mathbf{F}_p$ and a solvable subgroup $G\subset\mathbf{GL}(E)$ for which $E$ is an irreducible representation. Do the groups $H^1(G,E)$ and $H^2(G,E)$ both vanish ?

I have a finite-dimensional vector space $E$ over the finite prime field $\mathbf{F}_p$ and a solvable subgroup $G\subset\mathbf{GL}(E)$ for which $E$ is an irreducible representation. Do the groups $H^1(G,E)$ and $H^2(G,E)$ both vanish ?

I have a finite-dimensional vector space $E$ over $\mathbf{F}_p$ and a solvable subgroup $G\subset\mathrm{GL}(E)$ for which $E$ is an irreducible representation. Do the groups $H^1(G,E)$ and $H^2(G,E)$ both vanish ?

I have a finite-dimensional vector space $E$ over $\mathbf{F}_p$ and a solvable subgroup $G\subset\mathbf{GL}(E)$ for which $E$ is an irreducible representation. Do the groups $H^1(G,E)$ and $H^2(G,E)$ both vanish ?