Finding Electric Field outside a Charged Cylinder

I'm trying to solve a problem that involves finding the electric field due to a uniformly cylinder of radius $r$, length $L$ and total charge $Q$. Well, my thought was: if I am to use Gauss' Law, I'll have to use a gaussian surface enclosing the cylinder.

Then here arises my doubt: My try was to enclose the cylinder with another cylinder with radius $R > r$, parametrize it and find it's normal $\hat{n}$ at each point. Then, I would need to calculate $\mathbf{E}\cdot \hat{n}$ and integrate. This would only be good to calculate $\mathbf{E}$ if it's magnitude is constant on the gaussian cylinder so that the magnitude would drop out of the integral. Also, I would need $\mathbf{E}$ parallel to the normal. It seems that both those things are true, but how can I see it and justify/prove it?

I just want suggestions, hints, in how can I see/prove that $\mathbf{E}$ has constant magnitude on the gaussian cylinder and how can I see/prove that $\mathbf{E}$ is parallel to the normal at each point. I also thought on some argument arround symmetry but I didn't find any consistent one.

Symmetries! Your system system has radial symmetry. You can use cylindrical coordinates, with the $z$ axis being the charged cylinder axis; you will easily observe that, due to the symmetry of the system, every quantity depends only on $r=x^2+y^2$ and $z$, rather than $x$, $y$, $z$. The answers to your questions can be easily derived from this fact.