  We compute the first explicit polynomials with Galois groups $G=P\Gamma
L_3(4)$, $PGL_3(4)$, $PSL_3(4)$ and $PSL_5(2)$ over $\mathbb{Q}(t)$.
Furthermore we compute the first examples of totally real polynomials with
Galois groups $PGL_2(11)$, $PSL_3(3)$, $M_{22}$ and $Aut(M_{22})$ over
$\mathbb{Q}$. All these examples make use of families of covers of the
projective line ramified over four or more points, and therefore use techniques
of explicit computations of Hurwitz spaces. Similar techniques were used
previously e.g. by Malle, Couveignes, Granboulan and Hallouin. Unlike previous
examples, however, some of our computations show the existence of rational
points on Hurwitz spaces that would not have been obvious from theoretical
arguments.
