Let $p>3$ be a prime number and let $\mathbb{F}_{p}(t)$ be the field of rational functions over $\mathbb{F}_{p}$.
Is there a curve $C$ such that the Galois group $\operatorname{Gal}(\mathbb{F}_{p}(C)/\mathbb{F}_{p}(t))$ is isomorphic to the cyclic group $C_6$?
In a way, this should be some analogue of the cyclotomic extension $\mathbb{Q}(\zeta_{7})$, but I'm not really sure how to explicitly construct it.