Let $X$ be a curve of genus three which is not hyperelliptic. Then $X$ is trigonal, i.e., there exists a finite morphism $X \to \mathbf P^1$ of degree $3$.

Let $Y\to X \to \mathbf P^1$ be a Galois closure of a trigonal curve. Then, unless $X\to \mathbf P^1$ is the Klein curve, the map $Y\to \mathbf P^1$ is of degree $6$.

Now, surely the latter map could have non-abelian Galois group, but I don't know any explicit examples. Can somebody give me an explicit example for which it is clear that the Galois group over $\mathbf P^1$ of the Galois closure is non-abelian?

Can we describe all such trigonal curves in the moduli space? How big is the dimension of the locus of curves which are hyperelliptic OR have a trigonal map $X \to \mathbf P^1$ whose Galois closure is cyclic (of degree 6) over $\mathbf P^1$?