Tschirnhausen bundles of sextic covers of ${\mathbb{P}}^1$

A degree $d$ genus $g$ cover of the complex projective line by a smooth irreducible curve $C$ yields a vector bundle on the projective line by pushforward of the structure sheaf. We classify the bundles that arise this way when $d=6$. Interestingly, our methods show that all constraints on the pushforward are ``explained’’ by multiplication in an algebra. Finally, we show that all possible pushforwards are realized by covers with a nontrivial proper subcover.