On the Chow ring of the Hurwitz space of degree three covers of ${\bf P}^{1}$

We determine the Chow ring (with ${\bf Q}$-coefficients) of the Hurwitz space parametrizing degree three covers of ${\bf P}^{1}$. We also compute the rational Picard groups of auxiliary spaces of degree three maps with special marked points. In particular, we show that the Chow ring is tautological, answering a question posed by N. Penev and the second author.