  The Wiman-Edge pencil is a pencil of genus $6$ curves for which the generic
member has automorphism group the alternating group $\mathfrak{A}_5$. There is
a unique smooth member, the Wiman sextic, with automorphism group the symmetric
group $\mathfrak{S}_5$. Farb and Looijenga proved that the monodromy of the
Wiman-Edge pencil is commensurable with the Hilbert modular group
$\mathrm{SL}_2(\mathbb{Z}[\sqrt{5}])$. In this note, we give a complete
description of the monodromy by congruence conditions modulo $4$ and $5$. The
congruence condition modulo $4$ is new, and this answers a question of
Farb-Looijenga. We also show that the smooth resolution of the Baily-Borel
compactification of the locally symmetric manifold associated with the
monodromy is a projective surface of general type. Lastly, we give new
information about the image of the period map for the pencil.
