  Gareth Jones asked during the 2014 SIGMAP conference for examples of regular
dessins with nonabelian fields of moduli. In this paper, we first construct
dessins whose moduli fields are nonabelian Galois extensions of the form
$\mathbb{Q}(\zeta_p,\sqrt[p]{q})$, where $p$ is an odd prime and $\zeta_p$ is a
$p$th root of unity and $q\in\mathbb{Q}$ is not a $p$th power, and we then show
that their regular closures have the same moduli fields. Finally, in the
special case $p=q=3$ we give another example of a regular dessin with moduli
field $\mathbb{Q}(\zeta_3,\sqrt[3]{3})$ of degree $2^{19}\cdot3^4$ and genus
$14155777$.
