Presentations of the braid group of the complex reflection group G(d, d, n)

We show that the braid group associated to the complex reflection group G(d, d, n) is an index d subgroup of the braid group of the orbifold quotient of the complex numbers by a cyclic group of order d. We also give a compatible presentation of G(d, d, n) and its braid group for each tagged triangulation of the disk with n marked points on its boundary and an interior marked point (interpreted as a cone point of degree d) in such a way that the presentations of Broué–Malle–Rouquier correspond to a special tagged triangulation.