Representations of branched twist spins with a non-trivial center of order 2

It is known that a presentation of the knot group of a branched twist spin is obtained from a Wirtinger presentation of the original 1-knot group by adding a generator corresponding to a regular orbit of the circle action and a certain relator. In particular, the additional generator is an element of the center of the knot group. In this paper, we focus on ${\mathrm{S}\mathrm{L}}_2({\mathbb{Z}}_3)$-representations and dihedral group representations. For the former case, we give a sufficient condition for the existence of an ${\mathrm{S}\mathrm{L}}_2({\mathbb{Z}}_3)$-representation for a branched twist spin. For the latter case, we determine the number of even-ordered dihedral group representations of branched twist spins.