  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 ${\rm
SL}_2(\mathbb{Z}_3)$-representations and dihedral group representations. For
the former case, we give a sufficient condition for the existence of an ${\rm
SL}_2(\mathbb{Z}_3)$-representation for a branched twist spin. For the latter
case, we determine the number of even-ordered dihedral group representations of
branched twist spins.
