  Recently, the first author classified finite groups obtained as automorphism
groups of smooth plane curves of degree $d \ge 4$ into five types. He gave an
upper bound of the order of the automorphism group for each types. For one of
them, the type (a-ii), that is given by $\max \left\{ 2 d (d - 2), 60 d
\right\}$. In this article, we shall construct typical examples of smooth plane
curve $C$ by applying the method of Galois points, whose automorphism group has
order $60 d$. In fact, we determine the structure of the automorphism group of
those curves.
