  Kulkarni showed that, if g is greater than 3, a periodic map on an oriented
surface S_g of genus g with order more than or equal to 4g is uniquely
determined by its order, up to conjugation and power. In this paper, we show
that, if g is greater than 30, the same phenomenon happens for periodic maps on
the surfaces with orders more than 8g/3 and, for any integer N, there is g > N
such that there are periodic maps of S_g of order 8g/3 which are not conjugate
up to power each other. Moreover, as a byproduct of our argument, we provide a
short proof of Wiman's classical theorem: the maximal order of periodic maps of
S_g is 4g+2.
