genus g groups of diagonal type

A transitive subgroup g ≤ sn is called a genus g group if there exist non identity elements x1, ..., xr ∈ g satisfying g = ⟨x1, x2, ..., xr⟩, πr i=1 xi = 1 and σr i=1 ind xi = 2(n + g −1). the hurwitz space hin r,g(g) is the space of genus g covers of the riemann sphere p1c with r branch points and the monodromy group g. isomorphisms of such covers are in one to one correspondence with genus g groups. in this article, we show that g possesses genus one and two group if it is diagonal type and acts primitively on ω. furthermore, we study the connectedness of the hurwitz space hin r,g(g) for genus 1 and 2.