Connected Components of H_(r,g)^A (G)

The Hurwitz space   is the space of genus g covers of the Riemann sphere  with  branch points and the monodromy group . In this paper, we enumerate the connected components of the Hurwitz spaces  for a finite primitive group of degree 7 and genus zero except . We achieve this with the aid of the computer algebra system GAP and the MAPCLASS package.