The Hurwitz space ℋr in(G) is the space of genus g = 0 covers of the Riemann sphere ℙ 1 with r branch points and the monodromy group G. Let G be the symmetric group S7 . In this paper, we enumerate the connected components of ℋr in(S7). Our approach uses computational tools, relying on the computer algebra system GAP and the MAPCLASS package, to find the connected components of ℋr in(S7). This work gives us the complete classification of primitive genus zero symmetric group of degree seven.

The Hurwitz space ℋr in(G) is the space of genus g = 0 covers of the Riemann sphere ℙ 1 with r branch points and the monodromy group G. Let G be the symmetric group S7 . In this paper, we enumerate the connected components of ℋr in(S7). Our approach uses computational tools, relying on the com...