The fundamental group of compact Riemann surface

We derive presentation and relations for a group of compact Riemann surface that is given as branched cover of the sphere. In the case that one of the permutations is of full cycle of the form $(\mathrm{1...}n)$ we derive a straightforward process to obtain the standard presentation of the fundamental group of Algebraic curve in the form ${\prod }_{i=1}^g[a_i,b_i]=1$