Abstract. The moduli space M_g is the space of analytic equivalence classes of Riemann surfaces of a fixed genus g. The space M_g has a natural equisymmetric stratification, each stratum consists of the Riemann surfaces with an automorphisms group acting in a given topological way. Using Fuchsian groups we describe the equisimmetrical stratification for the moduli space of surfaces of genus 4. This is a joint work with Antonio F. Costa.