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To draw a fundamental polygon (covering the quotient sphere) for a given signature:
sage: FundamentalPolygon([5,3,3,2]).draw()
Now for the fundamental region covering the curve on which a group is acting:
For a graphical representation:
Information on the identifications of the edges on the border, starting with the one painted red can be obtained with
[E1, -E8, E2, -E24, E3, -E23, E4, -E34, E5, -E21, E6, -E20, E7, -E13, 
   E8, -E1, E9, -E18, E10, -E17, E11, -E33, E12, -E32, E13, -E7, E14, 
   -E30, E15, -E29, E16, -E22, E17, -E10, E18, -E27, E19, -E26, E20, 
   -E6, E21, -E5, E22, -E16, E23, -E3, E24, -E2, E25, -E31, E26, -E19, 
   E27, -E36, E28, -E35, E29, -E15, E30, -E14, E31, -E25, E32, -E12, E33, 
   -E11, E34, -E4, E35, -E28, E36, -E9]
Here edge Ei is identified with edge -Ei
For the images of the group generators under the symplectic representation: