Hurwitz groups are non-trivial quotients ofVarious families of groups have been thoroughly investigated as candidates for Hurwitz groups. In this paper we are interested in parametrizing the actions of ▵(2,3,7) on the projective lines over finite fields Fq, with the help of coset diagrams. We are interested to know also the values of q for which there is a natural homomorphism induced from ▵(2,3,7) and there exist vertices on the vertical line of symmetry in the diagram depicting these actions. Also we use the Cebotarev's Density Theorem and Galois Theory to answer the question about the frequency with which such situations and certain fragments of the diagram occur.