Coset Diagrams for a Homomorphic Image of Δ (3, 3, k)

Let q be a prime power. By PL(Fq) the authors mean a projective line over the finite field Fq with the additional point ∞. In this article, the authors parametrize the conjugacy classes of nondegenerate homomorphisms which represent actions of Δ (3, 3,k) =〈 u, v:u3=v3=(uv)k=1〉 on PL(Fq), where q≡ ± 1(mod k). Also, for various values of k, they find the conditions for the existence of coset diagrams depicting the permutation actions of Δ (3, 3,k) on PL(Fq). The conditions are polynomials with integer coefficients and the diagrams are such that every vertex in them is fixed by . In this way, they get Δ (3, 3,k) as permutation groups on PL(Fq).