Summary: The representation of the action of \(\mathrm{PGL}(2, \mathbb{Z})\) on \(F_t\cup\{\infty\}\) in a graphical format is labeled as coset diagram. These finite graphs are acquired by the contraction of the circuits in infinite coset diagrams. A circuit in a coset diagram is a closed path of edges and triangles. If one vertex of the circuit is fixed by \((pq)^{\Delta_1}(pq^{-1})^{\Delta_2}(pq)^{\Delta_3}\dots (pq^{-1})^{\Delta_m}\in\mathrm{PSL}(2, \mathbb{Z})\), then this circuit is titled to be a length-\(m\) circuit, denoted by \((\Delta_1, \Delta_2, \Delta_3, \dots, \Delta_m)\). In this manuscript, we consider a circuit \(\Delta\) of length 6 as \((\Delta_1, \Delta_2, \Delta_3, \Delta_4, \Delta_5, \Delta_6)\) with vertical axis of symmetry, that is, \(\Delta_2 = \Delta_6\), \(\Delta_3 = \Delta_5\). Let \(\Gamma_1\) and \(\Gamma_2\) be the homomorphic images of \(\Delta\) acquired by contracting the vertices \(a,u\) and \(b,v\), respectively, then it is not necessary that \(\Gamma_1\) and \(\Gamma_2\) are different. In this study, we will find the total number of distinct homomorphic images of \(\Delta\) by contracting its all pairs of vertices with the condition \(\Delta_1 > \Delta_2 > \Delta_3 > \Delta_4\). The homomorphic images are obtained in this way having versatile applications in coding theory and cryptography. One can attain maximum nonlinearity factor using this in the encryption process.