In this article coset diagrams of the action of PSL(2, Z) on a \(PL(F_p)\) are obtained, through parametrization, which yields one of the eight finite generalized triangle groups which are homomorphic images or quotients of PSL(2, Z). Other than this we analyzed the coset diagrams for the parameter for three finite generalized triangle groups. One of the most dependable methods for achieving data security has been the block cipher. S-Boxes constructed using algebraic structure have gained popularity recently because of their advantageous cryptographic properties and high non-linearity have been found in these structures, which attract researchers. With the help of these parametrized actions, a novel algebraic method to create \(2^8\) S-Boxe was established. The S-Box provides strong cryptographic qualities of nonlinearity 112, differential uniformity 6, linear approximation probability of 0.0576 and differential attack probability of 0.0039. To assess the practical applicability of our S-box, we integrate it into an image encryption scheme and present experimental results to showcase its efficacy in real-world scenarios. When used with an image encryption framework, the following results were obtained: NPCR = 0.9959, UACI = 0.3348, and approx. Entropy 7.98. Therefore, GTG based parametrisation has been shown to be an effective and secure alternative to traditional algebraic construction method for S-Boxes.
