This paper presents a quantification of the complex geometry of symmetric A4-graph constructed from Conway group Co3  and a particular category of Co3 by calculating its complete topological descriptors. The primary objective is to characterize this graph using molecular topology by calculating a suite of significant topological indices. Furthermore, we derive the Hosoya and Schultz polynomials, which serve as generating functions for the full distance graph and degree spectra. A principal result is the establishment of closed-form relationships between these topological descriptors and fundamental group characteristics, notably conjugacy-class cardinality and vertex-degree distribution, revealing that the graph’s metric architecture is predetermined by its algebraic origin.