J Math Chem https://doi.org/10.1007/s10910-018-0900-y ORIGINAL PAPER PSL (2, 7) and carbon allotrope D168 Schwarzite Qaiser Mushtaq 1 · Nighat Mumtaz 2 Received: 5 November 2017 / Accepted: 20 March 2018 © Springer International Publishing AG, part of Springer Nature 2018 Abstract We investigate actions of the modular group PSL (2, Z) on the projective line over finite fields PL (F 7 n ) and find interesting relation between the coset diagram of orbits and the carbon allotrope with negative curvature D168 Shewarzite. We also highlighted some topological aspects of these diagrams. Keywords Projective special linear groups · Heptakisoctahedral group · Coset diagrams · Genus · Euler’s characteristics Mathematics Subject Classification Primary 05C38 · 15A15; Secondary 05A15 · 15A18 1 Introduction The point groups in chemistry portray the spatial symmetry of molecules [11, 12]. In this context the groups of the regular polyhedra are specifically noteworthy because of their point symmetry. It is discussed in [16] that these regular polyhedral groups are subgroups of larger permutation groups, which themselves are subgroups of the corresponding symmetric groups S n . Of specific pertinence to chemists in [15] is that these groups may be utilized to depict carbon allotrope structures with negative curvature built from hexagons and heptagons of sp2-hybridized carbon atoms [7, 15, B Nighat Mumtaz nighat_282@hotmail.com Qaiser Mushtaq pir_qmushtaq@yahoo.com 1 The Islamia University of Bahawalpur, Bahawalpur, Pakistan 2 Quaid-i-Azam University, Islamabad, Pakistan 123