Unusual Permutation Groups in Negative Curvature Carbon and Boron Nitride Structures

The concept of symmetry point groups for regular polyhedra can be generalized to special permutation groups to describe negative curvature polygonal networks that can be expanded to possible carbon and boron nitride structures through leapfrog transformations, which triple the number of vertices. Thus a D surface with 24 heptagons and 56 hexagons in the unit cell can be generated by a leapfrog transformation from the Klein figure consisting only of the 24 heptagons. The permutational symmetry of the Klein figure can be described by the simple PSL(2,7) (or heptakisoctahedral) group of order 168 with the conjugacy class structure E + 24C7 + 24C7 3 + 56C3 + 21C2 + 42C4. Analogous methods can be used to generate a D surface with 12 octagons and 32 hexagons by a leapfrog transformation from the Dyck figure consisting only of the 12 octagons. The permutational symmetry of the Dyck figure can be described by a group of order 96 and the conjugacy class structure E + 24S8 + 6C4 + 3C4 2 + 32C3 + 12C2 + 18S4. This group is not a simple group since it has a normal subgroup chain leading to the trivial group C1 through subgroups of order 48 and 16 not related to the octahedral or tetrahedral groups.