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The heptakisoctahedral group and its relevance to carbon allotropes with negative curvature. (English) Zbl 0992.92042
Summary: We introduce the pollakispolyhedral groups and describe in detail the representational structure of PSL(2,7) or $$^{7}$$O, the automorphism group of the Klein graph composed of 56 trivalent vertices arranged in 24 heptagonal faces. Leapfrog and quadruple transformations of the graph are described and their eigenvalue spectra derived. Considered as carbon frameworks on the ‘plumber’s nightmare’ surface these chiral structures exhibit significant steric strain which prevents the molecular realisation of the Klein symmetry.

##### MSC:
 9.2e+11 Molecular structure (graph-theoretic methods, methods of differential topology, etc.) 2e+100 Structure and classification of infinite or finite groups
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