Abstract
The PCI (Partial-Cycle-Index) method of Fujita’s USCI (Unit-Subduced-CycleIndex) approach has been applied to symmetry-itemized enumerations of cubane derivatives, where groups for specifying three-aspects of symmetry, i.e., the point group \(\mathbf{O}_h\) for chirality/achirality, the RS-stereogenic group \(\mathbf{O}_{\widehat{\sigma}}\) for RS-stereogenicity/RS-astereogenicity, and the LR-permutation group \(\mathbf{O}_{\widehat{I}}\) for sclerality/ascrelarity are considered as the subgroups of the RS-stereoisomeric group \(\mathbf{O}_{h\widehat{\sigma}\widehat{I}}\). Five types of stereoisograms are adopted as diagrammatical expressions of \(\mathbf{O}_{h\widehat{\sigma}\widehat{I}}\), after combined-permutation representations (CPR) are created as new tools for treating various groups according to Fujita’s stereoisogram approach. The use of CPRs under the GAP (Groups, Algorithms and Programming) system has provided new GAP functions for promoting symmetry-itemized enumerations. The type indices for characterizing stereoisograms (e.g., \([a, −, −]\) for a type-V stereoisogram) have been sophisticated into RS-stereoisomeric indices (e.g., \([[\mathbf{C}_s', \tilde{\mathbf{C}'}_s, \mathbf{C}_1]]\) for a cubane derivative with the composition \(\text{H}_5\text{Ap}\bar{\text{p}}\)). The type-V stereoisograms for cubane derivatives with the composition \(\text{H}_5\text{Ap}\bar{\text{p}}\) are discussed under extended pseudoasymmetry as a new concept.
