Symmetry-itemized enumeration of RS-stereoisomers of allenes. I. The fixed-point matrix method of the USCI approach combined with the stereoisogram approach

After the RS-stereoisomeric group (mathbf{D}_{2dwidetilde{sigma }widehat{I}}) of order 16 has been defined by starting point group (mathbf{D}_{2d}) of order 8, the isomorphism between (mathbf{D}_{2dwidetilde{sigma }widehat{I}}) and the point group (mathbf{D}_{4h}) of order 16 is thoroughly discussed. The non-redundant set of subgroups (SSG) of (mathbf{D}_{2dwidetilde{sigma }widehat{I}}) is obtained by referring to the non-redundant set of subgroups of (mathbf{D}_{4h}) . The coset representation for characterizing the orbit of the four positions of an allene skeleton is clarified to be (mathbf{D}_{2dwidetilde{sigma }widehat{I}}(/mathbf{C}_{swidetilde{sigma }widehat{I}})) , which is closely related to the (mathbf{D}_{4h}(/mathbf{C}_{2v}^{prime prime prime })) . According to the unit-subduced-cycle-index (USCI) approach (Fujita, Symmetry and combinatorial enumeration of chemistry. Springer, Berlin 1991), the subduction of (mathbf{D}_{2dwidetilde{sigma }widehat{I}}(/mathbf{C}_{swidetilde{sigma }widehat{I}})) is examined so as to generate unit subduced cycle indices with chirality fittingness (USCI-CFs). Then, the fixed-point matrix method of the USCI approach is applied to the USCI-CFs. Thereby, the numbers of quadruplets are calculated in an itemized fashion with respect to the subgroups of (mathbf{D}_{2dwidetilde{sigma }widehat{I}}) . After the subgroups of (mathbf{D}_{2dwidetilde{sigma }widehat{I}}) are categorized into types I–V, type-itemized enumeration of quadruplets is conducted to illustrate the versatility of the stereoisogram approach.