Regular Representations and Coset Representations Combined with a Mirror-Permutation. Concordant Construction of the Mark Table and the USCI-CF Table of the Point Group T_d

A combined-permutation representation (CPR) of degree 26 (= 24 + 2) for a regular representation (RR) of degree 24 is derived algebraically from a multiplication table of the point group T-d, where reflections are explicitly considered in the form of a mirror-permutation of degree 2. Thereby, the standard mark table and the standard USCI-CF table (unit-subduced-cycle-index-with-chirality-fittingness table) are concordantly generated by using the GAP functions MarkTablef orUSCI and constructUSClTable, which have been developed by Fujita for the purpose of systematizing the concordant construction. A CPR for each coset representation (CR) (G(i)\)T-d is obtained algebraically by means of the GAP function CosetRepCF developed by Fujita (Appendix A). On the other hand, CPRs for CRs are obtained geometrically as permutation groups by considering appropriate skeletons, where the point group T-d acts on an orbit of vertical bar T-d vertical bar/vertical bar G(i)vertical bar positions to be equivalent in a given skeleton so as to generate the CPR of degree vertical bar T-d vertical bar/vertical bar G(i)vertical bar. An RR as a CPR is obtained by considering a regular body (RB), the vertical bar T-d vertical bar positions of which are considered to be governed by RR (C-1\)T-d. These geometrically derived CPRs as groups are compared with the corresponding CPRs obtained algebraically.