Combined-permutation representations (CPRs) for characterizing $${\varvec{I}}_{h}$$-skeletons (a dodecahedral skeleton 1 having 20 vertices, an icosahedral skeleton 2 having 12 vertices, and a $$\hbox {C}_{{60}}$$-fullerene skeleton 3 having 60 vertices) are constructed by starting from respective sets of generators, where the permutation of each generator is combined with a mirror-permutation of 2-cycle to give the CPR of degree 22 (= 20 + 2) for 1, the CPR of degree 14 (= 12 + 2) for 2, and the CPR of degree 62 (= 60 + 2) for 3. Mark tables (tables of marks) of these CPRs are different in the sequence of subgroups from each other when they are produced as primary mark tables by the GAP system. On the other hand, the GAP functions MarkTableforUSCI and constructUSCITable, which have been previously developed to systematize the concordant construction of a standard mark table and a standard USCI-CF (unit-subduced-cycle-index-with-chirality-fittingness) table, are capable of constructing the standard mark table and the standard USCI-CF table even if we start from any of these CPRs. After a set of PCI-CFs (partial cycle indices with chirality fittingness) is calculated for each skeleton, symmetry-itemized combinatorial enumeration is conducted by means of the PCI method of Fujita’s USCI approach (Fujita in Symmetry and combinatorial enumeration in chemistry, Springer, Berlin, 1991). Construction of the CPR of degree 7 (= 5 + 2) for characterizing the $${\varvec{I}}_{h}$$ group is also discussed by starting from the alternating group $$\hbox {A}_{{5}}$$ isomorphic to the point group $${\varvec{I}}$$.

中文翻译：

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通过组合置换表示和新开发的 GAP 函数，从不同的 $${\varvec{I}}_{h}$$-骨骼衍生的化合物的对称逐项枚举

用于表征 $${\varvec{I}}_{h}$$-骨骼（具有 20 个顶点的十二面体骨架 1、具有 12 个顶点的二十面体骨架 2 和 $$\hbox { C}_{{60}}$$-富勒烯骨架 3（具有 60 个顶点）是通过从各自的生成器组开始构建的，其中每个生成器的排列与 2 周期的镜像排列相结合，给出 CPR 22 度 (= 20 + 2) 为 1，14 度 (= 12 + 2) 为 2，CPR 为 62 度 (= 60 + 2) 为 3。标记这些 CPR 的表（标记表）当 GAP 系统将它们作为主要标记表生成时，它们的子组序列彼此不同。另一方面，GAP 函数 MarkTableforUSCI 和constructUSCITable，之前已经开发用于系统化标准标记表和标准 USCI-CF（单位衰减周期指数与手性拟合）表的协调构建，能够构建标准标记表和标准 USCI -CF 表，即使我们从这些 CPR 中的任何一个开始。为每个骨架计算一组PCI-CFs（具有手性拟合度的部分循环指数）后，利用藤田USCI方法的PCI方法（Fujita in Symmetry and combinatorial enumeration in chemistry，Springer，柏林，1991 年）。用于表征 $${\varvec{I}}_{h}$$ 组的 7 (= 5 + 2) 次 CPR 的构造也通过从交替组 $$\hbox {A}_{ 开始讨论{5}}$$ 与点群 $${\varvec{I}}$$ 同构。