Comments on the classification of the finite subgroups of SU(3)

TL;DR

This paper revisits the classification of finite SU(3) subgroups, clarifying the structure of the less-understood series (C) and (D), and their relation to known series like Delta(3n^2) and Delta(6n^2).

Contribution

It provides explicit structural descriptions of the (C) and (D) subgroup series and clarifies their representation-theoretic relationships to Delta series.

Findings

(C) rom (Z_m rom Z_m') times Z_3

(D) rom (Z_n rom Z_n') times S_3

(C) groups relate to irreducible representations of Delta(3n^2)

Abstract

Many finite subgroups of SU(3) are commonly used in particle physics. The classification of the finite subgroups of SU(3) began with the work of H.F. Blichfeldt at the beginning of the 20th century. In Blichfeldt's work the two series (C) and (D) of finite subgroups of SU(3) are defined. While the group series Delta(3n^2) and Delta(6n^2) (which are subseries of (C) and (D), respectively) have been intensively studied, there is not much knowledge about the group series (C) and (D). In this work we will show that (C) and (D) have the structures (C) \cong (Z_m x Z_m') \rtimes Z_3 and (D) \cong (Z_n x Z_n') \rtimes S_3, respectively. Furthermore we will show that, while the (C)-groups can be interpreted as irreducible representations of Delta(3n^2), the (D)-groups can in general not be interpreted as irreducible representations of Delta(6n^2).