On the finite subgroups of U(3) of order smaller than 512

TL;DR

This paper systematically classifies finite subgroups of U(3) with order less than 512, identifies fundamental building blocks for models, and introduces a theorem to generate infinite series of such groups.

Contribution

It provides a comprehensive list of small finite subgroups of U(3), identifies those that are indecomposable, and presents a new theorem for constructing infinite series of these groups.

Findings

Identified all finite subgroups of U(3) under 512 with faithful three-dimensional irreducible representations.

Extracted fundamental building blocks for models based on these groups.

Developed a theorem to generate infinite series of finite subgroups of U(3).

Abstract

We use the SmallGroups Library to find the finite subgroups of U(3) of order smaller than 512 which possess a faithful three-dimensional irreducible representation. From the resulting list of groups we extract those groups that can not be written as direct products with cyclic groups. These groups are the basic building blocks for models based on finite subgroups of U(3). All resulting finite subgroups of SU(3) can be identified using the well known list of finite subgroups of SU(3) derived by Miller, Blichfeldt and Dickson at the beginning of the 20th century. Furthermore we prove a theorem which allows to construct infinite series of finite subgroups of U(3) from a special type of finite subgroups of U(3). This theorem is used to construct some new series of finite subgroups of U(3). The first members of these series can be found in the derived list of finite subgroups of U(3) of order smaller than 512. In the last part of this work we analyse some interesting finite subgroups of U(3), especially the group S_4(2)\cong A_4\rtimes Z_4, which is closely related to the important SU(3)-subgroup S_4.