# GAP listing of the finite subgroups of U(3) of order smaller than 2000

## TL;DR

This paper systematically classifies finite subgroups of U(3) with order less than 2000, identifying those with faithful three-dimensional irreducible representations and distinguishing subgroups of SU(3) from other U(3) subgroups.

## Contribution

It provides a comprehensive GAP-based listing and classification of finite subgroups of U(3) under 2000, including generators and structural series, expanding understanding of these groups.

## Key findings

- Identified all finite subgroups of U(3) under 2000 with specific properties.
- Classified subgroups of SU(3) within the larger U(3) groups.
- Generated explicit generators for non-SU(3) subgroups.

## Abstract

We have sorted the SmallGroups library of all the finite groups of order smaller than 2000 to identify the groups that possess a faithful three-dimensional irreducible representation (`irrep') and cannot be written as the direct product of a smaller group times a cyclic group. Using the computer algebra system GAP, we have scanned all the three-dimensional irreps of each of those groups to identify those that are subgroups of SU(3); we have labelled each of those subgroups of SU(3) by using the extant complete classification of the finite subgroups of SU(3). Turning to the subgroups of U(3) that are not subgroups of SU(3), we have found the generators of all of them and classified most of them in series according to their generators and structure.