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

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 cannot 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 us 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 analyze some interesting finite subgroups of U(3), especially the group , which is closely related to the important SU(3)-subgroup S4.