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 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.

36 pages, 1 figure; errors corrected; some comments and references added

References in corpus (27)

• Stringy origin of non-Abelian discrete flavor symmetries
• Neutrino tri-bi-maximal mixing from a non-Abelian discrete family symmetry
• The Unique Horizontal Symmetry of Leptons
• The Horizontal Symmetry for Neutrino Mixing
• Tri-Bimaximal Neutrino Mixing and the Family Symmetry Z_7 x Z_3
• Symmetry of Lepton Mixing
• A Simplest A4 Model for Tri-Bimaximal Neutrino Mixing
• The Flavor Group Delta(3n^2)
• Fermion Masses and Mixings from Dihedral Flavor Symmetries with Preserved Subgroups
• Near Tribimaximal Neutrino Mixing with Delta(27) Symmetry
• The Flavor Group Delta(6n^2)
• Is S4 the horizontal symmetry of tri-bimaximal lepton mixing?
• Lepton Mixing and Cancellation of the Dirac Mass Hierarchy in SO(10) GUTs with Flavor Symmetries T7 and Sigma(81)
• S_4 Flavor Symmetry Embedded into SU(3) and Lepton Masses and Mixing
• An SO(10) GUT Model with $S4$ Flavor Symmetry
• Simple Finite Non-Abelian Flavor Groups
• Flavor S_4 times Z_2 symmetry and neutrino mixing
• A new family symmetry for SO(10) GUTs
• Tri-bimaximal lepton mixing from symmetry only
• Intermediate left-right gauge symmetry, unification of couplings and fermion masses in SUSY $SO(10)\times S_4$
• New Lepton Family Symmetry and Neutrino Tribimaximal Mixing
• Gauge boson families in grand unified theories of fermion masses: E_6^4 x S_4
• Lepton Family Symmetry and Possible Application to the Koide Mass Formula
• A direct link between neutrinoless double beta decay and leptogenesis in a seesaw model with $S_4$ symmetry
• Overview of Neutrino Mixing Models and Ways to Differentiate among Them
• Neutrino Mass: Mechanisms and Models
• An Update on Models of Neutrino Masses and Mixings

Cited by in corpus (11)

• Tribimaximal Mixing From Small Groups
• Discrete flavour symmetries for degenerate solar neutrino pair and their predictions
• Seeking SUSY fixed points in the $4-\epsilon$ expansion
• Neutrino Mixing and Discrete Symmetries
• Residual $Z_2$ symmetries and leptonic mixing patterns from finite discrete subgroups of $U(3)$
• Frustrated symmetries in multi-Higgs-doublet models
• Cancellation Mechanism of FCNCs in S_4 x Z_2 Flavor Symmetric Extra U(1) Model
• Fully Constrained Majorana Neutrino Mass Matrices Using $\Sigma (72\times 3)$
• Systematic analysis of 3HDM symmetries
• Classifying irreducible fixed points of five scalar fields in perturbation theory
• Group-theoretical search for rows or columns of the lepton mixing matrix