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.
Highlights
Many high-energy physicists are thrilled by the prospect that the numerical entries of the leptonic mixing matrix (PMNS matrix) might be related to some small finite group
There is no complete classification of the finite subgroups of U(3),1 though a few series of those subgroups have been derived in ref
A full theoretical study of each individual group can always be undertaken, for large groups such a study becomes impractical and it is convenient to have recourse to the computer algebra system GAP, which is tailored to deal with finite groups and can readily furnish the structure, irreducible representations (‘irreps’), character table, and so on, of each of them
Summary
Many high-energy physicists are thrilled by the prospect that the numerical entries of the leptonic mixing matrix (PMNS matrix) might be related to some small (or maybe not so small) finite group. The first step in this work was to survey the whole SmallGroups list of groups of order smaller than 2 000 in order to identify the ones that have at least one faithful three-dimensional irreducible representation; cannot be written as the direct product of a smaller group and a cyclic group. We feel that having a complete listing of all those subgroups of order less than 2 000, together with their generators, may be a useful step towards achieving such a classification; at the very least, it allows one to get a feeling for what it might look like. In an appendix we provide tables of all the finite subgroups of U(3) that have a faithful three-dimensional irrep and are not isomorphic to the direct product of a smaller group and a cyclic group. We order the groups according to their SmallGroups classification, viz. in increasing order first of o and of j in their [o, j] identifiers
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