A new set of generators and a physical interpretation for the SU (3) finite subgroup D(9,1,1;2,1,1)

After 100 years of effort, the classification of all the finite subgroups of $SU(3)$ is yet incomplete. The most recently updated list can be found in Ludl (J Phys A Math Theory 44:255204, 2011), where the structure of the series $(C)$ and $(D)$ of $SU(3)$ -subgroups is studied. We provide a minimal set of generators for one of these groups which has order $162$ . These generators appear up to phase as the image of an irreducible unitary braid group representation issued from the Jones–Kauffman version of $SU(2)$ Chern–Simons theory at level $4$ . In light of these new generators, we study the structure of the group in detail and recover the fact that it is isomorphic to the semidirect product $\mathbb Z _9\times \mathbb Z _3\rtimes S_3$ with respect to conjugation.