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Spectral Measures for $G_2$ II: finite subgroups

Joint spectral measures associated to the rank two Lie group $G_2$, including the representation graphs for the irreducible representations of $G_2$ and its maximal torus, nimrep graphs associated to the $G_2$ modular invariants have been studied. In this paper we study the joint spectral measures for the McKay graphs (or representation graphs) of finite subgroups of $G_2$. Using character theoretic methods we classify all non-conjugate embeddings of each subgroup into the fundamental representation of $G_2$ and present their McKay graphs, some of which are new.