Applications of a unique group embedding of internal symmetry groups andSU(2): spin-extensions for G2 and for the Sp(6)-Trion model

A recently proposed method of embeddingSU (2) and an internal symmetry groupG into a bigger groupG is applied to construct a spin extension ofG ₂ andSp (6). BecauseG ₂ andSp (6) possess a generalized quark model the embedding group can be proved to be unique and to be given bySp (14) resp.O (12) forG ₂ resp.Sp (6). For a particle classification splittings are calculated and tabulated forSp (14) ↓G ₂ xSU (2) andO (12) ↓Sp (6) xSU (2). The identification of low dimensional irreducible representations ofO (12) is quite satisfactory whereas an unreasonable number of unobserved particles are needed to fill up the supermultiplets of the spin extensionSp (14) of G₂.