# Quark model extension to all six flavors

Gell-Mann's $SU(3)$ quark model is extremely successful at describing the bound states of the three light quarks $u,d,s$. The bound states fall neatly into the irreducible representations of $\mathfrak{su}(3)$. With the recent discovery of the doubly charmed baryon $\Xi_{cc}^{++}$ I have been thinking about how this "eightfold way" may be extended to include all six flavors. Is it as simple as extending the flavor symmetry group to $SU(6)$? I am a little rusty on my group theory, but if I remember correctly $SU(3)$ is not isomorphic to a subgroup of $SU(6)$. So how could this extension preserve the highly successful theory of $SU(3)$ for the lighter quarks?