No, the gauge group is independent of flavour - the quotient comes from the fact that the quantum numbers of the SM particles match up nicely so that transforming the fields by the $\mathbb Z_2\times\mathbb Z_3$ centre of $\mathrm{SU}(2)\times\mathrm{SU}(3)$ can always be undone with an appropriate $\mathrm U(1)$ rotation. So there's a $\mathbb Z_6$ invariance here, but nothing to suggest that we should quotient out by it or one of its subgroups.
This is because correlators of local operators in SU(N) gauge theory remain the same when you mod by a central discrete factor - you have to use …