# How the number of charges (colors) and the number of photons (gluons) is connected?

This question is a continuation of "Can a third type of electrical charge exist?" and specifically this comment.

I know the common knowledge that there is 1 kind of electric charge and thus 1 kind of photon; there are 2 kinds of weak isospin and thus 3 kinds of weak bosons; there are 3 kinds of color charge and thus 8 kinds of gluons. But I wonder if this is fixed.

How I understand this: if there are $n$ possible charges then there are $n$ identical fermion fields each carrying its own kind of charge. Then the gauge field acts on them changing their phases (only! to conserve the probability, probably). So its group should be some group acting on $\mathbb{C}^n$ and conserving the norm. Usually the maximal such group is considered, that is $\mathrm{U}(1)$, $\mathrm{SU}(2)$ and $\mathrm{SU}(3)$ respectively.

1 . May some subgroup of the maximal one be used instead? For example, $\mathrm{SO}(n)$ or $\mathrm{SU}(m)$ for $m<n$?

2 . May fermions make up not the fundamental representation of the gauge group but some other representation?

Next, I notice that the maximal group for several charges is $\mathrm{U}(n)$, but in weak and color interactions $\mathrm{SU}(n)$'s are used instead. $\mathrm{U}(n)=\mathrm{U}(1)\rtimes\mathrm{SU}(n)$, and as I understand, the common phase factor cannot belong to every interaction in the Standard Model, so it belongs only to the electromagnetic interaction, and other interactions lose it, and so their maximal groups are restricted to $\mathrm{SU}(n)$'s.
If yes, then in the case we imagine more kinds of electric charge, then the electromagnetic gauge group can be $\mathrm{U}(n)$ instead of $\mathrm{SU}(n)$, and the number of photons $n^2$ instead of $n^2-1$ respectively.