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It is known that the Lie algebra of the SM is $$ \mathfrak{su}(3)\oplus \mathfrak{su}(2)\oplus \mathbb{R}\,, $$ so that the Lie group is $$ G_{\text{SM}} = \dfrac{SU(3)\times SU(2) \times U(1)}{\Gamma}\,, $$ where $\Gamma\in\{\mathbb{Z}_6, \mathbb{Z}_3, \mathbb{Z}_2, \mathbb{1}\}$. The reason why these elections of $\Gamma$ are allowed is because such redefinitions of the fields leave them invariant (see this SE post).

Now, my question is, does the election of the $\Gamma$ have any effect on the Physics? In other words, is the $\Gamma$ measurable? Of course, these must be non-local effects (if they exist!).

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The gauge group of the Standard Model is the quotient by $\mathbb Z_6$. Larger groups may be physically relevant, but it would take the discovery of new particles not in representations of the Standard Model group to settle the question.

We were in an analogous situation in the past when only electromagnetic $U(1)$ was known and all known particles had charges that were multiples of the electron charge. Then it was possible that the correct gauge group was the $U(1)$ whose base charge was the electron charge, or that that was the quotient of a larger $U(1)$ by $Z_n$ for any positive integer $n$, or even that the gauge group was $\mathbb R$. It turned out that $n=3$, but it took the discovery of quarks to show that (or at least to show that $n$ is a multiple of $3$). There is nothing you can do using only particles with a charge of $n$ times the fundamental charge to show that a smaller fundamental charge exists.

The possibilities for extending the Standard Model group aren't limited to larger subgroups of $SU(3)\times SU(2)\times U(1)$. I think that the most popular GUT groups contain only the quotient by $\mathbb Z_6$.

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