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!).