The gauge group of the strong and electroweak interactions in Nature could be any of the four that share the same Lie algebra: $\frac{SU(3)_c \times SU(2)_L \times U(1)_Y} {Z_p} \equiv G_p$, $p=6,3,2,1$, with each group leading to the same perturbative dynamics. However, their global structure does differ and alters the representation theory for each group. Each of these cases allows in its spectrum for the matter fields of the SM but also for new distinctive representations. For instance, under the assumption that $q_L$ possesses the minimum possible hypercharge in Nature, $G_p$ allows for particles with a multiple of $p e/6$ for electric charge. In this talk, we will discuss the possibility that new discoveries in the particle spectrum could be used to tell the SM gauge group apart.