There are thirteen connected Lie groups' with the same Lie algebra as $SU(3)\times SU(2) \times U(1)$. [...] We can eliminate choices 1—4 by demanding that the gauge group be compact. Choices 9—13 may be removed from consideration by using the fact that the true non Abelian group of the quarks and leptons in the standard model is $SU(3) \times SU(2)$, since color triplet and weak doublet representations exist in nature.Note that the simply connected universal covering group of all 13 groups above is $SU(3) \times SU(2) \times R$, while $SU(3) \times SU(2) \times U(1)$ is the covering group for groups 5—13. As discussed previously by O'Raifeartaigh, there then remain four possible true symmetry groups for the standard model: $$SU(3) \times SU(2) \times U(1)$$ $$ SU(3) \times U(2) $$ $$ U(3) \times SU(2) $$ $$ S(U(3) \times U(2)) \, . $$ [..] It should be understood from the beginning that the groups $U(3) \times SU(2)$, $SU(3) \times U(2)$, and $S(U(3) \times U(2) )$ yield the same perturbative quantum field theory as $SU(3) \times SU(2) \times U(1)$, since perturbative effects depend only on the Lie algebra. Whether or not the four groups lead to different nonperturbative effects is to the author's knowledge still an open question. [[http://journals.aps.org/prd/pdf/10.1103/PhysRevD.43.2709|Global structure of the standard model, anomalies, and charge quantization]] by Joseph Hucks

What people usually call the gauge group of the Standard Model: SU(3) × SU(2) × U(1) actually has a bit of flab in it: there's a normal subgroup that acts trivially on all known particles. This subgroup is isomorphic to Z/6. If we mod out by this, we get the "true" gauge group of the Standard Model: G = (SU(3) × SU(2) × U(1))/(Z/6) And, this turns out to have a neat description. It's isomorphic to the subgroup of SU(5) consisting of matrices like this: (g 0) (0 h) where g is a 3×3 block and h is a 2×2 block. For obvious reasons, I call this group S(U(3) × U(2)) If you want some intuition for this, think of the 3×3 block as related to the strong force, and the 2×2 block as related to the electroweak force. A 3×3 matrix can mix up the 3 "colors" that quarks come in - red, green, and blue - and that's what the strong force is all about. Similarly, a 2×2 matrix can mix up the 2 "isospins" that quarks and leptons come in - up and down - and that's part of what the electroweak force is about. [...] l assume that one way or another, you're happy with the idea of S(U(3) × U(2)) as the true gauge group of the Standard Model! Maybe you understand it, maybe you're just willing to nod your head and accept it. http://math.ucr.edu/home/baez/week253.html

r. Correlation functions of local operators in R1,3 depend only on the Lie algebra of the gauge group and are unaffected by global issues such as the choice of Γ. This means that no current experiment can distinguish between the four possibilities. Nonetheless, the physics in flat space can depend in subtle ways on Γ (and in more dramatic ways when spacetime has interesting topology). The purpose of this paper is to describe the crudest differences between the theories: the spectrum of line operators and the periodicities of theta angles.https://arxiv.org/pdf/1705.01853.pdf

As a further example, consider the standard electroweak theory. In this case, the gauge group was identified in Section 1.4 as $U(2)$ or $[SU(2)\times U(1)]/Z_2$, by which we meant that $U(2)$ can be obtained from $SU(2) \times U(1)$ by identifying certain pairs of elements as explained in the paragraph after 1.4.9. The group $SU(2) \times U(1)$ itself consists of couples of elements from respectively the groups $SU(2)$ and $U(1)$, which we mai represent symbolically as points inside the rectangle in Figure 2.8 where the vertical axis represents the group $SU(2)$ and the horizontal axis $U(1)$, and the parallel edges of the rectangle are understood to be identified so as to make the rectangle into a "hyper torus". page 26 in Some Elementary Gauge Theory Concepts by Hong-Mo Chan, Sheung Tsun Tsou

There are thirteen connected Lie groups' with the same Lie algebra as $SU(3)\times SU(2) \times U(1)$. [...] We can eliminate choices 1—4 by demanding that the gauge group be compact. Choices 9—13 may be removed from consideration by using the fact that the true non Abelian group of the quarks and leptons in the standard model is $SU(3) \times SU(2)$, since color triplet and weak doublet representations exist in nature.Note that the simply connected universal covering group of all 13 groups above is $SU(3) \times SU(2) \times R$, while $SU(3) \times SU(2) \times U(1)$ is the covering group for groups 5—13. As discussed previously by O'Raifeartaigh, there then remain four possible true symmetry groups for the standard model: $$SU(3) \times SU(2) \times U(1)$$ $$ SU(3) \times U(2) $$ $$ U(3) \times SU(2) $$ $$ S(U(3) \times U(2)) \, . $$ [..] It should be understood from the beginning that the groups $U(3) \times SU(2)$, $SU(3) \times U(2)$, and $S(U(3) \times U(2) )$ yield the same perturbative quantum field theory as $SU(3) \times SU(2) \times U(1)$, since perturbative effects depend only on the Lie algebra. Whether or not the four groups lead to different nonperturbative effects is to the author's knowledge still an open question. [[http://journals.aps.org/prd/pdf/10.1103/PhysRevD.43.2709|Global structure of the standard model, anomalies, and charge quantization]] by Joseph Hucks

The Lie algebra of the Standard Model is the Lie algebra of SU(3)xSU(2)xU(1). But, the Lie group corresponding to this Lie algebra is not unique. The true Lie group of the Standard Model can be determined by requiring that the corresponding group representations are faithful. A mathematical discussion of this point can be found in John C. Baez, The True Internal Symmetry Group of the Standard Model. Given the particle content of the the Standard Model, the Lie group of the Standard Model is in fact SU(3)xSU(2)xU(1)/Z6. Details can be found in: B.L.G. Bakker, A.I. Veselov and M.A. Aubkov, Phys. Lett. B 583 (2004) 379. [[[http://www.sciencedirect.com/science/article/pii/S0370269304000759|HTML]]] M.A. Zubkov, Phys. Lett. B 649 (2007) 91. [[[http://www.sciencedirect.com/science/article/pii/S0370269307004303|HTML]]] http://scipp.ucsc.edu/~haber/ph218/