I don't know if this question is a duplicate, so I'll delete if is.
Well, I'm in the very beginning of the study of contemporary topics such as gauge theories, I would say that I'm in a "science communication + General Relativity mathematical/physical background" level.
So, in the book $[1]$, in chapter 1 introduction, the author wrote:
From a mathematical point of view, continuous symmetry groups can be conceptualised as Lie groups. By definition, Lie groups are groups in an algebraic sense which are at the same time smooth manifolds, so that both structures – algebraic and differentiable – are compatible. $\tag{1}$
and then he stated:
The currently accepted Standard Model of elementary particles, for instance, is a gauge theory with Lie group:
$$ SU(3) \times SU(2) \times U(1) \tag{2} $$
Now, in GR, the spacetime is a manifold and the points of spacetime are events. Intuitively, an event is a 4-dimensional "point" where a particle can be at.
Well, based on $(1)$, we can say that Lie groups are manifolds; based on $(2)$, we can say that $SU(3) \times SU(2) \times U(1)$ is a Lie group and therefore a manifold.
So, what is supposed to mean (physically) a point on the manifold $SU(3) \times SU(2) \times U(1)$?
$$ * * * $$
$[1]$ HAMILTON. J.D.M, Mathematical Gauge Theory With Applications to the Standard Model of Particle Physics, Springer, 2017.