I see constantly written that the gauge group of the SM is SU(3)XSU(2)XU(1), but what operation is that represented by X? Given that all the 3 terms are Lie groups, I think that is possible that is Group Direct Product?

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Yes, it is a direct product. Basically all particles must be representations of these symmetry groups, but the symmetry groups are essentially unconnected. For example, you can perform a SU(3) transformation on the system and get the same physics back, without needing to do anything with the SU(2) and U(1).

 

Technically, saying it is SU(3)xSU(2)xU(1) is true only above energies of about 174GeV. Below this scale the Higgs mechanism breaks the electroweak part of the symmetry down to U(1), ie. SU(2)xU(1) -> U(1), where the last U(1) which is a result of the breaking is electromagnetism, and is different from the U(1) in 'SU(2)xU(1)'.