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Role of the SU($2$) and SU($3$) subgroups in observing confinement in the G($2$) gauge group

Applying the domain model to the G($2$) gauge group and the thick center vortex model to the SU($2$) and SU($3$) subgroups of G($2$), we calculate the potentials between static sources, as well as some parameters of the vortex profile. Comparing the results obtained from G($2$) and its subgroups, we argue that SU($2$) and SU($3$) gauge groups have important roles in observing confinement in the G($2$) gauge group.