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The effect of $SU(4)$ group factor in the thick center vortex potentials

The group factor for the higher representations of the $SU(4)$ gauge group is investigated. For the zero N-ality representations, the minimum values of the real part of the group factor which appear at large distances, are located at the points where $50%$ of the maximum vortex flux is in the Wilson loop. This amount is equal to $60%$ for the representations which belong to N-ality=$1$ class except the fundamental representation. The value of the real part of the group factor at the minimum points is equal to the value of center of the group factor at intermediate distances. The appearance of these points is due to the difference between the asymptotic and intermediate string tensions. The group factor passes the phase factor to create the larger intermediate string tension than the asymptotic one.