The remarkable attention which fair clustering has received in the last few years has resulted in a significant number of different notions of fairness. Despite the fact that these notions are well-justified, they are often motivated and studied in a disjoint manner where one fairness desideratum is considered exclusively in isolation from the others. This leaves the understanding of the relations between different fairness notions as an important open problem in fair clustering. In this paper, we take the first step in this direction. Specifically, we consider the two most prominent demographic representation fairness notions in clustering: (1) Group Fairness ($\textbf{GF}$), where the different demographic groups are supposed to have close to population-level representation in each cluster and (2) Diversity in Center Selection ($\textbf{DS}$), where the selected centers are supposed to have close to population-level representation of each group. We show that given a constant approximation algorithm for one constraint ($\textbf{GF}$ or $\textbf{DS}$ only) we can obtain a constant approximation solution that satisfies both constraints simultaneously. Interestingly, we prove that any given solution that satisfies the $\textbf{GF}$ constraint can always be post-processed at a bounded degradation to the clustering cost to additionally satisfy the $\textbf{DS}$ constraint while the same statement is not true given a solution that satisfies $\textbf{DS}$ instead. Furthermore, we show that both $\textbf{GF}$ and $\textbf{DS}$ are incompatible (having an empty feasibility set in the worst case) with a collection of other distance-based fairness notions. Finally, we carry experiments to validate our theoretical findings.