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We give a method to compute presentations of saturated cluster modulargroups. Using this, we obtain finite presentations of the saturated clustermodular groups of finite mutation type $X_6$ and $X_7$. We verify that thecluster modular groups of finite mutation type $\widetilde{E}_6$,$\widetilde{E}_7$, $\widetilde{E}_8$, $G_2^{(*,*)}$, $X_6$ and $X_7$ arevirtually generated by cluster Dehn twists.

Presentations of Cluster Modular Groups and Generation by Cluster Dehn Twists