Presentations of Cluster Modular Groups and Generation by Cluster Dehn Twists

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