One of the restrictions used in all of the works done on phylogenetic invariants for group based models has been that the group be abelian. In my thesis, I aim to generalize the method of invariants for group-based models of DNA sequence evolution to include nonabelian groups. By using a nonabelian group to act one the nucleotides, one could capture the structure of the symmetric model for DNA sequence evolution. If successful, this line of research would unify the two separated strands of active research in the area today: Allman and Rhodes’s invariants for the symmetric model and Strumfels and Sullivant’s toric ideals of phylogenetic invariants. Furthermore, I want to look at the statistical properties of polynomial invariants to get a better understanding of how they behave when used with real, “noisy” data.