The goal of this thesis is to investigate a conjecture about Mirror Symmetry for Landau Ginzburg (LG) models with non-abelian gauge groups. The conjecture predicts that the LG A-model for a polynomial-group pair $(W,G)$ is equivalent to the LG B-model for the dual pair $(W^*, G^*)$. In particular, the A-model and B-model include the construction of a Frobenius algebra. The LG mirror symmetry conjecture predicts that the A-model Frobenius algebra for $(W,G)$ will be isomorphic to the B-model Frobenius algebra for the dual pair $(W^*,G^*)$. Part of the conjecture includes a rule describing how to construct the dual pair. Until now, no examples of this phenomenon have been verified. In this thesis we will verify the conjecture for the polynomial $W(x_1,x_2,x_3,x_4) = x_1^4+x_2^4+x_3^4+x_4^4$ with a maximal admissible non-abelian group. I present a supplementary guide along with a worked example to compute the state spaces of each of the A and B models with non-abelian groups. This includes formalizing G-actions to take invariants, computing each state space, formalizing the product on each state space, and as the main result, showing there indeed exists an isomorphism of Graded Frobenius Algebras between the LG A-model and dual LG B-model.