In this work, we investigate isomorphisms of graphs associated with the 216 maximal self-complementary \(C^3\) -codes over the genetic alphabet \(\{A,C,G,T\}\) . Such codes play an important role in maintaining the correct reading frame during the translational process in the ribosome and have been classified into 27 equivalence classes under the action of the dihedral group \(D_4\) . Naturally, this group action induces graph isomorphisms between the graphs associated with maximal self-complementary \(C^3\) -codes, as shown in Fimmel et al. (2016). However, we demonstrate here that these induced isomorphisms of the associated graphs are not the only graph isomorphisms between such codes. Specifically, we calculate the largely non-trivial automorphism groups of all the 216 graphs associated to maximal self-complementary \(C^3\) -codes and we show that no isomorphism exists between maximal self-complementary \(C^3\) -codes belonging to different equivalence classes. Finally, we provide examples illustrating that the assumptions of maximality, self-complementarity, or the \(C^3\) -property can not be omitted.
