Group graph of the genetic code

The genetic code doublets can be divided into two octets of completely degenerate and ambiguous coding dinucleotides. These two octets have the algebraic property of lying on continuously connected planes on the group graph (a tesseract) of the Cartesian product of two Klein 4-groups of nucleotide exchange operators. The K X K group can also be broken into four cosets, one of which has completely degenerate coding elements, and another that has completely ambiguous coding elements. The two octets of coding doublets have the further algebraic property that the product of their internal exchange operators naturally divide into two exactly equivalent sets. These properties of the genetic code are relevant to unraveling error-detecting and error-correcting (proof-reading) aspects of the genetic code and may be helpful in understanding the context-sensitive grammar of genetic language.