The Matrix Quaternion Group of Rotational Symmetries in the Genetic Code

Herein, a matrix representation of the Hamilton quaternion group by 4 × 4 square matrices with entries equal to −1, 0, or 1 is defined. It is proven that this group, denoted as QM,∘, is a group of rotational symmetries of the four-dimensional hypercube ℤ24, that is, a subgroup of the special orthogonal group SO4ℤ. As a consequence, QM,∘ is a group of rotational symmetries for each of the biological hypercubes RNY, YNY, YNR, and RNR. It is also proven that QM,∘ is a group of permutations of the eight cubes contained in the four-dimensional hypercube ℤ24. The latter is a novel result. It is also proven that the matrix quaternion group QM,∘ is a normal subgroup of SO4ℤ and that the latter is a semidirect product of QM,∘ with a copy of the special orthogonal group SO3ℤ, also called an octahedral group because it is a group of rotational symmetries of a regular octahedron or of a three-dimensional cube.