Axial isotropy types of the full cube symmetry group ($D_4,C_2\times C_2,S_3$)

The full symmetry group of a cube acts as all signed permutation matrices. The stabilizer subgroups of $(1,0,0)$, $(1,1,0)$ and $(1,1,1)$ have orders eight, four and six, respectively, and their fixed-point subspaces of a group action are the lines through those vectors. The three group orbits have sizes six, twelve and eight. They therefore give distinct axial types for the equivariant branching lemma. More generally the fixed-space dimension of a vector stabilizer subgroup equals the number of distinct nonzero absolute coordinate values, so these are precisely its one-dimensional types.

// Destination: dynamical-systems.bigb