= Axial isotropy types of the full cube symmetry group
{title2=$D_4,\ C_2\times C_2,\ S_3$}
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
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