= Axis stabilizer in the full octahedral symmetry group
{title2=$G_{\mathbb Re_3}\cong D_8\times C_2$}
An opposite-vertex line in the <full octahedral symmetry group> is an unoriented coordinate axis. Its <stabilizer subgroup> has the block form $\operatorname{diag}(A,\varepsilon)$, where $A$ is a signed two-dimensional permutation matrix and $\varepsilon=\pm1$. The eight possibilities for $A$ form the <dihedral group> of a square, of order eight; the independent axial sign gives a <cyclic group> of order two. Thus the stabilizer is their direct product, of order sixteen. It is larger than the stabilizer of an individual vertex, because exchanging the two endpoints still fixes the line.
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