Past exam of the mathematics course of the University of Cambridge / 2024 / ia / Paper 3 / 8D / b / i / Solution

The group acts transitively on the eight vertices. At a chosen vertex, an isometry fixing it may permute its three incident edges arbitrarily, and each of the six permutations is realized by an isometry. Thus the vertex stabilizer is isomorphic to $S_3$ and has order six. Orbit-stabilizer givesThis begins the three stabilizer descriptions in the symmetry group of a cube.