Q 2. Consider the octahedron with its six vertices (i.e. corners) labelled as shown. In this question we will consider the symmetry group G of rotations of the octahedron. We will not consider reflections. The rotational 3D symmetries correspond to permutations of the vertices, and can be represented as elements of S6.

(a) Write down an example of a rotation ρ that fixes vertex 3, as a permutation written in cycle notation. Do not choose ρ = id.

(b) Write down an example of a rotation µ that does not fix any vertex, as a permutation written in cycle notation.

(c) Write down the orbit of vertex 3 under the action of G.

(d) Write down the stabiliser of vertex 3 under the action of G.

(e) Calculate the order of G.

(f) Show that G is non-Abelian.

(g) Find a subgroup of G of order 6. List the permutations in the subgroup in cycle notation and determine whether it is isomorphic to Z6 or S3.