[Refer to Figure 57.2 for this problem. It is part of Example 57.3, which involves other

Chapter 8, Problem 8.17

(choose chapter or problem)

[Refer to Figure 57.2 for this problem. It is part of Example 57.3, which involves other ideasthat may be ignored here.] The group G of all rotations of a cube has order 24. The elements ofG are of five kinds, and are listed in Example 57.3. Each element of G corresponds to a uniquepermutation of the vertices of the cube. For example, rotation of 1800 about the segment ijcorresponds to (ah)(de)(bg)(cf).(a) Find the permutation of the vertices corresponding to each of the six 1800 rotations aboutlines joining midpoints of opposite edges, such as kl.(b) Find the permutation of the vertices corresponding to each of the eight 1200 rotations aboutlines joining opposite vertices, such as ag.(c) Show that G has a subgroup of order 12.