(25 pts) (a) (10 pts) Find the symmetric group G about the vertices 1, 2, 3, 4, 5, 6 of the regular hexagon (6 sided polygon) by listing its all members in cycle notations. (b) (5 pts) Find out the cycle index of the group G by (a). (c) (5 pts) Find the pattern inventory of the G-invariant vertex colorings of the hexagon by three colors Blue, Green and Red. (d) (5 pts) Given 10 distinct colors. Find the number of G-invariant vertex colorings of the hexagon by the 10 colors.