(Solved): 1 G=(A,B,C|A²,B4, C2, BCB = C, AB=BA, AC = CA) To ensure that you are working with correct informa ...

1 G=(A,B,C|A²,B4, C2, BCB = C, AB=BA, AC = CA) To ensure that you are working with correct information about this group, a correct link table is given below: Number Element A B ? 2 5 2 9 2 B 6 3 10 3 B2 7 4 11 4 8 12 5 ? 6 13 6 BA 2 7 14 7 B2A 3 8 15 8 BA 4 5 16 9 ? 13 12 1 10 BC 14 9 2 11 BPC 15 10 3 12 B?C 16 11 4 13 AC 9 16 5 14 BAC 10 13 6 15 B2C 11 14 7 16 BAC 12 15 8 You are to find out the following information about G. Provide your solutions first, and attach any relevant working afterwards. (a) A table listing the order, the square and the inverse of each element. (b) A list of each of the subgroups of G, indicating whether the subgroup is abelian, cyclic or normal; and a diagram showing the inclusions between these subgroups. (c) A table listing the centraliser subgroup of eah element. (d) For each non-trivial normal subgroup N, the group table for the quotient group G/N. (e) A list of the conjugacy classes of G. (f) The centre Z(G) and the centre Z(G/Z(G)). (g) The character table for G, with a clear explanation of how you found it.