Thread: How do you create character tables for groups S3 and S4?

Hi

I'm studying character tables in my representation theory module and don't understand how to create them for the cyclic groups S3 or S4. COuld anyone help?

I understand:
- that the set of each type of cycle forms a conjugacy class
eg. for S3: {1-cycles in S3}, {2-cycles in S3}, {3-cycles in S3} these sets are conjugacy classes

- the size of the centraliser for each conjugacy class is the size of the group divided by the size of the conjugacy class:
eg. |C(g)| = |G|/|x^G| for all g in G (x^G is the conjugacy class for x in G)

- the square of the entries in each column of the character table, sum to the value of the centraliser for the conjugacy class

- the cols and rows of the table are normal to each other

- the number of characters in the table is equal to the number of conjugacy classes

- for the linear character of degree 1, X1, the character of each conjugacy class is 1, so the top row is all ones in the table.

For S3, I get as far as:

conjugacy classes: | 1-cycles | 2-cycles | 3-cycles
|centraliser| | 6 | 2 | 3
X1 | 1 | 1 | 1
X2 | | |
X3 | | |

(This is meant to be a table...Sorry. Rows 2 and 3 are empty)

I can see that for the 1-cycle that a +/-2 and +/-1 is missing.
Also the other entries in the column of the 3-cycles should be +/-1 and +/-1.
The entries in the column of the 2-cycle should be 0 and +/-1.

How do I know which way round to place them? Is there a trick to it? Is there a way to verify the numbers in the rows?

For S4 I got as far as:

conjugacy classes: | 1-cycles | 2-cycles | 3-cycles | 4-cycles | 2-cycle,2-cycle
|centraliser| | 24 | 4 | 3 | 4 | 8
X1 | 1 | 1 | 1 | 1 | 1
X2 | | | | |
X3 | | | | |
X4 | | | | |
X5 | | | | |

Thanks for your help.