#### GAP QA #11

# Classes for a character table

**Keywords:**
Character table, Conjugacy Classes, Symmetric group
## Respondent:

Max Neunhöffer (
Max.Neunhoeffer@Math.RWTH-Aachen.DE)

Alexander Hulpke (
hulpke@math.colostate.edu)
## Question:

Suppose I have the character table fo Sym4 generated by
gap> tbl:=CharacterTable("Symmetric",4);

How can I get the conjugacy classes from this table ( in the right

order) (I don't want to have to compute them again using

ConjugacyClasses(Sym4))?
## Answer:

The above table has an Attribute "ClassParameters":
gap> ClassParameters(tbl);
[ [ 1, [ 1, 1, 1, 1 ] ], [ 1, [ 2, 1, 1 ] ], [ 1, [ 2, 2 ] ], [ 1, [ 3, 1 ] ],
[ 1, [ 4 ] ] ]

The partitions in the second components describe the mapping to the

conjugacy classes in the symmetric group, i.e. the third class ([2,2]) is

the class of (1,2)(3,4)

Note that for character tables of other groups than symmetric groups

"ClassParameters" do not exist. The following method works in general:

Create the table from the group:

gap> Sym4 := SymmetricGroup(4);;
gap> tbl := CharacterTable(Sym4);;
gap> Irr(tbl);;

For symmetric groups this essentially only is a lookup of

known information and does not take significantly longer than

<tt>CharacterTable("Symmetric",4);</tt> would, for other groups it will

actually calculate the character table from scratch. However in many cases

this will be still faster than trying to identify classes by hand.

You can now get the conjugacy classes in the right order from the character

table:

gap> ConjugacyClasses(tbl);
[ ()^G, (1,2)^G, (1,2)(3,4)^G, (1,2,3)^G, (1,2,3,4)^G ]

Though it is the case in this example, it is not guaranteed in general

that the arrangement of classes in the table is the same as in the group.

You can get the identification by
gap> IdentificationOfConjugacyClasses(tbl);
[ 1 .. 5 ]

which in this case guarantees that the order of the (already computed) conjugacy

classes in the object Sym4 is the same as the order of the classes

in the character table object tbl.

This approach will also work for other groups than symmetric groups,

provided that GAP is able to calculate the character table.

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