The CharacterTable( G ) command computes the (ordinary) character table of the finite group G, which must be either a permutation group or a Cayley table group.

The CharacterTable command uses an implementation of the Burnside-Dixon-Schneider algorithm to compute the character table.

Character tables are printed using a Matrix so their display is sensitive to the value of the rtablesize interface option.  (See the Examples section.)

Character tables support the following methods, in which ct denotes a character table returned by the CharacterTable command.

The ClassNumber( ct ) command returns the class number of the underlying group of the character table ct, which is the number of conjugacy classes of the group. In the case when the group is Abelian, this is simply the order of the group but, for non-Abelian groups, the number of conjugacy classes is less than the order of the group.

The ConjugateRank( ct ) command returns the conjugate rank of the underlying group of the character table ct. This is the number of distinct non-trivial class lengths (those other than $1$).

The ClassNames( ct ) command returns an ordered list of the names given to the conjugacy classes.  These have the form of an integer, indicating the order of the class representative (and hence, of all of its members) followed by one or more letters used to distinguish classes whose elements have the same order.  For example, the class names "4a" and "4b" distinguish two distinct conjugacy classes of elements of order $4$.

The ClassRepresentatives( ct ) command returns an Array containing the (ordered) class representatives for the conjugacy classes of the underlying group of the character table ct.

The CharacterDegrees( ct ) command returns a sorted list of pairs [d, m] of the degrees d of the irreducible characters and their respective multiplicities m.

The Exponent( ct ) command returns the exponent of the underlying group, using data stored in the character table.

The GetMatrix( ct ) command returns the character values of the character table ct in a Matrix.

For a prime number p, the PowerMap( ct, p ) command returns the $p$th power map of the character table ct.  This is an Array a with $k$ elements, where $k$ is the class number of the underlying group of ct, such that the class of the $p$th power of an element of the $i$th class is in the class a[ i ], for i = 1 .. k.

The Display( ct, opts ) command prints the character table in a nicer format, and optionally allows you to include additional information, such as the Frobenius-Schur indicator values of the characters (using the showindicator option) and power maps (using the showpowermaps option) for the prime divisors of the group order.

$\mathrm{with}\left(\mathrm{GroupTheory}\right)\:$

$G≔\mathrm{Symm}\left(3\right)$

$G≔{\mathbf{S}}_3$

$\mathrm{ct}≔\mathrm{CharacterTable}\left(G\right)$

$\mathrm{ct}≔\left[\right]$

$\mathrm{Display}\left(\mathrm{ct}\right)$

$\mathrm{ClassNumber}\left(\mathrm{ct}\right)$

$3$

$\mathrm{ConjugateRank}\left(\mathrm{ct}\right)$

$2$

$\mathrm{ClassNames}\left(\mathrm{ct}\right)$

$\left[''1a''\,''2a''\,''3a''\right]$

$\mathrm{ct}≔\mathrm{CharacterTable}\left(\mathrm{CyclicGroup}\left(4\right)\right)$

$\mathrm{ct}≔\left[\right]$

$\mathrm{Display}\left(\mathrm{ct}\right)$

$\mathrm{Display}\left(\mathrm{CharacterTable}\left(\mathrm{Alt}\left(4\right)\right)\,'\mathrm{showindicator}'\right)$

$\mathrm{Display}\left(\mathrm{CharacterTable}\left(\mathrm{Alt}\left(4\right)\right)\,'\mathrm{showpowermaps}'\right)$

$\mathrm{Display}\left(\mathrm{CharacterTable}\left(\mathrm{PSL}\left(2\,7\right)\right)\right)$

$\mathrm{ct}≔\mathrm{CharacterTable}\left(\mathrm{DihedralGroup}\left(12\right)\right)\:$

$\mathrm{CharacterDegrees}\left(\mathrm{ct}\right)$

$\left[\left[1\,4\right]\,\left[2\,5\right]\right]$

$\mathrm{pmap}≔\mathrm{PowerMap}\left(\mathrm{ct}\,2\right)$

$\mathrm{pmap}≔\left[1\,1\,1\,1\,5\,2\,5\,7\,7\right]$

$\mathrm{cc}≔\mathrm{ConjugacyClasses}\left(\mathrm{ct}\right)\:$

$g≔\mathrm{Representative}\left(\mathrm{cc}\left[7\right]\right)$

$g≔\left(1\,3\,5\,7\,9\,11\right)\left(2\,4\,6\,8\,10\,12\right)$

$\mathrm{evalb}\left(g^2\mathbf{in}\mathrm{cc}\left[5\right]\right)$

$\mathrm{true}$

$\mathrm{ct}≔\mathrm{CharacterTable}\left(\mathrm{ElementaryGroup}\left(2\,5\right)\right)$

$\mathrm{interface}\left('\mathrm{rtablesize}'=35\right)\:$

$\mathrm{ct}$

The GroupTheory[CharacterTable] command was introduced in Maple 2017.

For more information on Maple 2017 changes, see Updates in Maple 2017.