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GroupTheory

 CharacterTable
 construct the character table of a finite group

 Calling Sequence CharacterTable( G )

Parameters

 G - finite group

Description

 • 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.)

Methods

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

 ClassNumber( ct ) returns the class number of the group for which this is a character table ClassNames( ct ) returns the (ordered) list of conjugacy class names ClassRepresentatives( ct ) returns the (ordered) list of conjugacy class representatives CharacterDegrees( ct ) returns a sorted list of pairs [d,m] of the degrees d of the irreducible characters and their respective multiplicities m Exponent( ct ) returns the exponent of the group GetMatrix( ct ) returns the matrix of character values PowerMap( ct, p ) returns the $p$th power map Display( ct, opts ) prints the character table in a more readable format

 • 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 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.

Examples

 > $\mathrm{with}\left(\mathrm{GroupTheory}\right):$
 > $G≔\mathrm{Symm}\left(3\right)$
 ${\mathrm{GroupTheory}}{:-}{\mathrm{SymmetricGroup}}{}\left({3}\right)$ (1)
 > $\mathrm{ct}≔\mathrm{CharacterTable}\left(G\right)$
 ${\mathrm{GroupTheory}}{:-}{\mathrm{CharacterTable}}{}\left({\mathbf{module}}\left({}\right)\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}{...}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}{\mathbf{end module}}\right)$ (2)
 > $\mathrm{Display}\left(\mathrm{ct}\right)$

 C 1a 2a 3a |C| 1 3 2 $\mathrm{χ__1}$ $1$ $1$ $1$ $\mathrm{χ__2}$ $1$ $-1$ $1$ $\mathrm{χ__3}$ $2$ $0$ $-1$

 > $\mathrm{ClassNumber}\left(\mathrm{ct}\right)$
 ${3}$ (3)
 > $\mathrm{ClassNames}\left(\mathrm{ct}\right)$
 $\left[{"1a"}{,}{"2a"}{,}{"3a"}\right]$ (4)
 > $\mathrm{ct}≔\mathrm{CharacterTable}\left(\mathrm{CyclicGroup}\left(4\right)\right)$
 ${\mathrm{GroupTheory}}{:-}{\mathrm{CharacterTable}}{}\left({\mathbf{module}}\left({}\right)\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}{...}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}{\mathbf{end module}}\right)$ (5)
 > $\mathrm{Display}\left(\mathrm{ct}\right)$

 C 1a 2a 4a 4b |C| 1 1 1 1 $\mathrm{χ__1}$ $1$ $1$ $1$ $1$ $\mathrm{χ__2}$ $1$ $-1$ $I$ $-I$ $\mathrm{χ__3}$ $1$ $-1$ $-I$ $I$ $\mathrm{χ__4}$ $1$ $1$ $-1$ $-1$

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

 C 1a 2a 3a 3b |C| 1 3 4 4 ${\mathrm{\nu }}_{2}$ 1 $\mathrm{χ__1}$ $1$ $1$ $1$ $1$ 0 $\mathrm{χ__2}$ $1$ $1$ $-\frac{1}{2}-\frac{I\sqrt{3}}{2}$ $-\frac{1}{2}+\frac{I\sqrt{3}}{2}$ 0 $\mathrm{χ__3}$ $1$ $1$ $-\frac{1}{2}+\frac{I\sqrt{3}}{2}$ $-\frac{1}{2}-\frac{I\sqrt{3}}{2}$ 1 $\mathrm{χ__4}$ $3$ $-1$ $0$ $0$

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

 C 1a 2a 3a 3b |C| 1 3 4 4 ${C}^{2}$ 1a 1a 3b 3a ${C}^{3}$ 1a 2a 1a 1a $\mathrm{χ__1}$ $1$ $1$ $1$ $1$ $\mathrm{χ__2}$ $1$ $1$ $-\frac{1}{2}-\frac{I\sqrt{3}}{2}$ $-\frac{1}{2}+\frac{I\sqrt{3}}{2}$ $\mathrm{χ__3}$ $1$ $1$ $-\frac{1}{2}+\frac{I\sqrt{3}}{2}$ $-\frac{1}{2}-\frac{I\sqrt{3}}{2}$ $\mathrm{χ__4}$ $3$ $-1$ $0$ $0$

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

 C 1a 2a 3a 4a 7a 7b |C| 1 21 56 42 24 24 $\mathrm{χ__1}$ $1$ $1$ $1$ $1$ $1$ $1$ $\mathrm{χ__2}$ $3$ $-1$ $0$ $1$ $-{\left(-1\right)}^{4}{7}}-{\left(-1\right)}^{2}{7}}+{\left(-1\right)}^{1}{7}}-1$ ${\left(-1\right)}^{2}{7}}+{\left(-1\right)}^{4}{7}}-{\left(-1\right)}^{1}{7}}$ $\mathrm{χ__3}$ $3$ $-1$ $0$ $1$ ${\left(-1\right)}^{2}{7}}+{\left(-1\right)}^{4}{7}}-{\left(-1\right)}^{1}{7}}$ $-{\left(-1\right)}^{4}{7}}-{\left(-1\right)}^{2}{7}}+{\left(-1\right)}^{1}{7}}-1$ $\mathrm{χ__4}$ $6$ $2$ $0$ $0$ $-1$ $-1$ $\mathrm{χ__5}$ $7$ $-1$ $1$ $-1$ $0$ $0$ $\mathrm{χ__6}$ $8$ $0$ $-1$ $0$ $1$ $1$

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

The following command indicates that the dihedral group of degree $12$ has four linear characters (of degree $1$) and $5$ irreducible characters of degree $2$.

 > $\mathrm{CharacterDegrees}\left(\mathrm{ct}\right)$
 $\left[\left[{1}{,}{4}\right]{,}\left[{2}{,}{5}\right]\right]$ (6)
 > $\mathrm{pmap}≔\mathrm{PowerMap}\left(\mathrm{ct},2\right)$
 ${\mathrm{pmap}}{≔}\left[{1}{,}{1}{,}{1}{,}{1}{,}{5}{,}{2}{,}{5}{,}{7}{,}{7}\right]$ (7)
 > $\mathrm{cc}≔\mathrm{ConjugacyClasses}\left(\mathrm{ct}\right):$
 > $g≔\mathrm{Representative}\left(\mathrm{cc}\left[7\right]\right)$
 ${\mathrm{Perm}}{}\left(\left[\left[{1}{,}{3}{,}{5}{,}{7}{,}{9}{,}{11}\right]{,}\left[{2}{,}{4}{,}{6}{,}{8}{,}{10}{,}{12}\right]\right]\right)$ (8)
 > $\mathrm{evalb}\left({g}^{2}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathbf{in}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{cc}\left[5\right]\right)$
 ${\mathrm{true}}$ (9)

It may be necessary to set the value of the rtablesize interface variable to see the character values for larger character tables.

 > $\mathrm{ct}≔\mathrm{CharacterTable}\left(\mathrm{ElementaryGroup}\left(2,5\right)\right)$
 ${\mathrm{GroupTheory}}{:-}{\mathrm{CharacterTable}}{}\left({\mathbf{module}}\left({}\right)\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}{...}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}{\mathbf{end module}}\right)$ (10)
 > $\mathrm{interface}\left('\mathrm{rtablesize}'=35\right):$
 > $\mathrm{ct}$
 ${\mathrm{GroupTheory}}{:-}{\mathrm{CharacterTable}}{}\left({\mathbf{module}}\left({}\right)\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}{...}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}{\mathbf{end module}}\right)$ (11)
 > 

Compatibility

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