# Character Ring Operations

## Related Structures

### `Parent(R): AlgChtr -> Pow`

### `Category(R): AlgChtr -> Cat`

### `NumberOfClasses(R): AlgChtr -> RngIntElt`

### `Nclasses(R): AlgChtr -> RngIntElt`

Given the ring $R$ of class functions, return the number of conjugacy classes of the finite group underlying $R$.

### `ClassesData(R): AlgChtr -> SeqEnum[Tup]`

Given the ring $R$ of class functions, return a sequence of pairs, one pair `<o,n>` for each conjugacy class of the underlying group, where $o$ is the order of the group elements in the class, and $n$ is the length of (number of group elements in) the class.

### `Group(R): AlgChtr -> Grp`

Given the ring $R$ of class functions on a finite group $G$, return $G$. This will cause an error if $R$ does not have a group attached. If unsure whether or not $R$ has a group attached, check the `Group` attribute of $R$ using `assigned R`Group`.

### `PowerMap(R): AlgChtr -> Map`

Given a character ring $R$, return the associated class power map. This will cause an error unless $R$ has a power map assigned, or has a group attached. The `PowerMap` is an attribute of $R$, as is `Group`, and its presence can be checked using `assigned R`PowerMap`.

If the power map of $R$ is not already assigned, and there is an assigned group, this function will compute the power map using operations within the group.

### `Kernel(x): AlgChtrElt -> Grp`

The kernel of the character $x$ of $G$, i.e. the normal subgroup of $G$ consisting of those elements $g$ for which $x(g) = x(1)$.

### `Centre(x): AlgChtrElt -> Grp`

The centre of the character $x$ of $G$, i.e. the subgroup of $G$ consisting of those classes $C$ of $G$ for which $|x(g)|$, $g$ in $C$, is equal to the degree of $x$.

### `CoefficientField(x): AlgChtrElt -> Rng`

The minimal cyclotomic field containing all values of the class function $x$.

### `CharacterField(x): AlgChtrElt -> Rng`

The subfield of the coefficient field of the class function $x$ that is generated by the values of $x$.

### `DegreeOfCharacterField(x): AlgChtrElt -> RngIntElt`

The degree of the character field of the class function $x$ as an extension of the rational numbers.