# Characters and Character Tables

Facilities for working with group characters can be found in Chapter [Characters of Finite Groups](../CharactersOfFiniteGroups/index-characters-of-finite-groups.md#chapchtr). In this section we repeat a small number of character intrinsics that are used frequently when computing with $K[G]$-modules.

## Ordinary Characters

In this section the various intrinsics for computing the table of absolutely irreducible complex characters for a finite group are described.

### `Character(M): ModGrp -> AlgChtrElt`

Given a $K[G]$-module $M$ where $K$ is the field of rationals or a number field, the character for $M$ over the field $K$ is returned.

### `CharacterTable(G): Chtr -> SeqEnum`

The table of irreducible complex characters for the group $G$ is constructed.

### `SymmetricCharacterTable(n): RngIntElt -> SeqEnum`

### `AlternatingCharacterTable(n): RngIntElt -> SeqEnum`

The table of irreducible complex characters for the symmetric (alternating) group of degree $n$ is constructed.

### `RationalCharacterTable(G): Chtr -> SeqEnum`

The table of irreducible rational characters for the group $G$ is constructed.

## Brauer Characters

In this section the intrinsic for computing the table of absolutely irreducible Brauer characters of a finite group are described.

### `BrauerCharacterTable(G, p): Chtr -> SeqEnum`

Construct the table of irreducible Brauer characters in characteristic $p$ for the group $G$. For soluble groups this is deduced from the ordinary character table. For non-soluble groups the absolutely irreducible $p$-modular representations are constructed.

### `Character(M): ModGrp -> AlgChtrElt`

Given a $K[G]$-module $M$, where $K$ is a finite field of characteristic $p$ a prime, the $p$-modular Brauer character of $M$ is constructed.