---
tag: P16AFZT3K0
kind: concept
name: Character table
msc: [20C15]
sources:
  - url: https://en.wikipedia.org/wiki/Character_table
same_as:
  wikidata: Q11498021
  wikipedia: Character table
---

## Summary

The character table of a finite group lists the values of its irreducible complex [[characters|RMWMYBY8TC]] on its [[conjugacy classes|0DEBXAJ5SA]].
It is a square matrix, because the number of irreducible characters equals the number of conjugacy classes.
Much information about the group can be read from it, such as its order, its normal subgroups and whether it is simple or abelian.

## Explanations

### L1 human

A character table is a compact summary of all the ways a finite group can act by matrices.
Each row belongs to one irreducible representation, each column to one class of conjugate elements, and each entry is a trace.
For the cyclic group of order $2$ the table has rows $(1, 1)$ and $(1, -1)$.
Chemists use character tables of symmetry groups of molecules to classify vibrations and orbitals.