Calculator Inputs
Example Data Table
| Group | Order | Classes | Degree Pattern | Typical Study Focus |
|---|---|---|---|---|
| S₃ | 6 | 3 | 1, 1, 2 | First non-abelian character table. |
| D₄ | 8 | 5 | 1, 1, 1, 1, 2 | Square symmetry and reflection classes. |
| Q₈ | 8 | 5 | 1, 1, 1, 1, 2 | Quaternion structure and central elements. |
| A₄ | 12 | 4 | 1, 1, 1, 3 | Complex characters and 3-cycles. |
This example table helps compare standard finite groups before generating a full character table.
Formula Used
Group Order
The order of the group is the total of conjugacy class sizes: |G| = Σ |Ck|.
Degree Identity
The irreducible character degrees satisfy Σ di2 = |G|, where di = χi(e).
Row Orthogonality
⟨χi, χj⟩ = (1/|G|) Σ |Ck| χi(Ck) χ̄j(Ck) = δij.
Column Orthogonality
For representatives from conjugacy classes Cr and Cs, Σ χi(Cr)χ̄i(Cs) = 0 when the classes differ, and |G| / |Cr| on the diagonal.
Interpretation
Character tables summarize class values of all irreducible representations. They reveal symmetry, degree structure, class behavior, and decomposition rules for class functions.
How to Use This Calculator
1. Select a Group
Choose a preset finite group such as S₃, D₄, Q₈, or A₄.
2. Pick Irreps
Set a primary irrep and a comparison irrep for the weighted inner product check.
3. Choose Output Style
Select algebraic or numeric display, then adjust precision and optional diagnostics.
4. Generate and Export
Submit the form to view the table, matrices, graph, and download options.
FAQs
1. What does a character table show?
It lists irreducible characters across conjugacy classes. Each row represents an irreducible representation, and each column represents a class. The table summarizes deep structural information about the group.
2. Why are conjugacy classes used instead of single elements?
Characters are constant on conjugacy classes. Using classes compresses the data and matches the natural symmetry of representation theory, making orthogonality relations practical and meaningful.
3. What is the meaning of χ(e)?
The value of a character at the identity equals the degree of the representation. Those degrees are central because their squares add up to the order of the group.
4. Why do some groups use complex entries?
Certain irreducible representations produce roots of unity such as ω or i. These are normal in finite group theory and still obey the same orthogonality rules.
5. What does row orthogonality verify?
It checks whether irreducible characters behave like an orthonormal basis under the weighted inner product. Diagonal entries should be 1, and off-diagonal entries should be 0.
6. What does column orthogonality verify?
It confirms the relationship between class columns. Matching columns produce a diagonal value tied to the centralizer size, while distinct columns should collapse to zero.
7. When is a group abelian in this calculator?
A preset is treated as abelian when every conjugacy class has size one. In that case, each irreducible character has degree one as well.
8. Can I use this for study and verification?
Yes. It is helpful for coursework, notes, and quick checks of standard finite groups. It is especially useful when reviewing degrees, classes, and orthogonality identities.