Character table

The character table of a finite group lists its irreducible complex characters on its conjugacy classes. The number of rows equals the number of conjugacy classes, and the squared row degrees sum to the group order.

Table of contents
- Irreducible representations of G6n Character table
  - Character table of G6n Irreducible representations of G6n
- Irreducible character degree Character table

Irreducible representations of G6n ( $G_{6 n}$ )

 0  0

For

G_{6 n} = ⟨ a, b : a^{2 n} = b^{3} = 1, a^{- 1} ba = b^{- 1} ⟩,

(1)

there are

2 n

linear characters, given by

a \mapsto ξ^{j}

b \mapsto 1

for a primitive

(2 n)

th root of unity

ξ

, and

n

two-dimensional irreducible representations

a \mapsto (0 ξ^{k} ξ^{k} 0), b \mapsto (ω 0 0 ω^{2}), 0 \leq k < n,

(2)

where

ω^{3} = 1

and

ω \neq = 1

. The two-dimensional representations parametrized by

ε

and

- ε

are isomorphic. The sum of squares of irreducible degrees is

2 n + 4 n = 6 n

, so this list is complete.

Character table of G6n ( $G_{6 n}$ )

 0  0

The conjugacy classes of

G_{6 n}

are

C_{r} = {a^{2 r}}, D_{r} = {a^{2 r} b, a^{2 r} b^{2}}, E_{r} = {a^{2 r + 1}, a^{2 r + 1} b, a^{2 r + 1} b^{2}},

(1)

for

0 \leq r < n

. With

0 \leq j < 2 n

and

0 \leq k < n

, the linear character

λ_{j}

and two-dimensional character

χ_{k}

have values

λ_{j} χ_{k} C_{r} ξ^{2 j r} 2 ξ^{2 k r} D_{r} ξ^{2 j r} - ξ^{2 k r} E_{r} ξ^{j (2 r + 1)} 0.

(2)

Irreducible character degree

 0  0

The degree of an irreducible character

χ

χ (1)

, the dimension of the corresponding irreducible representation.

Irreducible characters of degree coprime to p ( $Irr_{p^{'}} (G)$ )

 0  0

For a finite group

G

and prime number

p

, the notation

Irr_{p^{'}} (G) = {χ \in Irr (G) : p ∤ χ (1)}

(1)

denotes the irreducible characters whose irreducible character degree is coprime to

p

Linear character

 0  0

A linear character is the character of a one-dimensional representation and therefore has degree one.

Nonlinear irreducible character

 0  0

A nonlinear irreducible character has degree greater than one.

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A **character table** is a mathematical tool used in the field of group theory, a branch of abstract algebra. It provides a compact way to represent the irreducible representations of a finite group. The character table of a group includes the following key components: 1. **Irreducible Representations**: Each row of the character table corresponds to an irreducible representation (a representation that cannot be decomposed into smaller representations) of the group.

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