# nLab GL(2,3)

Contents

### Context

#### Group Theory

group theory

Classical groups

Finite groups

Group schemes

Topological groups

Lie groups

Super-Lie groups

Higher groups

Cohomology and Extensions

Related concepts

# Contents

## Idea

The finite general linear group $GL(2,\mathbb{3})$ for coefficients the prime field $\mathbb{F}_3$.

## Properties

### Character table

The character table for $GL(2,\mathbb{F}_3)$ over the complex numbers looks like that of the binary octahedral group. The only difference is the Schur indices. Hence the character tables over the real numbers do differ.

linear representation theory of the finite general linear group GL(2,3)

$\,$

group order: ${\vert GL(2,3)\vert} = 48$

conjugacy classes:12A2B3468A8B
their cardinality:111286866

character table over the complex numbers $\mathbb{C}$

irrep12A2B3468A8B
$\rho_1$11111111
$\rho_2$11-1111-1-1
$\rho_3$220-12-100
$\rho_4$2-20-101$-\sqrt{-2}$$\sqrt{-2}$
$\rho_5$2-20-101$\sqrt{-2}$$-\sqrt{-2}$
$\rho_6$33-10-1011
$\rho_7$3310-10-1-1
$\rho_8$4-4010-100

character table over the rational numbers $\mathbb{Q}$ and real numbers $\mathbb{R}$

irrep12A2B3468A8B
$\rho_1$11111111
$\rho_2$11-1111-1-1
$\rho_3$220-12-100
$\rho_4 \oplus \rho_5$4-40-20200
$\rho_6$33-10-1011
$\rho_7$3310-10-1-1
$\rho_8$4-4010-100

References

## References

Last revised on September 2, 2021 at 08:42:27. See the history of this page for a list of all contributions to it.