Contents

group theory

# 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