nLab GL(2,3)

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:	1	2A	2B	3	4	6	8A	8B
their cardinality:	1	1	12	8	6	8	6	6

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

irrep	1	2A	2B	3	4	6	8A	8B
$\rho_1$	1	1	1	1	1	1	1	1
$\rho_2$	1	1	-1	1	1	1	-1	-1
$\rho_3$	2	2	0	-1	2	-1	0	0
$\rho_4$	2	-2	0	-1	0	1	$-\sqrt{-2}$	$\sqrt{-2}$
$\rho_5$	2	-2	0	-1	0	1	$\sqrt{-2}$	$-\sqrt{-2}$
$\rho_6$	3	3	-1	0	-1	0	1	1
$\rho_7$	3	3	1	0	-1	0	-1	-1
$\rho_8$	4	-4	0	1	0	-1	0	0

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

irrep	1	2A	2B	3	4	6	8A	8B
$\rho_1$	1	1	1	1	1	1	1	1
$\rho_2$	1	1	-1	1	1	1	-1	-1
$\rho_3$	2	2	0	-1	2	-1	0	0
$\rho_4 \oplus \rho_5$	4	-4	0	-2	0	2	0	0
$\rho_6$	3	3	-1	0	-1	0	1	1
$\rho_7$	3	3	1	0	-1	0	-1	-1
$\rho_8$	4	-4	0	1	0	-1	0	0

References

• GroupNames, GL(2,3)

category: character tables

Related concepts

• SL(2,3)

References

• GroupNames, GL(2,3)