nLab character table of 2O

linear representation theory of binary octahedral group $2 O$

$\,$

group order: ${\vert 2O\vert} = 48$

conjugacy classes:	1	-1	$i$	a	c	e	f	g
their cardinality:	1	1	6	8	8	6	6	12

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

irrep	1	-1	$i$	a	c	e	f	g
$\rho_1$	1	1	1	1	1	1	1	1
$\rho_2$	1	1	1	1	1	-1	-1	-1
$\rho_3$	2	2	2	-1	-1	0	0	0
$\rho_4$	3	3	-1	0	0	1	1	-1
$\rho_5$	3	3	-1	0	0	-1	-1	1
$\rho_6$	2	-2	0	1	-1	$\sqrt{2}$	$-\sqrt{2}$	0
$\rho_7$	2	-2	0	1	-1	$-\sqrt{2}$	$\sqrt{2}$	0
$\rho_8$	4	-4	0	-1	1	0	0	0

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

irrep	1	-1	$i$	a	c	e	f	g
$\rho_1$	1	1	1	1	1	1	1	1
$\rho_2$	1	1	1	1	1	-1	-1	-1
$\rho_3$	2	2	2	-1	-1	0	0	0
$\rho_4$	3	3	-1	0	0	1	1	-1
$\rho_5$	3	3	-1	0	0	-1	-1	1
$\rho_6 \oplus \rho_6$	4	-4	0	2	-2	$2 \sqrt{2}$	$-2 \sqrt{2}$	0
$\rho_7 \oplus \rho_7$	4	-4	0	2	-2	$-2 \sqrt{2}$	$2 \sqrt{2}$	0
$\rho_8 \oplus \rho_8$	8	-8	0	-2	2	0	0	0

References

• GroupNames, CSU(2,3)

• James Montaldi, Real representations – Binary cubic – 2O

• Groupprops, Linear representation theory of binary octahedral group

• Bockland, Character tables and McKay quivers (pdf)

category: character tables