nLab character table of 2D4=Dic2=Q8

linear representation theory of binary dihedral group $2 D_4$

$=$ dicyclic group $Dic_2$ $=$ quaternion group $Q_8$

$\,$

group order: ${\vert 2D_4\vert} = 8$

conjugacy classes:	1	2	4A	4B	4C
their cardinality:	1	1	2	2	2

$\,$

splitting field	$\mathbb{Q}(\alpha, \beta)$ with $\alpha^2 + \beta^2 = -1$
field generated by characters	$\mathbb{Q}$

character table over splitting field $\mathbb{Q}(\alpha,\beta)$/complex numbers $\mathbb{C}$

irrep	1	2	4A	4B	4C	Schur index
$\rho_1$	1	1	1	1	1	1
$\rho_2$	1	1	-1	1	-1	1
$\rho_3$	1	1	1	-1	-1	1
$\rho_4$	1	1	-1	-1	1	1
$\rho_5$	2	-2	0	0	0	2

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

irrep	1	2	4A	4B	4C
$\rho_1$	1	1	1	1	1
$\rho_2$	1	1	-1	1	-1
$\rho_3$	1	1	1	-1	-1
$\rho_4$	1	1	-1	-1	1
$\rho_5 \oplus \rho_5$	4	-4	0	0	0

References

• GroupNames, Q8,

• Groupprops, Linear representation theory of dicyclic groups

• James Montaldi, Real representations – Binary cubic – Q8

category: character tables