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
Last revised on October 8, 2018 at 06:21:59. See the history of this page for a list of all contributions to it.