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
Groupprops, Linear representation theory of binary octahedral group
Bockland, Character tables and McKay quivers (pdf)
Last revised on September 2, 2021 at 08:43:40. See the history of this page for a list of all contributions to it.