Group table for the character group for $\textrm{Gal}(K/\mathbb{Q})$
$K$ is the global number field defined by \( x^{8} - x^{7} + x^{5} - x^{4} + x^{3} - x + 1 \)
| $\times$ | \(\chi_{ 15 } ( 1, ·)\) | \(\chi_{ 15 } ( 2, ·)\) | \(\chi_{ 15 } ( 4, ·)\) | \(\chi_{ 15 } ( 7, ·)\) | \(\chi_{ 15 } ( 8, ·)\) | \(\chi_{ 15 } ( 11, ·)\) | \(\chi_{ 15 } ( 13, ·)\) | \(\chi_{ 15 } ( 14, ·)\) |
|---|---|---|---|---|---|---|---|---|
| \(\chi_{ 15 }(1, ·)\) | \(\chi_{ 15 } ( 1, ·)\) | \(\chi_{ 15 } ( 2, ·)\) | \(\chi_{ 15 } ( 4, ·)\) | \(\chi_{ 15 } ( 7, ·)\) | \(\chi_{ 15 } ( 8, ·)\) | \(\chi_{ 15 } ( 11, ·)\) | \(\chi_{ 15 } ( 13, ·)\) | \(\chi_{ 15 } ( 14, ·)\) |
| \(\chi_{ 15 }(2, ·)\) | \(\chi_{ 15 } ( 2, ·)\) | \(\chi_{ 15 } ( 4, ·)\) | \(\chi_{ 15 } ( 8, ·)\) | \(\chi_{ 15 } ( 14, ·)\) | \(\chi_{ 15 } ( 1, ·)\) | \(\chi_{ 15 } ( 7, ·)\) | \(\chi_{ 15 } ( 11, ·)\) | \(\chi_{ 15 } ( 13, ·)\) |
| \(\chi_{ 15 }(4, ·)\) | \(\chi_{ 15 } ( 4, ·)\) | \(\chi_{ 15 } ( 8, ·)\) | \(\chi_{ 15 } ( 1, ·)\) | \(\chi_{ 15 } ( 13, ·)\) | \(\chi_{ 15 } ( 2, ·)\) | \(\chi_{ 15 } ( 14, ·)\) | \(\chi_{ 15 } ( 7, ·)\) | \(\chi_{ 15 } ( 11, ·)\) |
| \(\chi_{ 15 }(7, ·)\) | \(\chi_{ 15 } ( 7, ·)\) | \(\chi_{ 15 } ( 14, ·)\) | \(\chi_{ 15 } ( 13, ·)\) | \(\chi_{ 15 } ( 4, ·)\) | \(\chi_{ 15 } ( 11, ·)\) | \(\chi_{ 15 } ( 2, ·)\) | \(\chi_{ 15 } ( 1, ·)\) | \(\chi_{ 15 } ( 8, ·)\) |
| \(\chi_{ 15 }(8, ·)\) | \(\chi_{ 15 } ( 8, ·)\) | \(\chi_{ 15 } ( 1, ·)\) | \(\chi_{ 15 } ( 2, ·)\) | \(\chi_{ 15 } ( 11, ·)\) | \(\chi_{ 15 } ( 4, ·)\) | \(\chi_{ 15 } ( 13, ·)\) | \(\chi_{ 15 } ( 14, ·)\) | \(\chi_{ 15 } ( 7, ·)\) |
| \(\chi_{ 15 }(11, ·)\) | \(\chi_{ 15 } ( 11, ·)\) | \(\chi_{ 15 } ( 7, ·)\) | \(\chi_{ 15 } ( 14, ·)\) | \(\chi_{ 15 } ( 2, ·)\) | \(\chi_{ 15 } ( 13, ·)\) | \(\chi_{ 15 } ( 1, ·)\) | \(\chi_{ 15 } ( 8, ·)\) | \(\chi_{ 15 } ( 4, ·)\) |
| \(\chi_{ 15 }(13, ·)\) | \(\chi_{ 15 } ( 13, ·)\) | \(\chi_{ 15 } ( 11, ·)\) | \(\chi_{ 15 } ( 7, ·)\) | \(\chi_{ 15 } ( 1, ·)\) | \(\chi_{ 15 } ( 14, ·)\) | \(\chi_{ 15 } ( 8, ·)\) | \(\chi_{ 15 } ( 4, ·)\) | \(\chi_{ 15 } ( 2, ·)\) |
| \(\chi_{ 15 }(14, ·)\) | \(\chi_{ 15 } ( 14, ·)\) | \(\chi_{ 15 } ( 13, ·)\) | \(\chi_{ 15 } ( 11, ·)\) | \(\chi_{ 15 } ( 8, ·)\) | \(\chi_{ 15 } ( 7, ·)\) | \(\chi_{ 15 } ( 4, ·)\) | \(\chi_{ 15 } ( 2, ·)\) | \(\chi_{ 15 } ( 1, ·)\) |