Group table for the character group for $\textrm{Gal}(K/\mathbb{Q})$
$K$ is the global number field defined by \( x^{6} - x^{5} + 7 x^{4} - 8 x^{3} + 43 x^{2} - 42 x + 49 \)
| $\times$ | \(\chi_{ 57 } ( 1, ·)\) | \(\chi_{ 57 } ( 20, ·)\) | \(\chi_{ 57 } ( 49, ·)\) | \(\chi_{ 57 } ( 7, ·)\) | \(\chi_{ 57 } ( 26, ·)\) | \(\chi_{ 57 } ( 11, ·)\) |
|---|---|---|---|---|---|---|
| \(\chi_{ 57 }(1, ·)\) | \(\chi_{ 57 } ( 1, ·)\) | \(\chi_{ 57 } ( 20, ·)\) | \(\chi_{ 57 } ( 49, ·)\) | \(\chi_{ 57 } ( 7, ·)\) | \(\chi_{ 57 } ( 26, ·)\) | \(\chi_{ 57 } ( 11, ·)\) |
| \(\chi_{ 57 }(20, ·)\) | \(\chi_{ 57 } ( 20, ·)\) | \(\chi_{ 57 } ( 1, ·)\) | \(\chi_{ 57 } ( 11, ·)\) | \(\chi_{ 57 } ( 26, ·)\) | \(\chi_{ 57 } ( 7, ·)\) | \(\chi_{ 57 } ( 49, ·)\) |
| \(\chi_{ 57 }(49, ·)\) | \(\chi_{ 57 } ( 49, ·)\) | \(\chi_{ 57 } ( 11, ·)\) | \(\chi_{ 57 } ( 7, ·)\) | \(\chi_{ 57 } ( 1, ·)\) | \(\chi_{ 57 } ( 20, ·)\) | \(\chi_{ 57 } ( 26, ·)\) |
| \(\chi_{ 57 }(7, ·)\) | \(\chi_{ 57 } ( 7, ·)\) | \(\chi_{ 57 } ( 26, ·)\) | \(\chi_{ 57 } ( 1, ·)\) | \(\chi_{ 57 } ( 49, ·)\) | \(\chi_{ 57 } ( 11, ·)\) | \(\chi_{ 57 } ( 20, ·)\) |
| \(\chi_{ 57 }(26, ·)\) | \(\chi_{ 57 } ( 26, ·)\) | \(\chi_{ 57 } ( 7, ·)\) | \(\chi_{ 57 } ( 20, ·)\) | \(\chi_{ 57 } ( 11, ·)\) | \(\chi_{ 57 } ( 49, ·)\) | \(\chi_{ 57 } ( 1, ·)\) |
| \(\chi_{ 57 }(11, ·)\) | \(\chi_{ 57 } ( 11, ·)\) | \(\chi_{ 57 } ( 49, ·)\) | \(\chi_{ 57 } ( 26, ·)\) | \(\chi_{ 57 } ( 20, ·)\) | \(\chi_{ 57 } ( 1, ·)\) | \(\chi_{ 57 } ( 7, ·)\) |