Group table for the character group for $\textrm{Gal}(K/\mathbb{Q})$
$K$ is the global number field defined by \(x^{6} \) \(\mathstrut -\mathstrut 3 x^{5} \) \(\mathstrut +\mathstrut 6 x^{4} \) \(\mathstrut -\mathstrut 5 x^{3} \) \(\mathstrut +\mathstrut 33 x^{2} \) \(\mathstrut -\mathstrut 54 x \) \(\mathstrut +\mathstrut 111 \)
| $\times$ | \(\chi_{ 99 } ( 1, ·)\) | \(\chi_{ 99 } ( 34, ·)\) | \(\chi_{ 99 } ( 67, ·)\) | \(\chi_{ 99 } ( 10, ·)\) | \(\chi_{ 99 } ( 43, ·)\) | \(\chi_{ 99 } ( 76, ·)\) |
|---|---|---|---|---|---|---|
| \(\chi_{ 99 }(1, ·)\) | \(\chi_{ 99 } ( 1, ·)\) | \(\chi_{ 99 } ( 34, ·)\) | \(\chi_{ 99 } ( 67, ·)\) | \(\chi_{ 99 } ( 10, ·)\) | \(\chi_{ 99 } ( 43, ·)\) | \(\chi_{ 99 } ( 76, ·)\) |
| \(\chi_{ 99 }(34, ·)\) | \(\chi_{ 99 } ( 34, ·)\) | \(\chi_{ 99 } ( 67, ·)\) | \(\chi_{ 99 } ( 1, ·)\) | \(\chi_{ 99 } ( 43, ·)\) | \(\chi_{ 99 } ( 76, ·)\) | \(\chi_{ 99 } ( 10, ·)\) |
| \(\chi_{ 99 }(67, ·)\) | \(\chi_{ 99 } ( 67, ·)\) | \(\chi_{ 99 } ( 1, ·)\) | \(\chi_{ 99 } ( 34, ·)\) | \(\chi_{ 99 } ( 76, ·)\) | \(\chi_{ 99 } ( 10, ·)\) | \(\chi_{ 99 } ( 43, ·)\) |
| \(\chi_{ 99 }(10, ·)\) | \(\chi_{ 99 } ( 10, ·)\) | \(\chi_{ 99 } ( 43, ·)\) | \(\chi_{ 99 } ( 76, ·)\) | \(\chi_{ 99 } ( 1, ·)\) | \(\chi_{ 99 } ( 34, ·)\) | \(\chi_{ 99 } ( 67, ·)\) |
| \(\chi_{ 99 }(43, ·)\) | \(\chi_{ 99 } ( 43, ·)\) | \(\chi_{ 99 } ( 76, ·)\) | \(\chi_{ 99 } ( 10, ·)\) | \(\chi_{ 99 } ( 34, ·)\) | \(\chi_{ 99 } ( 67, ·)\) | \(\chi_{ 99 } ( 1, ·)\) |
| \(\chi_{ 99 }(76, ·)\) | \(\chi_{ 99 } ( 76, ·)\) | \(\chi_{ 99 } ( 10, ·)\) | \(\chi_{ 99 } ( 43, ·)\) | \(\chi_{ 99 } ( 67, ·)\) | \(\chi_{ 99 } ( 1, ·)\) | \(\chi_{ 99 } ( 34, ·)\) |