Group table for the character group for $\textrm{Gal}(K/\mathbb{Q})$

$K$ is the global number field defined by \( x^{4} - x^{3} - 4 x^{2} + 4 x + 1 \)

$\times$ \(\chi_{ 15 } ( 1, ·)\) \(\chi_{ 15 } ( 8, ·)\) \(\chi_{ 15 } ( 2, ·)\) \(\chi_{ 15 } ( 4, ·)\)
\(\chi_{ 15 }(1, ·)\) \(\chi_{ 15 } ( 1, ·)\) \(\chi_{ 15 } ( 8, ·)\) \(\chi_{ 15 } ( 2, ·)\) \(\chi_{ 15 } ( 4, ·)\)
\(\chi_{ 15 }(8, ·)\) \(\chi_{ 15 } ( 8, ·)\) \(\chi_{ 15 } ( 4, ·)\) \(\chi_{ 15 } ( 1, ·)\) \(\chi_{ 15 } ( 2, ·)\)
\(\chi_{ 15 }(2, ·)\) \(\chi_{ 15 } ( 2, ·)\) \(\chi_{ 15 } ( 1, ·)\) \(\chi_{ 15 } ( 4, ·)\) \(\chi_{ 15 } ( 8, ·)\)
\(\chi_{ 15 }(4, ·)\) \(\chi_{ 15 } ( 4, ·)\) \(\chi_{ 15 } ( 2, ·)\) \(\chi_{ 15 } ( 8, ·)\) \(\chi_{ 15 } ( 1, ·)\)