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

$K$ is the global number field defined by <span class="tset-container"><span class="tset-raw" id="tset-raw-349" raw="x^4 + 1" israw="0" ondblclick="ondouble(349)">\(x^{4} + 1\)</span>&nbsp;&nbsp;<span onclick="iconrawtset(349)"><img alt="Toggle raw display" src="/static/images/t2r.png" class="tset-icon" id="tset-raw-icon-349" style="position:relative;top: 2px"></span></span>

$\times$ \(\chi_{ 8 } ( 1, ·)\) \(\chi_{ 8 } ( 3, ·)\) \(\chi_{ 8 } ( 5, ·)\) \(\chi_{ 8 } ( 7, ·)\)
\(\chi_{ 8 }(1, ·)\) \(\chi_{ 8 } ( 1, ·)\) \(\chi_{ 8 } ( 3, ·)\) \(\chi_{ 8 } ( 5, ·)\) \(\chi_{ 8 } ( 7, ·)\)
\(\chi_{ 8 }(3, ·)\) \(\chi_{ 8 } ( 3, ·)\) \(\chi_{ 8 } ( 1, ·)\) \(\chi_{ 8 } ( 7, ·)\) \(\chi_{ 8 } ( 5, ·)\)
\(\chi_{ 8 }(5, ·)\) \(\chi_{ 8 } ( 5, ·)\) \(\chi_{ 8 } ( 7, ·)\) \(\chi_{ 8 } ( 1, ·)\) \(\chi_{ 8 } ( 3, ·)\)
\(\chi_{ 8 }(7, ·)\) \(\chi_{ 8 } ( 7, ·)\) \(\chi_{ 8 } ( 5, ·)\) \(\chi_{ 8 } ( 3, ·)\) \(\chi_{ 8 } ( 1, ·)\)