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Properties

Label 2.3.6.30a
Base 2.1.1.0a1.1
Degree \(18\)
e \(6\)
f \(3\)
c \(30\)

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Defining polynomial over unramified subextension

$x^{6} + 2 a_{5} x^{5} + 4 c_{10} x^{4} + 4 b_{9} x^{3} + 4 b_{7} x + 2$

Invariants

Residue field characteristic: $2$
Degree: $18$
Base field: $\Q_{2}$
Ramification index $e$: $6$
Residue field degree $f$: $3$
Discriminant exponent $c$: $30$
Artin slopes: $[\frac{8}{3}]$
Swan slopes: $[\frac{5}{3}]$
Means: $\langle\frac{5}{6}\rangle$
Rams: $(5)$
Field count: $304$ (complete)
Ambiguity: $6$
Mass: $448$
Absolute Mass: $448/3$

Diagrams

Varying

Indices of inseparability: $[5,0]$
Associated inertia: $[2,1]$
Jump Set: $[3,9]$

Galois groups and Hidden Artin slopes

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Fields

Galois group Galois Artin Indices
Assoc. Inertia Jump Set Fields per packet
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Showing 1-50 of 304

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Label Packet size Polynomial Galois group Galois degree $\#\Aut(K/\Q_p)$ Artin slope content Swan slope content Hidden Artin slopes Hidden Swan slopes Ind. of Insep. Assoc. Inertia Resid. Poly Jump Set
2.3.6.30a1.1 4 $( x^{3} + x + 1 )^{6} + 2 ( x^{3} + x + 1 )^{5} + 2$ $C_6\times S_4$ (as 18T61) $144$ $6$ $[2, \frac{8}{3}, \frac{8}{3}]_{3}^{6}$ $[1,\frac{5}{3},\frac{5}{3}]_{3}^{6}$ $[2,\frac{8}{3}]^{2}$ $[1,\frac{5}{3}]^{2}$ $[5, 0]$ $[2, 1]$ $z^4 + z^2 + 1,z + t^2$ $[3, 9]$
2.3.6.30a1.2 4 $( x^{3} + x + 1 )^{6} + 2 ( x^{3} + x + 1 )^{5} + 4 ( x^{3} + x + 1 )^{4} + 2$ $C_6\times S_4$ (as 18T61) $144$ $6$ $[2, \frac{8}{3}, \frac{8}{3}]_{3}^{6}$ $[1,\frac{5}{3},\frac{5}{3}]_{3}^{6}$ $[2,\frac{8}{3}]^{2}$ $[1,\frac{5}{3}]^{2}$ $[5, 0]$ $[2, 1]$ $z^4 + z^2 + 1,z + t^2$ $[3, 9]$
2.3.6.30a1.3 4 $( x^{3} + x + 1 )^{6} + 2 ( x^{3} + x + 1 )^{5} + 4 x ( x^{3} + x + 1 )^{3} + 2$ $C_2^4:(C_6\times S_4)$ (as 18T367) $2304$ $2$ $[\frac{4}{3}, \frac{4}{3}, \frac{4}{3}, \frac{4}{3}, 2, \frac{8}{3}, \frac{8}{3}]_{3}^{6}$ $[\frac{1}{3},\frac{1}{3},\frac{1}{3},\frac{1}{3},1,\frac{5}{3},\frac{5}{3}]_{3}^{6}$ $[\frac{4}{3},\frac{4}{3},\frac{4}{3},\frac{4}{3},2,\frac{8}{3}]^{2}$ $[\frac{1}{3},\frac{1}{3},\frac{1}{3},\frac{1}{3},1,\frac{5}{3}]^{2}$ $[5, 0]$ $[2, 1]$ $z^4 + z^2 + 1,z + t^2$ $[3, 9]$
2.3.6.30a1.4 4 $( x^{3} + x + 1 )^{6} + 2 ( x^{3} + x + 1 )^{5} + 4 ( x^{3} + x + 1 )^{4} + 4 x ( x^{3} + x + 1 )^{3} + 2$ $C_2^4:(C_6\times S_4)$ (as 18T367) $2304$ $2$ $[\frac{4}{3}, \frac{4}{3}, \frac{4}{3}, \frac{4}{3}, 2, \frac{8}{3}, \frac{8}{3}]_{3}^{6}$ $[\frac{1}{3},\frac{1}{3},\frac{1}{3},\frac{1}{3},1,\frac{5}{3},\frac{5}{3}]_{3}^{6}$ $[\frac{4}{3},\frac{4}{3},\frac{4}{3},\frac{4}{3},2,\frac{8}{3}]^{2}$ $[\frac{1}{3},\frac{1}{3},\frac{1}{3},\frac{1}{3},1,\frac{5}{3}]^{2}$ $[5, 0]$ $[2, 1]$ $z^4 + z^2 + 1,z + t^2$ $[3, 9]$
2.3.6.30a1.5 4 $( x^{3} + x + 1 )^{6} + 2 ( x^{3} + x + 1 )^{5} + 4 ( x^{3} + x + 1 )^{3} + 2$ $C_6\times S_4$ (as 18T61) $144$ $6$ $[2, \frac{8}{3}, \frac{8}{3}]_{3}^{6}$ $[1,\frac{5}{3},\frac{5}{3}]_{3}^{6}$ $[2,\frac{8}{3}]^{2}$ $[1,\frac{5}{3}]^{2}$ $[5, 0]$ $[2, 1]$ $z^4 + z^2 + 1,z + t^2$ $[3, 9]$
2.3.6.30a1.6 4 $( x^{3} + x + 1 )^{6} + 2 ( x^{3} + x + 1 )^{5} + 4 ( x^{3} + x + 1 )^{4} + 4 ( x^{3} + x + 1 )^{3} + 2$ $C_6\times S_4$ (as 18T61) $144$ $6$ $[2, \frac{8}{3}, \frac{8}{3}]_{3}^{6}$ $[1,\frac{5}{3},\frac{5}{3}]_{3}^{6}$ $[2,\frac{8}{3}]^{2}$ $[1,\frac{5}{3}]^{2}$ $[5, 0]$ $[2, 1]$ $z^4 + z^2 + 1,z + t^2$ $[3, 9]$
2.3.6.30a1.7 4 $( x^{3} + x + 1 )^{6} + 2 ( x^{3} + x + 1 )^{5} + \left(4 x + 4\right) ( x^{3} + x + 1 )^{3} + 2$ $C_2^4:(C_6\times S_4)$ (as 18T367) $2304$ $2$ $[\frac{4}{3}, \frac{4}{3}, \frac{4}{3}, \frac{4}{3}, 2, \frac{8}{3}, \frac{8}{3}]_{3}^{6}$ $[\frac{1}{3},\frac{1}{3},\frac{1}{3},\frac{1}{3},1,\frac{5}{3},\frac{5}{3}]_{3}^{6}$ $[\frac{4}{3},\frac{4}{3},\frac{4}{3},\frac{4}{3},2,\frac{8}{3}]^{2}$ $[\frac{1}{3},\frac{1}{3},\frac{1}{3},\frac{1}{3},1,\frac{5}{3}]^{2}$ $[5, 0]$ $[2, 1]$ $z^4 + z^2 + 1,z + t^2$ $[3, 9]$
2.3.6.30a1.8 4 $( x^{3} + x + 1 )^{6} + 2 ( x^{3} + x + 1 )^{5} + 4 ( x^{3} + x + 1 )^{4} + \left(4 x + 4\right) ( x^{3} + x + 1 )^{3} + 2$ $C_2^4:(C_6\times S_4)$ (as 18T367) $2304$ $2$ $[\frac{4}{3}, \frac{4}{3}, \frac{4}{3}, \frac{4}{3}, 2, \frac{8}{3}, \frac{8}{3}]_{3}^{6}$ $[\frac{1}{3},\frac{1}{3},\frac{1}{3},\frac{1}{3},1,\frac{5}{3},\frac{5}{3}]_{3}^{6}$ $[\frac{4}{3},\frac{4}{3},\frac{4}{3},\frac{4}{3},2,\frac{8}{3}]^{2}$ $[\frac{1}{3},\frac{1}{3},\frac{1}{3},\frac{1}{3},1,\frac{5}{3}]^{2}$ $[5, 0]$ $[2, 1]$ $z^4 + z^2 + 1,z + t^2$ $[3, 9]$
2.3.6.30a1.9 12 $( x^{3} + x + 1 )^{6} + 2 ( x^{3} + x + 1 )^{5} + 4 x ( x^{3} + x + 1 ) + 2$ $C_2^5.(A_4\times S_4)$ (as 18T544) $9216$ $2$ $[\frac{4}{3}, \frac{4}{3}, \frac{4}{3}, \frac{4}{3}, 2, 2, 2, \frac{8}{3}, \frac{8}{3}]_{3}^{6}$ $[\frac{1}{3},\frac{1}{3},\frac{1}{3},\frac{1}{3},1,1,1,\frac{5}{3},\frac{5}{3}]_{3}^{6}$ $[\frac{4}{3},\frac{4}{3},\frac{4}{3},\frac{4}{3},2,2,2,\frac{8}{3}]^{2}$ $[\frac{1}{3},\frac{1}{3},\frac{1}{3},\frac{1}{3},1,1,1,\frac{5}{3}]^{2}$ $[5, 0]$ $[2, 1]$ $z^4 + z^2 + 1,z + t^2$ $[3, 9]$
2.3.6.30a1.10 12 $( x^{3} + x + 1 )^{6} + 2 ( x^{3} + x + 1 )^{5} + 4 ( x^{3} + x + 1 )^{4} + 4 x ( x^{3} + x + 1 ) + 2$ $C_2^5.(A_4\times S_4)$ (as 18T544) $9216$ $2$ $[\frac{4}{3}, \frac{4}{3}, \frac{4}{3}, \frac{4}{3}, 2, 2, 2, \frac{8}{3}, \frac{8}{3}]_{3}^{6}$ $[\frac{1}{3},\frac{1}{3},\frac{1}{3},\frac{1}{3},1,1,1,\frac{5}{3},\frac{5}{3}]_{3}^{6}$ $[\frac{4}{3},\frac{4}{3},\frac{4}{3},\frac{4}{3},2,2,2,\frac{8}{3}]^{2}$ $[\frac{1}{3},\frac{1}{3},\frac{1}{3},\frac{1}{3},1,1,1,\frac{5}{3}]^{2}$ $[5, 0]$ $[2, 1]$ $z^4 + z^2 + 1,z + t^2$ $[3, 9]$
2.3.6.30a1.11 12 $( x^{3} + x + 1 )^{6} + 2 ( x^{3} + x + 1 )^{5} + 4 x^{2} ( x^{3} + x + 1 )^{3} + 4 x ( x^{3} + x + 1 ) + 2$ $C_2^5.(A_4\times S_4)$ (as 18T544) $9216$ $2$ $[\frac{4}{3}, \frac{4}{3}, \frac{4}{3}, \frac{4}{3}, 2, 2, 2, \frac{8}{3}, \frac{8}{3}]_{3}^{6}$ $[\frac{1}{3},\frac{1}{3},\frac{1}{3},\frac{1}{3},1,1,1,\frac{5}{3},\frac{5}{3}]_{3}^{6}$ $[\frac{4}{3},\frac{4}{3},\frac{4}{3},\frac{4}{3},2,2,2,\frac{8}{3}]^{2}$ $[\frac{1}{3},\frac{1}{3},\frac{1}{3},\frac{1}{3},1,1,1,\frac{5}{3}]^{2}$ $[5, 0]$ $[2, 1]$ $z^4 + z^2 + 1,z + t^2$ $[3, 9]$
2.3.6.30a1.12 12 $( x^{3} + x + 1 )^{6} + 2 ( x^{3} + x + 1 )^{5} + 4 ( x^{3} + x + 1 )^{4} + 4 x^{2} ( x^{3} + x + 1 )^{3} + 4 x ( x^{3} + x + 1 ) + 2$ $C_2^5.(A_4\times S_4)$ (as 18T544) $9216$ $2$ $[\frac{4}{3}, \frac{4}{3}, \frac{4}{3}, \frac{4}{3}, 2, 2, 2, \frac{8}{3}, \frac{8}{3}]_{3}^{6}$ $[\frac{1}{3},\frac{1}{3},\frac{1}{3},\frac{1}{3},1,1,1,\frac{5}{3},\frac{5}{3}]_{3}^{6}$ $[\frac{4}{3},\frac{4}{3},\frac{4}{3},\frac{4}{3},2,2,2,\frac{8}{3}]^{2}$ $[\frac{1}{3},\frac{1}{3},\frac{1}{3},\frac{1}{3},1,1,1,\frac{5}{3}]^{2}$ $[5, 0]$ $[2, 1]$ $z^4 + z^2 + 1,z + t^2$ $[3, 9]$
2.3.6.30a1.13 12 $( x^{3} + x + 1 )^{6} + 2 ( x^{3} + x + 1 )^{5} + 4 x ( x^{3} + x + 1 )^{3} + 4 x ( x^{3} + x + 1 ) + 2$ $C_2^5.(A_4\times S_4)$ (as 18T544) $9216$ $2$ $[\frac{4}{3}, \frac{4}{3}, \frac{4}{3}, \frac{4}{3}, 2, 2, 2, \frac{8}{3}, \frac{8}{3}]_{3}^{6}$ $[\frac{1}{3},\frac{1}{3},\frac{1}{3},\frac{1}{3},1,1,1,\frac{5}{3},\frac{5}{3}]_{3}^{6}$ $[\frac{4}{3},\frac{4}{3},\frac{4}{3},\frac{4}{3},2,2,2,\frac{8}{3}]^{2}$ $[\frac{1}{3},\frac{1}{3},\frac{1}{3},\frac{1}{3},1,1,1,\frac{5}{3}]^{2}$ $[5, 0]$ $[2, 1]$ $z^4 + z^2 + 1,z + t^2$ $[3, 9]$
2.3.6.30a1.14 12 $( x^{3} + x + 1 )^{6} + 2 ( x^{3} + x + 1 )^{5} + 4 ( x^{3} + x + 1 )^{4} + 4 x ( x^{3} + x + 1 )^{3} + 4 x ( x^{3} + x + 1 ) + 2$ $C_2^5.(A_4\times S_4)$ (as 18T544) $9216$ $2$ $[\frac{4}{3}, \frac{4}{3}, \frac{4}{3}, \frac{4}{3}, 2, 2, 2, \frac{8}{3}, \frac{8}{3}]_{3}^{6}$ $[\frac{1}{3},\frac{1}{3},\frac{1}{3},\frac{1}{3},1,1,1,\frac{5}{3},\frac{5}{3}]_{3}^{6}$ $[\frac{4}{3},\frac{4}{3},\frac{4}{3},\frac{4}{3},2,2,2,\frac{8}{3}]^{2}$ $[\frac{1}{3},\frac{1}{3},\frac{1}{3},\frac{1}{3},1,1,1,\frac{5}{3}]^{2}$ $[5, 0]$ $[2, 1]$ $z^4 + z^2 + 1,z + t^2$ $[3, 9]$
2.3.6.30a1.15 4 $( x^{3} + x + 1 )^{6} + 2 ( x^{3} + x + 1 )^{5} + \left(4 x^{2} + 4 x\right) ( x^{3} + x + 1 )^{3} + 4 x ( x^{3} + x + 1 ) + 2$ $A_4^2:C_2^2$ (as 18T176) $576$ $2$ $[2, 2, 2, \frac{8}{3}, \frac{8}{3}]_{3}^{6}$ $[1,1,1,\frac{5}{3},\frac{5}{3}]_{3}^{6}$ $[2,2,2,\frac{8}{3}]^{2}$ $[1,1,1,\frac{5}{3}]^{2}$ $[5, 0]$ $[2, 1]$ $z^4 + z^2 + 1,z + t^2$ $[3, 9]$
2.3.6.30a1.16 4 $( x^{3} + x + 1 )^{6} + 2 ( x^{3} + x + 1 )^{5} + 4 ( x^{3} + x + 1 )^{4} + \left(4 x^{2} + 4 x\right) ( x^{3} + x + 1 )^{3} + 4 x ( x^{3} + x + 1 ) + 2$ $A_4^2:C_2^2$ (as 18T176) $576$ $2$ $[2, 2, 2, \frac{8}{3}, \frac{8}{3}]_{3}^{6}$ $[1,1,1,\frac{5}{3},\frac{5}{3}]_{3}^{6}$ $[2,2,2,\frac{8}{3}]^{2}$ $[1,1,1,\frac{5}{3}]^{2}$ $[5, 0]$ $[2, 1]$ $z^4 + z^2 + 1,z + t^2$ $[3, 9]$
2.3.6.30a1.17 12 $( x^{3} + x + 1 )^{6} + 2 ( x^{3} + x + 1 )^{5} + 4 ( x^{3} + x + 1 )^{3} + 4 x ( x^{3} + x + 1 ) + 2$ $C_2^5.(A_4\times S_4)$ (as 18T544) $9216$ $2$ $[\frac{4}{3}, \frac{4}{3}, \frac{4}{3}, \frac{4}{3}, 2, 2, 2, \frac{8}{3}, \frac{8}{3}]_{3}^{6}$ $[\frac{1}{3},\frac{1}{3},\frac{1}{3},\frac{1}{3},1,1,1,\frac{5}{3},\frac{5}{3}]_{3}^{6}$ $[\frac{4}{3},\frac{4}{3},\frac{4}{3},\frac{4}{3},2,2,2,\frac{8}{3}]^{2}$ $[\frac{1}{3},\frac{1}{3},\frac{1}{3},\frac{1}{3},1,1,1,\frac{5}{3}]^{2}$ $[5, 0]$ $[2, 1]$ $z^4 + z^2 + 1,z + t^2$ $[3, 9]$
2.3.6.30a1.18 12 $( x^{3} + x + 1 )^{6} + 2 ( x^{3} + x + 1 )^{5} + 4 ( x^{3} + x + 1 )^{4} + 4 ( x^{3} + x + 1 )^{3} + 4 x ( x^{3} + x + 1 ) + 2$ $C_2^5.(A_4\times S_4)$ (as 18T544) $9216$ $2$ $[\frac{4}{3}, \frac{4}{3}, \frac{4}{3}, \frac{4}{3}, 2, 2, 2, \frac{8}{3}, \frac{8}{3}]_{3}^{6}$ $[\frac{1}{3},\frac{1}{3},\frac{1}{3},\frac{1}{3},1,1,1,\frac{5}{3},\frac{5}{3}]_{3}^{6}$ $[\frac{4}{3},\frac{4}{3},\frac{4}{3},\frac{4}{3},2,2,2,\frac{8}{3}]^{2}$ $[\frac{1}{3},\frac{1}{3},\frac{1}{3},\frac{1}{3},1,1,1,\frac{5}{3}]^{2}$ $[5, 0]$ $[2, 1]$ $z^4 + z^2 + 1,z + t^2$ $[3, 9]$
2.3.6.30a1.19 12 $( x^{3} + x + 1 )^{6} + 2 ( x^{3} + x + 1 )^{5} + \left(4 x^{2} + 4\right) ( x^{3} + x + 1 )^{3} + 4 x ( x^{3} + x + 1 ) + 2$ $C_2^5.(A_4\times S_4)$ (as 18T544) $9216$ $2$ $[\frac{4}{3}, \frac{4}{3}, \frac{4}{3}, \frac{4}{3}, 2, 2, 2, \frac{8}{3}, \frac{8}{3}]_{3}^{6}$ $[\frac{1}{3},\frac{1}{3},\frac{1}{3},\frac{1}{3},1,1,1,\frac{5}{3},\frac{5}{3}]_{3}^{6}$ $[\frac{4}{3},\frac{4}{3},\frac{4}{3},\frac{4}{3},2,2,2,\frac{8}{3}]^{2}$ $[\frac{1}{3},\frac{1}{3},\frac{1}{3},\frac{1}{3},1,1,1,\frac{5}{3}]^{2}$ $[5, 0]$ $[2, 1]$ $z^4 + z^2 + 1,z + t^2$ $[3, 9]$
2.3.6.30a1.20 12 $( x^{3} + x + 1 )^{6} + 2 ( x^{3} + x + 1 )^{5} + 4 ( x^{3} + x + 1 )^{4} + \left(4 x^{2} + 4\right) ( x^{3} + x + 1 )^{3} + 4 x ( x^{3} + x + 1 ) + 2$ $C_2^5.(A_4\times S_4)$ (as 18T544) $9216$ $2$ $[\frac{4}{3}, \frac{4}{3}, \frac{4}{3}, \frac{4}{3}, 2, 2, 2, \frac{8}{3}, \frac{8}{3}]_{3}^{6}$ $[\frac{1}{3},\frac{1}{3},\frac{1}{3},\frac{1}{3},1,1,1,\frac{5}{3},\frac{5}{3}]_{3}^{6}$ $[\frac{4}{3},\frac{4}{3},\frac{4}{3},\frac{4}{3},2,2,2,\frac{8}{3}]^{2}$ $[\frac{1}{3},\frac{1}{3},\frac{1}{3},\frac{1}{3},1,1,1,\frac{5}{3}]^{2}$ $[5, 0]$ $[2, 1]$ $z^4 + z^2 + 1,z + t^2$ $[3, 9]$
2.3.6.30a1.21 12 $( x^{3} + x + 1 )^{6} + 2 ( x^{3} + x + 1 )^{5} + \left(4 x + 4\right) ( x^{3} + x + 1 )^{3} + 4 x ( x^{3} + x + 1 ) + 2$ $C_2^5.(A_4\times S_4)$ (as 18T544) $9216$ $2$ $[\frac{4}{3}, \frac{4}{3}, \frac{4}{3}, \frac{4}{3}, 2, 2, 2, \frac{8}{3}, \frac{8}{3}]_{3}^{6}$ $[\frac{1}{3},\frac{1}{3},\frac{1}{3},\frac{1}{3},1,1,1,\frac{5}{3},\frac{5}{3}]_{3}^{6}$ $[\frac{4}{3},\frac{4}{3},\frac{4}{3},\frac{4}{3},2,2,2,\frac{8}{3}]^{2}$ $[\frac{1}{3},\frac{1}{3},\frac{1}{3},\frac{1}{3},1,1,1,\frac{5}{3}]^{2}$ $[5, 0]$ $[2, 1]$ $z^4 + z^2 + 1,z + t^2$ $[3, 9]$
2.3.6.30a1.22 12 $( x^{3} + x + 1 )^{6} + 2 ( x^{3} + x + 1 )^{5} + 4 ( x^{3} + x + 1 )^{4} + \left(4 x + 4\right) ( x^{3} + x + 1 )^{3} + 4 x ( x^{3} + x + 1 ) + 2$ $C_2^5.(A_4\times S_4)$ (as 18T544) $9216$ $2$ $[\frac{4}{3}, \frac{4}{3}, \frac{4}{3}, \frac{4}{3}, 2, 2, 2, \frac{8}{3}, \frac{8}{3}]_{3}^{6}$ $[\frac{1}{3},\frac{1}{3},\frac{1}{3},\frac{1}{3},1,1,1,\frac{5}{3},\frac{5}{3}]_{3}^{6}$ $[\frac{4}{3},\frac{4}{3},\frac{4}{3},\frac{4}{3},2,2,2,\frac{8}{3}]^{2}$ $[\frac{1}{3},\frac{1}{3},\frac{1}{3},\frac{1}{3},1,1,1,\frac{5}{3}]^{2}$ $[5, 0]$ $[2, 1]$ $z^4 + z^2 + 1,z + t^2$ $[3, 9]$
2.3.6.30a1.23 4 $( x^{3} + x + 1 )^{6} + 2 ( x^{3} + x + 1 )^{5} + \left(4 x^{2} + 4 x + 4\right) ( x^{3} + x + 1 )^{3} + 4 x ( x^{3} + x + 1 ) + 2$ $A_4^2:C_2^2$ (as 18T176) $576$ $2$ $[2, 2, 2, \frac{8}{3}, \frac{8}{3}]_{3}^{6}$ $[1,1,1,\frac{5}{3},\frac{5}{3}]_{3}^{6}$ $[2,2,2,\frac{8}{3}]^{2}$ $[1,1,1,\frac{5}{3}]^{2}$ $[5, 0]$ $[2, 1]$ $z^4 + z^2 + 1,z + t^2$ $[3, 9]$
2.3.6.30a1.24 4 $( x^{3} + x + 1 )^{6} + 2 ( x^{3} + x + 1 )^{5} + 4 ( x^{3} + x + 1 )^{4} + \left(4 x^{2} + 4 x + 4\right) ( x^{3} + x + 1 )^{3} + 4 x ( x^{3} + x + 1 ) + 2$ $A_4^2:C_2^2$ (as 18T176) $576$ $2$ $[2, 2, 2, \frac{8}{3}, \frac{8}{3}]_{3}^{6}$ $[1,1,1,\frac{5}{3},\frac{5}{3}]_{3}^{6}$ $[2,2,2,\frac{8}{3}]^{2}$ $[1,1,1,\frac{5}{3}]^{2}$ $[5, 0]$ $[2, 1]$ $z^4 + z^2 + 1,z + t^2$ $[3, 9]$
2.3.6.30a1.25 2 $( x^{3} + x + 1 )^{6} + 2 ( x^{3} + x + 1 )^{5} + 4 ( x^{3} + x + 1 ) + 2$ $C_3\times S_4$ (as 18T30) $72$ $6$ $[\frac{8}{3}, \frac{8}{3}]_{3}^{6}$ $[\frac{5}{3},\frac{5}{3}]_{3}^{6}$ $[\frac{8}{3}]^{2}$ $[\frac{5}{3}]^{2}$ $[5, 0]$ $[2, 1]$ $z^4 + z^2 + 1,z + t^2$ $[3, 9]$
2.3.6.30a1.26 2 $( x^{3} + x + 1 )^{6} + 2 ( x^{3} + x + 1 )^{5} + 4 ( x^{3} + x + 1 )^{4} + 4 ( x^{3} + x + 1 ) + 2$ $C_3\times S_4$ (as 18T33) $72$ $6$ $[\frac{8}{3}, \frac{8}{3}]_{3}^{6}$ $[\frac{5}{3},\frac{5}{3}]_{3}^{6}$ $[\frac{8}{3}]^{2}$ $[\frac{5}{3}]^{2}$ $[5, 0]$ $[2, 1]$ $z^4 + z^2 + 1,z + t^2$ $[3, 9]$
2.3.6.30a1.27 2 $( x^{3} + x + 1 )^{6} + 2 ( x^{3} + x + 1 )^{5} + 4 x ( x^{3} + x + 1 )^{3} + 4 ( x^{3} + x + 1 ) + 2$ $C_2^4:(C_3\times S_4)$ (as 18T270) $1152$ $2$ $[\frac{4}{3}, \frac{4}{3}, \frac{4}{3}, \frac{4}{3}, \frac{8}{3}, \frac{8}{3}]_{3}^{6}$ $[\frac{1}{3},\frac{1}{3},\frac{1}{3},\frac{1}{3},\frac{5}{3},\frac{5}{3}]_{3}^{6}$ $[\frac{4}{3},\frac{4}{3},\frac{4}{3},\frac{4}{3},\frac{8}{3}]^{2}$ $[\frac{1}{3},\frac{1}{3},\frac{1}{3},\frac{1}{3},\frac{5}{3}]^{2}$ $[5, 0]$ $[2, 1]$ $z^4 + z^2 + 1,z + t^2$ $[3, 9]$
2.3.6.30a1.28 2 $( x^{3} + x + 1 )^{6} + 2 ( x^{3} + x + 1 )^{5} + 4 ( x^{3} + x + 1 )^{4} + 4 x ( x^{3} + x + 1 )^{3} + 4 ( x^{3} + x + 1 ) + 2$ $C_2^4:(C_3\times S_4)$ (as 18T269) $1152$ $2$ $[\frac{4}{3}, \frac{4}{3}, \frac{4}{3}, \frac{4}{3}, \frac{8}{3}, \frac{8}{3}]_{3}^{6}$ $[\frac{1}{3},\frac{1}{3},\frac{1}{3},\frac{1}{3},\frac{5}{3},\frac{5}{3}]_{3}^{6}$ $[\frac{4}{3},\frac{4}{3},\frac{4}{3},\frac{4}{3},\frac{8}{3}]^{2}$ $[\frac{1}{3},\frac{1}{3},\frac{1}{3},\frac{1}{3},\frac{5}{3}]^{2}$ $[5, 0]$ $[2, 1]$ $z^4 + z^2 + 1,z + t^2$ $[3, 9]$
2.3.6.30a1.29 2 $( x^{3} + x + 1 )^{6} + 2 ( x^{3} + x + 1 )^{5} + 4 ( x^{3} + x + 1 )^{3} + 4 ( x^{3} + x + 1 ) + 2$ $C_3\times S_4$ (as 18T30) $72$ $6$ $[\frac{8}{3}, \frac{8}{3}]_{3}^{6}$ $[\frac{5}{3},\frac{5}{3}]_{3}^{6}$ $[\frac{8}{3}]^{2}$ $[\frac{5}{3}]^{2}$ $[5, 0]$ $[2, 1]$ $z^4 + z^2 + 1,z + t^2$ $[3, 9]$
2.3.6.30a1.30 2 $( x^{3} + x + 1 )^{6} + 2 ( x^{3} + x + 1 )^{5} + 4 ( x^{3} + x + 1 )^{4} + 4 ( x^{3} + x + 1 )^{3} + 4 ( x^{3} + x + 1 ) + 2$ $C_3\times S_4$ (as 18T33) $72$ $6$ $[\frac{8}{3}, \frac{8}{3}]_{3}^{6}$ $[\frac{5}{3},\frac{5}{3}]_{3}^{6}$ $[\frac{8}{3}]^{2}$ $[\frac{5}{3}]^{2}$ $[5, 0]$ $[2, 1]$ $z^4 + z^2 + 1,z + t^2$ $[3, 9]$
2.3.6.30a1.31 2 $( x^{3} + x + 1 )^{6} + 2 ( x^{3} + x + 1 )^{5} + \left(4 x + 4\right) ( x^{3} + x + 1 )^{3} + 4 ( x^{3} + x + 1 ) + 2$ $C_2^4:(C_3\times S_4)$ (as 18T270) $1152$ $2$ $[\frac{4}{3}, \frac{4}{3}, \frac{4}{3}, \frac{4}{3}, \frac{8}{3}, \frac{8}{3}]_{3}^{6}$ $[\frac{1}{3},\frac{1}{3},\frac{1}{3},\frac{1}{3},\frac{5}{3},\frac{5}{3}]_{3}^{6}$ $[\frac{4}{3},\frac{4}{3},\frac{4}{3},\frac{4}{3},\frac{8}{3}]^{2}$ $[\frac{1}{3},\frac{1}{3},\frac{1}{3},\frac{1}{3},\frac{5}{3}]^{2}$ $[5, 0]$ $[2, 1]$ $z^4 + z^2 + 1,z + t^2$ $[3, 9]$
2.3.6.30a1.32 2 $( x^{3} + x + 1 )^{6} + 2 ( x^{3} + x + 1 )^{5} + 4 ( x^{3} + x + 1 )^{4} + \left(4 x + 4\right) ( x^{3} + x + 1 )^{3} + 4 ( x^{3} + x + 1 ) + 2$ $C_2^4:(C_3\times S_4)$ (as 18T269) $1152$ $2$ $[\frac{4}{3}, \frac{4}{3}, \frac{4}{3}, \frac{4}{3}, \frac{8}{3}, \frac{8}{3}]_{3}^{6}$ $[\frac{1}{3},\frac{1}{3},\frac{1}{3},\frac{1}{3},\frac{5}{3},\frac{5}{3}]_{3}^{6}$ $[\frac{4}{3},\frac{4}{3},\frac{4}{3},\frac{4}{3},\frac{8}{3}]^{2}$ $[\frac{1}{3},\frac{1}{3},\frac{1}{3},\frac{1}{3},\frac{5}{3}]^{2}$ $[5, 0]$ $[2, 1]$ $z^4 + z^2 + 1,z + t^2$ $[3, 9]$
2.3.6.30a1.33 6 $( x^{3} + x + 1 )^{6} + 2 ( x^{3} + x + 1 )^{5} + \left(4 x + 4\right) ( x^{3} + x + 1 ) + 2$ $C_2^4:(A_4\times S_4)$ (as 18T462) $4608$ $2$ $[\frac{4}{3}, \frac{4}{3}, \frac{4}{3}, \frac{4}{3}, 2, 2, \frac{8}{3}, \frac{8}{3}]_{3}^{6}$ $[\frac{1}{3},\frac{1}{3},\frac{1}{3},\frac{1}{3},1,1,\frac{5}{3},\frac{5}{3}]_{3}^{6}$ $[\frac{4}{3},\frac{4}{3},\frac{4}{3},\frac{4}{3},2,2,\frac{8}{3}]^{2}$ $[\frac{1}{3},\frac{1}{3},\frac{1}{3},\frac{1}{3},1,1,\frac{5}{3}]^{2}$ $[5, 0]$ $[2, 1]$ $z^4 + z^2 + 1,z + t^2$ $[3, 9]$
2.3.6.30a1.34 6 $( x^{3} + x + 1 )^{6} + 2 ( x^{3} + x + 1 )^{5} + 4 ( x^{3} + x + 1 )^{4} + \left(4 x + 4\right) ( x^{3} + x + 1 ) + 2$ $C_2^4:(A_4\times S_4)$ (as 18T463) $4608$ $2$ $[\frac{4}{3}, \frac{4}{3}, \frac{4}{3}, \frac{4}{3}, 2, 2, \frac{8}{3}, \frac{8}{3}]_{3}^{6}$ $[\frac{1}{3},\frac{1}{3},\frac{1}{3},\frac{1}{3},1,1,\frac{5}{3},\frac{5}{3}]_{3}^{6}$ $[\frac{4}{3},\frac{4}{3},\frac{4}{3},\frac{4}{3},2,2,\frac{8}{3}]^{2}$ $[\frac{1}{3},\frac{1}{3},\frac{1}{3},\frac{1}{3},1,1,\frac{5}{3}]^{2}$ $[5, 0]$ $[2, 1]$ $z^4 + z^2 + 1,z + t^2$ $[3, 9]$
2.3.6.30a1.35 6 $( x^{3} + x + 1 )^{6} + 2 ( x^{3} + x + 1 )^{5} + 4 x^{2} ( x^{3} + x + 1 )^{3} + \left(4 x + 4\right) ( x^{3} + x + 1 ) + 2$ $C_2^4:(A_4\times S_4)$ (as 18T462) $4608$ $2$ $[\frac{4}{3}, \frac{4}{3}, \frac{4}{3}, \frac{4}{3}, 2, 2, \frac{8}{3}, \frac{8}{3}]_{3}^{6}$ $[\frac{1}{3},\frac{1}{3},\frac{1}{3},\frac{1}{3},1,1,\frac{5}{3},\frac{5}{3}]_{3}^{6}$ $[\frac{4}{3},\frac{4}{3},\frac{4}{3},\frac{4}{3},2,2,\frac{8}{3}]^{2}$ $[\frac{1}{3},\frac{1}{3},\frac{1}{3},\frac{1}{3},1,1,\frac{5}{3}]^{2}$ $[5, 0]$ $[2, 1]$ $z^4 + z^2 + 1,z + t^2$ $[3, 9]$
2.3.6.30a1.36 6 $( x^{3} + x + 1 )^{6} + 2 ( x^{3} + x + 1 )^{5} + 4 ( x^{3} + x + 1 )^{4} + 4 x^{2} ( x^{3} + x + 1 )^{3} + \left(4 x + 4\right) ( x^{3} + x + 1 ) + 2$ $C_2^4:(A_4\times S_4)$ (as 18T463) $4608$ $2$ $[\frac{4}{3}, \frac{4}{3}, \frac{4}{3}, \frac{4}{3}, 2, 2, \frac{8}{3}, \frac{8}{3}]_{3}^{6}$ $[\frac{1}{3},\frac{1}{3},\frac{1}{3},\frac{1}{3},1,1,\frac{5}{3},\frac{5}{3}]_{3}^{6}$ $[\frac{4}{3},\frac{4}{3},\frac{4}{3},\frac{4}{3},2,2,\frac{8}{3}]^{2}$ $[\frac{1}{3},\frac{1}{3},\frac{1}{3},\frac{1}{3},1,1,\frac{5}{3}]^{2}$ $[5, 0]$ $[2, 1]$ $z^4 + z^2 + 1,z + t^2$ $[3, 9]$
2.3.6.30a1.37 6 $( x^{3} + x + 1 )^{6} + 2 ( x^{3} + x + 1 )^{5} + 4 x ( x^{3} + x + 1 )^{3} + \left(4 x + 4\right) ( x^{3} + x + 1 ) + 2$ $C_2^4:(A_4\times S_4)$ (as 18T462) $4608$ $2$ $[\frac{4}{3}, \frac{4}{3}, \frac{4}{3}, \frac{4}{3}, 2, 2, \frac{8}{3}, \frac{8}{3}]_{3}^{6}$ $[\frac{1}{3},\frac{1}{3},\frac{1}{3},\frac{1}{3},1,1,\frac{5}{3},\frac{5}{3}]_{3}^{6}$ $[\frac{4}{3},\frac{4}{3},\frac{4}{3},\frac{4}{3},2,2,\frac{8}{3}]^{2}$ $[\frac{1}{3},\frac{1}{3},\frac{1}{3},\frac{1}{3},1,1,\frac{5}{3}]^{2}$ $[5, 0]$ $[2, 1]$ $z^4 + z^2 + 1,z + t^2$ $[3, 9]$
2.3.6.30a1.38 6 $( x^{3} + x + 1 )^{6} + 2 ( x^{3} + x + 1 )^{5} + 4 ( x^{3} + x + 1 )^{4} + 4 x ( x^{3} + x + 1 )^{3} + \left(4 x + 4\right) ( x^{3} + x + 1 ) + 2$ $C_2^4:(A_4\times S_4)$ (as 18T463) $4608$ $2$ $[\frac{4}{3}, \frac{4}{3}, \frac{4}{3}, \frac{4}{3}, 2, 2, \frac{8}{3}, \frac{8}{3}]_{3}^{6}$ $[\frac{1}{3},\frac{1}{3},\frac{1}{3},\frac{1}{3},1,1,\frac{5}{3},\frac{5}{3}]_{3}^{6}$ $[\frac{4}{3},\frac{4}{3},\frac{4}{3},\frac{4}{3},2,2,\frac{8}{3}]^{2}$ $[\frac{1}{3},\frac{1}{3},\frac{1}{3},\frac{1}{3},1,1,\frac{5}{3}]^{2}$ $[5, 0]$ $[2, 1]$ $z^4 + z^2 + 1,z + t^2$ $[3, 9]$
2.3.6.30a1.39 2 $( x^{3} + x + 1 )^{6} + 2 ( x^{3} + x + 1 )^{5} + \left(4 x^{2} + 4 x\right) ( x^{3} + x + 1 )^{3} + \left(4 x + 4\right) ( x^{3} + x + 1 ) + 2$ $A_4\times S_4$ (as 18T114) $288$ $2$ $[2, 2, \frac{8}{3}, \frac{8}{3}]_{3}^{6}$ $[1,1,\frac{5}{3},\frac{5}{3}]_{3}^{6}$ $[2,2,\frac{8}{3}]^{2}$ $[1,1,\frac{5}{3}]^{2}$ $[5, 0]$ $[2, 1]$ $z^4 + z^2 + 1,z + t^2$ $[3, 9]$
2.3.6.30a1.40 2 $( x^{3} + x + 1 )^{6} + 2 ( x^{3} + x + 1 )^{5} + 4 ( x^{3} + x + 1 )^{4} + \left(4 x^{2} + 4 x\right) ( x^{3} + x + 1 )^{3} + \left(4 x + 4\right) ( x^{3} + x + 1 ) + 2$ $A_4\times S_4$ (as 18T115) $288$ $2$ $[2, 2, \frac{8}{3}, \frac{8}{3}]_{3}^{6}$ $[1,1,\frac{5}{3},\frac{5}{3}]_{3}^{6}$ $[2,2,\frac{8}{3}]^{2}$ $[1,1,\frac{5}{3}]^{2}$ $[5, 0]$ $[2, 1]$ $z^4 + z^2 + 1,z + t^2$ $[3, 9]$
2.3.6.30a1.41 6 $( x^{3} + x + 1 )^{6} + 2 ( x^{3} + x + 1 )^{5} + 4 ( x^{3} + x + 1 )^{3} + \left(4 x + 4\right) ( x^{3} + x + 1 ) + 2$ $C_2^4:(A_4\times S_4)$ (as 18T462) $4608$ $2$ $[\frac{4}{3}, \frac{4}{3}, \frac{4}{3}, \frac{4}{3}, 2, 2, \frac{8}{3}, \frac{8}{3}]_{3}^{6}$ $[\frac{1}{3},\frac{1}{3},\frac{1}{3},\frac{1}{3},1,1,\frac{5}{3},\frac{5}{3}]_{3}^{6}$ $[\frac{4}{3},\frac{4}{3},\frac{4}{3},\frac{4}{3},2,2,\frac{8}{3}]^{2}$ $[\frac{1}{3},\frac{1}{3},\frac{1}{3},\frac{1}{3},1,1,\frac{5}{3}]^{2}$ $[5, 0]$ $[2, 1]$ $z^4 + z^2 + 1,z + t^2$ $[3, 9]$
2.3.6.30a1.42 6 $( x^{3} + x + 1 )^{6} + 2 ( x^{3} + x + 1 )^{5} + 4 ( x^{3} + x + 1 )^{4} + 4 ( x^{3} + x + 1 )^{3} + \left(4 x + 4\right) ( x^{3} + x + 1 ) + 2$ $C_2^4:(A_4\times S_4)$ (as 18T463) $4608$ $2$ $[\frac{4}{3}, \frac{4}{3}, \frac{4}{3}, \frac{4}{3}, 2, 2, \frac{8}{3}, \frac{8}{3}]_{3}^{6}$ $[\frac{1}{3},\frac{1}{3},\frac{1}{3},\frac{1}{3},1,1,\frac{5}{3},\frac{5}{3}]_{3}^{6}$ $[\frac{4}{3},\frac{4}{3},\frac{4}{3},\frac{4}{3},2,2,\frac{8}{3}]^{2}$ $[\frac{1}{3},\frac{1}{3},\frac{1}{3},\frac{1}{3},1,1,\frac{5}{3}]^{2}$ $[5, 0]$ $[2, 1]$ $z^4 + z^2 + 1,z + t^2$ $[3, 9]$
2.3.6.30a1.43 6 $( x^{3} + x + 1 )^{6} + 2 ( x^{3} + x + 1 )^{5} + \left(4 x^{2} + 4\right) ( x^{3} + x + 1 )^{3} + \left(4 x + 4\right) ( x^{3} + x + 1 ) + 2$ $C_2^4:(A_4\times S_4)$ (as 18T462) $4608$ $2$ $[\frac{4}{3}, \frac{4}{3}, \frac{4}{3}, \frac{4}{3}, 2, 2, \frac{8}{3}, \frac{8}{3}]_{3}^{6}$ $[\frac{1}{3},\frac{1}{3},\frac{1}{3},\frac{1}{3},1,1,\frac{5}{3},\frac{5}{3}]_{3}^{6}$ $[\frac{4}{3},\frac{4}{3},\frac{4}{3},\frac{4}{3},2,2,\frac{8}{3}]^{2}$ $[\frac{1}{3},\frac{1}{3},\frac{1}{3},\frac{1}{3},1,1,\frac{5}{3}]^{2}$ $[5, 0]$ $[2, 1]$ $z^4 + z^2 + 1,z + t^2$ $[3, 9]$
2.3.6.30a1.44 6 $( x^{3} + x + 1 )^{6} + 2 ( x^{3} + x + 1 )^{5} + 4 ( x^{3} + x + 1 )^{4} + \left(4 x^{2} + 4\right) ( x^{3} + x + 1 )^{3} + \left(4 x + 4\right) ( x^{3} + x + 1 ) + 2$ $C_2^4:(A_4\times S_4)$ (as 18T463) $4608$ $2$ $[\frac{4}{3}, \frac{4}{3}, \frac{4}{3}, \frac{4}{3}, 2, 2, \frac{8}{3}, \frac{8}{3}]_{3}^{6}$ $[\frac{1}{3},\frac{1}{3},\frac{1}{3},\frac{1}{3},1,1,\frac{5}{3},\frac{5}{3}]_{3}^{6}$ $[\frac{4}{3},\frac{4}{3},\frac{4}{3},\frac{4}{3},2,2,\frac{8}{3}]^{2}$ $[\frac{1}{3},\frac{1}{3},\frac{1}{3},\frac{1}{3},1,1,\frac{5}{3}]^{2}$ $[5, 0]$ $[2, 1]$ $z^4 + z^2 + 1,z + t^2$ $[3, 9]$
2.3.6.30a1.45 6 $( x^{3} + x + 1 )^{6} + 2 ( x^{3} + x + 1 )^{5} + \left(4 x + 4\right) ( x^{3} + x + 1 )^{3} + \left(4 x + 4\right) ( x^{3} + x + 1 ) + 2$ $C_2^4:(A_4\times S_4)$ (as 18T462) $4608$ $2$ $[\frac{4}{3}, \frac{4}{3}, \frac{4}{3}, \frac{4}{3}, 2, 2, \frac{8}{3}, \frac{8}{3}]_{3}^{6}$ $[\frac{1}{3},\frac{1}{3},\frac{1}{3},\frac{1}{3},1,1,\frac{5}{3},\frac{5}{3}]_{3}^{6}$ $[\frac{4}{3},\frac{4}{3},\frac{4}{3},\frac{4}{3},2,2,\frac{8}{3}]^{2}$ $[\frac{1}{3},\frac{1}{3},\frac{1}{3},\frac{1}{3},1,1,\frac{5}{3}]^{2}$ $[5, 0]$ $[2, 1]$ $z^4 + z^2 + 1,z + t^2$ $[3, 9]$
2.3.6.30a1.46 6 $( x^{3} + x + 1 )^{6} + 2 ( x^{3} + x + 1 )^{5} + 4 ( x^{3} + x + 1 )^{4} + \left(4 x + 4\right) ( x^{3} + x + 1 )^{3} + \left(4 x + 4\right) ( x^{3} + x + 1 ) + 2$ $C_2^4:(A_4\times S_4)$ (as 18T463) $4608$ $2$ $[\frac{4}{3}, \frac{4}{3}, \frac{4}{3}, \frac{4}{3}, 2, 2, \frac{8}{3}, \frac{8}{3}]_{3}^{6}$ $[\frac{1}{3},\frac{1}{3},\frac{1}{3},\frac{1}{3},1,1,\frac{5}{3},\frac{5}{3}]_{3}^{6}$ $[\frac{4}{3},\frac{4}{3},\frac{4}{3},\frac{4}{3},2,2,\frac{8}{3}]^{2}$ $[\frac{1}{3},\frac{1}{3},\frac{1}{3},\frac{1}{3},1,1,\frac{5}{3}]^{2}$ $[5, 0]$ $[2, 1]$ $z^4 + z^2 + 1,z + t^2$ $[3, 9]$
2.3.6.30a1.47 2 $( x^{3} + x + 1 )^{6} + 2 ( x^{3} + x + 1 )^{5} + \left(4 x^{2} + 4 x + 4\right) ( x^{3} + x + 1 )^{3} + \left(4 x + 4\right) ( x^{3} + x + 1 ) + 2$ $A_4\times S_4$ (as 18T114) $288$ $2$ $[2, 2, \frac{8}{3}, \frac{8}{3}]_{3}^{6}$ $[1,1,\frac{5}{3},\frac{5}{3}]_{3}^{6}$ $[2,2,\frac{8}{3}]^{2}$ $[1,1,\frac{5}{3}]^{2}$ $[5, 0]$ $[2, 1]$ $z^4 + z^2 + 1,z + t^2$ $[3, 9]$
2.3.6.30a1.48 2 $( x^{3} + x + 1 )^{6} + 2 ( x^{3} + x + 1 )^{5} + 4 ( x^{3} + x + 1 )^{4} + \left(4 x^{2} + 4 x + 4\right) ( x^{3} + x + 1 )^{3} + \left(4 x + 4\right) ( x^{3} + x + 1 ) + 2$ $A_4\times S_4$ (as 18T115) $288$ $2$ $[2, 2, \frac{8}{3}, \frac{8}{3}]_{3}^{6}$ $[1,1,\frac{5}{3},\frac{5}{3}]_{3}^{6}$ $[2,2,\frac{8}{3}]^{2}$ $[1,1,\frac{5}{3}]^{2}$ $[5, 0]$ $[2, 1]$ $z^4 + z^2 + 1,z + t^2$ $[3, 9]$
2.3.6.30a2.1 24 $( x^{3} + x + 1 )^{6} + 2 x ( x^{3} + x + 1 )^{5} + 2$ $C_2^4:(A_4\times S_4)$ (as 18T463) $4608$ $2$ $[2, 2, \frac{8}{3}, \frac{8}{3}, \frac{8}{3}, \frac{8}{3}, \frac{8}{3}, \frac{8}{3}]_{3}^{6}$ $[1,1,\frac{5}{3},\frac{5}{3},\frac{5}{3},\frac{5}{3},\frac{5}{3},\frac{5}{3}]_{3}^{6}$ $[2,2,\frac{8}{3},\frac{8}{3},\frac{8}{3},\frac{8}{3},\frac{8}{3}]^{2}$ $[1,1,\frac{5}{3},\frac{5}{3},\frac{5}{3},\frac{5}{3},\frac{5}{3}]^{2}$ $[5, 0]$ $[2, 1]$ $z^4 + z^2 + 1,z + (t + 1)$ $[3, 9]$
2.3.6.30a2.2 24 $( x^{3} + x + 1 )^{6} + 2 x ( x^{3} + x + 1 )^{5} + 4 ( x^{3} + x + 1 )^{4} + 2$ $C_2^4:(A_4\times S_4)$ (as 18T462) $4608$ $2$ $[2, 2, \frac{8}{3}, \frac{8}{3}, \frac{8}{3}, \frac{8}{3}, \frac{8}{3}, \frac{8}{3}]_{3}^{6}$ $[1,1,\frac{5}{3},\frac{5}{3},\frac{5}{3},\frac{5}{3},\frac{5}{3},\frac{5}{3}]_{3}^{6}$ $[2,2,\frac{8}{3},\frac{8}{3},\frac{8}{3},\frac{8}{3},\frac{8}{3}]^{2}$ $[1,1,\frac{5}{3},\frac{5}{3},\frac{5}{3},\frac{5}{3},\frac{5}{3}]^{2}$ $[5, 0]$ $[2, 1]$ $z^4 + z^2 + 1,z + (t + 1)$ $[3, 9]$
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