# Properties

 Label 6.33230963375.12t108.b.a Dimension 6 Group $V_4^2:(S_3\times C_2)$ Conductor $5^{3} \cdot 643^{3}$ Root number 1 Frobenius-Schur indicator 1

# Related objects

## Basic invariants

 Dimension: $6$ Group: $V_4^2:(S_3\times C_2)$ Conductor: $33230963375= 5^{3} \cdot 643^{3}$ Artin number field: Splitting field of 8.4.258405625.1 defined by $f= x^{8} - 3 x^{7} + 4 x^{6} - 5 x^{5} + x^{4} + x^{3} + 8 x^{2} - 7 x + 1$ over $\Q$ Size of Galois orbit: 1 Smallest containing permutation representation: 12T108 Parity: Odd Determinant: 1.3215.2t1.a.a Projective image: $C_2^2:S_4:C_2$ Projective field: Galois closure of 8.4.258405625.1

## Galois action

### Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 29 }$ to precision 29.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 29 }$: $x^{3} + 2 x + 27$
Roots:
 $r_{ 1 }$ $=$ $16 a^{2} + 12 a + 19 + \left(4 a^{2} + 22 a + 22\right)\cdot 29 + \left(25 a^{2} + 23 a + 8\right)\cdot 29^{2} + \left(9 a^{2} + 14 a + 9\right)\cdot 29^{3} + \left(13 a^{2} + 10 a + 23\right)\cdot 29^{4} + \left(9 a^{2} + 4 a + 9\right)\cdot 29^{5} + \left(18 a^{2} + 25 a + 20\right)\cdot 29^{6} + \left(12 a^{2} + 11\right)\cdot 29^{7} + \left(18 a^{2} + 21 a + 15\right)\cdot 29^{8} + \left(25 a^{2} + 6 a\right)\cdot 29^{9} + \left(23 a^{2} + 3\right)\cdot 29^{10} + \left(13 a^{2} + 14 a + 11\right)\cdot 29^{11} + \left(15 a^{2} + 16 a + 12\right)\cdot 29^{12} + \left(4 a^{2} + 17 a + 16\right)\cdot 29^{13} + \left(2 a^{2} + 18 a + 21\right)\cdot 29^{14} + \left(18 a^{2} + 4 a\right)\cdot 29^{15} + \left(18 a^{2} + 4 a + 6\right)\cdot 29^{16} + \left(4 a^{2} + 19 a + 22\right)\cdot 29^{17} + \left(4 a^{2} + 12 a\right)\cdot 29^{18} + \left(7 a^{2} + 15 a + 20\right)\cdot 29^{19} + \left(12 a^{2} + 25 a + 21\right)\cdot 29^{20} + \left(24 a^{2} + 8 a + 22\right)\cdot 29^{21} + \left(19 a^{2} + a + 11\right)\cdot 29^{22} + \left(26 a^{2} + 27 a + 4\right)\cdot 29^{23} + \left(12 a^{2} + 11 a + 8\right)\cdot 29^{24} + \left(27 a^{2} + 28 a + 23\right)\cdot 29^{25} + \left(21 a^{2} + 3 a + 1\right)\cdot 29^{26} + \left(22 a^{2} + 6 a + 14\right)\cdot 29^{27} + \left(5 a^{2} + 23 a + 1\right)\cdot 29^{28} +O\left(29^{ 29 }\right)$ $r_{ 2 }$ $=$ $25 a^{2} + 7 a + 2 + \left(5 a^{2} + 26 a + 5\right)\cdot 29 + \left(6 a^{2} + 13 a + 22\right)\cdot 29^{2} + \left(3 a^{2} + 22 a + 19\right)\cdot 29^{3} + \left(23 a^{2} + 2 a + 26\right)\cdot 29^{4} + \left(25 a^{2} + 14 a + 21\right)\cdot 29^{5} + \left(21 a^{2} + a + 5\right)\cdot 29^{6} + \left(23 a^{2} + 14 a + 7\right)\cdot 29^{7} + \left(10 a^{2} + 20 a + 5\right)\cdot 29^{8} + \left(5 a^{2} + 20 a + 12\right)\cdot 29^{9} + \left(20 a^{2} + 28 a + 17\right)\cdot 29^{10} + \left(15 a + 22\right)\cdot 29^{11} + \left(7 a^{2} + 10\right)\cdot 29^{12} + \left(20 a^{2} + 19 a + 8\right)\cdot 29^{13} + \left(10 a^{2} + 19 a + 23\right)\cdot 29^{14} + \left(28 a^{2} + 8 a + 4\right)\cdot 29^{15} + \left(25 a^{2} + 21 a + 6\right)\cdot 29^{16} + \left(26 a^{2} + 27 a + 13\right)\cdot 29^{17} + \left(6 a^{2} + 27 a + 4\right)\cdot 29^{18} + \left(3 a^{2} + 17 a + 5\right)\cdot 29^{19} + \left(23 a^{2} + 12 a + 7\right)\cdot 29^{20} + \left(13 a^{2} + 26 a + 18\right)\cdot 29^{21} + \left(26 a^{2} + 20 a + 20\right)\cdot 29^{22} + \left(17 a^{2} + 8 a + 21\right)\cdot 29^{23} + \left(4 a^{2} + 3 a + 6\right)\cdot 29^{24} + \left(12 a^{2} + 3 a + 22\right)\cdot 29^{25} + \left(24 a^{2} + a + 14\right)\cdot 29^{26} + \left(14 a^{2} + 8 a + 3\right)\cdot 29^{27} + \left(12 a^{2} + 12 a + 20\right)\cdot 29^{28} +O\left(29^{ 29 }\right)$ $r_{ 3 }$ $=$ $27 a + 28 + \left(9 a^{2} + 3 a + 13\right)\cdot 29 + \left(20 a^{2} + 19 a + 25\right)\cdot 29^{2} + \left(19 a^{2} + 18 a + 18\right)\cdot 29^{3} + \left(20 a + 24\right)\cdot 29^{4} + \left(27 a^{2} + 14 a + 4\right)\cdot 29^{5} + \left(22 a^{2} + 12 a\right)\cdot 29^{6} + \left(8 a^{2} + 13 a + 22\right)\cdot 29^{7} + \left(26 a^{2} + 26 a + 23\right)\cdot 29^{8} + \left(22 a^{2} + 17 a + 8\right)\cdot 29^{9} + \left(22 a^{2} + 16 a + 5\right)\cdot 29^{10} + \left(3 a^{2} + a + 2\right)\cdot 29^{11} + \left(18 a^{2} + 15 a + 14\right)\cdot 29^{12} + \left(19 a^{2} + 27 a + 7\right)\cdot 29^{13} + \left(14 a^{2} + 25\right)\cdot 29^{14} + \left(23 a^{2} + 13 a + 10\right)\cdot 29^{15} + \left(24 a^{2} + 28 a + 21\right)\cdot 29^{16} + \left(10 a^{2} + 28 a + 1\right)\cdot 29^{17} + \left(23 a^{2} + 10 a + 25\right)\cdot 29^{18} + \left(13 a^{2} + 12 a + 6\right)\cdot 29^{19} + \left(22 a^{2} + 28 a + 27\right)\cdot 29^{20} + \left(19 a^{2} + 15 a + 28\right)\cdot 29^{21} + \left(13 a^{2} + 15 a + 11\right)\cdot 29^{22} + \left(8 a^{2} + 13 a + 14\right)\cdot 29^{23} + \left(24 a^{2} + 2 a + 15\right)\cdot 29^{24} + \left(26 a^{2} + 20 a + 24\right)\cdot 29^{25} + \left(17 a^{2} + 28 a + 5\right)\cdot 29^{26} + \left(20 a^{2} + 20 a + 20\right)\cdot 29^{27} + \left(11 a^{2} + 13 a + 23\right)\cdot 29^{28} +O\left(29^{ 29 }\right)$ $r_{ 4 }$ $=$ $9 a^{2} + 11 + \left(2 a^{2} + 3 a + 24\right)\cdot 29 + \left(9 a^{2} + 9 a\right)\cdot 29^{2} + \left(22 a^{2} + 15 a + 3\right)\cdot 29^{3} + \left(7 a^{2} + 8 a + 5\right)\cdot 29^{4} + \left(10 a^{2} + 20 a + 21\right)\cdot 29^{5} + \left(8 a^{2} + 9 a + 9\right)\cdot 29^{6} + \left(11 a^{2} + 6 a + 25\right)\cdot 29^{7} + \left(10 a^{2} + 25 a + 21\right)\cdot 29^{8} + \left(5 a^{2} + 11 a + 4\right)\cdot 29^{9} + \left(a^{2} + 7 a + 15\right)\cdot 29^{10} + \left(13 a^{2} + 23 a + 14\right)\cdot 29^{11} + \left(3 a^{2} + 26 a + 23\right)\cdot 29^{12} + \left(12 a^{2} + 16\right)\cdot 29^{13} + \left(4 a^{2} + 18 a + 11\right)\cdot 29^{14} + \left(27 a^{2} + 12 a + 25\right)\cdot 29^{15} + \left(26 a^{2} + 19 a + 4\right)\cdot 29^{16} + \left(16 a^{2} + 23 a\right)\cdot 29^{17} + \left(27 a^{2} + 21 a + 21\right)\cdot 29^{18} + \left(13 a^{2} + 20 a + 16\right)\cdot 29^{19} + \left(6 a^{2} + 11 a + 15\right)\cdot 29^{20} + \left(5 a^{2} + 3 a + 9\right)\cdot 29^{21} + \left(27 a^{2} + 19 a + 20\right)\cdot 29^{22} + \left(a^{2} + 22 a + 5\right)\cdot 29^{23} + \left(14 a^{2} + 20 a + 21\right)\cdot 29^{24} + \left(15 a^{2} + 16 a + 28\right)\cdot 29^{25} + \left(18 a^{2} + 10 a + 25\right)\cdot 29^{26} + \left(16 a^{2} + 2 a + 14\right)\cdot 29^{27} + \left(21 a^{2} + 8 a + 17\right)\cdot 29^{28} +O\left(29^{ 29 }\right)$ $r_{ 5 }$ $=$ $20 a^{2} + 2 a + 16 + \left(17 a^{2} + 22 a + 25\right)\cdot 29 + \left(28 a^{2} + 26\right)\cdot 29^{2} + \left(15 a^{2} + 24 a + 13\right)\cdot 29^{3} + \left(20 a^{2} + 28 a + 12\right)\cdot 29^{4} + \left(20 a^{2} + 22 a + 25\right)\cdot 29^{5} + \left(26 a^{2} + 6 a + 14\right)\cdot 29^{6} + \left(8 a^{2} + 9 a + 12\right)\cdot 29^{7} + \left(21 a^{2} + 6 a + 7\right)\cdot 29^{8} + \left(28 a + 8\right)\cdot 29^{9} + \left(5 a^{2} + 4 a + 20\right)\cdot 29^{10} + \left(12 a^{2} + 4 a + 3\right)\cdot 29^{11} + \left(7 a^{2} + 16 a + 19\right)\cdot 29^{12} + \left(26 a^{2} + 6\right)\cdot 29^{13} + \left(9 a^{2} + 10 a + 9\right)\cdot 29^{14} + \left(7 a^{2} + 3 a + 18\right)\cdot 29^{15} + \left(6 a^{2} + 10 a + 25\right)\cdot 29^{16} + \left(a^{2} + 5 a + 17\right)\cdot 29^{17} + \left(7 a^{2} + 25 a + 22\right)\cdot 29^{18} + \left(a^{2} + 24 a + 28\right)\cdot 29^{19} + \left(17 a + 6\right)\cdot 29^{20} + \left(4 a^{2} + 9 a + 27\right)\cdot 29^{21} + \left(17 a^{2} + 23 a + 6\right)\cdot 29^{22} + \left(18 a^{2} + 21 a + 18\right)\cdot 29^{23} + \left(19 a^{2} + 5 a + 28\right)\cdot 29^{24} + \left(15 a^{2} + 21 a + 28\right)\cdot 29^{25} + \left(21 a^{2} + 18 a\right)\cdot 29^{26} + \left(20 a^{2} + 5 a + 1\right)\cdot 29^{27} + \left(24 a^{2} + 7 a + 12\right)\cdot 29^{28} +O\left(29^{ 29 }\right)$ $r_{ 6 }$ $=$ $14 + 21\cdot 29^{2} + 10\cdot 29^{3} + 27\cdot 29^{4} + 9\cdot 29^{5} + 11\cdot 29^{6} + 15\cdot 29^{7} + 5\cdot 29^{8} + 23\cdot 29^{9} + 3\cdot 29^{10} + 16\cdot 29^{11} + 18\cdot 29^{12} + 10\cdot 29^{13} + 2\cdot 29^{14} + 22\cdot 29^{15} + 10\cdot 29^{16} + 9\cdot 29^{17} + 10\cdot 29^{18} + 25\cdot 29^{19} + 13\cdot 29^{20} + 29^{21} + 19\cdot 29^{22} + 9\cdot 29^{23} + 11\cdot 29^{24} + 20\cdot 29^{25} + 24\cdot 29^{26} + 18\cdot 29^{27} + 20\cdot 29^{28} +O\left(29^{ 29 }\right)$ $r_{ 7 }$ $=$ $28 + 29 + 23\cdot 29^{3} + 5\cdot 29^{5} + 5\cdot 29^{6} + 27\cdot 29^{7} + 26\cdot 29^{8} + 26\cdot 29^{9} + 12\cdot 29^{10} + 14\cdot 29^{11} + 7\cdot 29^{12} + 14\cdot 29^{13} + 11\cdot 29^{14} + 22\cdot 29^{15} + 22\cdot 29^{16} + 9\cdot 29^{17} + 22\cdot 29^{18} + 6\cdot 29^{19} + 7\cdot 29^{20} + 20\cdot 29^{21} + 14\cdot 29^{22} + 16\cdot 29^{23} + 8\cdot 29^{24} + 24\cdot 29^{25} + 24\cdot 29^{26} + 22\cdot 29^{27} + 2\cdot 29^{28} +O\left(29^{ 29 }\right)$ $r_{ 8 }$ $=$ $17 a^{2} + 10 a + 1 + \left(18 a^{2} + 9 a + 22\right)\cdot 29 + \left(26 a^{2} + 20 a + 10\right)\cdot 29^{2} + \left(15 a^{2} + 20 a + 17\right)\cdot 29^{3} + \left(21 a^{2} + 15 a + 24\right)\cdot 29^{4} + \left(22 a^{2} + 10 a + 17\right)\cdot 29^{5} + \left(17 a^{2} + 2 a + 19\right)\cdot 29^{6} + \left(21 a^{2} + 14 a + 23\right)\cdot 29^{7} + \left(28 a^{2} + 16 a + 9\right)\cdot 29^{8} + \left(26 a^{2} + a + 2\right)\cdot 29^{9} + \left(13 a^{2} + 9\right)\cdot 29^{10} + \left(14 a^{2} + 28 a + 2\right)\cdot 29^{11} + \left(6 a^{2} + 11 a + 10\right)\cdot 29^{12} + \left(4 a^{2} + 21 a + 6\right)\cdot 29^{13} + \left(16 a^{2} + 19 a + 11\right)\cdot 29^{14} + \left(11 a^{2} + 15 a + 11\right)\cdot 29^{15} + \left(13 a^{2} + 3 a + 18\right)\cdot 29^{16} + \left(26 a^{2} + 11 a + 12\right)\cdot 29^{17} + \left(17 a^{2} + 17 a + 9\right)\cdot 29^{18} + \left(18 a^{2} + 24 a + 6\right)\cdot 29^{19} + \left(22 a^{2} + 19 a + 16\right)\cdot 29^{20} + \left(19 a^{2} + 22 a + 16\right)\cdot 29^{21} + \left(11 a^{2} + 6 a + 10\right)\cdot 29^{22} + \left(13 a^{2} + 22 a + 25\right)\cdot 29^{23} + \left(11 a^{2} + 13 a + 15\right)\cdot 29^{24} + \left(18 a^{2} + 26 a + 1\right)\cdot 29^{25} + \left(11 a^{2} + 23 a + 17\right)\cdot 29^{26} + \left(20 a^{2} + 14 a + 20\right)\cdot 29^{27} + \left(10 a^{2} + 22 a + 17\right)\cdot 29^{28} +O\left(29^{ 29 }\right)$

### Generators of the action on the roots $r_1, \ldots, r_{ 8 }$

 Cycle notation $(5,7)(6,8)$ $(1,2)(6,8)$ $(3,5)(4,7)$ $(1,4)(2,5)(3,8)(6,7)$ $(3,7)(4,5)$ $(1,8)(2,6)$ $(2,6,8)(3,7,5)$

### Character values on conjugacy classes

 Size Order Action on $r_1, \ldots, r_{ 8 }$ Character value $1$ $1$ $()$ $6$ $3$ $2$ $(1,6)(2,8)(3,5)(4,7)$ $-2$ $4$ $2$ $(1,4)(2,3)(5,8)(6,7)$ $0$ $6$ $2$ $(1,2)(6,8)$ $-2$ $6$ $2$ $(1,2)(3,7)(4,5)(6,8)$ $2$ $12$ $2$ $(1,4)(2,5)(3,8)(6,7)$ $-2$ $12$ $2$ $(5,7)(6,8)$ $0$ $32$ $3$ $(1,6,2)(3,7,5)$ $0$ $12$ $4$ $(1,7,6,4)(2,3,8,5)$ $2$ $12$ $4$ $(1,8,2,6)(3,5,4,7)$ $0$ $12$ $4$ $(1,7,6,4)(2,5,8,3)$ $0$ $24$ $4$ $(1,5,2,4)(3,8,7,6)$ $0$ $24$ $4$ $(1,8,2,6)(5,7)$ $0$ $32$ $6$ $(1,4)(2,5,6,3,8,7)$ $0$
The blue line marks the conjugacy class containing complex conjugation.