# Properties

 Label 4.22784.6t13.b Dimension 4 Group $C_3^2:D_4$ Conductor $2^{8} \cdot 89$ Frobenius-Schur indicator 1

# Related objects

## Basic invariants

 Dimension: $4$ Group: $C_3^2:D_4$ Conductor: $22784= 2^{8} \cdot 89$ Artin number field: Splitting field of $f= x^{6} - 2 x^{5} + x^{4} - 3 x^{2} + 2 x - 1$ over $\Q$ Size of Galois orbit: 1 Smallest containing permutation representation: $C_3^2:D_4$ Parity: Even Projective image: $S_3\wr C_2$ Projective field: Galois closure of 6.2.182272.1

## Galois action

### Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 73 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 73 }$: $x^{2} + 70 x + 5$
Roots:
 $r_{ 1 }$ $=$ $26 a + 61 + \left(42 a + 9\right)\cdot 73 + 46 a\cdot 73^{2} + \left(51 a + 27\right)\cdot 73^{3} + \left(7 a + 64\right)\cdot 73^{4} +O\left(73^{ 5 }\right)$ $r_{ 2 }$ $=$ $4 + 55\cdot 73 + 22\cdot 73^{2} + 55\cdot 73^{4} +O\left(73^{ 5 }\right)$ $r_{ 3 }$ $=$ $47 a + 66 + \left(30 a + 37\right)\cdot 73 + \left(26 a + 24\right)\cdot 73^{2} + \left(21 a + 62\right)\cdot 73^{3} + \left(65 a + 35\right)\cdot 73^{4} +O\left(73^{ 5 }\right)$ $r_{ 4 }$ $=$ $22 a + 18 + \left(25 a + 72\right)\cdot 73 + 73^{2} + \left(48 a + 42\right)\cdot 73^{3} + \left(39 a + 24\right)\cdot 73^{4} +O\left(73^{ 5 }\right)$ $r_{ 5 }$ $=$ $61 + 63\cdot 73 + 45\cdot 73^{2} + 47\cdot 73^{3} + 16\cdot 73^{4} +O\left(73^{ 5 }\right)$ $r_{ 6 }$ $=$ $51 a + 11 + \left(47 a + 53\right)\cdot 73 + \left(72 a + 50\right)\cdot 73^{2} + \left(24 a + 39\right)\cdot 73^{3} + \left(33 a + 22\right)\cdot 73^{4} +O\left(73^{ 5 }\right)$

### Generators of the action on the roots $r_1, \ldots, r_{ 6 }$

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

### Character values on conjugacy classes

 Size Order Action on $r_1, \ldots, r_{ 6 }$ Character values $c1$ $1$ $1$ $()$ $4$ $6$ $2$ $(1,2)(3,4)(5,6)$ $0$ $6$ $2$ $(3,5)$ $2$ $9$ $2$ $(3,5)(4,6)$ $0$ $4$ $3$ $(1,3,5)(2,4,6)$ $-2$ $4$ $3$ $(1,3,5)$ $1$ $18$ $4$ $(1,2)(3,6,5,4)$ $0$ $12$ $6$ $(1,4,3,6,5,2)$ $0$ $12$ $6$ $(2,4,6)(3,5)$ $-1$
The blue line marks the conjugacy class containing complex conjugation.