# Properties

 Label 2.751.24t22.a Dimension 2 Group $\textrm{GL(2,3)}$ Conductor $751$ Frobenius-Schur indicator 0

# Related objects

## Basic invariants

 Dimension: $2$ Group: $\textrm{GL(2,3)}$ Conductor: $751$ Artin number field: Splitting field of $f= x^{8} - x^{7} - 2 x^{6} + 4 x^{5} - 9 x^{4} + 17 x^{3} - 21 x^{2} + 7 x - 4$ over $\Q$ Size of Galois orbit: 2 Smallest containing permutation representation: 24T22 Parity: Odd Projective image: $S_4$ Projective field: Galois closure of 4.2.751.1

## Galois action

### Roots of defining polynomial

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

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

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

### Character values on conjugacy classes

 Size Order Action on $r_1, \ldots, r_{ 8 }$ Character values $c1$ $c2$ $1$ $1$ $()$ $2$ $2$ $1$ $2$ $(1,7)(2,5)(3,6)(4,8)$ $-2$ $-2$ $12$ $2$ $(1,4)(2,5)(7,8)$ $0$ $0$ $8$ $3$ $(1,8,2)(4,5,7)$ $-1$ $-1$ $6$ $4$ $(1,2,7,5)(3,8,6,4)$ $0$ $0$ $8$ $6$ $(1,4,2,7,8,5)(3,6)$ $1$ $1$ $6$ $8$ $(1,5,4,3,7,2,8,6)$ $-\zeta_{8}^{3} - \zeta_{8}$ $\zeta_{8}^{3} + \zeta_{8}$ $6$ $8$ $(1,2,4,6,7,5,8,3)$ $\zeta_{8}^{3} + \zeta_{8}$ $-\zeta_{8}^{3} - \zeta_{8}$
The blue line marks the conjugacy class containing complex conjugation.