# Properties

 Label 3.2e8_11e3.42t37.1 Dimension 3 Group $\GL(3,2)$ Conductor $2^{8} \cdot 11^{3}$ Frobenius-Schur indicator 0

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## Basic invariants

 Dimension: $3$ Group: $\GL(3,2)$ Conductor: $340736= 2^{8} \cdot 11^{3}$ Artin number field: Splitting field of $f= x^{7} - x^{6} + 2 x^{5} - 12 x^{4} - 14 x^{3} + 10 x^{2} + 10 x - 2$ over $\Q$ Size of Galois orbit: 2 Smallest containing permutation representation: $\PSL(2,7)$ Parity: Even

## Galois action

### Roots of defining polynomial

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

### Generators of the action on the roots $r_1, \ldots, r_{ 7 }$

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

### Character values on conjugacy classes

 Size Order Action on $r_1, \ldots, r_{ 7 }$ Character values $c1$ $c2$ $1$ $1$ $()$ $3$ $3$ $21$ $2$ $(2,6)(3,4)$ $-1$ $-1$ $56$ $3$ $(1,4,5)(2,7,3)$ $0$ $0$ $42$ $4$ $(1,7,4,2)(5,6)$ $1$ $1$ $24$ $7$ $(1,7,3,4,6,5,2)$ $\zeta_{7}^{4} + \zeta_{7}^{2} + \zeta_{7}$ $-\zeta_{7}^{4} - \zeta_{7}^{2} - \zeta_{7} - 1$ $24$ $7$ $(1,4,2,3,5,7,6)$ $-\zeta_{7}^{4} - \zeta_{7}^{2} - \zeta_{7} - 1$ $\zeta_{7}^{4} + \zeta_{7}^{2} + \zeta_{7}$
The blue line marks the conjugacy class containing complex conjugation.