Basic invariants
| Dimension: | $3$ |
| Group: | $S_4\times C_2$ |
| Conductor: | \(1672\)\(\medspace = 2^{3} \cdot 11 \cdot 19 \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 6.0.254144.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $S_4\times C_2$ |
| Parity: | even |
| Determinant: | 1.1672.2t1.a.a |
| Projective image: | $S_4$ |
| Projective stem field: | Galois closure of 4.2.18392.2 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} - x^{5} + 2x^{4} + x^{3} + x^{2} + 2x + 2 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 37 }$ to precision 7.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 37 }$:
\( x^{2} + 33x + 2 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 19 a + 5 + \left(36 a + 9\right)\cdot 37 + \left(16 a + 15\right)\cdot 37^{2} + \left(20 a + 4\right)\cdot 37^{3} + \left(2 a + 31\right)\cdot 37^{4} + \left(33 a + 1\right)\cdot 37^{5} + 15 a\cdot 37^{6} +O(37^{7})\)
|
| $r_{ 2 }$ | $=$ |
\( 27 a + 5 + \left(12 a + 20\right)\cdot 37 + \left(31 a + 29\right)\cdot 37^{2} + \left(4 a + 31\right)\cdot 37^{3} + \left(12 a + 7\right)\cdot 37^{4} + 9 a\cdot 37^{5} + \left(3 a + 12\right)\cdot 37^{6} +O(37^{7})\)
|
| $r_{ 3 }$ | $=$ |
\( 18 a + 7 + 25\cdot 37 + \left(20 a + 9\right)\cdot 37^{2} + \left(16 a + 32\right)\cdot 37^{3} + \left(34 a + 20\right)\cdot 37^{4} + \left(3 a + 20\right)\cdot 37^{5} + \left(21 a + 30\right)\cdot 37^{6} +O(37^{7})\)
|
| $r_{ 4 }$ | $=$ |
\( 10 a + 2 + \left(24 a + 7\right)\cdot 37 + \left(5 a + 31\right)\cdot 37^{2} + \left(32 a + 19\right)\cdot 37^{3} + \left(24 a + 14\right)\cdot 37^{4} + \left(27 a + 25\right)\cdot 37^{5} + \left(33 a + 15\right)\cdot 37^{6} +O(37^{7})\)
|
| $r_{ 5 }$ | $=$ |
\( 5 + 19\cdot 37 + 35\cdot 37^{4} + 20\cdot 37^{5} + 3\cdot 37^{6} +O(37^{7})\)
|
| $r_{ 6 }$ | $=$ |
\( 14 + 30\cdot 37 + 24\cdot 37^{2} + 22\cdot 37^{3} + 37^{4} + 5\cdot 37^{5} + 12\cdot 37^{6} +O(37^{7})\)
|
Generators of the action on the roots $r_1, \ldots, r_{ 6 }$
| Cycle notation |
Character values on conjugacy classes
| Size | Order | Action on $r_1, \ldots, r_{ 6 }$ | Character value |
| $1$ | $1$ | $()$ | $3$ |
| $1$ | $2$ | $(1,3)(2,4)(5,6)$ | $-3$ |
| $3$ | $2$ | $(1,3)$ | $1$ |
| $3$ | $2$ | $(1,3)(5,6)$ | $-1$ |
| $6$ | $2$ | $(2,5)(4,6)$ | $1$ |
| $6$ | $2$ | $(1,3)(2,5)(4,6)$ | $-1$ |
| $8$ | $3$ | $(1,5,2)(3,6,4)$ | $0$ |
| $6$ | $4$ | $(1,6,3,5)$ | $1$ |
| $6$ | $4$ | $(1,6,3,5)(2,4)$ | $-1$ |
| $8$ | $6$ | $(1,6,4,3,5,2)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.