# Properties

 Label 10.11555266180939776.30t164.a Dimension 10 Group $S_6$ Conductor $2^{28} \cdot 3^{16}$ Frobenius-Schur indicator 1

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## Basic invariants

 Dimension: $10$ Group: $S_6$ Conductor: $11555266180939776= 2^{28} \cdot 3^{16}$ Artin number field: Splitting field of $f= x^{6} - 6 x^{4} - 4 x^{3} + 6 x^{2} - 6$ over $\Q$ Size of Galois orbit: 1 Smallest containing permutation representation: 30T164 Parity: Even Projective image: $S_6$ Projective field: Galois closure of 6.2.13436928.5

## Galois action

### Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 167 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 167 }$: $x^{2} + 166 x + 5$
Roots:
 $r_{ 1 }$ $=$ $125 + 13\cdot 167 + 71\cdot 167^{2} + 8\cdot 167^{3} + 52\cdot 167^{4} +O\left(167^{ 5 }\right)$ $r_{ 2 }$ $=$ $141 + 151\cdot 167 + 21\cdot 167^{2} + 13\cdot 167^{3} + 133\cdot 167^{4} +O\left(167^{ 5 }\right)$ $r_{ 3 }$ $=$ $67 a + 128 + \left(164 a + 103\right)\cdot 167 + \left(112 a + 44\right)\cdot 167^{2} + \left(59 a + 111\right)\cdot 167^{3} + \left(a + 25\right)\cdot 167^{4} +O\left(167^{ 5 }\right)$ $r_{ 4 }$ $=$ $100 a + 28 + \left(2 a + 34\right)\cdot 167 + \left(54 a + 160\right)\cdot 167^{2} + \left(107 a + 57\right)\cdot 167^{3} + \left(165 a + 134\right)\cdot 167^{4} +O\left(167^{ 5 }\right)$ $r_{ 5 }$ $=$ $131 a + 141 + \left(44 a + 141\right)\cdot 167 + \left(132 a + 57\right)\cdot 167^{2} + \left(58 a + 108\right)\cdot 167^{3} + \left(17 a + 98\right)\cdot 167^{4} +O\left(167^{ 5 }\right)$ $r_{ 6 }$ $=$ $36 a + 105 + \left(122 a + 55\right)\cdot 167 + \left(34 a + 145\right)\cdot 167^{2} + \left(108 a + 34\right)\cdot 167^{3} + \left(149 a + 57\right)\cdot 167^{4} +O\left(167^{ 5 }\right)$

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

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

### Character values on conjugacy classes

 Size Order Action on $r_1, \ldots, r_{ 6 }$ Character values $c1$ $1$ $1$ $()$ $10$ $15$ $2$ $(1,2)(3,4)(5,6)$ $2$ $15$ $2$ $(1,2)$ $-2$ $45$ $2$ $(1,2)(3,4)$ $-2$ $40$ $3$ $(1,2,3)(4,5,6)$ $1$ $40$ $3$ $(1,2,3)$ $1$ $90$ $4$ $(1,2,3,4)(5,6)$ $0$ $90$ $4$ $(1,2,3,4)$ $0$ $144$ $5$ $(1,2,3,4,5)$ $0$ $120$ $6$ $(1,2,3,4,5,6)$ $-1$ $120$ $6$ $(1,2,3)(4,5)$ $1$
The blue line marks the conjugacy class containing complex conjugation.