# Properties

 Label 34T6 Degree $34$ Order $136$ Cyclic no Abelian no Solvable yes Primitive no $p$-group no Group: $C_{17}:C_8$

## Group action invariants

 Degree $n$: $34$ Transitive number $t$: $6$ Group: $C_{17}:C_8$ Parity: $-1$ Primitive: no Nilpotency class: $-1$ (not nilpotent) $|\Aut(F/K)|$: $2$ Generators: (1,28,24,21,4,12,15,17)(2,27,23,22,3,11,16,18)(5,30,7,14,34,9,31,25)(6,29,8,13,33,10,32,26)(19,20), (1,20)(2,19)(3,17)(4,18)(5,16)(6,15)(7,13)(8,14)(9,11)(10,12)(21,34)(22,33)(23,31)(24,32)(25,30)(26,29)

## Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$
$4$:  $C_4$
$8$:  $C_8$

Resolvents shown for degrees $\leq 47$

## Subfields

Degree 2: $C_2$

Degree 17: $C_{17}:C_{8}$

## Low degree siblings

17T4

Siblings are shown with degree $\leq 47$

A number field with this Galois group has no arithmetically equivalent fields.

## Conjugacy classes

 Cycle Type Size Order Representative $1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1$ $1$ $1$ $()$ $4, 4, 4, 4, 4, 4, 4, 4, 1, 1$ $17$ $4$ $( 3,10,34,28)( 4, 9,33,27)( 5,17,31,19)( 6,18,32,20)( 7,26,29,12)( 8,25,30,11) (13,16,23,21)(14,15,24,22)$ $4, 4, 4, 4, 4, 4, 4, 4, 1, 1$ $17$ $4$ $( 3,28,34,10)( 4,27,33, 9)( 5,19,31,17)( 6,20,32,18)( 7,12,29,26)( 8,11,30,25) (13,21,23,16)(14,22,24,15)$ $2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1$ $17$ $2$ $( 3,34)( 4,33)( 5,31)( 6,32)( 7,29)( 8,30)( 9,27)(10,28)(11,25)(12,26)(13,23) (14,24)(15,22)(16,21)(17,19)(18,20)$ $8, 8, 8, 8, 2$ $17$ $8$ $( 1, 2)( 3, 6,10,18,34,32,28,20)( 4, 5, 9,17,33,31,27,19)( 7,14,26,15,29,24, 12,22)( 8,13,25,16,30,23,11,21)$ $8, 8, 8, 8, 2$ $17$ $8$ $( 1, 2)( 3,18,28, 6,34,20,10,32)( 4,17,27, 5,33,19, 9,31)( 7,15,12,14,29,22, 26,24)( 8,16,11,13,30,21,25,23)$ $8, 8, 8, 8, 2$ $17$ $8$ $( 1, 2)( 3,20,28,32,34,18,10, 6)( 4,19,27,31,33,17, 9, 5)( 7,22,12,24,29,15, 26,14)( 8,21,11,23,30,16,25,13)$ $8, 8, 8, 8, 2$ $17$ $8$ $( 1, 2)( 3,32,10,20,34, 6,28,18)( 4,31, 9,19,33, 5,27,17)( 7,24,26,22,29,14, 12,15)( 8,23,25,21,30,13,11,16)$ $17, 17$ $8$ $17$ $( 1, 4, 6, 8, 9,11,14,15,18,20,22,24,25,27,30,32,33)( 2, 3, 5, 7,10,12,13,16, 17,19,21,23,26,28,29,31,34)$ $17, 17$ $8$ $17$ $( 1, 8,14,20,25,32, 4, 9,15,22,27,33, 6,11,18,24,30)( 2, 7,13,19,26,31, 3,10, 16,21,28,34, 5,12,17,23,29)$

## Group invariants

 Order: $136=2^{3} \cdot 17$ Cyclic: no Abelian: no Solvable: yes GAP id: [136, 12]
 Character table:  2 3 3 3 3 3 3 3 3 . . 17 1 . . . . . . . 1 1 1a 4a 4b 2a 8a 8b 8c 8d 17a 17b 2P 1a 2a 2a 1a 4a 4b 4b 4a 17a 17b 3P 1a 4b 4a 2a 8b 8a 8d 8c 17b 17a 5P 1a 4a 4b 2a 8d 8c 8b 8a 17b 17a 7P 1a 4b 4a 2a 8c 8d 8a 8b 17b 17a 11P 1a 4b 4a 2a 8b 8a 8d 8c 17b 17a 13P 1a 4a 4b 2a 8d 8c 8b 8a 17a 17b 17P 1a 4a 4b 2a 8a 8b 8c 8d 1a 1a X.1 1 1 1 1 1 1 1 1 1 1 X.2 1 1 1 1 -1 -1 -1 -1 1 1 X.3 1 -1 -1 1 A -A -A A 1 1 X.4 1 -1 -1 1 -A A A -A 1 1 X.5 1 A -A -1 B -/B /B -B 1 1 X.6 1 A -A -1 -B /B -/B B 1 1 X.7 1 -A A -1 -/B B -B /B 1 1 X.8 1 -A A -1 /B -B B -/B 1 1 X.9 8 . . . . . . . C *C X.10 8 . . . . . . . *C C A = -E(4) = -Sqrt(-1) = -i B = -E(8)^3 C = E(17)^3+E(17)^5+E(17)^6+E(17)^7+E(17)^10+E(17)^11+E(17)^12+E(17)^14 = (-1-Sqrt(17))/2 = -1-b17