# Properties

 Label 34T2 Order $$34$$ n $$34$$ Cyclic No Abelian No Solvable Yes Primitive No $p$-group No Group: $D_{17}$

## Group action invariants

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

## Low degree resolvents

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

Resolvents shown for degrees $\leq 47$

## Subfields

Degree 2: $C_2$

Degree 17: $D_{17}$

## Low degree siblings

17T2

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

## Group invariants

 Order: $34=2 \cdot 17$ Cyclic: No Abelian: No Solvable: Yes GAP id: [34, 1]
 Character table:  2 1 1 . . . . . . . . 17 1 . 1 1 1 1 1 1 1 1 1a 2a 17a 17b 17c 17d 17e 17f 17g 17h 2P 1a 1a 17b 17d 17f 17h 17g 17e 17c 17a 3P 1a 2a 17c 17f 17h 17e 17b 17a 17d 17g 5P 1a 2a 17e 17g 17b 17c 17h 17d 17a 17f 7P 1a 2a 17g 17c 17d 17f 17a 17h 17b 17e 11P 1a 2a 17f 17e 17a 17g 17d 17b 17h 17c 13P 1a 2a 17d 17h 17e 17a 17c 17g 17f 17b 17P 1a 2a 1a 1a 1a 1a 1a 1a 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 2 . A E D B F G H C X.4 2 . B C F A D H G E X.5 2 . C A H E G D F B X.6 2 . D G C F E A B H X.7 2 . E B G C H F D A X.8 2 . F H E D C B A G X.9 2 . G F A H B E C D X.10 2 . H D B G A C E F A = E(17)^5+E(17)^12 B = E(17)^3+E(17)^14 C = E(17)^6+E(17)^11 D = E(17)^2+E(17)^15 E = E(17)^7+E(17)^10 F = E(17)^8+E(17)^9 G = E(17)^4+E(17)^13 H = E(17)+E(17)^16