Galois group: 34T2

• Label: 34T2
• Degree: $34$
• Order: $34$
• Cyclic: no
• Abelian: no
• Solvable: yes
• Primitive: no
• $p$-group: no
• Group: $D_{17}$

Group invariants

Abstract group:		$D_{17}$
Order:		$34=2 \cdot 17$
Cyclic:		no
Abelian:		no
Solvable:		yes
Nilpotency class:		not nilpotent

Group action invariants

Degree $n$:		$34$
Transitive number $t$:		$2$
Parity:		$-1$
Primitive:		no
$\card{\Aut(F/K)}$:		$34$
Generators:		$(1,32)(2,31)(3,29)(4,30)(5,27)(6,28)(7,26)(8,25)(9,23)(10,24)(11,22)(12,21)(13,19)(14,20)(15,17)(16,18)(33,34)$, $(1,9,18,26,34,8,15,24,31,6,14,22,29,4,12,19,27)(2,10,17,25,33,7,16,23,32,5,13,21,30,3,11,20,28)$

Low degree resolvents

$\card{(G/N)}$	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

Label	Cycle Type	Size	Order	Index	Representative
1A	$1^{34}$	$1$	$1$	$0$	$()$
2A	$2^{17}$	$17$	$2$	$17$	$( 1, 3)( 2, 4)( 5,34)( 6,33)( 7,31)( 8,32)( 9,30)(10,29)(11,27)(12,28)(13,26)(14,25)(15,23)(16,24)(17,22)(18,21)(19,20)$
17A1	$17^{2}$	$2$	$17$	$32$	$( 1, 4, 6, 8, 9,12,14,15,18,19,22,24,26,27,29,31,34)( 2, 3, 5, 7,10,11,13,16,17,20,21,23,25,28,30,32,33)$
17A2	$17^{2}$	$2$	$17$	$32$	$( 1, 6, 9,14,18,22,26,29,34, 4, 8,12,15,19,24,27,31)( 2, 5,10,13,17,21,25,30,33, 3, 7,11,16,20,23,28,32)$
17A3	$17^{2}$	$2$	$17$	$32$	$( 1, 8,14,19,26,31, 4, 9,15,22,27,34, 6,12,18,24,29)( 2, 7,13,20,25,32, 3,10,16,21,28,33, 5,11,17,23,30)$
17A4	$17^{2}$	$2$	$17$	$32$	$( 1, 9,18,26,34, 8,15,24,31, 6,14,22,29, 4,12,19,27)( 2,10,17,25,33, 7,16,23,32, 5,13,21,30, 3,11,20,28)$
17A5	$17^{2}$	$2$	$17$	$32$	$( 1,12,22,31, 8,18,27, 4,14,24,34, 9,19,29, 6,15,26)( 2,11,21,32, 7,17,28, 3,13,23,33,10,20,30, 5,16,25)$
17A6	$17^{2}$	$2$	$17$	$32$	$( 1,14,26, 4,15,27, 6,18,29, 8,19,31, 9,22,34,12,24)( 2,13,25, 3,16,28, 5,17,30, 7,20,32,10,21,33,11,23)$
17A7	$17^{2}$	$2$	$17$	$32$	$( 1,15,29, 9,24, 4,18,31,12,26, 6,19,34,14,27, 8,22)( 2,16,30,10,23, 3,17,32,11,25, 5,20,33,13,28, 7,21)$
17A8	$17^{2}$	$2$	$17$	$32$	$( 1,18,34,15,31,14,29,12,27, 9,26, 8,24, 6,22, 4,19)( 2,17,33,16,32,13,30,11,28,10,25, 7,23, 5,21, 3,20)$

Malle's constant $a(G)$:     $1/17$

Character table

		1A	2A	17A1	17A2	17A3	17A4	17A5	17A6	17A7	17A8
	Size	1	17	2	2	2	2	2	2	2	2
	2 P	1A	1A	17A2	17A4	17A6	17A8	17A7	17A5	17A3	17A1
	17 P	1A	2A	1A	1A	1A	1A	1A	1A	1A	1A
	Type
34.1.1a	R	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
34.1.1b	R	$1$	$-1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
34.1.2a1	R	$2$	$0$	${\zeta }_{17}^{-8}+{\zeta }_{17}^8$	${\zeta }_{17}^{-5}+{\zeta }_{17}^5$	${\zeta }_{17}^{-2}+{\zeta }_{17}^2$	${\zeta }_{17}^{-1}+{\zeta }_{17}$	${\zeta }_{17}^{-4}+{\zeta }_{17}^4$	${\zeta }_{17}^{-7}+{\zeta }_{17}^7$	${\zeta }_{17}^{-3}+{\zeta }_{17}^3$	${\zeta }_{17}^{-6}+{\zeta }_{17}^6$
34.1.2a2	R	$2$	$0$	${\zeta }_{17}^{-7}+{\zeta }_{17}^7$	${\zeta }_{17}^{-2}+{\zeta }_{17}^2$	${\zeta }_{17}^{-6}+{\zeta }_{17}^6$	${\zeta }_{17}^{-3}+{\zeta }_{17}^3$	${\zeta }_{17}^{-5}+{\zeta }_{17}^5$	${\zeta }_{17}^{-4}+{\zeta }_{17}^4$	${\zeta }_{17}^{-8}+{\zeta }_{17}^8$	${\zeta }_{17}^{-1}+{\zeta }_{17}$
34.1.2a3	R	$2$	$0$	${\zeta }_{17}^{-6}+{\zeta }_{17}^6$	${\zeta }_{17}^{-8}+{\zeta }_{17}^8$	${\zeta }_{17}^{-7}+{\zeta }_{17}^7$	${\zeta }_{17}^{-5}+{\zeta }_{17}^5$	${\zeta }_{17}^{-3}+{\zeta }_{17}^3$	${\zeta }_{17}^{-1}+{\zeta }_{17}$	${\zeta }_{17}^{-2}+{\zeta }_{17}^2$	${\zeta }_{17}^{-4}+{\zeta }_{17}^4$
34.1.2a4	R	$2$	$0$	${\zeta }_{17}^{-5}+{\zeta }_{17}^5$	${\zeta }_{17}^{-1}+{\zeta }_{17}$	${\zeta }_{17}^{-3}+{\zeta }_{17}^3$	${\zeta }_{17}^{-7}+{\zeta }_{17}^7$	${\zeta }_{17}^{-6}+{\zeta }_{17}^6$	${\zeta }_{17}^{-2}+{\zeta }_{17}^2$	${\zeta }_{17}^{-4}+{\zeta }_{17}^4$	${\zeta }_{17}^{-8}+{\zeta }_{17}^8$
34.1.2a5	R	$2$	$0$	${\zeta }_{17}^{-4}+{\zeta }_{17}^4$	${\zeta }_{17}^{-6}+{\zeta }_{17}^6$	${\zeta }_{17}^{-1}+{\zeta }_{17}$	${\zeta }_{17}^{-8}+{\zeta }_{17}^8$	${\zeta }_{17}^{-2}+{\zeta }_{17}^2$	${\zeta }_{17}^{-5}+{\zeta }_{17}^5$	${\zeta }_{17}^{-7}+{\zeta }_{17}^7$	${\zeta }_{17}^{-3}+{\zeta }_{17}^3$
34.1.2a6	R	$2$	$0$	${\zeta }_{17}^{-3}+{\zeta }_{17}^3$	${\zeta }_{17}^{-4}+{\zeta }_{17}^4$	${\zeta }_{17}^{-5}+{\zeta }_{17}^5$	${\zeta }_{17}^{-6}+{\zeta }_{17}^6$	${\zeta }_{17}^{-7}+{\zeta }_{17}^7$	${\zeta }_{17}^{-8}+{\zeta }_{17}^8$	${\zeta }_{17}^{-1}+{\zeta }_{17}$	${\zeta }_{17}^{-2}+{\zeta }_{17}^2$
34.1.2a7	R	$2$	$0$	${\zeta }_{17}^{-2}+{\zeta }_{17}^2$	${\zeta }_{17}^{-3}+{\zeta }_{17}^3$	${\zeta }_{17}^{-8}+{\zeta }_{17}^8$	${\zeta }_{17}^{-4}+{\zeta }_{17}^4$	${\zeta }_{17}^{-1}+{\zeta }_{17}$	${\zeta }_{17}^{-6}+{\zeta }_{17}^6$	${\zeta }_{17}^{-5}+{\zeta }_{17}^5$	${\zeta }_{17}^{-7}+{\zeta }_{17}^7$
34.1.2a8	R	$2$	$0$	${\zeta }_{17}^{-1}+{\zeta }_{17}$	${\zeta }_{17}^{-7}+{\zeta }_{17}^7$	${\zeta }_{17}^{-4}+{\zeta }_{17}^4$	${\zeta }_{17}^{-2}+{\zeta }_{17}^2$	${\zeta }_{17}^{-8}+{\zeta }_{17}^8$	${\zeta }_{17}^{-3}+{\zeta }_{17}^3$	${\zeta }_{17}^{-6}+{\zeta }_{17}^6$	${\zeta }_{17}^{-5}+{\zeta }_{17}^5$

Regular extensions

Data not computed