Galois group: 17T2

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

Group action invariants

Degree $n$:		$17$
Transitive number $t$:		$2$
Group:		$D_{17}$
Parity:		$1$
Primitive:		yes
$\card{\Aut(F/K)}$:		$1$
Generators:		(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17), (2,17)(3,16)(4,15)(5,14)(6,13)(7,12)(8,11)(9,10)

Low degree resolvents

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

Resolvents shown for degrees $\leq 47$

Subfields

Prime degree - none

Low degree siblings

There are no siblings with degree $\leq 47$

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

Conjugacy classes

Label	Cycle Type	Size	Order	Representative
	$ 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, 1 $	$17$	$2$	$( 2,17)( 3,16)( 4,15)( 5,14)( 6,13)( 7,12)( 8,11)( 9,10)$
	$ 17 $	$2$	$17$	$( 1, 2, 3, 4, 5, 6, 7, 8, 9,10,11,12,13,14,15,16,17)$
	$ 17 $	$2$	$17$	$( 1, 3, 5, 7, 9,11,13,15,17, 2, 4, 6, 8,10,12,14,16)$
	$ 17 $	$2$	$17$	$( 1, 4, 7,10,13,16, 2, 5, 8,11,14,17, 3, 6, 9,12,15)$
	$ 17 $	$2$	$17$	$( 1, 5, 9,13,17, 4, 8,12,16, 3, 7,11,15, 2, 6,10,14)$
	$ 17 $	$2$	$17$	$( 1, 6,11,16, 4, 9,14, 2, 7,12,17, 5,10,15, 3, 8,13)$
	$ 17 $	$2$	$17$	$( 1, 7,13, 2, 8,14, 3, 9,15, 4,10,16, 5,11,17, 6,12)$
	$ 17 $	$2$	$17$	$( 1, 8,15, 5,12, 2, 9,16, 6,13, 3,10,17, 7,14, 4,11)$
	$ 17 $	$2$	$17$	$( 1, 9,17, 8,16, 7,15, 6,14, 5,13, 4,12, 3,11, 2,10)$

Group invariants

Order:		$34=2 \cdot 17$
Cyclic:		no
Abelian:		no
Solvable:		yes
Nilpotency class:		not nilpotent
Label:		34.1
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	17A5	17A2	17A4	17A6	17A3	17A8	17A1	17A7
	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	$2$	$0$	${\zeta }_{17}^{-8}+{\zeta }_{17}^8$	${\zeta }_{17}^{-1}+{\zeta }_{17}$	${\zeta }_{17}^{-7}+{\zeta }_{17}^7$	${\zeta }_{17}^{-2}+{\zeta }_{17}^2$	${\zeta }_{17}^{-6}+{\zeta }_{17}^6$	${\zeta }_{17}^{-3}+{\zeta }_{17}^3$	${\zeta }_{17}^{-5}+{\zeta }_{17}^5$	${\zeta }_{17}^{-4}+{\zeta }_{17}^4$
34.1.2a2	R	$2$	$0$	${\zeta }_{17}^{-7}+{\zeta }_{17}^7$	${\zeta }_{17}^{-3}+{\zeta }_{17}^3$	${\zeta }_{17}^{-4}+{\zeta }_{17}^4$	${\zeta }_{17}^{-6}+{\zeta }_{17}^6$	${\zeta }_{17}^{-1}+{\zeta }_{17}$	${\zeta }_{17}^{-8}+{\zeta }_{17}^8$	${\zeta }_{17}^{-2}+{\zeta }_{17}^2$	${\zeta }_{17}^{-5}+{\zeta }_{17}^5$
34.1.2a3	R	$2$	$0$	${\zeta }_{17}^{-6}+{\zeta }_{17}^6$	${\zeta }_{17}^{-5}+{\zeta }_{17}^5$	${\zeta }_{17}^{-1}+{\zeta }_{17}$	${\zeta }_{17}^{-7}+{\zeta }_{17}^7$	${\zeta }_{17}^{-4}+{\zeta }_{17}^4$	${\zeta }_{17}^{-2}+{\zeta }_{17}^2$	${\zeta }_{17}^{-8}+{\zeta }_{17}^8$	${\zeta }_{17}^{-3}+{\zeta }_{17}^3$
34.1.2a4	R	$2$	$0$	${\zeta }_{17}^{-5}+{\zeta }_{17}^5$	${\zeta }_{17}^{-7}+{\zeta }_{17}^7$	${\zeta }_{17}^{-2}+{\zeta }_{17}^2$	${\zeta }_{17}^{-3}+{\zeta }_{17}^3$	${\zeta }_{17}^{-8}+{\zeta }_{17}^8$	${\zeta }_{17}^{-4}+{\zeta }_{17}^4$	${\zeta }_{17}^{-1}+{\zeta }_{17}$	${\zeta }_{17}^{-6}+{\zeta }_{17}^6$
34.1.2a5	R	$2$	$0$	${\zeta }_{17}^{-4}+{\zeta }_{17}^4$	${\zeta }_{17}^{-8}+{\zeta }_{17}^8$	${\zeta }_{17}^{-5}+{\zeta }_{17}^5$	${\zeta }_{17}^{-1}+{\zeta }_{17}$	${\zeta }_{17}^{-3}+{\zeta }_{17}^3$	${\zeta }_{17}^{-7}+{\zeta }_{17}^7$	${\zeta }_{17}^{-6}+{\zeta }_{17}^6$	${\zeta }_{17}^{-2}+{\zeta }_{17}^2$
34.1.2a6	R	$2$	$0$	${\zeta }_{17}^{-3}+{\zeta }_{17}^3$	${\zeta }_{17}^{-6}+{\zeta }_{17}^6$	${\zeta }_{17}^{-8}+{\zeta }_{17}^8$	${\zeta }_{17}^{-5}+{\zeta }_{17}^5$	${\zeta }_{17}^{-2}+{\zeta }_{17}^2$	${\zeta }_{17}^{-1}+{\zeta }_{17}$	${\zeta }_{17}^{-4}+{\zeta }_{17}^4$	${\zeta }_{17}^{-7}+{\zeta }_{17}^7$
34.1.2a7	R	$2$	$0$	${\zeta }_{17}^{-2}+{\zeta }_{17}^2$	${\zeta }_{17}^{-4}+{\zeta }_{17}^4$	${\zeta }_{17}^{-6}+{\zeta }_{17}^6$	${\zeta }_{17}^{-8}+{\zeta }_{17}^8$	${\zeta }_{17}^{-7}+{\zeta }_{17}^7$	${\zeta }_{17}^{-5}+{\zeta }_{17}^5$	${\zeta }_{17}^{-3}+{\zeta }_{17}^3$	${\zeta }_{17}^{-1}+{\zeta }_{17}$
34.1.2a8	R	$2$	$0$	${\zeta }_{17}^{-1}+{\zeta }_{17}$	${\zeta }_{17}^{-2}+{\zeta }_{17}^2$	${\zeta }_{17}^{-3}+{\zeta }_{17}^3$	${\zeta }_{17}^{-4}+{\zeta }_{17}^4$	${\zeta }_{17}^{-5}+{\zeta }_{17}^5$	${\zeta }_{17}^{-6}+{\zeta }_{17}^6$	${\zeta }_{17}^{-7}+{\zeta }_{17}^7$	${\zeta }_{17}^{-8}+{\zeta }_{17}^8$