Galois group: 17T2

• Label: 17T2
• Degree: $17$
• Order: $34$
• Cyclic: no
• Abelian: no
• Solvable: yes
• Primitive: yes
• $p$-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 $n$:		$17$
Transitive number $t$:		$2$
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

$\card{(G/N)}$	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	Index	Representative
1A	$1^{17}$	$1$	$1$	$0$	$()$
2A	$2^{8},1$	$17$	$2$	$8$	$( 1,15)( 2,14)( 3,13)( 4,12)( 5,11)( 6,10)( 7, 9)(16,17)$
17A1	$17$	$2$	$17$	$16$	$( 1,17,16,15,14,13,12,11,10, 9, 8, 7, 6, 5, 4, 3, 2)$
17A2	$17$	$2$	$17$	$16$	$( 1,16,14,12,10, 8, 6, 4, 2,17,15,13,11, 9, 7, 5, 3)$
17A3	$17$	$2$	$17$	$16$	$( 1,15,12, 9, 6, 3,17,14,11, 8, 5, 2,16,13,10, 7, 4)$
17A4	$17$	$2$	$17$	$16$	$( 1,14,10, 6, 2,15,11, 7, 3,16,12, 8, 4,17,13, 9, 5)$
17A5	$17$	$2$	$17$	$16$	$( 1,13, 8, 3,15,10, 5,17,12, 7, 2,14, 9, 4,16,11, 6)$
17A6	$17$	$2$	$17$	$16$	$( 1,12, 6,17,11, 5,16,10, 4,15, 9, 3,14, 8, 2,13, 7)$
17A7	$17$	$2$	$17$	$16$	$( 1,11, 4,14, 7,17,10, 3,13, 6,16, 9, 2,12, 5,15, 8)$
17A8	$17$	$2$	$17$	$16$	$( 1,10, 2,11, 3,12, 4,13, 5,14, 6,15, 7,16, 8,17, 9)$

Malle's constant $a(G)$:     $1/8$

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