Galois group: 34T4

• Label: 34T4
• Degree: $34$
• Order: $68$
• Cyclic: no
• Abelian: no
• Solvable: yes
• Primitive: no
• $p$-group: no
• Group: $C_{17}:C_4$

Group action invariants

Degree $n$:		$34$
Transitive number $t$:		$4$
Group:		$C_{17}:C_4$
Parity:		$-1$
Primitive:		no
$\card{\Aut(F/K)}$:		$2$
Generators:		(1,15,4,23)(2,16,3,24)(5,31,34,8)(6,32,33,7)(9,14,29,25)(10,13,30,26)(11,21,28,18)(12,22,27,17)(19,20), (1,14,25,4,16,27,5,18,30,7,19,32,10,21,34,12,24)(2,13,26,3,15,28,6,17,29,8,20,31,9,22,33,11,23)

Low degree resolvents

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

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 17: $C_{17}:C_{4}$

Low degree siblings

17T3

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

Group invariants

Order:		$68=2^{2} \cdot 17$
Cyclic:		no
Abelian:		no
Solvable:		yes
Nilpotency class:		not nilpotent
Label:		68.3
Character table:

		1A	2A	4A1	4A-1	17A1	17A2	17A3	17A6
	Size	1	17	17	17	4	4	4	4
	2 P	1A	1A	2A	2A	17A3	17A2	17A6	17A1
	17 P	1A	2A	4A1	4A-1	1A	1A	1A	1A
	Type
68.3.1a	R	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
68.3.1b	R	$1$	$1$	$-1$	$-1$	$1$	$1$	$1$	$1$
68.3.1c1	C	$1$	$-1$	$-i$	$i$	$1$	$1$	$1$	$1$
68.3.1c2	C	$1$	$-1$	$i$	$-i$	$1$	$1$	$1$	$1$
68.3.4a1	R	$4$	$0$	$0$	$0$	${\zeta }_{17}^{-7}+{\zeta }_{17}^{-6}+{\zeta }_{17}^6+{\zeta }_{17}^7$	${\zeta }_{17}^{-5}+{\zeta }_{17}^{-3}+{\zeta }_{17}^3+{\zeta }_{17}^5$	${\zeta }_{17}^{-4}+{\zeta }_{17}^{-1}+{\zeta }_{17}+{\zeta }_{17}^4$	${\zeta }_{17}^{-8}+{\zeta }_{17}^{-2}+{\zeta }_{17}^2+{\zeta }_{17}^8$
68.3.4a2	R	$4$	$0$	$0$	$0$	${\zeta }_{17}^{-8}+{\zeta }_{17}^{-2}+{\zeta }_{17}^2+{\zeta }_{17}^8$	${\zeta }_{17}^{-4}+{\zeta }_{17}^{-1}+{\zeta }_{17}+{\zeta }_{17}^4$	${\zeta }_{17}^{-7}+{\zeta }_{17}^{-6}+{\zeta }_{17}^6+{\zeta }_{17}^7$	${\zeta }_{17}^{-5}+{\zeta }_{17}^{-3}+{\zeta }_{17}^3+{\zeta }_{17}^5$
68.3.4a3	R	$4$	$0$	$0$	$0$	${\zeta }_{17}^{-5}+{\zeta }_{17}^{-3}+{\zeta }_{17}^3+{\zeta }_{17}^5$	${\zeta }_{17}^{-7}+{\zeta }_{17}^{-6}+{\zeta }_{17}^6+{\zeta }_{17}^7$	${\zeta }_{17}^{-8}+{\zeta }_{17}^{-2}+{\zeta }_{17}^2+{\zeta }_{17}^8$	${\zeta }_{17}^{-4}+{\zeta }_{17}^{-1}+{\zeta }_{17}+{\zeta }_{17}^4$
68.3.4a4	R	$4$	$0$	$0$	$0$	${\zeta }_{17}^{-4}+{\zeta }_{17}^{-1}+{\zeta }_{17}+{\zeta }_{17}^4$	${\zeta }_{17}^{-8}+{\zeta }_{17}^{-2}+{\zeta }_{17}^2+{\zeta }_{17}^8$	${\zeta }_{17}^{-5}+{\zeta }_{17}^{-3}+{\zeta }_{17}^3+{\zeta }_{17}^5$	${\zeta }_{17}^{-7}+{\zeta }_{17}^{-6}+{\zeta }_{17}^6+{\zeta }_{17}^7$