Galois group: 17T4

• Label: 17T4
• Degree: $17$
• Order: $136$
• Cyclic: no
• Abelian: no
• Solvable: yes
• Primitive: yes
• $p$-group: no
• Group: $C_{17}:C_{8}$

Group action invariants

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

Low degree resolvents

|G/N|	Galois groups for stem field(s)
$2$:	$C_2$
$4$:	$C_4$
$8$:	$C_8$

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
	$1^{17}$	$1$	$1$	$()$
	$8^{2},1$	$17$	$8$	$( 2, 3, 5, 9,17,16,14,10)( 4, 7,13, 8,15,12, 6,11)$
	$4^{4},1$	$17$	$4$	$( 2, 5,17,14)( 3, 9,16,10)( 4,13,15, 6)( 7, 8,12,11)$
	$8^{2},1$	$17$	$8$	$( 2, 9,14, 3,17,10, 5,16)( 4, 8, 6, 7,15,11,13,12)$
	$8^{2},1$	$17$	$8$	$( 2,10,14,16,17, 9, 5, 3)( 4,11, 6,12,15, 8,13, 7)$
	$4^{4},1$	$17$	$4$	$( 2,14,17, 5)( 3,10,16, 9)( 4, 6,15,13)( 7,11,12, 8)$
	$8^{2},1$	$17$	$8$	$( 2,16, 5,10,17, 3,14, 9)( 4,12,13,11,15, 7, 6, 8)$
	$2^{8},1$	$17$	$2$	$( 2,17)( 3,16)( 4,15)( 5,14)( 6,13)( 7,12)( 8,11)( 9,10)$
	$17$	$8$	$17$	$( 1, 2, 3, 4, 5, 6, 7, 8, 9,10,11,12,13,14,15,16,17)$
	$17$	$8$	$17$	$( 1, 4, 7,10,13,16, 2, 5, 8,11,14,17, 3, 6, 9,12,15)$

Group invariants

Order:		$136=2^{3} \cdot 17$
Cyclic:		no
Abelian:		no
Solvable:		yes
Nilpotency class:		not nilpotent
Label:		136.12
Character table:

		1A	2A	4A1	4A-1	8A1	8A-1	8A3	8A-3	17A1	17A3
	Size	1	17	17	17	17	17	17	17	8	8
	2 P	1A	1A	2A	2A	4A1	4A-1	4A-1	4A1	17A1	17A3
	17 P	1A	2A	4A1	4A-1	8A1	8A-1	8A3	8A-3	1A	1A
	Type
136.12.1a	R	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
136.12.1b	R	$1$	$1$	$1$	$1$	$-1$	$-1$	$-1$	$-1$	$1$	$1$
136.12.1c1	C	$1$	$1$	$-1$	$-1$	$-i$	$i$	$i$	$-i$	$1$	$1$
136.12.1c2	C	$1$	$1$	$-1$	$-1$	$i$	$-i$	$-i$	$i$	$1$	$1$
136.12.1d1	C	$1$	$-1$	$-{\zeta }_8^2$	${\zeta }_8^2$	${\zeta }_8^3$	$-{\zeta }_8$	${\zeta }_8$	$-{\zeta }_8^3$	$1$	$1$
136.12.1d2	C	$1$	$-1$	${\zeta }_8^2$	$-{\zeta }_8^2$	$-{\zeta }_8$	${\zeta }_8^3$	$-{\zeta }_8^3$	${\zeta }_8$	$1$	$1$
136.12.1d3	C	$1$	$-1$	$-{\zeta }_8^2$	${\zeta }_8^2$	$-{\zeta }_8^3$	${\zeta }_8$	$-{\zeta }_8$	${\zeta }_8^3$	$1$	$1$
136.12.1d4	C	$1$	$-1$	${\zeta }_8^2$	$-{\zeta }_8^2$	${\zeta }_8$	$-{\zeta }_8^3$	${\zeta }_8^3$	$-{\zeta }_8$	$1$	$1$
136.12.8a1	R	$8$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	${\zeta }_{17}^{-7}+{\zeta }_{17}^{-6}+{\zeta }_{17}^{-5}+{\zeta }_{17}^{-3}+{\zeta }_{17}^3+{\zeta }_{17}^5+{\zeta }_{17}^6+{\zeta }_{17}^7$	$-{\zeta }_{17}^{-7}-{\zeta }_{17}^{-6}-{\zeta }_{17}^{-5}-{\zeta }_{17}^{-3}-1-{\zeta }_{17}^3-{\zeta }_{17}^5-{\zeta }_{17}^6-{\zeta }_{17}^7$
136.12.8a2	R	$8$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$-{\zeta }_{17}^{-7}-{\zeta }_{17}^{-6}-{\zeta }_{17}^{-5}-{\zeta }_{17}^{-3}-1-{\zeta }_{17}^3-{\zeta }_{17}^5-{\zeta }_{17}^6-{\zeta }_{17}^7$	${\zeta }_{17}^{-7}+{\zeta }_{17}^{-6}+{\zeta }_{17}^{-5}+{\zeta }_{17}^{-3}+{\zeta }_{17}^3+{\zeta }_{17}^5+{\zeta }_{17}^6+{\zeta }_{17}^7$