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 invariants

Abstract group:		$C_{17}:C_{8}$
Order:		$136=2^{3} \cdot 17$
Cyclic:		no
Abelian:		no
Solvable:		yes
Nilpotency class:		not nilpotent

Group action invariants

Degree $n$:		$17$
Transitive number $t$:		$4$
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

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

Malle's constant $a(G)$:     $1/8$

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	$8$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$-{\zeta }_{17}^{-7}-{\zeta }_{17}^{-6}-{\zeta }_{17}^{-5}-{\zeta }_{17}^{-3}-1-{\zeta }_{17}^3-{\zeta }_{17}^5-{\zeta }_{17}^6-{\zeta }_{17}^7$	${\zeta }_{17}^{-7}+{\zeta }_{17}^{-6}+{\zeta }_{17}^{-5}+{\zeta }_{17}^{-3}+{\zeta }_{17}^3+{\zeta }_{17}^5+{\zeta }_{17}^6+{\zeta }_{17}^7$

Regular extensions

Data not computed