Galois group: 17T3

• Label: 17T3
• Degree: $17$
• Order: $68$
• Cyclic: no
• Abelian: no
• Solvable: yes
• Primitive: yes
• $p$-group: no
• Group: $C_{17}:C_{4}$

Group invariants

Abstract group:		$C_{17}:C_{4}$
Order:		$68=2^{2} \cdot 17$
Cyclic:		no
Abelian:		no
Solvable:		yes
Nilpotency class:		not nilpotent

Group action invariants

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

Low degree resolvents

$\card{(G/N)}$	Galois groups for stem field(s)
$2$:	$C_2$
$4$:	$C_4$

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	$2^{8},1$	$17$	$2$	$8$	$( 1, 2)( 3,17)( 4,16)( 5,15)( 6,14)( 7,13)( 8,12)( 9,11)$
4A1	$4^{4},1$	$17$	$4$	$12$	$( 1,12, 2, 8)( 3, 4,17,16)( 5,13,15, 7)( 6, 9,14,11)$
4A-1	$4^{4},1$	$17$	$4$	$12$	$( 1, 8, 2,12)( 3,16,17, 4)( 5, 7,15,13)( 6,11,14, 9)$
17A1	$17$	$4$	$17$	$16$	$( 1,10, 2,11, 3,12, 4,13, 5,14, 6,15, 7,16, 8,17, 9)$
17A2	$17$	$4$	$17$	$16$	$( 1, 2, 3, 4, 5, 6, 7, 8, 9,10,11,12,13,14,15,16,17)$
17A3	$17$	$4$	$17$	$16$	$( 1,11, 4,14, 7,17,10, 3,13, 6,16, 9, 2,12, 5,15, 8)$
17A6	$17$	$4$	$17$	$16$	$( 1, 4, 7,10,13,16, 2, 5, 8,11,14,17, 3, 6, 9,12,15)$

Malle's constant $a(G)$:     $1/8$

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	17A2	17A1	17A6	17A3
	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	$4$	$0$	$0$	$0$	${\zeta }_{17}^{-7}+{\zeta }_{17}^{-6}+{\zeta }_{17}^6+{\zeta }_{17}^7$	${\zeta }_{17}^{-5}+{\zeta }_{17}^{-3}+{\zeta }_{17}^3+{\zeta }_{17}^5$	${\zeta }_{17}^{-8}+{\zeta }_{17}^{-2}+{\zeta }_{17}^2+{\zeta }_{17}^8$	${\zeta }_{17}^{-4}+{\zeta }_{17}^{-1}+{\zeta }_{17}+{\zeta }_{17}^4$
68.3.4a2	R	$4$	$0$	$0$	$0$	${\zeta }_{17}^{-8}+{\zeta }_{17}^{-2}+{\zeta }_{17}^2+{\zeta }_{17}^8$	${\zeta }_{17}^{-4}+{\zeta }_{17}^{-1}+{\zeta }_{17}+{\zeta }_{17}^4$	${\zeta }_{17}^{-5}+{\zeta }_{17}^{-3}+{\zeta }_{17}^3+{\zeta }_{17}^5$	${\zeta }_{17}^{-7}+{\zeta }_{17}^{-6}+{\zeta }_{17}^6+{\zeta }_{17}^7$
68.3.4a3	R	$4$	$0$	$0$	$0$	${\zeta }_{17}^{-5}+{\zeta }_{17}^{-3}+{\zeta }_{17}^3+{\zeta }_{17}^5$	${\zeta }_{17}^{-7}+{\zeta }_{17}^{-6}+{\zeta }_{17}^6+{\zeta }_{17}^7$	${\zeta }_{17}^{-4}+{\zeta }_{17}^{-1}+{\zeta }_{17}+{\zeta }_{17}^4$	${\zeta }_{17}^{-8}+{\zeta }_{17}^{-2}+{\zeta }_{17}^2+{\zeta }_{17}^8$
68.3.4a4	R	$4$	$0$	$0$	$0$	${\zeta }_{17}^{-4}+{\zeta }_{17}^{-1}+{\zeta }_{17}+{\zeta }_{17}^4$	${\zeta }_{17}^{-8}+{\zeta }_{17}^{-2}+{\zeta }_{17}^2+{\zeta }_{17}^8$	${\zeta }_{17}^{-7}+{\zeta }_{17}^{-6}+{\zeta }_{17}^6+{\zeta }_{17}^7$	${\zeta }_{17}^{-5}+{\zeta }_{17}^{-3}+{\zeta }_{17}^3+{\zeta }_{17}^5$

Regular extensions

Data not computed