Galois group: 34T5

• Label: 34T5
• Degree: $34$
• Order: $136$
• Cyclic: no
• Abelian: no
• Solvable: yes
• Primitive: no
• $p$-group: no
• Group: $C_{34}:C_4$

Group invariants

Abstract group:		$C_{34}:C_4$
Order:		$136=2^{3} \cdot 17$
Cyclic:		no
Abelian:		no
Solvable:		yes
Nilpotency class:		not nilpotent

Group action invariants

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

Low degree resolvents

$\card{(G/N)}$	Galois groups for stem field(s)
$2$:	$C_2$ x 3
$4$:	$C_4$ x 2, $C_2^2$
$8$:	$C_4\times C_2$
$68$:	$C_{17}:C_{4}$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

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

Low degree siblings

34T5

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	Index	Representative
1A	$1^{34}$	$1$	$1$	$0$	$()$
2A	$2^{17}$	$1$	$2$	$17$	$( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22)(23,24)(25,26)(27,28)(29,30)(31,32)(33,34)$
2B	$2^{16},1^{2}$	$17$	$2$	$16$	$( 3,34)( 4,33)( 5,31)( 6,32)( 7,29)( 8,30)( 9,27)(10,28)(11,25)(12,26)(13,23)(14,24)(15,22)(16,21)(17,19)(18,20)$
2C	$2^{17}$	$17$	$2$	$17$	$( 1, 6)( 2, 5)( 3, 4)( 7,33)( 8,34)( 9,32)(10,31)(11,29)(12,30)(13,27)(14,28)(15,25)(16,26)(17,24)(18,23)(19,21)(20,22)$
4A1	$4^{8},1^{2}$	$17$	$4$	$24$	$( 3,10,34,28)( 4, 9,33,27)( 5,18,31,20)( 6,17,32,19)( 7,26,29,12)( 8,25,30,11)(13,15,23,22)(14,16,24,21)$
4A-1	$4^{8},1^{2}$	$17$	$4$	$24$	$( 3,28,34,10)( 4,27,33, 9)( 5,20,31,18)( 6,19,32,17)( 7,12,29,26)( 8,11,30,25)(13,22,23,15)(14,21,24,16)$
4B1	$4^{8},2$	$17$	$4$	$25$	$( 1,19,24, 6)( 2,20,23, 5)( 3,27,22,31)( 4,28,21,32)( 7, 9,17,16)( 8,10,18,15)(11,26,14,34)(12,25,13,33)(29,30)$
4B-1	$4^{8},2$	$17$	$4$	$25$	$( 1,17,21, 6)( 2,18,22, 5)( 3, 9,19,14)( 4,10,20,13)( 7,27,15,30)( 8,28,16,29)(11,12)(23,31,34,25)(24,32,33,26)$
17A1	$17^{2}$	$4$	$17$	$32$	$( 1,11,21,31, 8,18,27, 4,14,24,33, 9,20,30, 5,16,25)( 2,12,22,32, 7,17,28, 3,13,23,34,10,19,29, 6,15,26)$
17A2	$17^{2}$	$4$	$17$	$32$	$( 1,21, 8,27,14,33,20, 5,25,11,31,18, 4,24, 9,30,16)( 2,22, 7,28,13,34,19, 6,26,12,32,17, 3,23,10,29,15)$
17A3	$17^{2}$	$4$	$17$	$32$	$( 1,27,20,11, 4,30,21,14, 5,31,24,16, 8,33,25,18, 9)( 2,28,19,12, 3,29,22,13, 6,32,23,15, 7,34,26,17,10)$
17A6	$17^{2}$	$4$	$17$	$32$	$( 1,20, 4,21, 5,24, 8,25, 9,27,11,30,14,31,16,33,18)( 2,19, 3,22, 6,23, 7,26,10,28,12,29,13,32,15,34,17)$
34A1	$34$	$4$	$34$	$33$	$( 1, 7,14,19,25,32, 4,10,16,22,27,34, 5,12,18,23,30, 2, 8,13,20,26,31, 3, 9,15,21,28,33, 6,11,17,24,29)$
34A3	$34$	$4$	$34$	$33$	$( 1,34,31,29,27,26,24,22,20,17,16,13,11,10, 8, 6, 4, 2,33,32,30,28,25,23,21,19,18,15,14,12, 9, 7, 5, 3)$
34A7	$34$	$4$	$34$	$33$	$( 1,32,27,23,20,15,11, 7, 4,34,30,26,21,17,14,10, 5, 2,31,28,24,19,16,12, 8, 3,33,29,25,22,18,13, 9, 6)$
34A9	$34$	$4$	$34$	$33$	$( 1,23,11,34,21,10,31,19, 8,29,18, 6,27,15, 4,26,14, 2,24,12,33,22, 9,32,20, 7,30,17, 5,28,16, 3,25,13)$

Malle's constant $a(G)$:     $1/16$

Character table

		1A	2A	2B	2C	4A1	4A-1	4B1	4B-1	17A1	17A2	17A3	17A6	34A1	34A3	34A7	34A9
	Size	1	1	17	17	17	17	17	17	4	4	4	4	4	4	4	4
	2 P	1A	1A	1A	1A	2B	2B	2B	2B	17A6	17A3	17A2	17A1	17A6	17A2	17A1	17A3
	17 P	1A	2A	2B	2C	4A1	4A-1	4B1	4B-1	1A	1A	1A	1A	2A	2A	2A	2A
	Type
136.13.1a	R	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
136.13.1b	R	$1$	$-1$	$1$	$-1$	$-1$	$-1$	$1$	$1$	$1$	$1$	$1$	$1$	$-1$	$-1$	$-1$	$-1$
136.13.1c	R	$1$	$-1$	$1$	$-1$	$1$	$1$	$-1$	$-1$	$1$	$1$	$1$	$1$	$-1$	$-1$	$-1$	$-1$
136.13.1d	R	$1$	$1$	$1$	$1$	$-1$	$-1$	$-1$	$-1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
136.13.1e1	C	$1$	$-1$	$-1$	$1$	$-i$	$i$	$i$	$-i$	$1$	$1$	$1$	$1$	$-1$	$-1$	$-1$	$-1$
136.13.1e2	C	$1$	$-1$	$-1$	$1$	$i$	$-i$	$-i$	$i$	$1$	$1$	$1$	$1$	$-1$	$-1$	$-1$	$-1$
136.13.1f1	C	$1$	$1$	$-1$	$-1$	$-i$	$i$	$-i$	$i$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
136.13.1f2	C	$1$	$1$	$-1$	$-1$	$i$	$-i$	$i$	$-i$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
136.13.4a1	R	$4$	$4$	$0$	$0$	$0$	$0$	$0$	$0$	${\zeta }_{17}^{-7}+{\zeta }_{17}^{-6}+{\zeta }_{17}^6+{\zeta }_{17}^7$	${\zeta }_{17}^{-5}+{\zeta }_{17}^{-3}+{\zeta }_{17}^3+{\zeta }_{17}^5$	${\zeta }_{17}^{-4}+{\zeta }_{17}^{-1}+{\zeta }_{17}+{\zeta }_{17}^4$	${\zeta }_{17}^{-8}+{\zeta }_{17}^{-2}+{\zeta }_{17}^2+{\zeta }_{17}^8$	${\zeta }_{17}^{-5}+{\zeta }_{17}^{-3}+{\zeta }_{17}^3+{\zeta }_{17}^5$	${\zeta }_{17}^{-8}+{\zeta }_{17}^{-2}+{\zeta }_{17}^2+{\zeta }_{17}^8$	${\zeta }_{17}^{-4}+{\zeta }_{17}^{-1}+{\zeta }_{17}+{\zeta }_{17}^4$	${\zeta }_{17}^{-7}+{\zeta }_{17}^{-6}+{\zeta }_{17}^6+{\zeta }_{17}^7$
136.13.4a2	R	$4$	$4$	$0$	$0$	$0$	$0$	$0$	$0$	${\zeta }_{17}^{-8}+{\zeta }_{17}^{-2}+{\zeta }_{17}^2+{\zeta }_{17}^8$	${\zeta }_{17}^{-4}+{\zeta }_{17}^{-1}+{\zeta }_{17}+{\zeta }_{17}^4$	${\zeta }_{17}^{-7}+{\zeta }_{17}^{-6}+{\zeta }_{17}^6+{\zeta }_{17}^7$	${\zeta }_{17}^{-5}+{\zeta }_{17}^{-3}+{\zeta }_{17}^3+{\zeta }_{17}^5$	${\zeta }_{17}^{-4}+{\zeta }_{17}^{-1}+{\zeta }_{17}+{\zeta }_{17}^4$	${\zeta }_{17}^{-5}+{\zeta }_{17}^{-3}+{\zeta }_{17}^3+{\zeta }_{17}^5$	${\zeta }_{17}^{-7}+{\zeta }_{17}^{-6}+{\zeta }_{17}^6+{\zeta }_{17}^7$	${\zeta }_{17}^{-8}+{\zeta }_{17}^{-2}+{\zeta }_{17}^2+{\zeta }_{17}^8$
136.13.4a3	R	$4$	$4$	$0$	$0$	$0$	$0$	$0$	$0$	${\zeta }_{17}^{-5}+{\zeta }_{17}^{-3}+{\zeta }_{17}^3+{\zeta }_{17}^5$	${\zeta }_{17}^{-7}+{\zeta }_{17}^{-6}+{\zeta }_{17}^6+{\zeta }_{17}^7$	${\zeta }_{17}^{-8}+{\zeta }_{17}^{-2}+{\zeta }_{17}^2+{\zeta }_{17}^8$	${\zeta }_{17}^{-4}+{\zeta }_{17}^{-1}+{\zeta }_{17}+{\zeta }_{17}^4$	${\zeta }_{17}^{-7}+{\zeta }_{17}^{-6}+{\zeta }_{17}^6+{\zeta }_{17}^7$	${\zeta }_{17}^{-4}+{\zeta }_{17}^{-1}+{\zeta }_{17}+{\zeta }_{17}^4$	${\zeta }_{17}^{-8}+{\zeta }_{17}^{-2}+{\zeta }_{17}^2+{\zeta }_{17}^8$	${\zeta }_{17}^{-5}+{\zeta }_{17}^{-3}+{\zeta }_{17}^3+{\zeta }_{17}^5$
136.13.4a4	R	$4$	$4$	$0$	$0$	$0$	$0$	$0$	$0$	${\zeta }_{17}^{-4}+{\zeta }_{17}^{-1}+{\zeta }_{17}+{\zeta }_{17}^4$	${\zeta }_{17}^{-8}+{\zeta }_{17}^{-2}+{\zeta }_{17}^2+{\zeta }_{17}^8$	${\zeta }_{17}^{-5}+{\zeta }_{17}^{-3}+{\zeta }_{17}^3+{\zeta }_{17}^5$	${\zeta }_{17}^{-7}+{\zeta }_{17}^{-6}+{\zeta }_{17}^6+{\zeta }_{17}^7$	${\zeta }_{17}^{-8}+{\zeta }_{17}^{-2}+{\zeta }_{17}^2+{\zeta }_{17}^8$	${\zeta }_{17}^{-7}+{\zeta }_{17}^{-6}+{\zeta }_{17}^6+{\zeta }_{17}^7$	${\zeta }_{17}^{-5}+{\zeta }_{17}^{-3}+{\zeta }_{17}^3+{\zeta }_{17}^5$	${\zeta }_{17}^{-4}+{\zeta }_{17}^{-1}+{\zeta }_{17}+{\zeta }_{17}^4$
136.13.4b1	R	$4$	$-4$	$0$	$0$	$0$	$0$	$0$	$0$	${\zeta }_{17}^{-7}+{\zeta }_{17}^{-6}+{\zeta }_{17}^6+{\zeta }_{17}^7$	${\zeta }_{17}^{-5}+{\zeta }_{17}^{-3}+{\zeta }_{17}^3+{\zeta }_{17}^5$	${\zeta }_{17}^{-4}+{\zeta }_{17}^{-1}+{\zeta }_{17}+{\zeta }_{17}^4$	${\zeta }_{17}^{-8}+{\zeta }_{17}^{-2}+{\zeta }_{17}^2+{\zeta }_{17}^8$	$-{\zeta }_{17}^{-5}-{\zeta }_{17}^{-3}-{\zeta }_{17}^3-{\zeta }_{17}^5$	$-{\zeta }_{17}^{-8}-{\zeta }_{17}^{-2}-{\zeta }_{17}^2-{\zeta }_{17}^8$	$-{\zeta }_{17}^{-4}-{\zeta }_{17}^{-1}-{\zeta }_{17}-{\zeta }_{17}^4$	$-{\zeta }_{17}^{-7}-{\zeta }_{17}^{-6}-{\zeta }_{17}^6-{\zeta }_{17}^7$
136.13.4b2	R	$4$	$-4$	$0$	$0$	$0$	$0$	$0$	$0$	${\zeta }_{17}^{-8}+{\zeta }_{17}^{-2}+{\zeta }_{17}^2+{\zeta }_{17}^8$	${\zeta }_{17}^{-4}+{\zeta }_{17}^{-1}+{\zeta }_{17}+{\zeta }_{17}^4$	${\zeta }_{17}^{-7}+{\zeta }_{17}^{-6}+{\zeta }_{17}^6+{\zeta }_{17}^7$	${\zeta }_{17}^{-5}+{\zeta }_{17}^{-3}+{\zeta }_{17}^3+{\zeta }_{17}^5$	$-{\zeta }_{17}^{-4}-{\zeta }_{17}^{-1}-{\zeta }_{17}-{\zeta }_{17}^4$	$-{\zeta }_{17}^{-5}-{\zeta }_{17}^{-3}-{\zeta }_{17}^3-{\zeta }_{17}^5$	$-{\zeta }_{17}^{-7}-{\zeta }_{17}^{-6}-{\zeta }_{17}^6-{\zeta }_{17}^7$	$-{\zeta }_{17}^{-8}-{\zeta }_{17}^{-2}-{\zeta }_{17}^2-{\zeta }_{17}^8$
136.13.4b3	R	$4$	$-4$	$0$	$0$	$0$	$0$	$0$	$0$	${\zeta }_{17}^{-5}+{\zeta }_{17}^{-3}+{\zeta }_{17}^3+{\zeta }_{17}^5$	${\zeta }_{17}^{-7}+{\zeta }_{17}^{-6}+{\zeta }_{17}^6+{\zeta }_{17}^7$	${\zeta }_{17}^{-8}+{\zeta }_{17}^{-2}+{\zeta }_{17}^2+{\zeta }_{17}^8$	${\zeta }_{17}^{-4}+{\zeta }_{17}^{-1}+{\zeta }_{17}+{\zeta }_{17}^4$	$-{\zeta }_{17}^{-7}-{\zeta }_{17}^{-6}-{\zeta }_{17}^6-{\zeta }_{17}^7$	$-{\zeta }_{17}^{-4}-{\zeta }_{17}^{-1}-{\zeta }_{17}-{\zeta }_{17}^4$	$-{\zeta }_{17}^{-8}-{\zeta }_{17}^{-2}-{\zeta }_{17}^2-{\zeta }_{17}^8$	$-{\zeta }_{17}^{-5}-{\zeta }_{17}^{-3}-{\zeta }_{17}^3-{\zeta }_{17}^5$
136.13.4b4	R	$4$	$-4$	$0$	$0$	$0$	$0$	$0$	$0$	${\zeta }_{17}^{-4}+{\zeta }_{17}^{-1}+{\zeta }_{17}+{\zeta }_{17}^4$	${\zeta }_{17}^{-8}+{\zeta }_{17}^{-2}+{\zeta }_{17}^2+{\zeta }_{17}^8$	${\zeta }_{17}^{-5}+{\zeta }_{17}^{-3}+{\zeta }_{17}^3+{\zeta }_{17}^5$	${\zeta }_{17}^{-7}+{\zeta }_{17}^{-6}+{\zeta }_{17}^6+{\zeta }_{17}^7$	$-{\zeta }_{17}^{-8}-{\zeta }_{17}^{-2}-{\zeta }_{17}^2-{\zeta }_{17}^8$	$-{\zeta }_{17}^{-7}-{\zeta }_{17}^{-6}-{\zeta }_{17}^6-{\zeta }_{17}^7$	$-{\zeta }_{17}^{-5}-{\zeta }_{17}^{-3}-{\zeta }_{17}^3-{\zeta }_{17}^5$	$-{\zeta }_{17}^{-4}-{\zeta }_{17}^{-1}-{\zeta }_{17}-{\zeta }_{17}^4$

Regular extensions

Data not computed