Galois Group: 34T3

• Label: 34T3
• Order: \(68\)
• n: \(34\)
• Cyclic: No
• Abelian: No
• Solvable: Yes
• Primitive: No
• $p$-group: No
• Group: $D_{34}$

Group action invariants

Degree $n$ :		$34$
Transitive number $t$ :		$3$
Group :		$D_{34}$
Parity:		$-1$
Primitive:		No
Nilpotency class:		$-1$ (not nilpotent)
Generators:		(1,5,10,14,17,22,25,29,33,3,8,11,16,20,23,27,32,2,6,9,13,18,21,26,30,34,4,7,12,15,19,24,28,31), (1,24)(2,23)(3,21)(4,22)(5,19)(6,20)(7,17)(8,18)(9,16)(10,15)(11,13)(12,14)(25,34)(26,33)(27,32)(28,31)(29,30)
$|\Aut(F/K)|$:		$2$

Low degree resolvents

|G/N|	Galois groups for stem field(s)
2:	$C_2$ x 3
4:	$C_2^2$
34:	$D_{17}$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 17: $D_{17}$

Low degree siblings

34T3

Siblings are shown with degree $\leq 47$

A number field with this Galois group has no arithmetically equivalent fields.

Conjugacy Classes

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

Group invariants

Order:		$68=2^{2} \cdot 17$
Cyclic:		No
Abelian:		No
Solvable:		Yes
GAP id:		[68, 4]

Character table:

2 2 2 2 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 17 1 . 1 . 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1a 2a 2b 2c 34a 17a 34b 17b 34c 17c 34d 17d 34e 17e 17f 34f 34g 17g 2P 1a 1a 1a 1a 17b 17b 17d 17d 17f 17f 17h 17h 17g 17g 17e 17e 17c 17c 3P 1a 2a 2b 2c 34c 17c 34f 17f 34h 17h 34e 17e 34b 17b 17a 34a 34d 17d 5P 1a 2a 2b 2c 34e 17e 34g 17g 34b 17b 34c 17c 34h 17h 17d 34d 34a 17a 7P 1a 2a 2b 2c 34g 17g 34c 17c 34d 17d 34f 17f 34a 17a 17h 34h 34b 17b 11P 1a 2a 2b 2c 34f 17f 34e 17e 34a 17a 34g 17g 34d 17d 17b 34b 34h 17h 13P 1a 2a 2b 2c 34d 17d 34h 17h 34e 17e 34a 17a 34c 17c 17g 34g 34f 17f 17P 1a 2a 2b 2c 2b 1a 2b 1a 2b 1a 2b 1a 2b 1a 1a 2b 2b 1a 19P 1a 2a 2b 2c 34b 17b 34d 17d 34f 17f 34h 17h 34g 17g 17e 34e 34c 17c 23P 1a 2a 2b 2c 34f 17f 34e 17e 34a 17a 34g 17g 34d 17d 17b 34b 34h 17h 29P 1a 2a 2b 2c 34e 17e 34g 17g 34b 17b 34c 17c 34h 17h 17d 34d 34a 17a 31P 1a 2a 2b 2c 34c 17c 34f 17f 34h 17h 34e 17e 34b 17b 17a 34a 34d 17d X.1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 X.2 1 -1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 1 1 -1 -1 1 X.3 1 -1 1 -1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 X.4 1 1 -1 -1 -1 1 -1 1 -1 1 -1 1 -1 1 1 -1 -1 1 X.5 2 . 2 . A A E E D D B B F F G G H H X.6 2 . 2 . B B C C F F A A D D H H G G X.7 2 . 2 . C C A A H H E E G G D D F F X.8 2 . 2 . D D G G C C F F E E A A B B X.9 2 . 2 . E E B B G G C C H H F F D D X.10 2 . 2 . F F H H E E D D C C B B A A X.11 2 . 2 . G G F F A A H H B B E E C C X.12 2 . 2 . H H D D B B G G A A C C E E X.13 2 . -2 . -A A -E E -D D -B B -F F G -G -H H X.14 2 . -2 . -B B -C C -F F -A A -D D H -H -G G X.15 2 . -2 . -C C -A A -H H -E E -G G D -D -F F X.16 2 . -2 . -D D -G G -C C -F F -E E A -A -B B X.17 2 . -2 . -E E -B B -G G -C C -H H F -F -D D X.18 2 . -2 . -F F -H H -E E -D D -C C B -B -A A X.19 2 . -2 . -G G -F F -A A -H H -B B E -E -C C X.20 2 . -2 . -H H -D D -B B -G G -A A C -C -E E 2 1 1 17 1 1 17h 34h 2P 17a 17a 3P 17g 34g 5P 17f 34f 7P 17e 34e 11P 17c 34c 13P 17b 34b 17P 1a 2b 19P 17a 34a 23P 17c 34c 29P 17f 34f 31P 17g 34g X.1 1 1 X.2 1 -1 X.3 1 1 X.4 1 -1 X.5 C C X.6 E E X.7 B B X.8 H H X.9 A A X.10 G G X.11 D D X.12 F F X.13 C -C X.14 E -E X.15 B -B X.16 H -H X.17 A -A X.18 G -G X.19 D -D X.20 F -F A = E(17)^5+E(17)^12 B = E(17)^3+E(17)^14 C = E(17)^6+E(17)^11 D = E(17)^2+E(17)^15 E = E(17)^7+E(17)^10 F = E(17)^8+E(17)^9 G = E(17)^4+E(17)^13 H = E(17)+E(17)^16