Galois Group: 14T27

• Label: 14T27
• Order: \(896\)
• n: \(14\)
• Cyclic: No
• Abelian: No
• Solvable: Yes
• Primitive: No
• $p$-group: No

Group action invariants

Degree $n$ :		$14$
Transitive number $t$ :		$27$
CHM label :		$2^{7}[1/2]D(7)$
Parity:		$-1$
Primitive:		No
Nilpotency class:		$-1$ (not nilpotent)
Generators:		(1,13)(2,12)(3,11)(4,10)(5,9)(6,8)(7,14), (2,9)(7,14), (1,3,5,7,9,11,13)(2,4,6,8,10,12,14)
$|\Aut(F/K)|$:		$2$

Low degree resolvents

|G/N|	Galois groups for stem field(s)
2:	$C_2$
14:	$D_{7}$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 7: $D_{7}$

Low degree siblings

14T27 x 6, 14T28 x 7, 16T1078, 28T98, 28T105 x 7, 28T106 x 21, 28T107 x 21, 28T108 x 7, 28T109 x 7

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 $	$1$	$1$	$()$
$ 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $	$7$	$2$	$( 2, 9)( 5,12)$
$ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1 $	$7$	$2$	$( 2, 9)( 4,11)( 5,12)( 7,14)$
$ 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $	$7$	$2$	$( 3,10)( 4,11)$
$ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1 $	$7$	$2$	$( 2, 9)( 3,10)( 4,11)( 5,12)$
$ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1 $	$14$	$2$	$( 3,10)( 5,12)( 6,13)( 7,14)$
$ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1 $	$7$	$2$	$( 2, 9)( 3,10)( 6,13)( 7,14)$
$ 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $	$7$	$2$	$( 1, 8)( 6,13)$
$ 2, 2, 2, 2, 2, 2, 1, 1 $	$7$	$2$	$( 1, 8)( 2, 9)( 4,11)( 5,12)( 6,13)( 7,14)$
$ 7, 7 $	$128$	$7$	$( 1,14, 6, 5,11,10, 2)( 3, 9, 8, 7,13,12, 4)$
$ 7, 7 $	$128$	$7$	$( 1, 5, 2, 6,10,14,11)( 3, 7, 4, 8,12, 9,13)$
$ 7, 7 $	$128$	$7$	$( 1, 6,11, 2,14, 5,10)( 3, 8,13, 4, 9, 7,12)$
$ 4, 2, 2, 2, 2, 1, 1 $	$56$	$4$	$( 1, 3)( 4,14)( 5,13,12, 6)( 7,11)( 8,10)$
$ 2, 2, 2, 2, 2, 2, 2 $	$56$	$2$	$( 1, 3)( 2, 9)( 4,14)( 5,13)( 6,12)( 7,11)( 8,10)$
$ 4, 4, 4, 1, 1 $	$56$	$4$	$( 1,10, 8, 3)( 4, 7,11,14)( 5,13,12, 6)$
$ 4, 4, 2, 2, 2 $	$56$	$4$	$( 1,10, 8, 3)( 2, 9)( 4, 7,11,14)( 5,13)( 6,12)$
$ 4, 2, 2, 2, 2, 1, 1 $	$56$	$4$	$( 1, 3, 8,10)( 4,14)( 5, 6)( 7,11)(12,13)$
$ 4, 4, 2, 2, 2 $	$56$	$4$	$( 1, 3, 8,10)( 2, 9)( 4,14)( 5, 6,12,13)( 7,11)$
$ 4, 2, 2, 2, 2, 1, 1 $	$56$	$4$	$( 1,10)( 3, 8)( 4, 7,11,14)( 5, 6)(12,13)$
$ 4, 4, 2, 2, 2 $	$56$	$4$	$( 1,10)( 2, 9)( 3, 8)( 4, 7,11,14)( 5, 6,12,13)$

Group invariants

Order:		$896=2^{7} \cdot 7$
Cyclic:		No
Abelian:		No
Solvable:		Yes
GAP id:		[896, 19344]

Character table:

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