Galois Group: 14T32

• Label: 14T32
• Order: \(1176\)
• n: \(14\)
• Cyclic: No
• Abelian: No
• Solvable: Yes
• Primitive: No
• $p$-group: No

Group action invariants

Degree $n$ :		$14$
Transitive number $t$ :		$32$
CHM label :		$[D(7)^{2}:3]2$
Parity:		$-1$
Primitive:		No
Nilpotency class:		$-1$ (not nilpotent)
Generators:		(2,4,6,8,10,12,14), (2,12)(4,10)(6,8), (1,9,11)(2,4,8)(3,13,5)(6,12,10), (1,8)(2,9)(3,10)(4,11)(5,12)(6,13)(7,14)
$|\Aut(F/K)|$:		$1$

Low degree resolvents

|G/N|	Galois groups for stem field(s)
2:	$C_2$ x 3
3:	$C_3$
4:	$C_2^2$
6:	$C_6$ x 3
8:	$D_{4}$
12:	$C_6\times C_2$
24:	$D_4 \times C_3$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 7: None

Low degree siblings

14T32, 28T136 x 2, 28T137 x 2, 28T138 x 2, 28T143, 42T195 x 2

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 $	$1$	$1$	$()$
$ 7, 1, 1, 1, 1, 1, 1, 1 $	$12$	$7$	$( 2, 4, 6, 8,10,12,14)$
$ 7, 7 $	$12$	$7$	$( 1, 3, 5, 7, 9,11,13)( 2, 4, 6, 8,10,12,14)$
$ 7, 7 $	$24$	$7$	$( 1, 3, 5, 7, 9,11,13)( 2, 8,14, 6,12, 4,10)$
$ 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $	$14$	$2$	$( 4,14)( 6,12)( 8,10)$
$ 7, 2, 2, 2, 1 $	$84$	$14$	$( 1, 3, 5, 7, 9,11,13)( 4,14)( 6,12)( 8,10)$
$ 2, 2, 2, 2, 2, 2, 1, 1 $	$49$	$2$	$( 3,13)( 4,14)( 5,11)( 6,12)( 7, 9)( 8,10)$
$ 3, 3, 3, 3, 1, 1 $	$49$	$3$	$( 3, 5, 9)( 4, 6,10)( 7,13,11)( 8,14,12)$
$ 6, 3, 3, 1, 1 $	$98$	$6$	$( 3, 5, 9)( 4,12,10,14, 6, 8)( 7,13,11)$
$ 6, 6, 1, 1 $	$49$	$6$	$( 3,11, 9,13, 5, 7)( 4,12,10,14, 6, 8)$
$ 3, 3, 3, 3, 1, 1 $	$49$	$3$	$( 3, 9, 5)( 4,10, 6)( 7,11,13)( 8,12,14)$
$ 6, 3, 3, 1, 1 $	$98$	$6$	$( 3, 9, 5)( 4, 8, 6,14,10,12)( 7,11,13)$
$ 6, 6, 1, 1 $	$49$	$6$	$( 3, 7, 5,13, 9,11)( 4, 8, 6,14,10,12)$
$ 2, 2, 2, 2, 2, 2, 2 $	$14$	$2$	$( 1, 8)( 2, 9)( 3,10)( 4,11)( 5,12)( 6,13)( 7,14)$
$ 14 $	$84$	$14$	$( 1,10, 3,12, 5,14, 7, 2, 9, 4,11, 6,13, 8)$
$ 4, 4, 4, 2 $	$98$	$4$	$( 1,10, 3, 8)( 2, 9)( 4,11,14, 7)( 5, 6,13,12)$
$ 6, 6, 2 $	$98$	$6$	$( 1,14,13,10, 5, 8)( 2, 3, 4, 7,12, 9)( 6,11)$
$ 12, 2 $	$98$	$12$	$( 1, 4, 7, 6,11,12, 9, 2, 3,14,13, 8)( 5,10)$
$ 6, 6, 2 $	$98$	$6$	$( 1,12, 3, 6, 7, 8)( 2, 5,14,11,10, 9)( 4,13)$
$ 12, 2 $	$98$	$12$	$( 1, 6, 7,10, 9, 2, 5, 4,13,14,11, 8)( 3,12)$

Group invariants

Order:		$1176=2^{3} \cdot 3 \cdot 7^{2}$
Cyclic:		No
Abelian:		No
Solvable:		Yes
GAP id:		[1176, 217]

Character table:

2 3 1 1 . 2 1 3 3 2 3 3 2 3 2 1 2 2 2 2 2 3 1 . . . 1 . 1 1 1 1 1 1 1 1 . 1 1 1 1 1 7 2 2 2 2 1 1 . . . . . . . 1 1 . . . . . 1a 7a 7b 7c 2a 14a 2b 3a 6a 6b 3b 6c 6d 2c 14b 4a 6e 12a 6f 12b 2P 1a 7a 7b 7c 1a 7a 1a 3b 3b 3b 3a 3a 3a 1a 7b 2b 3b 6d 3a 6b 3P 1a 7a 7b 7c 2a 14a 2b 1a 2a 2b 1a 2a 2b 2c 14b 4a 2c 4a 2c 4a 5P 1a 7a 7b 7c 2a 14a 2b 3b 6c 6d 3a 6a 6b 2c 14b 4a 6f 12b 6e 12a 7P 1a 1a 1a 1a 2a 2a 2b 3a 6a 6b 3b 6c 6d 2c 2c 4a 6e 12a 6f 12b 11P 1a 7a 7b 7c 2a 14a 2b 3b 6c 6d 3a 6a 6b 2c 14b 4a 6f 12b 6e 12a 13P 1a 7a 7b 7c 2a 14a 2b 3a 6a 6b 3b 6c 6d 2c 14b 4a 6e 12a 6f 12b 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 1 1 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 -1 -1 X.5 1 1 1 1 -1 -1 1 A -A A /A -/A /A -1 -1 1 -A A -/A /A X.6 1 1 1 1 -1 -1 1 /A -/A /A A -A A -1 -1 1 -/A /A -A A X.7 1 1 1 1 -1 -1 1 A -A A /A -/A /A 1 1 -1 A -A /A -/A X.8 1 1 1 1 -1 -1 1 /A -/A /A A -A A 1 1 -1 /A -/A A -A X.9 1 1 1 1 1 1 1 A A A /A /A /A -1 -1 -1 -A -A -/A -/A X.10 1 1 1 1 1 1 1 /A /A /A A A A -1 -1 -1 -/A -/A -A -A X.11 1 1 1 1 1 1 1 A A A /A /A /A 1 1 1 A A /A /A X.12 1 1 1 1 1 1 1 /A /A /A A A A 1 1 1 /A /A A A X.13 2 2 2 2 . . -2 2 . -2 2 . -2 . . . . . . . X.14 2 2 2 2 . . -2 B . -B /B . -/B . . . . . . . X.15 2 2 2 2 . . -2 /B . -/B B . -B . . . . . . . X.16 12 5 -2 -2 -6 1 . . . . . . . . . . . . . . X.17 12 5 -2 -2 6 -1 . . . . . . . . . . . . . . X.18 12 -2 5 -2 . . . . . . . . . -6 1 . . . . . X.19 12 -2 5 -2 . . . . . . . . . 6 -1 . . . . . X.20 24 -4 -4 3 . . . . . . . . . . . . . . . . A = E(3)^2 = (-1-Sqrt(-3))/2 = -1-b3 B = 2*E(3) = -1+Sqrt(-3) = 2b3