Galois Group: 14T24

• Label: 14T24
• Order: \(588\)
• n: \(14\)
• Cyclic: No
• Abelian: No
• Solvable: Yes
• Primitive: No
• $p$-group: No

Group action invariants

Degree $n$ :		$14$
Transitive number $t$ :		$24$
CHM label :		$[7^{2}:6]2$
Parity:		$-1$
Primitive:		No
Nilpotency class:		$-1$ (not nilpotent)
Generators:		(2,4,6,8,10,12,14), (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), (1,13)(2,12)(3,11)(4,10)(5,9)(6,8)
$|\Aut(F/K)|$:		$1$

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
12:	$C_6\times C_2$
42:	$F_7$ x 2
84:	$F_7 \times C_2$ x 2

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 7: None

Low degree siblings

14T24 x 2, 28T77 x 3, 42T121 x 3

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 $	$6$	$7$	$( 1, 3, 5, 7, 9,11,13)( 2, 4, 6, 8,10,12,14)$
$ 7, 7 $	$12$	$7$	$( 1, 3, 5, 7, 9,11,13)( 2, 8,14, 6,12, 4,10)$
$ 7, 7 $	$12$	$7$	$( 1, 3, 5, 7, 9,11,13)( 2, 6,10,14, 4, 8,12)$
$ 7, 7 $	$6$	$7$	$( 1, 3, 5, 7, 9,11,13)( 2,14,12,10, 8, 6, 4)$
$ 3, 3, 3, 3, 1, 1 $	$49$	$3$	$( 3, 5, 9)( 4, 6,10)( 7,13,11)( 8,14,12)$
$ 3, 3, 3, 3, 1, 1 $	$49$	$3$	$( 3, 9, 5)( 4,10, 6)( 7,11,13)( 8,12,14)$
$ 2, 2, 2, 2, 2, 2, 2 $	$7$	$2$	$( 1, 8)( 2, 9)( 3,10)( 4,11)( 5,12)( 6,13)( 7,14)$
$ 14 $	$42$	$14$	$( 1,10, 3,12, 5,14, 7, 2, 9, 4,11, 6,13, 8)$
$ 6, 6, 2 $	$49$	$6$	$( 1,14,13,10, 5, 8)( 2, 3, 4, 7,12, 9)( 6,11)$
$ 6, 6, 2 $	$49$	$6$	$( 1,12, 3, 6, 7, 8)( 2, 5,14,11,10, 9)( 4,13)$
$ 2, 2, 2, 2, 2, 2, 1, 1 $	$49$	$2$	$( 3,13)( 4,14)( 5,11)( 6,12)( 7, 9)( 8,10)$
$ 6, 6, 1, 1 $	$49$	$6$	$( 3,11, 9,13, 5, 7)( 4,12,10,14, 6, 8)$
$ 6, 6, 1, 1 $	$49$	$6$	$( 3, 7, 5,13, 9,11)( 4, 8, 6,14,10,12)$
$ 14 $	$42$	$14$	$( 1, 8, 3, 6, 5, 4, 7, 2, 9,14,11,12,13,10)$
$ 2, 2, 2, 2, 2, 2, 2 $	$7$	$2$	$( 1,10)( 2, 9)( 3, 8)( 4, 7)( 5, 6)(11,14)(12,13)$
$ 6, 6, 2 $	$49$	$6$	$( 1,14, 7, 2, 3,10)( 4,13)( 5, 6, 9,12,11, 8)$
$ 6, 6, 2 $	$49$	$6$	$( 1,12, 7, 2, 5,10)( 3, 4,11,14,13, 6)( 8, 9)$

Group invariants

Order:		$588=2^{2} \cdot 3 \cdot 7^{2}$
Cyclic:		No
Abelian:		No
Solvable:		Yes
GAP id:		[588, 37]

Character table:

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