Galois Group: 14T47

• Label: 14T47
• Order: \(5040\)
• n: \(14\)
• Cyclic: No
• Abelian: No
• Solvable: No
• Primitive: No
• $p$-group: No
• Group: $A_7\times C_2$

Group action invariants

Degree $n$ :		$14$
Transitive number $t$ :		$47$
Group :		$A_7\times C_2$
CHM label :		$2[x]A(7)$
Parity:		$-1$
Primitive:		No
Nilpotency class:		$-1$ (not nilpotent)
Generators:		(3,13,5)(6,12,10), (1,8)(2,9)(3,10)(4,11)(5,12)(6,13)(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:	$C_2$
2520:	$A_7$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 7: $A_7$

Low degree siblings

30T566 x 2, 42T409, 42T410

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, 2, 2, 2, 2, 2 $	$1$	$2$	$( 1, 8)( 2, 9)( 3,10)( 4,11)( 5,12)( 6,13)( 7,14)$
$ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1 $	$105$	$2$	$( 1, 9)( 2, 8)( 3,11)( 4,10)$
$ 2, 2, 2, 2, 2, 2, 2 $	$105$	$2$	$( 1, 2)( 3, 4)( 5,12)( 6,13)( 7,14)( 8, 9)(10,11)$
$ 3, 3, 1, 1, 1, 1, 1, 1, 1, 1 $	$70$	$3$	$( 1, 9, 3)( 2,10, 8)$
$ 6, 2, 2, 2, 2 $	$70$	$6$	$( 1, 2, 3, 8, 9,10)( 4,11)( 5,12)( 6,13)( 7,14)$
$ 3, 3, 2, 2, 2, 2 $	$210$	$6$	$( 1, 9, 3)( 2,10, 8)( 4,12)( 5,11)( 6,14)( 7,13)$
$ 6, 2, 2, 2, 2 $	$210$	$6$	$( 1, 2, 3, 8, 9,10)( 4, 5)( 6, 7)(11,12)(13,14)$
$ 3, 3, 3, 3, 1, 1 $	$280$	$3$	$( 1, 9, 3)( 2,10, 8)( 4,12, 6)( 5,13,11)$
$ 6, 6, 2 $	$280$	$6$	$( 1, 2, 3, 8, 9,10)( 4, 5, 6,11,12,13)( 7,14)$
$ 4, 4, 2, 2, 1, 1 $	$630$	$4$	$( 1, 9, 3,11)( 2,10, 4, 8)( 5,13)( 6,12)$
$ 4, 4, 2, 2, 2 $	$630$	$4$	$( 1, 2, 3, 4)( 5, 6)( 7,14)( 8, 9,10,11)(12,13)$
$ 5, 5, 1, 1, 1, 1 $	$504$	$5$	$( 1, 9, 3,11, 5)( 2,10, 4,12, 8)$
$ 10, 2, 2 $	$504$	$10$	$( 1, 2, 3, 4, 5, 8, 9,10,11,12)( 6,13)( 7,14)$
$ 7, 7 $	$360$	$7$	$( 1, 9, 3,11, 5,13, 7)( 2,10, 4,12, 6,14, 8)$
$ 14 $	$360$	$14$	$( 1, 2, 3, 4, 5, 6, 7, 8, 9,10,11,12,13,14)$
$ 7, 7 $	$360$	$7$	$( 1, 9, 3,11, 5, 7,13)( 2,10, 4,12,14, 6, 8)$
$ 14 $	$360$	$14$	$( 1, 2, 3, 4, 5,14,13, 8, 9,10,11,12, 7, 6)$

Group invariants

Order:		$5040=2^{4} \cdot 3^{2} \cdot 5 \cdot 7$
Cyclic:		No
Abelian:		No
Solvable:		No
GAP id:		Data not available

Character table:

2 4 4 4 4 3 3 3 3 1 1 3 3 1 1 1 1 1 1 3 2 2 1 1 2 2 1 1 2 2 . . . . . . . . 5 1 1 . . . . . . . . . . 1 1 . . . . 7 1 1 . . . . . . . . . . . . 1 1 1 1 1a 2a 2b 2c 3a 6a 6b 6c 3b 6d 4a 4b 5a 10a 7a 14a 7b 14b 2P 1a 1a 1a 1a 3a 3a 3a 3a 3b 3b 2b 2b 5a 5a 7a 7a 7b 7b 3P 1a 2a 2b 2c 1a 2a 2b 2c 1a 2a 4a 4b 5a 10a 7b 14b 7a 14a 5P 1a 2a 2b 2c 3a 6a 6b 6c 3b 6d 4a 4b 1a 2a 7b 14b 7a 14a 7P 1a 2a 2b 2c 3a 6a 6b 6c 3b 6d 4a 4b 5a 10a 1a 2a 1a 2a 11P 1a 2a 2b 2c 3a 6a 6b 6c 3b 6d 4a 4b 5a 10a 7a 14a 7b 14b 13P 1a 2a 2b 2c 3a 6a 6b 6c 3b 6d 4a 4b 5a 10a 7b 14b 7a 14a 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 6 6 2 2 3 3 -1 -1 . . . . 1 1 -1 -1 -1 -1 X.4 6 -6 2 -2 3 -3 -1 1 . . . . 1 -1 -1 1 -1 1 X.5 10 -10 -2 2 1 -1 1 -1 1 -1 . . . . A -A /A -/A X.6 10 -10 -2 2 1 -1 1 -1 1 -1 . . . . /A -/A A -A X.7 10 10 -2 -2 1 1 1 1 1 1 . . . . A A /A /A X.8 10 10 -2 -2 1 1 1 1 1 1 . . . . /A /A A A X.9 14 14 2 2 2 2 2 2 -1 -1 . . -1 -1 . . . . X.10 14 14 2 2 -1 -1 -1 -1 2 2 . . -1 -1 . . . . X.11 14 -14 2 -2 2 -2 2 -2 -1 1 . . -1 1 . . . . X.12 14 -14 2 -2 -1 1 -1 1 2 -2 . . -1 1 . . . . X.13 15 15 -1 -1 3 3 -1 -1 . . -1 -1 . . 1 1 1 1 X.14 15 -15 -1 1 3 -3 -1 1 . . -1 1 . . 1 -1 1 -1 X.15 21 21 1 1 -3 -3 1 1 . . -1 -1 1 1 . . . . X.16 21 -21 1 -1 -3 3 1 -1 . . -1 1 1 -1 . . . . X.17 35 35 -1 -1 -1 -1 -1 -1 -1 -1 1 1 . . . . . . X.18 35 -35 -1 1 -1 1 -1 1 -1 1 1 -1 . . . . . . A = E(7)^3+E(7)^5+E(7)^6 = (-1-Sqrt(-7))/2 = -1-b7