Galois group: 14T40

• Label: 14T40
• Degree: $14$
• Order: $2688$
• Cyclic: no
• Abelian: no
• Solvable: yes
• Primitive: no
• $p$-group: no
• Group: $C_2^6:F_7$

Group action invariants

Degree $n$:		$14$
Transitive number $t$:		$40$
Group:		$C_2^6:F_7$
CHM label:		$1/2[2^{7}]F_{42}(7)$
Parity:		$-1$
Primitive:		no
$\card{\Aut(F/K)}$:		$2$
Generators:		(1,9,11)(2,4,8)(3,13,5)(6,12,10), (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)

Low degree resolvents

|G/N|	Galois groups for stem field(s)
$2$:	$C_2$
$3$:	$C_3$
$6$:	$C_6$
$42$:	$F_7$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 7: $F_7$

Low degree siblings

14T41, 16T1502, 28T215, 28T227, 28T228, 28T237, 42T314, 42T315, 42T316, 42T317, 42T318, 42T319

Siblings are shown with degree $\leq 47$

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

Conjugacy classes

Label	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 $	$21$	$2$	$(1,8)(2,9)$
	$ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1 $	$21$	$2$	$( 1, 8)( 2, 9)( 5,12)( 6,13)$
	$ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1 $	$14$	$2$	$( 1, 8)( 2, 9)( 3,10)( 6,13)$
	$ 2, 2, 2, 2, 2, 2, 1, 1 $	$7$	$2$	$( 1, 8)( 2, 9)( 3,10)( 5,12)( 6,13)( 7,14)$
	$ 7, 7 $	$384$	$7$	$( 1,13, 4, 9, 7,12,10)( 2,14, 5, 3, 8, 6,11)$
	$ 3, 3, 3, 3, 1, 1 $	$112$	$3$	$( 2,10, 5)( 3,12, 9)( 4,14, 6)( 7,13,11)$
	$ 6, 3, 3, 2 $	$224$	$6$	$( 1, 8)( 2,10, 5, 9, 3,12)( 4,14, 6)( 7,13,11)$
	$ 6, 6, 1, 1 $	$112$	$6$	$( 2,10,12, 9, 3, 5)( 4,14,13,11, 7, 6)$
	$ 3, 3, 3, 3, 1, 1 $	$112$	$3$	$( 2, 5,10)( 3, 9,12)( 4, 6,14)( 7,11,13)$
	$ 6, 3, 3, 2 $	$224$	$6$	$( 1, 8)( 2, 5,10, 9,12, 3)( 4, 6,14)( 7,11,13)$
	$ 6, 6, 1, 1 $	$112$	$6$	$( 2,12, 3, 9, 5,10)( 4,13, 7,11, 6,14)$
	$ 4, 2, 2, 2, 2, 1, 1 $	$168$	$4$	$( 2, 7)( 3, 6)( 4,12,11, 5)( 9,14)(10,13)$
	$ 4, 4, 2, 2, 2 $	$168$	$4$	$( 1, 8)( 2, 7, 9,14)( 3, 6)( 4,12,11, 5)(10,13)$
	$ 4, 4, 4, 1, 1 $	$56$	$4$	$( 2,14, 9, 7)( 3,13,10, 6)( 4,12,11, 5)$
	$ 2, 2, 2, 2, 2, 2, 2 $	$56$	$2$	$( 1, 8)( 2,14)( 3, 6)( 4, 5)( 7, 9)(10,13)(11,12)$
	$ 12, 1, 1 $	$224$	$12$	$( 2,13, 5,14, 3, 4, 9, 6,12, 7,10,11)$
	$ 6, 6, 2 $	$224$	$6$	$( 1, 8)( 2,13, 5,14, 3, 4)( 6,12, 7,10,11, 9)$
	$ 12, 1, 1 $	$224$	$12$	$( 2,11,10, 7, 5, 6, 9, 4, 3,14,12,13)$
	$ 6, 6, 2 $	$224$	$6$	$( 1, 8)( 2,11,10, 7, 5, 6)( 3,14,12,13, 9, 4)$

Group invariants

Order:		$2688=2^{7} \cdot 3 \cdot 7$
Cyclic:		no
Abelian:		no
Solvable:		yes
Nilpotency class:		not nilpotent
Label:		2688.cb
Character table:

	Size
	2 P
	3 P
	7 P
	Type