Galois group: 14T25

• Label: 14T25
• Degree: $14$
• Order: $588$
• Cyclic: no
• Abelian: no
• Solvable: yes
• Primitive: no
• pp-group: no
• Group: $C_7^2:D_6$

Group action invariants

Degree $n$:		$14$
Transitive number $t$:		$25$
Group:		$C_7^2:D_6$
CHM label:		$[7^{2}:6_{3}]2$
Parity:		$-1$
Primitive:		no
$\card{\Aut(F/K)}$:		$1$
Generators:		(2,4,6,8,10,12,14), (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), (1,11,9)(2,4,8)(3,5,13)(6,12,10)

Low degree resolvents

$\card{(G/N)}$	Galois groups for stem field(s)
$2$:	$C_2$ x 3
$4$:	$C_2^2$
$6$:	$S_3$
$12$:	$D_{6}$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 7: None

Low degree siblings

21T23 x 2, 28T78, 42T110 x 2, 42T111 x 2, 42T112 x 2, 42T122

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	Index	Representative
1A	$1^{14}$	$1$	$1$	$0$	$()$
2A	$2^{7}$	$21$	$2$	$7$	$( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)$
2B	$2^{7}$	$21$	$2$	$7$	$( 1, 2)( 3,14)( 4,13)( 5,12)( 6,11)( 7,10)( 8, 9)$
2C	$2^{6},1^{2}$	$49$	$2$	$6$	$( 3,13)( 4,14)( 5,11)( 6,12)( 7, 9)( 8,10)$
3A	$3^{4},1^{2}$	$98$	$3$	$8$	$( 1, 5,13)( 2,10,14)( 3, 9, 7)( 6,12, 8)$
6A	$6^{2},1^{2}$	$98$	$6$	$10$	$( 1,11, 5, 3, 7,13)( 2, 8,12,10, 4,14)$
7A1	$7^{2}$	$6$	$7$	$12$	$( 1, 7,13, 5,11, 3, 9)( 2,14,12,10, 8, 6, 4)$
7A2	$7^{2}$	$6$	$7$	$12$	$( 1, 9, 3,11, 5,13, 7)( 2,10, 4,12, 6,14, 8)$
7A3	$7^{2}$	$6$	$7$	$12$	$( 1,13,11, 9, 7, 5, 3)( 2, 6,10,14, 4, 8,12)$
7B1	$7^{2}$	$6$	$7$	$12$	$( 1, 5, 9,13, 3, 7,11)( 2, 8,14, 6,12, 4,10)$
7B2	$7^{2}$	$6$	$7$	$12$	$( 1, 3, 5, 7, 9,11,13)( 2, 4, 6, 8,10,12,14)$
7B3	$7^{2}$	$6$	$7$	$12$	$( 1, 9, 3,11, 5,13, 7)( 2,14,12,10, 8, 6, 4)$
7C	$7,1^{7}$	$12$	$7$	$6$	$( 1,13,11, 9, 7, 5, 3)$
14A1	$14$	$42$	$14$	$13$	$( 1,12, 7, 4,13,10, 5, 2,11, 8, 3,14, 9, 6)$
14A3	$14$	$42$	$14$	$13$	$( 1, 4, 5, 8, 9,12,13, 2, 3, 6, 7,10,11,14)$
14A5	$14$	$42$	$14$	$13$	$( 1,10, 3,12, 5,14, 7, 2, 9, 4,11, 6,13, 8)$
14B1	$14$	$42$	$14$	$13$	$( 1, 8,11,12, 7, 2, 3, 6,13,10, 9,14, 5, 4)$
14B3	$14$	$42$	$14$	$13$	$( 1,12,13,14,11, 2, 9, 4, 7, 6, 5, 8, 3,10)$
14B5	$14$	$42$	$14$	$13$	$( 1, 4, 9,10, 3, 2,11, 8, 5,14,13, 6, 7,12)$

Malle's constant $a(G)$:     $1/6$

Group invariants

Order:		$588=2^{2} \cdot 3 \cdot 7^{2}$
Cyclic:		no
Abelian:		no
Solvable:		yes
Nilpotency class:		not nilpotent
Label:		588.35
Character table:

		1A	2A	2B	2C	3A	6A	7A1	7A2	7A3	7B1	7B2	7B3	7C	14A1	14A3	14A5	14B1	14B3	14B5
	Size	1	21	21	49	98	98	6	6	6	6	6	6	12	42	42	42	42	42	42
	2 P	1A	1A	1A	1A	3A	3A	7A2	7A3	7B3	7B1	7A1	7B2	7C	7A2	7A1	7A3	7B1	7B3	7B2
	3 P	1A	2A	2B	2C	1A	2C	7A3	7A1	7B1	7B2	7A2	7B3	7C	14A1	14A3	14A5	14B3	14B5	14B1
	7 P	1A	2A	2B	2C	3A	6A	1A	1A	1A	1A	1A	1A	1A	2A	2A	2A	2B	2B	2B
	Type
588.35.1a	R	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
588.35.1b	R	$1$	$-1$	$-1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$-1$	$-1$	$-1$	$-1$	$-1$	$-1$
588.35.1c	R	$1$	$-1$	$1$	$-1$	$1$	$-1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$-1$	$-1$	$-1$	$1$	$1$	$1$
588.35.1d	R	$1$	$1$	$-1$	$-1$	$1$	$-1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$-1$	$-1$	$-1$
588.35.2a	R	$2$	$0$	$0$	$2$	$-1$	$-1$	$2$	$2$	$2$	$2$	$2$	$2$	$2$	$0$	$0$	$0$	$0$	$0$	$0$
588.35.2b	R	$2$	$0$	$0$	$-2$	$-1$	$1$	$2$	$2$	$2$	$2$	$2$	$2$	$2$	$0$	$0$	$0$	$0$	$0$	$0$
588.35.6a1	R	$6$	$0$	$2$	$0$	$0$	$0$	$2{\zeta }_7^{-3}+2+2{\zeta }_7^3$	$2{\zeta }_7^{-1}+2+2{\zeta }_7$	$2{\zeta }_7^{-2}+2+2{\zeta }_7^2$	${\zeta }_7^{-3}-{\zeta }_7^{-2}-1-{\zeta }_7^2+{\zeta }_7^3$	$-2{\zeta }_7^{-3}-{\zeta }_7^{-2}-2-{\zeta }_7^2-2{\zeta }_7^3$	${\zeta }_7^{-3}+2{\zeta }_7^{-2}+2{\zeta }_7^2+{\zeta }_7^3$	$-1$	$0$	$0$	$0$	${\zeta }_7^{-3}+{\zeta }_7^3$	${\zeta }_7^{-2}+{\zeta }_7^2$	${\zeta }_7^{-1}+{\zeta }_7$
588.35.6a2	R	$6$	$0$	$2$	$0$	$0$	$0$	$2{\zeta }_7^{-2}+2+2{\zeta }_7^2$	$2{\zeta }_7^{-3}+2+2{\zeta }_7^3$	$2{\zeta }_7^{-1}+2+2{\zeta }_7$	${\zeta }_7^{-3}+2{\zeta }_7^{-2}+2{\zeta }_7^2+{\zeta }_7^3$	${\zeta }_7^{-3}-{\zeta }_7^{-2}-1-{\zeta }_7^2+{\zeta }_7^3$	$-2{\zeta }_7^{-3}-{\zeta }_7^{-2}-2-{\zeta }_7^2-2{\zeta }_7^3$	$-1$	$0$	$0$	$0$	${\zeta }_7^{-2}+{\zeta }_7^2$	${\zeta }_7^{-1}+{\zeta }_7$	${\zeta }_7^{-3}+{\zeta }_7^3$
588.35.6a3	R	$6$	$0$	$2$	$0$	$0$	$0$	$2{\zeta }_7^{-1}+2+2{\zeta }_7$	$2{\zeta }_7^{-2}+2+2{\zeta }_7^2$	$2{\zeta }_7^{-3}+2+2{\zeta }_7^3$	$-2{\zeta }_7^{-3}-{\zeta }_7^{-2}-2-{\zeta }_7^2-2{\zeta }_7^3$	${\zeta }_7^{-3}+2{\zeta }_7^{-2}+2{\zeta }_7^2+{\zeta }_7^3$	${\zeta }_7^{-3}-{\zeta }_7^{-2}-1-{\zeta }_7^2+{\zeta }_7^3$	$-1$	$0$	$0$	$0$	${\zeta }_7^{-1}+{\zeta }_7$	${\zeta }_7^{-3}+{\zeta }_7^3$	${\zeta }_7^{-2}+{\zeta }_7^2$
588.35.6b1	R	$6$	$2$	$0$	$0$	$0$	$0$	${\zeta }_7^{-3}+2{\zeta }_7^{-2}+2{\zeta }_7^2+{\zeta }_7^3$	${\zeta }_7^{-3}-{\zeta }_7^{-2}-1-{\zeta }_7^2+{\zeta }_7^3$	$-2{\zeta }_7^{-3}-{\zeta }_7^{-2}-2-{\zeta }_7^2-2{\zeta }_7^3$	$2{\zeta }_7^{-1}+2+2{\zeta }_7$	$2{\zeta }_7^{-2}+2+2{\zeta }_7^2$	$2{\zeta }_7^{-3}+2+2{\zeta }_7^3$	$-1$	${\zeta }_7^{-2}+{\zeta }_7^2$	${\zeta }_7^{-1}+{\zeta }_7$	${\zeta }_7^{-3}+{\zeta }_7^3$	$0$	$0$	$0$
588.35.6b2	R	$6$	$2$	$0$	$0$	$0$	$0$	${\zeta }_7^{-3}-{\zeta }_7^{-2}-1-{\zeta }_7^2+{\zeta }_7^3$	$-2{\zeta }_7^{-3}-{\zeta }_7^{-2}-2-{\zeta }_7^2-2{\zeta }_7^3$	${\zeta }_7^{-3}+2{\zeta }_7^{-2}+2{\zeta }_7^2+{\zeta }_7^3$	$2{\zeta }_7^{-2}+2+2{\zeta }_7^2$	$2{\zeta }_7^{-3}+2+2{\zeta }_7^3$	$2{\zeta }_7^{-1}+2+2{\zeta }_7$	$-1$	${\zeta }_7^{-3}+{\zeta }_7^3$	${\zeta }_7^{-2}+{\zeta }_7^2$	${\zeta }_7^{-1}+{\zeta }_7$	$0$	$0$	$0$
588.35.6b3	R	$6$	$2$	$0$	$0$	$0$	$0$	$-2{\zeta }_7^{-3}-{\zeta }_7^{-2}-2-{\zeta }_7^2-2{\zeta }_7^3$	${\zeta }_7^{-3}+2{\zeta }_7^{-2}+2{\zeta }_7^2+{\zeta }_7^3$	${\zeta }_7^{-3}-{\zeta }_7^{-2}-1-{\zeta }_7^2+{\zeta }_7^3$	$2{\zeta }_7^{-3}+2+2{\zeta }_7^3$	$2{\zeta }_7^{-1}+2+2{\zeta }_7$	$2{\zeta }_7^{-2}+2+2{\zeta }_7^2$	$-1$	${\zeta }_7^{-1}+{\zeta }_7$	${\zeta }_7^{-3}+{\zeta }_7^3$	${\zeta }_7^{-2}+{\zeta }_7^2$	$0$	$0$	$0$
588.35.6c1	R	$6$	$-2$	$0$	$0$	$0$	$0$	${\zeta }_7^{-3}+2{\zeta }_7^{-2}+2{\zeta }_7^2+{\zeta }_7^3$	${\zeta }_7^{-3}-{\zeta }_7^{-2}-1-{\zeta }_7^2+{\zeta }_7^3$	$-2{\zeta }_7^{-3}-{\zeta }_7^{-2}-2-{\zeta }_7^2-2{\zeta }_7^3$	$2{\zeta }_7^{-1}+2+2{\zeta }_7$	$2{\zeta }_7^{-2}+2+2{\zeta }_7^2$	$2{\zeta }_7^{-3}+2+2{\zeta }_7^3$	$-1$	$-{\zeta }_7^{-2}-{\zeta }_7^2$	$-{\zeta }_7^{-1}-{\zeta }_7$	$-{\zeta }_7^{-3}-{\zeta }_7^3$	$0$	$0$	$0$
588.35.6c2	R	$6$	$-2$	$0$	$0$	$0$	$0$	${\zeta }_7^{-3}-{\zeta }_7^{-2}-1-{\zeta }_7^2+{\zeta }_7^3$	$-2{\zeta }_7^{-3}-{\zeta }_7^{-2}-2-{\zeta }_7^2-2{\zeta }_7^3$	${\zeta }_7^{-3}+2{\zeta }_7^{-2}+2{\zeta }_7^2+{\zeta }_7^3$	$2{\zeta }_7^{-2}+2+2{\zeta }_7^2$	$2{\zeta }_7^{-3}+2+2{\zeta }_7^3$	$2{\zeta }_7^{-1}+2+2{\zeta }_7$	$-1$	$-{\zeta }_7^{-3}-{\zeta }_7^3$	$-{\zeta }_7^{-2}-{\zeta }_7^2$	$-{\zeta }_7^{-1}-{\zeta }_7$	$0$	$0$	$0$
588.35.6c3	R	$6$	$-2$	$0$	$0$	$0$	$0$	$-2{\zeta }_7^{-3}-{\zeta }_7^{-2}-2-{\zeta }_7^2-2{\zeta }_7^3$	${\zeta }_7^{-3}+2{\zeta }_7^{-2}+2{\zeta }_7^2+{\zeta }_7^3$	${\zeta }_7^{-3}-{\zeta }_7^{-2}-1-{\zeta }_7^2+{\zeta }_7^3$	$2{\zeta }_7^{-3}+2+2{\zeta }_7^3$	$2{\zeta }_7^{-1}+2+2{\zeta }_7$	$2{\zeta }_7^{-2}+2+2{\zeta }_7^2$	$-1$	$-{\zeta }_7^{-1}-{\zeta }_7$	$-{\zeta }_7^{-3}-{\zeta }_7^3$	$-{\zeta }_7^{-2}-{\zeta }_7^2$	$0$	$0$	$0$
588.35.6d1	R	$6$	$0$	$-2$	$0$	$0$	$0$	$2{\zeta }_7^{-3}+2+2{\zeta }_7^3$	$2{\zeta }_7^{-1}+2+2{\zeta }_7$	$2{\zeta }_7^{-2}+2+2{\zeta }_7^2$	${\zeta }_7^{-3}-{\zeta }_7^{-2}-1-{\zeta }_7^2+{\zeta }_7^3$	$-2{\zeta }_7^{-3}-{\zeta }_7^{-2}-2-{\zeta }_7^2-2{\zeta }_7^3$	${\zeta }_7^{-3}+2{\zeta }_7^{-2}+2{\zeta }_7^2+{\zeta }_7^3$	$-1$	$0$	$0$	$0$	$-{\zeta }_7^{-3}-{\zeta }_7^3$	$-{\zeta }_7^{-2}-{\zeta }_7^2$	$-{\zeta }_7^{-1}-{\zeta }_7$
588.35.6d2	R	$6$	$0$	$-2$	$0$	$0$	$0$	$2{\zeta }_7^{-2}+2+2{\zeta }_7^2$	$2{\zeta }_7^{-3}+2+2{\zeta }_7^3$	$2{\zeta }_7^{-1}+2+2{\zeta }_7$	${\zeta }_7^{-3}+2{\zeta }_7^{-2}+2{\zeta }_7^2+{\zeta }_7^3$	${\zeta }_7^{-3}-{\zeta }_7^{-2}-1-{\zeta }_7^2+{\zeta }_7^3$	$-2{\zeta }_7^{-3}-{\zeta }_7^{-2}-2-{\zeta }_7^2-2{\zeta }_7^3$	$-1$	$0$	$0$	$0$	$-{\zeta }_7^{-2}-{\zeta }_7^2$	$-{\zeta }_7^{-1}-{\zeta }_7$	$-{\zeta }_7^{-3}-{\zeta }_7^3$
588.35.6d3	R	$6$	$0$	$-2$	$0$	$0$	$0$	$2{\zeta }_7^{-1}+2+2{\zeta }_7$	$2{\zeta }_7^{-2}+2+2{\zeta }_7^2$	$2{\zeta }_7^{-3}+2+2{\zeta }_7^3$	$-2{\zeta }_7^{-3}-{\zeta }_7^{-2}-2-{\zeta }_7^2-2{\zeta }_7^3$	${\zeta }_7^{-3}+2{\zeta }_7^{-2}+2{\zeta }_7^2+{\zeta }_7^3$	${\zeta }_7^{-3}-{\zeta }_7^{-2}-1-{\zeta }_7^2+{\zeta }_7^3$	$-1$	$0$	$0$	$0$	$-{\zeta }_7^{-1}-{\zeta }_7$	$-{\zeta }_7^{-3}-{\zeta }_7^3$	$-{\zeta }_7^{-2}-{\zeta }_7^2$
588.35.12a	R	$12$	$0$	$0$	$0$	$0$	$0$	$-2$	$-2$	$-2$	$-2$	$-2$	$-2$	$5$	$0$	$0$	$0$	$0$	$0$	$0$