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 invariants

Abstract group:		$C_7^2:D_6$
Order:		$588=2^{2} \cdot 3 \cdot 7^{2}$
Cyclic:		no
Abelian:		no
Solvable:		yes
Nilpotency class:		not nilpotent

Group action invariants

Degree $n$:		$14$
Transitive number $t$:		$25$
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	$2^{7}$	$21$	$2$	$7$	$( 1, 8)( 2, 9)( 3,10)( 4,11)( 5,12)( 6,13)( 7,14)$
2B	$2^{7}$	$21$	$2$	$7$	$( 1,14)( 2,13)( 3,12)( 4,11)( 5,10)( 6, 9)( 7, 8)$
2C	$2^{6},1^{2}$	$49$	$2$	$6$	$( 1,11)( 2,10)( 3, 9)( 4, 8)( 5, 7)(12,14)$
3A	$3^{4},1^{2}$	$98$	$3$	$8$	$( 1, 3, 7)( 2, 4,12)( 5,11, 9)( 8,14,10)$
6A	$6^{2},1^{2}$	$98$	$6$	$10$	$( 1, 5, 3,11, 7, 9)( 2,14, 4,10,12, 8)$
7A1	$7^{2}$	$6$	$7$	$12$	$( 1,11, 7, 3,13, 9, 5)( 2,12, 8, 4,14,10, 6)$
7A2	$7^{2}$	$6$	$7$	$12$	$( 1, 7,13, 5,11, 3, 9)( 2, 8,14, 6,12, 4,10)$
7A3	$7^{2}$	$6$	$7$	$12$	$( 1, 3, 5, 7, 9,11,13)( 2, 4, 6, 8,10,12,14)$
7B1	$7^{2}$	$6$	$7$	$12$	$( 1, 5, 9,13, 3, 7,11)( 2,12, 8, 4,14,10, 6)$
7B2	$7^{2}$	$6$	$7$	$12$	$( 1, 9, 3,11, 5,13, 7)( 2, 8,14, 6,12, 4,10)$
7B3	$7^{2}$	$6$	$7$	$12$	$( 1,13,11, 9, 7, 5, 3)( 2, 4, 6, 8,10,12,14)$
7C	$7,1^{7}$	$12$	$7$	$6$	$( 2,14,12,10, 8, 6, 4)$
14A1	$14$	$42$	$14$	$13$	$( 1, 6,11, 2, 7,12, 3, 8,13, 4, 9,14, 5,10)$
14A3	$14$	$42$	$14$	$13$	$( 1, 2, 3, 4, 5, 6, 7, 8, 9,10,11,12,13,14)$
14A5	$14$	$42$	$14$	$13$	$( 1,12, 9, 6, 3,14,11, 8, 5, 2,13,10, 7, 4)$
14B1	$14$	$42$	$14$	$13$	$( 1,12, 5, 8, 9, 4,13,14, 3,10, 7, 6,11, 2)$
14B3	$14$	$42$	$14$	$13$	$( 1, 8,13,10,11,12, 9,14, 7, 2, 5, 4, 3, 6)$
14B5	$14$	$42$	$14$	$13$	$( 1, 4, 7,12,13, 6, 5,14,11, 8, 3, 2, 9,10)$

Malle's constant $a(G)$:     $1/6$

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	7A1	7B2	7B3	7B1	7C	7A1	7A3	7A2	7B1	7B3	7B2
	3 P	1A	2A	2B	2C	1A	2C	7A3	7A1	7A2	7B3	7B1	7B2	7C	14A3	14A5	14A1	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	$6$	$0$	$2$	$0$	$0$	$0$	$2{\zeta }_7^{-3}+2+2{\zeta }_7^3$	$2{\zeta }_7^{-1}+2+2{\zeta }_7$	${\zeta }_7^{-3}-{\zeta }_7^{-2}-1-{\zeta }_7^2+{\zeta }_7^3$	$-2{\zeta }_7^{-3}-{\zeta }_7^{-2}-2-{\zeta }_7^2-2{\zeta }_7^3$	$2{\zeta }_7^{-2}+2+2{\zeta }_7^2$	${\zeta }_7^{-3}+2{\zeta }_7^{-2}+2{\zeta }_7^2+{\zeta }_7^3$	$-1$	$0$	$0$	$0$	${\zeta }_7^{-2}+{\zeta }_7^2$	${\zeta }_7^{-1}+{\zeta }_7$	${\zeta }_7^{-3}+{\zeta }_7^3$
588.35.6a2	R	$6$	$0$	$2$	$0$	$0$	$0$	$2{\zeta }_7^{-2}+2+2{\zeta }_7^2$	$2{\zeta }_7^{-3}+2+2{\zeta }_7^3$	${\zeta }_7^{-3}+2{\zeta }_7^{-2}+2{\zeta }_7^2+{\zeta }_7^3$	${\zeta }_7^{-3}-{\zeta }_7^{-2}-1-{\zeta }_7^2+{\zeta }_7^3$	$2{\zeta }_7^{-1}+2+2{\zeta }_7$	$-2{\zeta }_7^{-3}-{\zeta }_7^{-2}-2-{\zeta }_7^2-2{\zeta }_7^3$	$-1$	$0$	$0$	$0$	${\zeta }_7^{-1}+{\zeta }_7$	${\zeta }_7^{-3}+{\zeta }_7^3$	${\zeta }_7^{-2}+{\zeta }_7^2$
588.35.6a3	R	$6$	$0$	$2$	$0$	$0$	$0$	$2{\zeta }_7^{-1}+2+2{\zeta }_7$	$2{\zeta }_7^{-2}+2+2{\zeta }_7^2$	$-2{\zeta }_7^{-3}-{\zeta }_7^{-2}-2-{\zeta }_7^2-2{\zeta }_7^3$	${\zeta }_7^{-3}+2{\zeta }_7^{-2}+2{\zeta }_7^2+{\zeta }_7^3$	$2{\zeta }_7^{-3}+2+2{\zeta }_7^3$	${\zeta }_7^{-3}-{\zeta }_7^{-2}-1-{\zeta }_7^2+{\zeta }_7^3$	$-1$	$0$	$0$	$0$	${\zeta }_7^{-3}+{\zeta }_7^3$	${\zeta }_7^{-2}+{\zeta }_7^2$	${\zeta }_7^{-1}+{\zeta }_7$
588.35.6b1	R	$6$	$2$	$0$	$0$	$0$	$0$	${\zeta }_7^{-3}+2{\zeta }_7^{-2}+2{\zeta }_7^2+{\zeta }_7^3$	${\zeta }_7^{-3}-{\zeta }_7^{-2}-1-{\zeta }_7^2+{\zeta }_7^3$	$2{\zeta }_7^{-1}+2+2{\zeta }_7$	$2{\zeta }_7^{-2}+2+2{\zeta }_7^2$	$-2{\zeta }_7^{-3}-{\zeta }_7^{-2}-2-{\zeta }_7^2-2{\zeta }_7^3$	$2{\zeta }_7^{-3}+2+2{\zeta }_7^3$	$-1$	${\zeta }_7^{-3}+{\zeta }_7^3$	${\zeta }_7^{-2}+{\zeta }_7^2$	${\zeta }_7^{-1}+{\zeta }_7$	$0$	$0$	$0$
588.35.6b2	R	$6$	$2$	$0$	$0$	$0$	$0$	${\zeta }_7^{-3}-{\zeta }_7^{-2}-1-{\zeta }_7^2+{\zeta }_7^3$	$-2{\zeta }_7^{-3}-{\zeta }_7^{-2}-2-{\zeta }_7^2-2{\zeta }_7^3$	$2{\zeta }_7^{-2}+2+2{\zeta }_7^2$	$2{\zeta }_7^{-3}+2+2{\zeta }_7^3$	${\zeta }_7^{-3}+2{\zeta }_7^{-2}+2{\zeta }_7^2+{\zeta }_7^3$	$2{\zeta }_7^{-1}+2+2{\zeta }_7$	$-1$	${\zeta }_7^{-1}+{\zeta }_7$	${\zeta }_7^{-3}+{\zeta }_7^3$	${\zeta }_7^{-2}+{\zeta }_7^2$	$0$	$0$	$0$
588.35.6b3	R	$6$	$2$	$0$	$0$	$0$	$0$	$-2{\zeta }_7^{-3}-{\zeta }_7^{-2}-2-{\zeta }_7^2-2{\zeta }_7^3$	${\zeta }_7^{-3}+2{\zeta }_7^{-2}+2{\zeta }_7^2+{\zeta }_7^3$	$2{\zeta }_7^{-3}+2+2{\zeta }_7^3$	$2{\zeta }_7^{-1}+2+2{\zeta }_7$	${\zeta }_7^{-3}-{\zeta }_7^{-2}-1-{\zeta }_7^2+{\zeta }_7^3$	$2{\zeta }_7^{-2}+2+2{\zeta }_7^2$	$-1$	${\zeta }_7^{-2}+{\zeta }_7^2$	${\zeta }_7^{-1}+{\zeta }_7$	${\zeta }_7^{-3}+{\zeta }_7^3$	$0$	$0$	$0$
588.35.6c1	R	$6$	$-2$	$0$	$0$	$0$	$0$	${\zeta }_7^{-3}+2{\zeta }_7^{-2}+2{\zeta }_7^2+{\zeta }_7^3$	${\zeta }_7^{-3}-{\zeta }_7^{-2}-1-{\zeta }_7^2+{\zeta }_7^3$	$2{\zeta }_7^{-1}+2+2{\zeta }_7$	$2{\zeta }_7^{-2}+2+2{\zeta }_7^2$	$-2{\zeta }_7^{-3}-{\zeta }_7^{-2}-2-{\zeta }_7^2-2{\zeta }_7^3$	$2{\zeta }_7^{-3}+2+2{\zeta }_7^3$	$-1$	$-{\zeta }_7^{-3}-{\zeta }_7^3$	$-{\zeta }_7^{-2}-{\zeta }_7^2$	$-{\zeta }_7^{-1}-{\zeta }_7$	$0$	$0$	$0$
588.35.6c2	R	$6$	$-2$	$0$	$0$	$0$	$0$	${\zeta }_7^{-3}-{\zeta }_7^{-2}-1-{\zeta }_7^2+{\zeta }_7^3$	$-2{\zeta }_7^{-3}-{\zeta }_7^{-2}-2-{\zeta }_7^2-2{\zeta }_7^3$	$2{\zeta }_7^{-2}+2+2{\zeta }_7^2$	$2{\zeta }_7^{-3}+2+2{\zeta }_7^3$	${\zeta }_7^{-3}+2{\zeta }_7^{-2}+2{\zeta }_7^2+{\zeta }_7^3$	$2{\zeta }_7^{-1}+2+2{\zeta }_7$	$-1$	$-{\zeta }_7^{-1}-{\zeta }_7$	$-{\zeta }_7^{-3}-{\zeta }_7^3$	$-{\zeta }_7^{-2}-{\zeta }_7^2$	$0$	$0$	$0$
588.35.6c3	R	$6$	$-2$	$0$	$0$	$0$	$0$	$-2{\zeta }_7^{-3}-{\zeta }_7^{-2}-2-{\zeta }_7^2-2{\zeta }_7^3$	${\zeta }_7^{-3}+2{\zeta }_7^{-2}+2{\zeta }_7^2+{\zeta }_7^3$	$2{\zeta }_7^{-3}+2+2{\zeta }_7^3$	$2{\zeta }_7^{-1}+2+2{\zeta }_7$	${\zeta }_7^{-3}-{\zeta }_7^{-2}-1-{\zeta }_7^2+{\zeta }_7^3$	$2{\zeta }_7^{-2}+2+2{\zeta }_7^2$	$-1$	$-{\zeta }_7^{-2}-{\zeta }_7^2$	$-{\zeta }_7^{-1}-{\zeta }_7$	$-{\zeta }_7^{-3}-{\zeta }_7^3$	$0$	$0$	$0$
588.35.6d1	R	$6$	$0$	$-2$	$0$	$0$	$0$	$2{\zeta }_7^{-3}+2+2{\zeta }_7^3$	$2{\zeta }_7^{-1}+2+2{\zeta }_7$	${\zeta }_7^{-3}-{\zeta }_7^{-2}-1-{\zeta }_7^2+{\zeta }_7^3$	$-2{\zeta }_7^{-3}-{\zeta }_7^{-2}-2-{\zeta }_7^2-2{\zeta }_7^3$	$2{\zeta }_7^{-2}+2+2{\zeta }_7^2$	${\zeta }_7^{-3}+2{\zeta }_7^{-2}+2{\zeta }_7^2+{\zeta }_7^3$	$-1$	$0$	$0$	$0$	$-{\zeta }_7^{-2}-{\zeta }_7^2$	$-{\zeta }_7^{-1}-{\zeta }_7$	$-{\zeta }_7^{-3}-{\zeta }_7^3$
588.35.6d2	R	$6$	$0$	$-2$	$0$	$0$	$0$	$2{\zeta }_7^{-2}+2+2{\zeta }_7^2$	$2{\zeta }_7^{-3}+2+2{\zeta }_7^3$	${\zeta }_7^{-3}+2{\zeta }_7^{-2}+2{\zeta }_7^2+{\zeta }_7^3$	${\zeta }_7^{-3}-{\zeta }_7^{-2}-1-{\zeta }_7^2+{\zeta }_7^3$	$2{\zeta }_7^{-1}+2+2{\zeta }_7$	$-2{\zeta }_7^{-3}-{\zeta }_7^{-2}-2-{\zeta }_7^2-2{\zeta }_7^3$	$-1$	$0$	$0$	$0$	$-{\zeta }_7^{-1}-{\zeta }_7$	$-{\zeta }_7^{-3}-{\zeta }_7^3$	$-{\zeta }_7^{-2}-{\zeta }_7^2$
588.35.6d3	R	$6$	$0$	$-2$	$0$	$0$	$0$	$2{\zeta }_7^{-1}+2+2{\zeta }_7$	$2{\zeta }_7^{-2}+2+2{\zeta }_7^2$	$-2{\zeta }_7^{-3}-{\zeta }_7^{-2}-2-{\zeta }_7^2-2{\zeta }_7^3$	${\zeta }_7^{-3}+2{\zeta }_7^{-2}+2{\zeta }_7^2+{\zeta }_7^3$	$2{\zeta }_7^{-3}+2+2{\zeta }_7^3$	${\zeta }_7^{-3}-{\zeta }_7^{-2}-1-{\zeta }_7^2+{\zeta }_7^3$	$-1$	$0$	$0$	$0$	$-{\zeta }_7^{-3}-{\zeta }_7^3$	$-{\zeta }_7^{-2}-{\zeta }_7^2$	$-{\zeta }_7^{-1}-{\zeta }_7$
588.35.12a	R	$12$	$0$	$0$	$0$	$0$	$0$	$-2$	$-2$	$-2$	$-2$	$-2$	$-2$	$5$	$0$	$0$	$0$	$0$	$0$	$0$

Regular extensions

Data not computed