Galois group: 28T16

• Label: 28T16
• Degree: $28$
• Order: $84$
• Cyclic: no
• Abelian: no
• Solvable: yes
• Primitive: no
• $p$-group: no
• Group: $C_7:A_4$

Group action invariants

Degree $n$:		$28$
Transitive number $t$:		$16$
Group:		$C_7:A_4$
Parity:		$1$
Primitive:		no
$\card{\Aut(F/K)}$:		$1$
Generators:		(1,11,16)(2,10,14)(3,12,13)(4,9,15)(5,28,24)(6,25,22)(7,27,21)(8,26,23)(17,19,20), (1,5,15)(2,7,14)(3,8,16)(4,6,13)(9,24,20)(10,22,17)(11,21,19)(12,23,18)(25,27,26)

Low degree resolvents

|G/N|	Galois groups for stem field(s)
$3$:	$C_3$
$12$:	$A_4$
$21$:	$C_7:C_3$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 4: $A_4$

Degree 7: $C_7:C_3$

Degree 14: None

Low degree siblings

42T8

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, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $	$1$	$1$	$()$
	$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 1 $	$28$	$3$	$( 2, 3, 4)( 5,10,18)( 6,12,19)( 7,11,17)( 8, 9,20)(13,28,21)(14,26,24) (15,25,22)(16,27,23)$
	$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 1 $	$28$	$3$	$( 2, 4, 3)( 5,18,10)( 6,19,12)( 7,17,11)( 8,20, 9)(13,21,28)(14,24,26) (15,22,25)(16,23,27)$
	$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $	$3$	$2$	$( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22) (23,24)(25,26)(27,28)$
	$ 14, 14 $	$3$	$14$	$( 1, 5,12,13,19,23,26, 2, 6,11,14,20,24,25)( 3, 7,10,15,17,21,28, 4, 8, 9,16, 18,22,27)$
	$ 7, 7, 7, 7 $	$3$	$7$	$( 1, 6,12,14,19,24,26)( 2, 5,11,13,20,23,25)( 3, 8,10,16,17,22,28) ( 4, 7, 9,15,18,21,27)$
	$ 14, 14 $	$3$	$14$	$( 1, 7,12,15,19,21,26, 4, 6, 9,14,18,24,27)( 2, 8,11,16,20,22,25, 3, 5,10,13, 17,23,28)$
	$ 14, 14 $	$3$	$14$	$( 1, 8,12,16,19,22,26, 3, 6,10,14,17,24,28)( 2, 7,11,15,20,21,25, 4, 5, 9,13, 18,23,27)$
	$ 14, 14 $	$3$	$14$	$( 1,13,26,11,24, 5,19, 2,14,25,12,23, 6,20)( 3,15,28, 9,22, 7,17, 4,16,27,10, 21, 8,18)$
	$ 7, 7, 7, 7 $	$3$	$7$	$( 1,14,26,12,24, 6,19)( 2,13,25,11,23, 5,20)( 3,16,28,10,22, 8,17) ( 4,15,27, 9,21, 7,18)$
	$ 14, 14 $	$3$	$14$	$( 1,15,26, 9,24, 7,19, 4,14,27,12,21, 6,18)( 2,16,25,10,23, 8,20, 3,13,28,11, 22, 5,17)$
	$ 14, 14 $	$3$	$14$	$( 1,16,26,10,24, 8,19, 3,14,28,12,22, 6,17)( 2,15,25, 9,23, 7,20, 4,13,27,11, 21, 5,18)$

Group invariants

Order:		$84=2^{2} \cdot 3 \cdot 7$
Cyclic:		no
Abelian:		no
Solvable:		yes
Nilpotency class:		not nilpotent
Label:		84.11
Character table:

1A

2A

3A1

3A-1

7A1

7A-1

14A1

14A-1

14A3

14A-3

14A5

14A-5

Size

1

3

28

28

3

3

3

3

3

3

3

3

2 P

1A

1A

3A-1

3A1

7A1

7A-1

7A1

7A-1

7A-1

7A1

7A-1

7A1

3 P

1A

2A

1A

1A

7A-1

7A1

14A5

14A-3

14A-5

14A-1

14A1

14A3

7 P

1A

2A

3A1

3A-1

1A

1A

2A

2A

2A

2A

2A

2A

Type

84.11.1a

R

$1$

$1$

$1$

$1$

$1$

$1$

$1$

$1$

$1$

$1$

$1$

$1$

84.11.1b1

C

$1$

$1$

${\zeta }_3^{-1}$

${\zeta }_3$

$1$

$1$

$1$

$1$

$1$

$1$

$1$

$1$

84.11.1b2

C

$1$

$1$

${\zeta }_3$

${\zeta }_3^{-1}$

$1$

$1$

$1$

$1$

$1$

$1$

$1$

$1$

84.11.3a

R

$3$

$-1$

$0$

$0$

$3$

$3$

$-1$

$-1$

$-1$

$-1$

$-1$

$-1$

84.11.3b1

C

$3$

$3$

$0$

$0$

$-{\zeta }_7^{-3}-1-{\zeta }_7-{\zeta }_7^2$

${\zeta }_7^{-3}+{\zeta }_7+{\zeta }_7^2$

$-{\zeta }_7^{-3}-1-{\zeta }_7-{\zeta }_7^2$

${\zeta }_7^{-3}+{\zeta }_7+{\zeta }_7^2$

${\zeta }_7^{-3}+{\zeta }_7+{\zeta }_7^2$

$-{\zeta }_7^{-3}-1-{\zeta }_7-{\zeta }_7^2$

${\zeta }_7^{-3}+{\zeta }_7+{\zeta }_7^2$

$-{\zeta }_7^{-3}-1-{\zeta }_7-{\zeta }_7^2$

84.11.3b2

C

$3$

$3$

$0$

$0$

${\zeta }_7^{-3}+{\zeta }_7+{\zeta }_7^2$

$-{\zeta }_7^{-3}-1-{\zeta }_7-{\zeta }_7^2$

${\zeta }_7^{-3}+{\zeta }_7+{\zeta }_7^2$

$-{\zeta }_7^{-3}-1-{\zeta }_7-{\zeta }_7^2$

$-{\zeta }_7^{-3}-1-{\zeta }_7-{\zeta }_7^2$

${\zeta }_7^{-3}+{\zeta }_7+{\zeta }_7^2$

$-{\zeta }_7^{-3}-1-{\zeta }_7-{\zeta }_7^2$

${\zeta }_7^{-3}+{\zeta }_7+{\zeta }_7^2$

84.11.3c1

C

$3$

$-1$

$0$

$0$

$-{\zeta }_7^{-3}-1-{\zeta }_7-{\zeta }_7^2$

${\zeta }_7^{-3}+{\zeta }_7+{\zeta }_7^2$

${\zeta }_7^{-3}+1+{\zeta }_7+{\zeta }_7^2+2{\zeta }_7^3$

${\zeta }_7^{-3}-{\zeta }_7-{\zeta }_7^2$

$-{\zeta }_7^{-3}-{\zeta }_7+{\zeta }_7^2$

${\zeta }_7^{-3}+2{\zeta }_7^{-2}+1+{\zeta }_7+{\zeta }_7^2$

$-{\zeta }_7^{-3}+{\zeta }_7-{\zeta }_7^2$

$-{\zeta }_7^{-3}-2{\zeta }_7^{-2}-1-{\zeta }_7-{\zeta }_7^2-2{\zeta }_7^3$

84.11.3c2

C

$3$

$-1$

$0$

$0$

${\zeta }_7^{-3}+{\zeta }_7+{\zeta }_7^2$

$-{\zeta }_7^{-3}-1-{\zeta }_7-{\zeta }_7^2$

${\zeta }_7^{-3}-{\zeta }_7-{\zeta }_7^2$

${\zeta }_7^{-3}+1+{\zeta }_7+{\zeta }_7^2+2{\zeta }_7^3$

${\zeta }_7^{-3}+2{\zeta }_7^{-2}+1+{\zeta }_7+{\zeta }_7^2$

$-{\zeta }_7^{-3}-{\zeta }_7+{\zeta }_7^2$

$-{\zeta }_7^{-3}-2{\zeta }_7^{-2}-1-{\zeta }_7-{\zeta }_7^2-2{\zeta }_7^3$

$-{\zeta }_7^{-3}+{\zeta }_7-{\zeta }_7^2$

84.11.3c3

C

$3$

$-1$

$0$

$0$

$-{\zeta }_7^{-3}-1-{\zeta }_7-{\zeta }_7^2$

${\zeta }_7^{-3}+{\zeta }_7+{\zeta }_7^2$

${\zeta }_7^{-3}+2{\zeta }_7^{-2}+1+{\zeta }_7+{\zeta }_7^2$

$-{\zeta }_7^{-3}-{\zeta }_7+{\zeta }_7^2$

$-{\zeta }_7^{-3}+{\zeta }_7-{\zeta }_7^2$

$-{\zeta }_7^{-3}-2{\zeta }_7^{-2}-1-{\zeta }_7-{\zeta }_7^2-2{\zeta }_7^3$

${\zeta }_7^{-3}-{\zeta }_7-{\zeta }_7^2$

${\zeta }_7^{-3}+1+{\zeta }_7+{\zeta }_7^2+2{\zeta }_7^3$

84.11.3c4

C

$3$

$-1$

$0$

$0$

${\zeta }_7^{-3}+{\zeta }_7+{\zeta }_7^2$

$-{\zeta }_7^{-3}-1-{\zeta }_7-{\zeta }_7^2$

$-{\zeta }_7^{-3}-{\zeta }_7+{\zeta }_7^2$

${\zeta }_7^{-3}+2{\zeta }_7^{-2}+1+{\zeta }_7+{\zeta }_7^2$

$-{\zeta }_7^{-3}-2{\zeta }_7^{-2}-1-{\zeta }_7-{\zeta }_7^2-2{\zeta }_7^3$

$-{\zeta }_7^{-3}+{\zeta }_7-{\zeta }_7^2$

${\zeta }_7^{-3}+1+{\zeta }_7+{\zeta }_7^2+2{\zeta }_7^3$

${\zeta }_7^{-3}-{\zeta }_7-{\zeta }_7^2$

84.11.3c5

C

$3$

$-1$

$0$

$0$

$-{\zeta }_7^{-3}-1-{\zeta }_7-{\zeta }_7^2$

${\zeta }_7^{-3}+{\zeta }_7+{\zeta }_7^2$

$-{\zeta }_7^{-3}-2{\zeta }_7^{-2}-1-{\zeta }_7-{\zeta }_7^2-2{\zeta }_7^3$

$-{\zeta }_7^{-3}+{\zeta }_7-{\zeta }_7^2$

${\zeta }_7^{-3}-{\zeta }_7-{\zeta }_7^2$

${\zeta }_7^{-3}+1+{\zeta }_7+{\zeta }_7^2+2{\zeta }_7^3$

$-{\zeta }_7^{-3}-{\zeta }_7+{\zeta }_7^2$

${\zeta }_7^{-3}+2{\zeta }_7^{-2}+1+{\zeta }_7+{\zeta }_7^2$

84.11.3c6

C

$3$

$-1$

$0$

$0$

${\zeta }_7^{-3}+{\zeta }_7+{\zeta }_7^2$

$-{\zeta }_7^{-3}-1-{\zeta }_7-{\zeta }_7^2$

$-{\zeta }_7^{-3}+{\zeta }_7-{\zeta }_7^2$

$-{\zeta }_7^{-3}-2{\zeta }_7^{-2}-1-{\zeta }_7-{\zeta }_7^2-2{\zeta }_7^3$

${\zeta }_7^{-3}+1+{\zeta }_7+{\zeta }_7^2+2{\zeta }_7^3$

${\zeta }_7^{-3}-{\zeta }_7-{\zeta }_7^2$

${\zeta }_7^{-3}+2{\zeta }_7^{-2}+1+{\zeta }_7+{\zeta }_7^2$

$-{\zeta }_7^{-3}-{\zeta }_7+{\zeta }_7^2$