Galois group: 27T8

• Label: 27T8
• Degree: $27$
• Order: $54$
• Cyclic: no
• Abelian: no
• Solvable: yes
• Primitive: no
• $p$-group: no
• Group: $D_{27}$

Group action invariants

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

Low degree resolvents

|G/N|	Galois groups for stem field(s)
$2$:	$C_2$
$6$:	$S_3$
$18$:	$D_{9}$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 3: $S_3$

Degree 9: $D_{9}$

Low degree siblings

There are no siblings 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	$1^{27}$	$1$	$1$	$0$	$()$
2A	$2^{13},1$	$27$	$2$	$13$	$( 2, 3)( 4,27)( 5,26)( 6,25)( 7,23)( 8,22)( 9,24)(10,20)(11,19)(12,21)(13,18)(14,17)(15,16)$
3A	$3^{9}$	$2$	$3$	$18$	$( 1, 2, 3)( 4, 5, 6)( 7, 8, 9)(10,11,12)(13,14,15)(16,17,18)(19,20,21)(22,23,24)(25,26,27)$
9A1	$9^{3}$	$2$	$9$	$24$	$( 1,20,11, 2,21,12, 3,19,10)( 4,24,14, 5,22,15, 6,23,13)( 7,25,16, 8,26,17, 9,27,18)$
9A2	$9^{3}$	$2$	$9$	$24$	$( 1,19,12, 2,20,10, 3,21,11)( 4,23,15, 5,24,13, 6,22,14)( 7,27,17, 8,25,18, 9,26,16)$
9A4	$9^{3}$	$2$	$9$	$24$	$( 1,21,10, 2,19,11, 3,20,12)( 4,22,13, 5,23,14, 6,24,15)( 7,26,18, 8,27,16, 9,25,17)$
27A1	$27$	$2$	$27$	$26$	$( 1,16, 5,19, 7,24,12,27,13, 2,17, 6,20, 8,22,10,25,14, 3,18, 4,21, 9,23,11,26,15)$
27A2	$27$	$2$	$27$	$26$	$( 1,18, 6,19, 9,22,12,26,14, 2,16, 4,20, 7,23,10,27,15, 3,17, 5,21, 8,24,11,25,13)$
27A4	$27$	$2$	$27$	$26$	$( 1,27,22,21,16,13,10, 9, 5, 2,25,23,19,17,14,11, 7, 6, 3,26,24,20,18,15,12, 8, 4)$
27A5	$27$	$2$	$27$	$26$	$( 1, 9,14,20,27, 5,11,18,22, 2, 7,15,21,25, 6,12,16,23, 3, 8,13,19,26, 4,10,17,24)$
27A7	$27$	$2$	$27$	$26$	$( 1,25,24,21,17,15,10, 7, 4, 2,26,22,19,18,13,11, 8, 5, 3,27,23,20,16,14,12, 9, 6)$
27A8	$27$	$2$	$27$	$26$	$( 1,26,23,21,18,14,10, 8, 6, 2,27,24,19,16,15,11, 9, 4, 3,25,22,20,17,13,12, 7, 5)$
27A10	$27$	$2$	$27$	$26$	$( 1,17, 4,19, 8,23,12,25,15, 2,18, 5,20, 9,24,10,26,13, 3,16, 6,21, 7,22,11,27,14)$
27A11	$27$	$2$	$27$	$26$	$( 1, 8,15,20,26, 6,11,17,23, 2, 9,13,21,27, 4,12,18,24, 3, 7,14,19,25, 5,10,16,22)$
27A13	$27$	$2$	$27$	$26$	$( 1, 7,13,20,25, 4,11,16,24, 2, 8,14,21,26, 5,12,17,22, 3, 9,15,19,27, 6,10,18,23)$

Malle's constant $a(G)$:     $1/13$

Group invariants

Order:		$54=2 \cdot 3^{3}$
Cyclic:		no
Abelian:		no
Solvable:		yes
Nilpotency class:		not nilpotent
Label:		54.1
Character table:

1A

2A

3A

9A1

9A2

9A4

27A1

27A2

27A4

27A5

27A7

27A8

27A10

27A11

27A13

Size

1

27

2

2

2

2

2

2

2

2

2

2

2

2

2

2 P

1A

1A

3A

9A2

9A4

9A1

27A5

27A13

27A8

27A2

27A1

27A10

27A4

27A11

27A7

3 P

1A

2A

1A

3A

3A

3A

9A2

9A2

9A4

9A1

9A4

9A4

9A2

9A1

9A1

Type

54.1.1a

R

$1$

$1$

$1$

$1$

$1$

$1$

$1$

$1$

$1$

$1$

$1$

$1$

$1$

$1$

$1$

54.1.1b

R

$1$

$-1$

$1$

$1$

$1$

$1$

$1$

$1$

$1$

$1$

$1$

$1$

$1$

$1$

$1$

54.1.2a

R

$2$

$0$

$2$

$2$

$2$

$2$

$-1$

$-1$

$-1$

$-1$

$-1$

$-1$

$-1$

$-1$

$-1$

54.1.2b1

R

$2$

$0$

$2$

$-1$

$-1$

$-1$

${\zeta }_9^{-4}+{\zeta }_9^4$

${\zeta }_9^{-1}+{\zeta }_9$

${\zeta }_9^{-2}+{\zeta }_9^2$

${\zeta }_9^{-2}+{\zeta }_9^2$

${\zeta }_9^{-1}+{\zeta }_9$

${\zeta }_9^{-4}+{\zeta }_9^4$

${\zeta }_9^{-4}+{\zeta }_9^4$

${\zeta }_9^{-1}+{\zeta }_9$

${\zeta }_9^{-2}+{\zeta }_9^2$

54.1.2b2

R

$2$

$0$

$2$

$-1$

$-1$

$-1$

${\zeta }_9^{-2}+{\zeta }_9^2$

${\zeta }_9^{-4}+{\zeta }_9^4$

${\zeta }_9^{-1}+{\zeta }_9$

${\zeta }_9^{-1}+{\zeta }_9$

${\zeta }_9^{-4}+{\zeta }_9^4$

${\zeta }_9^{-2}+{\zeta }_9^2$

${\zeta }_9^{-2}+{\zeta }_9^2$

${\zeta }_9^{-4}+{\zeta }_9^4$

${\zeta }_9^{-1}+{\zeta }_9$

54.1.2b3

R

$2$

$0$

$2$

$-1$

$-1$

$-1$

${\zeta }_9^{-1}+{\zeta }_9$

${\zeta }_9^{-2}+{\zeta }_9^2$

${\zeta }_9^{-4}+{\zeta }_9^4$

${\zeta }_9^{-4}+{\zeta }_9^4$

${\zeta }_9^{-2}+{\zeta }_9^2$

${\zeta }_9^{-1}+{\zeta }_9$

${\zeta }_9^{-1}+{\zeta }_9$

${\zeta }_9^{-2}+{\zeta }_9^2$

${\zeta }_9^{-4}+{\zeta }_9^4$

54.1.2c1

R

$2$

$0$

$-1$

${\zeta }_{27}^{-12}+{\zeta }_{27}^{12}$

${\zeta }_{27}^{-3}+{\zeta }_{27}^3$

${\zeta }_{27}^{-6}+{\zeta }_{27}^6$

${\zeta }_{27}^{-13}+{\zeta }_{27}^{13}$

${\zeta }_{27}^{-1}+{\zeta }_{27}$

${\zeta }_{27}^{-2}+{\zeta }_{27}^2$

${\zeta }_{27}^{-11}+{\zeta }_{27}^{11}$

${\zeta }_{27}^{-10}+{\zeta }_{27}^{10}$

${\zeta }_{27}^{-4}+{\zeta }_{27}^4$

${\zeta }_{27}^{-5}+{\zeta }_{27}^5$

${\zeta }_{27}^{-8}+{\zeta }_{27}^8$

${\zeta }_{27}^{-7}+{\zeta }_{27}^7$

54.1.2c2

R

$2$

$0$

$-1$

${\zeta }_{27}^{-12}+{\zeta }_{27}^{12}$

${\zeta }_{27}^{-3}+{\zeta }_{27}^3$

${\zeta }_{27}^{-6}+{\zeta }_{27}^6$

${\zeta }_{27}^{-5}+{\zeta }_{27}^5$

${\zeta }_{27}^{-10}+{\zeta }_{27}^{10}$

${\zeta }_{27}^{-7}+{\zeta }_{27}^7$

${\zeta }_{27}^{-2}+{\zeta }_{27}^2$

${\zeta }_{27}^{-8}+{\zeta }_{27}^8$

${\zeta }_{27}^{-13}+{\zeta }_{27}^{13}$

${\zeta }_{27}^{-4}+{\zeta }_{27}^4$

${\zeta }_{27}^{-1}+{\zeta }_{27}$

${\zeta }_{27}^{-11}+{\zeta }_{27}^{11}$

54.1.2c3

R

$2$

$0$

$-1$

${\zeta }_{27}^{-12}+{\zeta }_{27}^{12}$

${\zeta }_{27}^{-3}+{\zeta }_{27}^3$

${\zeta }_{27}^{-6}+{\zeta }_{27}^6$

${\zeta }_{27}^{-4}+{\zeta }_{27}^4$

${\zeta }_{27}^{-8}+{\zeta }_{27}^8$

${\zeta }_{27}^{-11}+{\zeta }_{27}^{11}$

${\zeta }_{27}^{-7}+{\zeta }_{27}^7$

${\zeta }_{27}^{-1}+{\zeta }_{27}$

${\zeta }_{27}^{-5}+{\zeta }_{27}^5$

${\zeta }_{27}^{-13}+{\zeta }_{27}^{13}$

${\zeta }_{27}^{-10}+{\zeta }_{27}^{10}$

${\zeta }_{27}^{-2}+{\zeta }_{27}^2$

54.1.2c4

R

$2$

$0$

$-1$

${\zeta }_{27}^{-6}+{\zeta }_{27}^6$

${\zeta }_{27}^{-12}+{\zeta }_{27}^{12}$

${\zeta }_{27}^{-3}+{\zeta }_{27}^3$

${\zeta }_{27}^{-11}+{\zeta }_{27}^{11}$

${\zeta }_{27}^{-5}+{\zeta }_{27}^5$

${\zeta }_{27}^{-10}+{\zeta }_{27}^{10}$

${\zeta }_{27}^{-1}+{\zeta }_{27}$

${\zeta }_{27}^{-4}+{\zeta }_{27}^4$

${\zeta }_{27}^{-7}+{\zeta }_{27}^7$

${\zeta }_{27}^{-2}+{\zeta }_{27}^2$

${\zeta }_{27}^{-13}+{\zeta }_{27}^{13}$

${\zeta }_{27}^{-8}+{\zeta }_{27}^8$

54.1.2c5

R

$2$

$0$

$-1$

${\zeta }_{27}^{-6}+{\zeta }_{27}^6$

${\zeta }_{27}^{-12}+{\zeta }_{27}^{12}$

${\zeta }_{27}^{-3}+{\zeta }_{27}^3$

${\zeta }_{27}^{-7}+{\zeta }_{27}^7$

${\zeta }_{27}^{-13}+{\zeta }_{27}^{13}$

${\zeta }_{27}^{-1}+{\zeta }_{27}$

${\zeta }_{27}^{-8}+{\zeta }_{27}^8$

${\zeta }_{27}^{-5}+{\zeta }_{27}^5$

${\zeta }_{27}^{-2}+{\zeta }_{27}^2$

${\zeta }_{27}^{-11}+{\zeta }_{27}^{11}$

${\zeta }_{27}^{-4}+{\zeta }_{27}^4$

${\zeta }_{27}^{-10}+{\zeta }_{27}^{10}$

54.1.2c6

R

$2$

$0$

$-1$

${\zeta }_{27}^{-6}+{\zeta }_{27}^6$

${\zeta }_{27}^{-12}+{\zeta }_{27}^{12}$

${\zeta }_{27}^{-3}+{\zeta }_{27}^3$

${\zeta }_{27}^{-2}+{\zeta }_{27}^2$

${\zeta }_{27}^{-4}+{\zeta }_{27}^4$

${\zeta }_{27}^{-8}+{\zeta }_{27}^8$

${\zeta }_{27}^{-10}+{\zeta }_{27}^{10}$

${\zeta }_{27}^{-13}+{\zeta }_{27}^{13}$

${\zeta }_{27}^{-11}+{\zeta }_{27}^{11}$

${\zeta }_{27}^{-7}+{\zeta }_{27}^7$

${\zeta }_{27}^{-5}+{\zeta }_{27}^5$

${\zeta }_{27}^{-1}+{\zeta }_{27}$

54.1.2c7

R

$2$

$0$

$-1$

${\zeta }_{27}^{-3}+{\zeta }_{27}^3$

${\zeta }_{27}^{-6}+{\zeta }_{27}^6$

${\zeta }_{27}^{-12}+{\zeta }_{27}^{12}$

${\zeta }_{27}^{-10}+{\zeta }_{27}^{10}$

${\zeta }_{27}^{-7}+{\zeta }_{27}^7$

${\zeta }_{27}^{-13}+{\zeta }_{27}^{13}$

${\zeta }_{27}^{-4}+{\zeta }_{27}^4$

${\zeta }_{27}^{-11}+{\zeta }_{27}^{11}$

${\zeta }_{27}^{-1}+{\zeta }_{27}$

${\zeta }_{27}^{-8}+{\zeta }_{27}^8$

${\zeta }_{27}^{-2}+{\zeta }_{27}^2$

${\zeta }_{27}^{-5}+{\zeta }_{27}^5$

54.1.2c8

R

$2$

$0$

$-1$

${\zeta }_{27}^{-3}+{\zeta }_{27}^3$

${\zeta }_{27}^{-6}+{\zeta }_{27}^6$

${\zeta }_{27}^{-12}+{\zeta }_{27}^{12}$

${\zeta }_{27}^{-8}+{\zeta }_{27}^8$

${\zeta }_{27}^{-11}+{\zeta }_{27}^{11}$

${\zeta }_{27}^{-5}+{\zeta }_{27}^5$

${\zeta }_{27}^{-13}+{\zeta }_{27}^{13}$

${\zeta }_{27}^{-2}+{\zeta }_{27}^2$

${\zeta }_{27}^{-10}+{\zeta }_{27}^{10}$

${\zeta }_{27}^{-1}+{\zeta }_{27}$

${\zeta }_{27}^{-7}+{\zeta }_{27}^7$

${\zeta }_{27}^{-4}+{\zeta }_{27}^4$

54.1.2c9

R

$2$

$0$

$-1$

${\zeta }_{27}^{-3}+{\zeta }_{27}^3$

${\zeta }_{27}^{-6}+{\zeta }_{27}^6$

${\zeta }_{27}^{-12}+{\zeta }_{27}^{12}$

${\zeta }_{27}^{-1}+{\zeta }_{27}$

${\zeta }_{27}^{-2}+{\zeta }_{27}^2$

${\zeta }_{27}^{-4}+{\zeta }_{27}^4$

${\zeta }_{27}^{-5}+{\zeta }_{27}^5$

${\zeta }_{27}^{-7}+{\zeta }_{27}^7$

${\zeta }_{27}^{-8}+{\zeta }_{27}^8$

${\zeta }_{27}^{-10}+{\zeta }_{27}^{10}$

${\zeta }_{27}^{-11}+{\zeta }_{27}^{11}$

${\zeta }_{27}^{-13}+{\zeta }_{27}^{13}$