Galois group: 27T27

• Label: 27T27
• Degree: $27$
• Order: $81$
• Cyclic: no
• Abelian: no
• Solvable: yes
• Primitive: no
• $p$-group: yes
• Group: $C_3\wr C_3$

Group action invariants

Degree $n$:		$27$
Transitive number $t$:		$27$
Group:		$C_3\wr C_3$
Parity:		$1$
Primitive:		no
$\card{\Aut(F/K)}$:		$9$
Generators:		(1,4,25)(2,5,26)(3,6,27)(7,8,9)(10,11,12)(13,14,15), (1,12,19)(2,10,20)(3,11,21)(4,15,22)(5,13,23)(6,14,24)(7,17,26)(8,18,27)(9,16,25)

Low degree resolvents

|G/N|	Galois groups for stem field(s)
$3$:	$C_3$ x 4
$9$:	$C_3^2$
$27$:	$C_3^2:C_3$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 3: $C_3$

Degree 9: $C_3^2:C_3$, $C_3 \wr C_3 $ x 2

Low degree siblings

9T17 x 3, 27T19, 27T21, 27T27 x 2

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

Group invariants

Order:		$81=3^{4}$
Cyclic:		no
Abelian:		no
Solvable:		yes
Nilpotency class:		$3$
Label:		81.7
Character table:

		1A	3A1	3A-1	3B1	3B-1	3C1	3C-1	3D1	3D-1	3E1	3E-1	3F1	3F-1	9A1	9A-1	9B1	9B-1
	Size	1	1	1	3	3	3	3	3	3	3	3	9	9	9	9	9	9
	3 P	1A	3A-1	3A1	3B1	3E1	3C-1	3C1	3D1	3E-1	3D-1	3B-1	3F-1	3F1	9A-1	9A1	9B-1	9B1
	Type
81.7.1a	R	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
81.7.1b1	C	$1$	$1$	$1$	${\zeta }_3^{-1}$	${\zeta }_3$	${\zeta }_3$	${\zeta }_3^{-1}$	$1$	$1$	${\zeta }_3$	${\zeta }_3^{-1}$	${\zeta }_3^{-1}$	${\zeta }_3$	${\zeta }_3^{-1}$	${\zeta }_3$	$1$	$1$
81.7.1b2	C	$1$	$1$	$1$	${\zeta }_3$	${\zeta }_3^{-1}$	${\zeta }_3^{-1}$	${\zeta }_3$	$1$	$1$	${\zeta }_3^{-1}$	${\zeta }_3$	${\zeta }_3$	${\zeta }_3^{-1}$	${\zeta }_3$	${\zeta }_3^{-1}$	$1$	$1$
81.7.1c1	C	$1$	$1$	$1$	${\zeta }_3^{-1}$	${\zeta }_3$	${\zeta }_3$	${\zeta }_3^{-1}$	$1$	$1$	${\zeta }_3$	${\zeta }_3^{-1}$	${\zeta }_3$	${\zeta }_3^{-1}$	$1$	$1$	${\zeta }_3$	${\zeta }_3^{-1}$
81.7.1c2	C	$1$	$1$	$1$	${\zeta }_3$	${\zeta }_3^{-1}$	${\zeta }_3^{-1}$	${\zeta }_3$	$1$	$1$	${\zeta }_3^{-1}$	${\zeta }_3$	${\zeta }_3^{-1}$	${\zeta }_3$	$1$	$1$	${\zeta }_3^{-1}$	${\zeta }_3$
81.7.1d1	C	$1$	$1$	$1$	${\zeta }_3^{-1}$	${\zeta }_3$	${\zeta }_3$	${\zeta }_3^{-1}$	$1$	$1$	${\zeta }_3$	${\zeta }_3^{-1}$	$1$	$1$	${\zeta }_3$	${\zeta }_3^{-1}$	${\zeta }_3^{-1}$	${\zeta }_3$
81.7.1d2	C	$1$	$1$	$1$	${\zeta }_3$	${\zeta }_3^{-1}$	${\zeta }_3^{-1}$	${\zeta }_3$	$1$	$1$	${\zeta }_3^{-1}$	${\zeta }_3$	$1$	$1$	${\zeta }_3^{-1}$	${\zeta }_3$	${\zeta }_3$	${\zeta }_3^{-1}$
81.7.1e1	C	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	${\zeta }_3^{-1}$	${\zeta }_3$	${\zeta }_3$	${\zeta }_3^{-1}$	${\zeta }_3$	${\zeta }_3^{-1}$
81.7.1e2	C	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	${\zeta }_3$	${\zeta }_3^{-1}$	${\zeta }_3^{-1}$	${\zeta }_3$	${\zeta }_3^{-1}$	${\zeta }_3$
81.7.3a1	C	$3$	$3$	$3$	$0$	$0$	$0$	$0$	$3{\zeta }_3^{-1}$	$3{\zeta }_3$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$
81.7.3a2	C	$3$	$3$	$3$	$0$	$0$	$0$	$0$	$3{\zeta }_3$	$3{\zeta }_3^{-1}$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$
81.7.3b1	C	$3$	$3{\zeta }_3^{-1}$	$3{\zeta }_3$	$-1+{\zeta }_3$	$-2-{\zeta }_3$	$1+2{\zeta }_3$	$-1-2{\zeta }_3$	$0$	$0$	$1-{\zeta }_3$	$2+{\zeta }_3$	$0$	$0$	$0$	$0$	$0$	$0$
81.7.3b2	C	$3$	$3{\zeta }_3$	$3{\zeta }_3^{-1}$	$-2-{\zeta }_3$	$-1+{\zeta }_3$	$-1-2{\zeta }_3$	$1+2{\zeta }_3$	$0$	$0$	$2+{\zeta }_3$	$1-{\zeta }_3$	$0$	$0$	$0$	$0$	$0$	$0$
81.7.3c1	C	$3$	$3{\zeta }_3^{-1}$	$3{\zeta }_3$	$-1-2{\zeta }_3$	$1+2{\zeta }_3$	$1-{\zeta }_3$	$2+{\zeta }_3$	$0$	$0$	$-2-{\zeta }_3$	$-1+{\zeta }_3$	$0$	$0$	$0$	$0$	$0$	$0$
81.7.3c2	C	$3$	$3{\zeta }_3$	$3{\zeta }_3^{-1}$	$1+2{\zeta }_3$	$-1-2{\zeta }_3$	$2+{\zeta }_3$	$1-{\zeta }_3$	$0$	$0$	$-1+{\zeta }_3$	$-2-{\zeta }_3$	$0$	$0$	$0$	$0$	$0$	$0$
81.7.3d1	C	$3$	$3{\zeta }_3^{-1}$	$3{\zeta }_3$	$2+{\zeta }_3$	$1-{\zeta }_3$	$-2-{\zeta }_3$	$-1+{\zeta }_3$	$0$	$0$	$1+2{\zeta }_3$	$-1-2{\zeta }_3$	$0$	$0$	$0$	$0$	$0$	$0$
81.7.3d2	C	$3$	$3{\zeta }_3$	$3{\zeta }_3^{-1}$	$1-{\zeta }_3$	$2+{\zeta }_3$	$-1+{\zeta }_3$	$-2-{\zeta }_3$	$0$	$0$	$-1-2{\zeta }_3$	$1+2{\zeta }_3$	$0$	$0$	$0$	$0$	$0$	$0$