Galois group: 27T20

• Label: 27T20
• Degree: $27$
• Order: $81$
• Cyclic: no
• Abelian: no
• Solvable: yes
• Primitive: no
• $p$-group: yes
• Group: $C_3.\He_3$

Group action invariants

Degree $n$:		$27$
Transitive number $t$:		$20$
Group:		$C_3.\He_3$
Parity:		$1$
Primitive:		no
$\card{\Aut(F/K)}$:		$3$
Generators:		(4,5,6)(7,15,10)(8,13,11)(9,14,12)(16,20,22)(17,21,23)(18,19,24)(25,27,26), (1,12,20,3,11,19,2,10,21)(4,13,23,6,15,22,5,14,24)(7,16,26,9,18,25,8,17,27)

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: $C_3^2:C_3$

Low degree siblings

27T26

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

Group invariants

Order:		$81=3^{4}$
Cyclic:		no
Abelian:		no
Solvable:		yes
Nilpotency class:		$3$
Label:		81.8
Character table:

		1A	3A1	3A-1	3B1	3B-1	3C1	3C-1	9A1	9A-1	9A2	9A-2	9A4	9A-4	9B1	9B-1	9C1	9C-1
	Size	1	1	1	3	3	9	9	3	3	3	3	3	3	9	9	9	9
	3 P	1A	3A-1	3A1	3B-1	3B1	3C-1	3C1	9A2	9A4	9A-2	9A1	9A-1	9A-4	9B-1	9B1	9C-1	9C1
	Type
81.8.1a	R	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
81.8.1b1	C	$1$	$1$	$1$	$1$	$1$	${\zeta }_3^{-1}$	${\zeta }_3$	${\zeta }_3$	${\zeta }_3^{-1}$	${\zeta }_3^{-1}$	${\zeta }_3$	${\zeta }_3$	${\zeta }_3^{-1}$	${\zeta }_3^{-1}$	${\zeta }_3$	$1$	$1$
81.8.1b2	C	$1$	$1$	$1$	$1$	$1$	${\zeta }_3$	${\zeta }_3^{-1}$	${\zeta }_3^{-1}$	${\zeta }_3$	${\zeta }_3$	${\zeta }_3^{-1}$	${\zeta }_3^{-1}$	${\zeta }_3$	${\zeta }_3$	${\zeta }_3^{-1}$	$1$	$1$
81.8.1c1	C	$1$	$1$	$1$	$1$	$1$	${\zeta }_3^{-1}$	${\zeta }_3$	${\zeta }_3^{-1}$	${\zeta }_3$	${\zeta }_3$	${\zeta }_3^{-1}$	${\zeta }_3^{-1}$	${\zeta }_3$	$1$	$1$	${\zeta }_3^{-1}$	${\zeta }_3$
81.8.1c2	C	$1$	$1$	$1$	$1$	$1$	${\zeta }_3$	${\zeta }_3^{-1}$	${\zeta }_3$	${\zeta }_3^{-1}$	${\zeta }_3^{-1}$	${\zeta }_3$	${\zeta }_3$	${\zeta }_3^{-1}$	$1$	$1$	${\zeta }_3$	${\zeta }_3^{-1}$
81.8.1d1	C	$1$	$1$	$1$	$1$	$1$	${\zeta }_3^{-1}$	${\zeta }_3$	$1$	$1$	$1$	$1$	$1$	$1$	${\zeta }_3$	${\zeta }_3^{-1}$	${\zeta }_3$	${\zeta }_3^{-1}$
81.8.1d2	C	$1$	$1$	$1$	$1$	$1$	${\zeta }_3$	${\zeta }_3^{-1}$	$1$	$1$	$1$	$1$	$1$	$1$	${\zeta }_3^{-1}$	${\zeta }_3$	${\zeta }_3^{-1}$	${\zeta }_3$
81.8.1e1	C	$1$	$1$	$1$	$1$	$1$	$1$	$1$	${\zeta }_3^{-1}$	${\zeta }_3$	${\zeta }_3$	${\zeta }_3^{-1}$	${\zeta }_3^{-1}$	${\zeta }_3$	${\zeta }_3^{-1}$	${\zeta }_3$	${\zeta }_3$	${\zeta }_3^{-1}$
81.8.1e2	C	$1$	$1$	$1$	$1$	$1$	$1$	$1$	${\zeta }_3$	${\zeta }_3^{-1}$	${\zeta }_3^{-1}$	${\zeta }_3$	${\zeta }_3$	${\zeta }_3^{-1}$	${\zeta }_3$	${\zeta }_3^{-1}$	${\zeta }_3^{-1}$	${\zeta }_3$
81.8.3a1	C	$3$	$3$	$3$	$3{\zeta }_3^{-1}$	$3{\zeta }_3$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$
81.8.3a2	C	$3$	$3$	$3$	$3{\zeta }_3$	$3{\zeta }_3^{-1}$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$
81.8.3b1	C	$3$	$3{\zeta }_9^{-3}$	$3{\zeta }_9^3$	$0$	$0$	$0$	$0$	$2{\zeta }_9^{-4}+{\zeta }_9^2$	$-{\zeta }_9+{\zeta }_9^4$	$2{\zeta }_9+{\zeta }_9^4$	$-{\zeta }_9^{-4}-2{\zeta }_9^2$	$-{\zeta }_9^{-4}+{\zeta }_9^2$	$-{\zeta }_9-2{\zeta }_9^4$	$0$	$0$	$0$	$0$
81.8.3b2	C	$3$	$3{\zeta }_9^3$	$3{\zeta }_9^{-3}$	$0$	$0$	$0$	$0$	$-{\zeta }_9+{\zeta }_9^4$	$2{\zeta }_9^{-4}+{\zeta }_9^2$	$-{\zeta }_9^{-4}-2{\zeta }_9^2$	$2{\zeta }_9+{\zeta }_9^4$	$-{\zeta }_9-2{\zeta }_9^4$	$-{\zeta }_9^{-4}+{\zeta }_9^2$	$0$	$0$	$0$	$0$
81.8.3b3	C	$3$	$3{\zeta }_9^{-3}$	$3{\zeta }_9^3$	$0$	$0$	$0$	$0$	$-{\zeta }_9^{-4}-2{\zeta }_9^2$	$2{\zeta }_9+{\zeta }_9^4$	$-{\zeta }_9-2{\zeta }_9^4$	$-{\zeta }_9^{-4}+{\zeta }_9^2$	$2{\zeta }_9^{-4}+{\zeta }_9^2$	$-{\zeta }_9+{\zeta }_9^4$	$0$	$0$	$0$	$0$
81.8.3b4	C	$3$	$3{\zeta }_9^3$	$3{\zeta }_9^{-3}$	$0$	$0$	$0$	$0$	$2{\zeta }_9+{\zeta }_9^4$	$-{\zeta }_9^{-4}-2{\zeta }_9^2$	$-{\zeta }_9^{-4}+{\zeta }_9^2$	$-{\zeta }_9-2{\zeta }_9^4$	$-{\zeta }_9+{\zeta }_9^4$	$2{\zeta }_9^{-4}+{\zeta }_9^2$	$0$	$0$	$0$	$0$
81.8.3b5	C	$3$	$3{\zeta }_9^{-3}$	$3{\zeta }_9^3$	$0$	$0$	$0$	$0$	$-{\zeta }_9^{-4}+{\zeta }_9^2$	$-{\zeta }_9-2{\zeta }_9^4$	$-{\zeta }_9+{\zeta }_9^4$	$2{\zeta }_9^{-4}+{\zeta }_9^2$	$-{\zeta }_9^{-4}-2{\zeta }_9^2$	$2{\zeta }_9+{\zeta }_9^4$	$0$	$0$	$0$	$0$
81.8.3b6	C	$3$	$3{\zeta }_9^3$	$3{\zeta }_9^{-3}$	$0$	$0$	$0$	$0$	$-{\zeta }_9-2{\zeta }_9^4$	$-{\zeta }_9^{-4}+{\zeta }_9^2$	$2{\zeta }_9^{-4}+{\zeta }_9^2$	$-{\zeta }_9+{\zeta }_9^4$	$2{\zeta }_9+{\zeta }_9^4$	$-{\zeta }_9^{-4}-2{\zeta }_9^2$	$0$	$0$	$0$	$0$