Galois group: 27T26

• Label: 27T26
• 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 $t$:		$26$
Group:		$C_3.\He_3$
Parity:		$1$
Primitive:		no
$\card{\Aut(F/K)}$:		$9$
Generators:		(1,4,7,2,5,8,3,6,9)(10,13,18,11,14,16,12,15,17)(19,22,26,20,23,27,21,24,25), (1,12,25)(2,10,26)(3,11,27)(4,13,20)(5,14,21)(6,15,19)(7,16,24)(8,17,22)(9,18,23)

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: $C_3$ x 4

Degree 9: $C_3^2$

Low degree siblings

27T20

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

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	$3$	$3{\zeta }_9^3$	$3{\zeta }_9^{-3}$	$0$	$0$	$0$	$0$	$-{\zeta }_9-2{\zeta }_9^4$	$-{\zeta }_9^{-4}+{\zeta }_9^2$	$2{\zeta }_9^{-4}+{\zeta }_9^2$	$-{\zeta }_9+{\zeta }_9^4$	$2{\zeta }_9+{\zeta }_9^4$	$-{\zeta }_9^{-4}-2{\zeta }_9^2$	$0$	$0$	$0$	$0$