Galois group: 27T24

• Label: 27T24
• Degree: $27$
• Order: $81$
• Cyclic: no
• Abelian: no
• Solvable: yes
• Primitive: no
• $p$-group: yes
• Group: $\He_3:C_3$

Group action invariants

Degree $n$:		$27$
Transitive number $t$:		$24$
Group:		$\He_3:C_3$
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)(19,23,25,20,24,26,21,22,27), (1,13,23)(2,14,24)(3,15,22)(4,16,25)(5,17,26)(6,18,27)(7,10,20)(8,11,21)(9,12,19)

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

27T23 x 3

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 $	$3$	$9$	$( 1, 4, 7, 2, 5, 8, 3, 6, 9)(10,13,18,11,14,16,12,15,17)(19,23,25,20,24,26,21, 22,27)$
	$ 9, 9, 9 $	$3$	$9$	$( 1, 4, 7, 2, 5, 8, 3, 6, 9)(10,15,16,11,13,17,12,14,18)(19,24,27,20,22,25,21, 23,26)$
	$ 9, 9, 9 $	$3$	$9$	$( 1, 5, 9, 2, 6, 7, 3, 4, 8)(10,13,18,11,14,16,12,15,17)(19,22,26,20,23,27,21, 24,25)$
	$ 9, 9, 9 $	$3$	$9$	$( 1, 7, 5, 3, 9, 4, 2, 8, 6)(10,16,13,12,18,15,11,17,14)(19,27,22,21,26,24,20, 25,23)$
	$ 9, 9, 9 $	$3$	$9$	$( 1, 7, 5, 3, 9, 4, 2, 8, 6)(10,17,15,12,16,14,11,18,13)(19,26,23,21,25,22,20, 27,24)$
	$ 9, 9, 9 $	$3$	$9$	$( 1, 8, 4, 3, 7, 6, 2, 9, 5)(10,16,13,12,18,15,11,17,14)(19,26,23,21,25,22,20, 27,24)$
	$ 3, 3, 3, 3, 3, 3, 3, 3, 3 $	$9$	$3$	$( 1,10,25)( 2,11,26)( 3,12,27)( 4,14,20)( 5,15,21)( 6,13,19)( 7,17,24) ( 8,18,22)( 9,16,23)$
	$ 3, 3, 3, 3, 3, 3, 3, 3, 3 $	$9$	$3$	$( 1,13,22)( 2,14,23)( 3,15,24)( 4,16,27)( 5,17,25)( 6,18,26)( 7,10,19) ( 8,11,20)( 9,12,21)$
	$ 3, 3, 3, 3, 3, 3, 3, 3, 3 $	$9$	$3$	$( 1,16,21)( 2,17,19)( 3,18,20)( 4,10,22)( 5,11,23)( 6,12,24)( 7,14,27) ( 8,15,25)( 9,13,26)$
	$ 3, 3, 3, 3, 3, 3, 3, 3, 3 $	$9$	$3$	$( 1,19,16)( 2,20,17)( 3,21,18)( 4,23,10)( 5,24,11)( 6,22,12)( 7,25,14) ( 8,26,15)( 9,27,13)$
	$ 3, 3, 3, 3, 3, 3, 3, 3, 3 $	$9$	$3$	$( 1,22,13)( 2,23,14)( 3,24,15)( 4,27,16)( 5,25,17)( 6,26,18)( 7,19,10) ( 8,20,11)( 9,21,12)$
	$ 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:		81.9
Character table:

		1A	3A1	3A-1	3B1	3B-1	3C1	3C-1	3D1	3D-1	3E1	3E-1	9A1	9A-1	9A2	9A-2	9A4	9A-4
	Size	1	1	1	3	3	9	9	9	9	9	9	3	3	3	3	3	3
	3 P	1A	3A-1	3A1	3B-1	3B1	3D1	3D-1	3E1	3E-1	3C-1	3C1	9A-4	9A4	9A2	9A1	9A-1	9A-2
	Type
81.9.1a	R	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
81.9.1b1	C	$1$	$1$	$1$	$1$	$1$	${\zeta }_3^{-1}$	${\zeta }_3$	${\zeta }_3$	${\zeta }_3^{-1}$	${\zeta }_3$	${\zeta }_3^{-1}$	$1$	$1$	$1$	$1$	$1$	$1$
81.9.1b2	C	$1$	$1$	$1$	$1$	$1$	${\zeta }_3$	${\zeta }_3^{-1}$	${\zeta }_3^{-1}$	${\zeta }_3$	${\zeta }_3^{-1}$	${\zeta }_3$	$1$	$1$	$1$	$1$	$1$	$1$
81.9.1c1	C	$1$	$1$	$1$	$1$	$1$	${\zeta }_3^{-1}$	${\zeta }_3$	${\zeta }_3^{-1}$	${\zeta }_3$	$1$	$1$	${\zeta }_3$	${\zeta }_3^{-1}$	${\zeta }_3^{-1}$	${\zeta }_3$	${\zeta }_3$	${\zeta }_3^{-1}$
81.9.1c2	C	$1$	$1$	$1$	$1$	$1$	${\zeta }_3$	${\zeta }_3^{-1}$	${\zeta }_3$	${\zeta }_3^{-1}$	$1$	$1$	${\zeta }_3^{-1}$	${\zeta }_3$	${\zeta }_3$	${\zeta }_3^{-1}$	${\zeta }_3^{-1}$	${\zeta }_3$
81.9.1d1	C	$1$	$1$	$1$	$1$	$1$	${\zeta }_3^{-1}$	${\zeta }_3$	$1$	$1$	${\zeta }_3^{-1}$	${\zeta }_3$	${\zeta }_3^{-1}$	${\zeta }_3$	${\zeta }_3$	${\zeta }_3^{-1}$	${\zeta }_3^{-1}$	${\zeta }_3$
81.9.1d2	C	$1$	$1$	$1$	$1$	$1$	${\zeta }_3$	${\zeta }_3^{-1}$	$1$	$1$	${\zeta }_3$	${\zeta }_3^{-1}$	${\zeta }_3$	${\zeta }_3^{-1}$	${\zeta }_3^{-1}$	${\zeta }_3$	${\zeta }_3$	${\zeta }_3^{-1}$
81.9.1e1	C	$1$	$1$	$1$	$1$	$1$	$1$	$1$	${\zeta }_3^{-1}$	${\zeta }_3$	${\zeta }_3$	${\zeta }_3^{-1}$	${\zeta }_3^{-1}$	${\zeta }_3$	${\zeta }_3$	${\zeta }_3^{-1}$	${\zeta }_3^{-1}$	${\zeta }_3$
81.9.1e2	C	$1$	$1$	$1$	$1$	$1$	$1$	$1$	${\zeta }_3$	${\zeta }_3^{-1}$	${\zeta }_3^{-1}$	${\zeta }_3$	${\zeta }_3$	${\zeta }_3^{-1}$	${\zeta }_3^{-1}$	${\zeta }_3$	${\zeta }_3$	${\zeta }_3^{-1}$
81.9.3a1	C	$3$	$3$	$3$	$3{\zeta }_3^{-1}$	$3{\zeta }_3$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$
81.9.3a2	C	$3$	$3$	$3$	$3{\zeta }_3$	$3{\zeta }_3^{-1}$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$
81.9.3b1	C	$3$	$3{\zeta }_9^{-3}$	$3{\zeta }_9^3$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	${\zeta }_9^{-4}-{\zeta }_9^2$	${\zeta }_9+2{\zeta }_9^4$	${\zeta }_9-{\zeta }_9^4$	$-2{\zeta }_9^{-4}-{\zeta }_9^2$	${\zeta }_9^{-4}+2{\zeta }_9^2$	$-2{\zeta }_9-{\zeta }_9^4$
81.9.3b2	C	$3$	$3{\zeta }_9^3$	$3{\zeta }_9^{-3}$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	${\zeta }_9+2{\zeta }_9^4$	${\zeta }_9^{-4}-{\zeta }_9^2$	$-2{\zeta }_9^{-4}-{\zeta }_9^2$	${\zeta }_9-{\zeta }_9^4$	$-2{\zeta }_9-{\zeta }_9^4$	${\zeta }_9^{-4}+2{\zeta }_9^2$
81.9.3b3	C	$3$	$3{\zeta }_9^{-3}$	$3{\zeta }_9^3$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	${\zeta }_9^{-4}+2{\zeta }_9^2$	$-2{\zeta }_9-{\zeta }_9^4$	${\zeta }_9+2{\zeta }_9^4$	${\zeta }_9^{-4}-{\zeta }_9^2$	$-2{\zeta }_9^{-4}-{\zeta }_9^2$	${\zeta }_9-{\zeta }_9^4$
81.9.3b4	C	$3$	$3{\zeta }_9^3$	$3{\zeta }_9^{-3}$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$-2{\zeta }_9-{\zeta }_9^4$	${\zeta }_9^{-4}+2{\zeta }_9^2$	${\zeta }_9^{-4}-{\zeta }_9^2$	${\zeta }_9+2{\zeta }_9^4$	${\zeta }_9-{\zeta }_9^4$	$-2{\zeta }_9^{-4}-{\zeta }_9^2$
81.9.3b5	C	$3$	$3{\zeta }_9^{-3}$	$3{\zeta }_9^3$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$-2{\zeta }_9^{-4}-{\zeta }_9^2$	${\zeta }_9-{\zeta }_9^4$	$-2{\zeta }_9-{\zeta }_9^4$	${\zeta }_9^{-4}+2{\zeta }_9^2$	${\zeta }_9^{-4}-{\zeta }_9^2$	${\zeta }_9+2{\zeta }_9^4$
81.9.3b6	C	$3$	$3{\zeta }_9^3$	$3{\zeta }_9^{-3}$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	${\zeta }_9-{\zeta }_9^4$	$-2{\zeta }_9^{-4}-{\zeta }_9^2$	${\zeta }_9^{-4}+2{\zeta }_9^2$	$-2{\zeta }_9-{\zeta }_9^4$	${\zeta }_9+2{\zeta }_9^4$	${\zeta }_9^{-4}-{\zeta }_9^2$