Galois group: 27T5

• Label: 27T5
• Degree: $27$
• Order: $27$
• Cyclic: no
• Abelian: no
• Solvable: yes
• Transitivity: $1$
• Primitive: no
• $p$-group: yes
• Group: $C_9:C_3$

Group invariants

Abstract group:		$C_9:C_3$
Order:		$27=3^{3}$
Cyclic:		no
Abelian:		no
Solvable:		yes
Nilpotency class:		$2$

Group action invariants

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

Low degree resolvents

$\card{(G/N)}$	Galois groups for stem field(s)
$3$:	$C_3$ x 4
$9$:	$C_3^2$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 3: $C_3$ x 4

Degree 9: $C_3^2$, $C_9:C_3$

Low degree siblings

9T6

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	Index	Representative
1A	$1^{27}$	$1$	$1$	$0$	$()$
3A1	$3^{9}$	$1$	$3$	$18$	$( 1,11,21)( 2,12,19)( 3,10,20)( 4,14,24)( 5,15,22)( 6,13,23)( 7,17,27)( 8,18,25)( 9,16,26)$
3A-1	$3^{9}$	$1$	$3$	$18$	$( 1,21,11)( 2,19,12)( 3,20,10)( 4,24,14)( 5,22,15)( 6,23,13)( 7,27,17)( 8,25,18)( 9,26,16)$
3B1	$3^{9}$	$3$	$3$	$18$	$( 1,20,12)( 2,21,10)( 3,19,11)( 4,13,22)( 5,14,23)( 6,15,24)( 7, 9, 8)(16,18,17)(25,27,26)$
3B-1	$3^{9}$	$3$	$3$	$18$	$( 1,12,20)( 2,10,21)( 3,11,19)( 4,22,13)( 5,23,14)( 6,24,15)( 7, 8, 9)(16,17,18)(25,26,27)$
9A1	$9^{3}$	$3$	$9$	$24$	$( 1, 4, 8,11,14,18,21,24,25)( 2, 5, 9,12,15,16,19,22,26)( 3, 6, 7,10,13,17,20,23,27)$
9A-1	$9^{3}$	$3$	$9$	$24$	$( 1, 8,14,21,25, 4,11,18,24)( 2, 9,15,19,26, 5,12,16,22)( 3, 7,13,20,27, 6,10,17,23)$
9B1	$9^{3}$	$3$	$9$	$24$	$( 1,23, 9,11, 6,16,21,13,26)( 2,24, 7,12, 4,17,19,14,27)( 3,22, 8,10, 5,18,20,15,25)$
9B-1	$9^{3}$	$3$	$9$	$24$	$( 1,16,23,21, 9,13,11,26, 6)( 2,17,24,19, 7,14,12,27, 4)( 3,18,22,20, 8,15,10,25, 5)$
9C1	$9^{3}$	$3$	$9$	$24$	$( 1,27, 5,21,17,22,11, 7,15)( 2,25, 6,19,18,23,12, 8,13)( 3,26, 4,20,16,24,10, 9,14)$
9C-1	$9^{3}$	$3$	$9$	$24$	$( 1,15, 7,11,22,17,21, 5,27)( 2,13, 8,12,23,18,19, 6,25)( 3,14, 9,10,24,16,20, 4,26)$

Malle's constant $a(G)$:     $1/18$

Character table

		1A	3A1	3A-1	3B1	3B-1	9A1	9A-1	9B1	9B-1	9C1	9C-1
	Size	1	1	1	3	3	3	3	3	3	3	3
	3 P	1A	3A-1	3A1	3B-1	3B1	9A-1	9A1	9B-1	9B1	9C-1	9C1
	Type
27.4.1a	R	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
27.4.1b1	C	$1$	$1$	$1$	${\zeta }_3^{-1}$	${\zeta }_3$	${\zeta }_3^{-1}$	${\zeta }_3$	${\zeta }_3$	${\zeta }_3^{-1}$	$1$	$1$
27.4.1b2	C	$1$	$1$	$1$	${\zeta }_3$	${\zeta }_3^{-1}$	${\zeta }_3$	${\zeta }_3^{-1}$	${\zeta }_3^{-1}$	${\zeta }_3$	$1$	$1$
27.4.1c1	C	$1$	$1$	$1$	${\zeta }_3^{-1}$	${\zeta }_3$	${\zeta }_3$	${\zeta }_3^{-1}$	$1$	$1$	${\zeta }_3$	${\zeta }_3^{-1}$
27.4.1c2	C	$1$	$1$	$1$	${\zeta }_3$	${\zeta }_3^{-1}$	${\zeta }_3^{-1}$	${\zeta }_3$	$1$	$1$	${\zeta }_3^{-1}$	${\zeta }_3$
27.4.1d1	C	$1$	$1$	$1$	${\zeta }_3^{-1}$	${\zeta }_3$	$1$	$1$	${\zeta }_3^{-1}$	${\zeta }_3$	${\zeta }_3^{-1}$	${\zeta }_3$
27.4.1d2	C	$1$	$1$	$1$	${\zeta }_3$	${\zeta }_3^{-1}$	$1$	$1$	${\zeta }_3$	${\zeta }_3^{-1}$	${\zeta }_3$	${\zeta }_3^{-1}$
27.4.1e1	C	$1$	$1$	$1$	$1$	$1$	${\zeta }_3^{-1}$	${\zeta }_3$	${\zeta }_3^{-1}$	${\zeta }_3$	${\zeta }_3$	${\zeta }_3^{-1}$
27.4.1e2	C	$1$	$1$	$1$	$1$	$1$	${\zeta }_3$	${\zeta }_3^{-1}$	${\zeta }_3$	${\zeta }_3^{-1}$	${\zeta }_3^{-1}$	${\zeta }_3$
27.4.3a1	C	$3$	$3{\zeta }_3^{-1}$	$3{\zeta }_3$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$
27.4.3a2	C	$3$	$3{\zeta }_3$	$3{\zeta }_3^{-1}$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$

Regular extensions

Data not computed