Galois group: 27T19

• Label: 27T19
• Degree: $27$
• Order: $81$
• Cyclic: no
• Abelian: no
• Solvable: yes
• Transitivity: $1$
• Primitive: no
• $p$-group: yes
• Group: $C_3\wr C_3$

Group invariants

Abstract group:		$C_3\wr C_3$
Order:		$81=3^{4}$
Cyclic:		no
Abelian:		no
Solvable:		yes
Nilpotency class:		$3$

Group action invariants

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

Low degree resolvents

$\card{(G/N)}$	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: $C_3^2$, $C_3 \wr C_3 $ x 3

Low degree siblings

9T17 x 3, 27T21, 27T27 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	Index	Representative
1A	$1^{27}$	$1$	$1$	$0$	$()$
3A1	$3^{9}$	$1$	$3$	$18$	$( 1,26, 6)( 2,27, 4)( 3,25, 5)( 7,14,12)( 8,15,10)( 9,13,11)(16,23,21)(17,24,19)(18,22,20)$
3A-1	$3^{9}$	$1$	$3$	$18$	$( 1, 6,26)( 2, 4,27)( 3, 5,25)( 7,12,14)( 8,10,15)( 9,11,13)(16,21,23)(17,19,24)(18,20,22)$
3B1	$3^{6},1^{9}$	$3$	$3$	$12$	$( 7,14,12)( 8,15,10)( 9,13,11)(16,21,23)(17,19,24)(18,20,22)$
3B-1	$3^{6},1^{9}$	$3$	$3$	$12$	$( 7,12,14)( 8,10,15)( 9,11,13)(16,23,21)(17,24,19)(18,22,20)$
3C1	$3^{9}$	$3$	$3$	$18$	$( 1,25, 4)( 2,26, 5)( 3,27, 6)( 7, 9, 8)(10,12,11)(13,15,14)(16,18,17)(19,21,20)(22,24,23)$
3C-1	$3^{9}$	$3$	$3$	$18$	$( 1, 4,25)( 2, 5,26)( 3, 6,27)( 7, 8, 9)(10,11,12)(13,14,15)(16,17,18)(19,20,21)(22,23,24)$
3D1	$3^{9}$	$3$	$3$	$18$	$( 1, 5,27)( 2, 6,25)( 3, 4,26)( 7,13,10)( 8,14,11)( 9,15,12)(16,22,19)(17,23,20)(18,24,21)$
3D-1	$3^{9}$	$3$	$3$	$18$	$( 1, 4,25)( 2, 5,26)( 3, 6,27)( 7,15,11)( 8,13,12)( 9,14,10)(16,19,22)(17,20,23)(18,21,24)$
3E1	$3^{9}$	$3$	$3$	$18$	$( 1, 3, 2)( 4, 6, 5)( 7,11,15)( 8,12,13)( 9,10,14)(16,20,24)(17,21,22)(18,19,23)(25,27,26)$
3E-1	$3^{9}$	$3$	$3$	$18$	$( 1, 2, 3)( 4, 5, 6)( 7,15,11)( 8,13,12)( 9,14,10)(16,24,20)(17,22,21)(18,23,19)(25,26,27)$
3F1	$3^{9}$	$9$	$3$	$18$	$( 1,23,10)( 2,24,11)( 3,22,12)( 4,17,13)( 5,18,14)( 6,16,15)( 7,25,20)( 8,26,21)( 9,27,19)$
3F-1	$3^{9}$	$9$	$3$	$18$	$( 1,10,21)( 2,11,19)( 3,12,20)( 4,13,24)( 5,14,22)( 6,15,23)( 7,18,25)( 8,16,26)( 9,17,27)$
9A1	$9^{3}$	$9$	$9$	$24$	$( 1, 7,17,26,14,24, 6,12,19)( 2, 8,18,27,15,22, 4,10,20)( 3, 9,16,25,13,23, 5,11,21)$
9A-1	$9^{3}$	$9$	$9$	$24$	$( 1,17,14, 6,19, 7,26,24,12)( 2,18,15, 4,20, 8,27,22,10)( 3,16,13, 5,21, 9,25,23,11)$
9B1	$9^{3}$	$9$	$9$	$24$	$( 1,13,22, 6, 9,18,26,11,20)( 2,14,23, 4, 7,16,27,12,21)( 3,15,24, 5, 8,17,25,10,19)$
9B-1	$9^{3}$	$9$	$9$	$24$	$( 1,20, 9,26,18,13, 6,22,11)( 2,21, 7,27,16,14, 4,23,12)( 3,19, 8,25,17,15, 5,24,10)$

Malle's constant $a(G)$:     $1/12$

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	3B-1	3B1	3C-1	3C1	3D-1	3D1	3E-1	3E1	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	$1$	$1$	$1$	${\zeta }_3^{-1}$	$1$	${\zeta }_3^{-1}$	${\zeta }_3$	${\zeta }_3^{-1}$	${\zeta }_3$	$1$	${\zeta }_3$	${\zeta }_3$	${\zeta }_3^{-1}$	${\zeta }_3$	${\zeta }_3^{-1}$	$1$	$1$
81.7.1b2	C	$1$	$1$	$1$	${\zeta }_3$	$1$	${\zeta }_3$	${\zeta }_3^{-1}$	${\zeta }_3$	${\zeta }_3^{-1}$	$1$	${\zeta }_3^{-1}$	${\zeta }_3^{-1}$	${\zeta }_3$	${\zeta }_3^{-1}$	${\zeta }_3$	$1$	$1$
81.7.1c1	C	$1$	$1$	$1$	${\zeta }_3^{-1}$	$1$	${\zeta }_3^{-1}$	${\zeta }_3$	${\zeta }_3^{-1}$	${\zeta }_3$	$1$	${\zeta }_3$	${\zeta }_3^{-1}$	${\zeta }_3$	$1$	$1$	${\zeta }_3^{-1}$	${\zeta }_3$
81.7.1c2	C	$1$	$1$	$1$	${\zeta }_3$	$1$	${\zeta }_3$	${\zeta }_3^{-1}$	${\zeta }_3$	${\zeta }_3^{-1}$	$1$	${\zeta }_3^{-1}$	${\zeta }_3$	${\zeta }_3^{-1}$	$1$	$1$	${\zeta }_3$	${\zeta }_3^{-1}$
81.7.1d1	C	$1$	$1$	$1$	${\zeta }_3^{-1}$	$1$	${\zeta }_3^{-1}$	${\zeta }_3$	${\zeta }_3^{-1}$	${\zeta }_3$	$1$	${\zeta }_3$	$1$	$1$	${\zeta }_3^{-1}$	${\zeta }_3$	${\zeta }_3$	${\zeta }_3^{-1}$
81.7.1d2	C	$1$	$1$	$1$	${\zeta }_3$	$1$	${\zeta }_3$	${\zeta }_3^{-1}$	${\zeta }_3$	${\zeta }_3^{-1}$	$1$	${\zeta }_3^{-1}$	$1$	$1$	${\zeta }_3$	${\zeta }_3^{-1}$	${\zeta }_3^{-1}$	${\zeta }_3$
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	$3$	$3$	$3$	$0$	$3{\zeta }_3^{-1}$	$0$	$0$	$0$	$0$	$3{\zeta }_3$	$0$	$0$	$0$	$0$	$0$	$0$	$0$
81.7.3a2	C	$3$	$3$	$3$	$0$	$3{\zeta }_3$	$0$	$0$	$0$	$0$	$3{\zeta }_3^{-1}$	$0$	$0$	$0$	$0$	$0$	$0$	$0$
81.7.3b1	C	$3$	$3{\zeta }_3^{-1}$	$3{\zeta }_3$	$-2-{\zeta }_3$	$0$	$1+2{\zeta }_3$	$-1-2{\zeta }_3$	$1-{\zeta }_3$	$-1+{\zeta }_3$	$0$	$2+{\zeta }_3$	$0$	$0$	$0$	$0$	$0$	$0$
81.7.3b2	C	$3$	$3{\zeta }_3$	$3{\zeta }_3^{-1}$	$-1+{\zeta }_3$	$0$	$-1-2{\zeta }_3$	$1+2{\zeta }_3$	$2+{\zeta }_3$	$-2-{\zeta }_3$	$0$	$1-{\zeta }_3$	$0$	$0$	$0$	$0$	$0$	$0$
81.7.3c1	C	$3$	$3{\zeta }_3^{-1}$	$3{\zeta }_3$	$1+2{\zeta }_3$	$0$	$1-{\zeta }_3$	$2+{\zeta }_3$	$-2-{\zeta }_3$	$-1-2{\zeta }_3$	$0$	$-1+{\zeta }_3$	$0$	$0$	$0$	$0$	$0$	$0$
81.7.3c2	C	$3$	$3{\zeta }_3$	$3{\zeta }_3^{-1}$	$-1-2{\zeta }_3$	$0$	$2+{\zeta }_3$	$1-{\zeta }_3$	$-1+{\zeta }_3$	$1+2{\zeta }_3$	$0$	$-2-{\zeta }_3$	$0$	$0$	$0$	$0$	$0$	$0$
81.7.3d1	C	$3$	$3{\zeta }_3^{-1}$	$3{\zeta }_3$	$1-{\zeta }_3$	$0$	$-2-{\zeta }_3$	$-1+{\zeta }_3$	$1+2{\zeta }_3$	$2+{\zeta }_3$	$0$	$-1-2{\zeta }_3$	$0$	$0$	$0$	$0$	$0$	$0$
81.7.3d2	C	$3$	$3{\zeta }_3$	$3{\zeta }_3^{-1}$	$2+{\zeta }_3$	$0$	$-1+{\zeta }_3$	$-2-{\zeta }_3$	$-1-2{\zeta }_3$	$1-{\zeta }_3$	$0$	$1+2{\zeta }_3$	$0$	$0$	$0$	$0$	$0$	$0$

Regular extensions

Data not computed