Galois group: 27T62

• Label: 27T62
• Degree: $27$
• Order: $162$
• Cyclic: no
• Abelian: no
• Solvable: yes
• Primitive: no
• $p$-group: no
• Group: $C_3^3:C_6$

Group invariants

Abstract group:		$C_3^3:C_6$
Order:		$162=2 \cdot 3^{4}$
Cyclic:		no
Abelian:		no
Solvable:		yes
Nilpotency class:		not nilpotent

Group action invariants

Degree $n$:		$27$
Transitive number $t$:		$62$
Parity:		$-1$
Primitive:		no
$\card{\Aut(F/K)}$:		$1$
Generators:		$(1,8,20,2,9,21,3,7,19)(4,10,24,5,11,22,6,12,23)(13,17,26,14,18,27,15,16,25)$, $(1,18)(2,17)(3,16)(4,22)(5,24)(6,23)(7,12)(8,11)(9,10)(14,15)(19,26)(20,25)(21,27)$

Low degree resolvents

$\card{(G/N)}$	Galois groups for stem field(s)
$2$:	$C_2$
$3$:	$C_3$
$6$:	$S_3$, $C_6$
$18$:	$S_3\times C_3$
$54$:	$C_3^2 : C_6$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 3: $S_3$

Degree 9: $C_3^2 : C_6$

Low degree siblings

9T22 x 3, 18T85 x 3, 27T53 x 3, 27T63

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

Malle's constant $a(G)$:     $1/13$

Character table

		1A	2A	3A	3B	3C	3D	3E	3F1	3F-1	6A1	6A-1	9A1	9A-1
	Size	1	27	2	6	6	6	6	9	9	27	27	18	18
	2 P	1A	1A	3A	3B	3C	3D	3E	3F-1	3F1	3F1	3F-1	9A-1	9A1
	3 P	1A	2A	1A	1A	1A	1A	1A	1A	1A	2A	2A	3A	3A
	Type
162.11.1a	R	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
162.11.1b	R	$1$	$-1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$-1$	$-1$	$1$	$1$
162.11.1c1	C	$1$	$1$	$1$	$1$	$1$	$1$	$1$	${\zeta }_3^{-1}$	${\zeta }_3$	${\zeta }_3$	${\zeta }_3^{-1}$	${\zeta }_3$	${\zeta }_3^{-1}$
162.11.1c2	C	$1$	$1$	$1$	$1$	$1$	$1$	$1$	${\zeta }_3$	${\zeta }_3^{-1}$	${\zeta }_3^{-1}$	${\zeta }_3$	${\zeta }_3^{-1}$	${\zeta }_3$
162.11.1d1	C	$1$	$-1$	$1$	$1$	$1$	$1$	$1$	${\zeta }_3^{-1}$	${\zeta }_3$	$-{\zeta }_3$	$-{\zeta }_3^{-1}$	${\zeta }_3$	${\zeta }_3^{-1}$
162.11.1d2	C	$1$	$-1$	$1$	$1$	$1$	$1$	$1$	${\zeta }_3$	${\zeta }_3^{-1}$	$-{\zeta }_3^{-1}$	$-{\zeta }_3$	${\zeta }_3^{-1}$	${\zeta }_3$
162.11.2a	R	$2$	$0$	$2$	$2$	$-1$	$-1$	$-1$	$2$	$2$	$0$	$0$	$-1$	$-1$
162.11.2b1	C	$2$	$0$	$2$	$2$	$-1$	$-1$	$-1$	$2{\zeta }_3^{-1}$	$2{\zeta }_3$	$0$	$0$	$-{\zeta }_3$	$-{\zeta }_3^{-1}$
162.11.2b2	C	$2$	$0$	$2$	$2$	$-1$	$-1$	$-1$	$2{\zeta }_3$	$2{\zeta }_3^{-1}$	$0$	$0$	$-{\zeta }_3^{-1}$	$-{\zeta }_3$
162.11.6a	R	$6$	$0$	$6$	$-3$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$
162.11.6b	R	$6$	$0$	$-3$	$0$	$-3$	$3$	$0$	$0$	$0$	$0$	$0$	$0$	$0$
162.11.6c	R	$6$	$0$	$-3$	$0$	$0$	$-3$	$3$	$0$	$0$	$0$	$0$	$0$	$0$
162.11.6d	R	$6$	$0$	$-3$	$0$	$3$	$0$	$-3$	$0$	$0$	$0$	$0$	$0$	$0$

Regular extensions

Data not computed