Galois group: 18T39

• Label: 18T39
• Degree: $18$
• Order: $72$
• Cyclic: no
• Abelian: no
• Solvable: yes
• Primitive: no
• $p$-group: no
• Group: $C_2^2:D_9$

Group action invariants

Degree $n$:		$18$
Transitive number $t$:		$39$
Group:		$C_2^2:D_9$
Parity:		$-1$
Primitive:		no
$\card{\Aut(F/K)}$:		$2$
Generators:		(1,10,13,5,8,18,3,11,16)(2,9,14,6,7,17,4,12,15), (1,18)(2,17)(3,16)(4,15)(5,13)(6,14)(7,10)(8,9)(11,12)

Low degree resolvents

|G/N|	Galois groups for stem field(s)
$2$:	$C_2$
$6$:	$S_3$
$18$:	$D_{9}$
$24$:	$S_4$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 3: $S_3$

Degree 6: $S_4$

Degree 9: $D_{9}$

Low degree siblings

18T38, 36T25, 36T57

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
	$1^{18}$	$1$	$1$	$()$
	$2^{6},1^{6}$	$3$	$2$	$( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)$
	$4^{3},2^{2},1^{2}$	$18$	$4$	$( 3, 5)( 4, 6)( 7,17, 8,18)( 9,15,10,16)(11,13,12,14)$
	$2^{9}$	$18$	$2$	$( 1, 2)( 3, 6)( 4, 5)( 7,17)( 8,18)( 9,15)(10,16)(11,13)(12,14)$
	$3^{6}$	$2$	$3$	$( 1, 3, 5)( 2, 4, 6)( 7, 9,12)( 8,10,11)(13,16,18)(14,15,17)$
	$6^{2},3^{2}$	$6$	$6$	$( 1, 3, 5)( 2, 4, 6)( 7,10,12, 8, 9,11)(13,15,18,14,16,17)$
	$9^{2}$	$8$	$9$	$( 1, 7,15, 5,12,14, 3, 9,17)( 2, 8,16, 6,11,13, 4,10,18)$
	$9^{2}$	$8$	$9$	$( 1, 9,14, 5, 7,17, 3,12,15)( 2,10,13, 6, 8,18, 4,11,16)$
	$9^{2}$	$8$	$9$	$( 1,11,17, 5,10,15, 3, 8,14)( 2,12,18, 6, 9,16, 4, 7,13)$

Group invariants

Order:		$72=2^{3} \cdot 3^{2}$
Cyclic:		no
Abelian:		no
Solvable:		yes
Nilpotency class:		not nilpotent
Label:		72.15
Character table:

		1A	2A	2B	3A	4A	6A	9A1	9A2	9A4
	Size	1	3	18	2	18	6	8	8	8
	2 P	1A	1A	1A	3A	2A	3A	9A2	9A4	9A1
	3 P	1A	2A	2B	1A	4A	2A	3A	3A	3A
	Type
72.15.1a	R	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
72.15.1b	R	$1$	$1$	$-1$	$1$	$-1$	$1$	$1$	$1$	$1$
72.15.2a	R	$2$	$2$	$0$	$2$	$0$	$2$	$-1$	$-1$	$-1$
72.15.2b1	R	$2$	$2$	$0$	$-1$	$0$	$-1$	${\zeta }_9^{-4}+{\zeta }_9^4$	${\zeta }_9^{-1}+{\zeta }_9$	${\zeta }_9^{-2}+{\zeta }_9^2$
72.15.2b2	R	$2$	$2$	$0$	$-1$	$0$	$-1$	${\zeta }_9^{-2}+{\zeta }_9^2$	${\zeta }_9^{-4}+{\zeta }_9^4$	${\zeta }_9^{-1}+{\zeta }_9$
72.15.2b3	R	$2$	$2$	$0$	$-1$	$0$	$-1$	${\zeta }_9^{-1}+{\zeta }_9$	${\zeta }_9^{-2}+{\zeta }_9^2$	${\zeta }_9^{-4}+{\zeta }_9^4$
72.15.3a	R	$3$	$-1$	$1$	$3$	$-1$	$-1$	$0$	$0$	$0$
72.15.3b	R	$3$	$-1$	$-1$	$3$	$1$	$-1$	$0$	$0$	$0$
72.15.6a	R	$6$	$-2$	$0$	$-3$	$0$	$1$	$0$	$0$	$0$