Galois group: 18T38

• Label: 18T38
• 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 $t$:		$38$
Group:		$C_2^2:D_9$
Parity:		$1$
Primitive:		no
$\card{\Aut(F/K)}$:		$2$
Generators:		(1,15)(2,16)(3,14)(4,13)(5,18)(6,17)(9,12)(10,11), (1,8,2,7)(3,12,4,11)(5,9,6,10)(13,14)(15,18)(16,17)

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

18T39, 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, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $	$1$	$1$	$()$
	$ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1 $	$3$	$2$	$( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)$
	$ 2, 2, 2, 2, 2, 2, 2, 2, 1, 1 $	$18$	$2$	$( 3, 5)( 4, 6)( 7,17)( 8,18)( 9,15)(10,16)(11,14)(12,13)$
	$ 4, 4, 4, 2, 2, 2 $	$18$	$4$	$( 1, 2)( 3, 6)( 4, 5)( 7,17, 8,18)( 9,15,10,16)(11,14,12,13)$
	$ 6, 6, 3, 3 $	$6$	$6$	$( 1, 3, 6, 2, 4, 5)( 7, 9,12)( 8,10,11)(13,16,17,14,15,18)$
	$ 3, 3, 3, 3, 3, 3 $	$2$	$3$	$( 1, 4, 6)( 2, 3, 5)( 7, 9,12)( 8,10,11)(13,15,17)(14,16,18)$
	$ 9, 9 $	$8$	$9$	$( 1, 7,15, 6,12,13, 4, 9,17)( 2, 8,16, 5,11,14, 3,10,18)$
	$ 9, 9 $	$8$	$9$	$( 1, 9,13, 6, 7,17, 4,12,15)( 2,10,14, 5, 8,18, 3,11,16)$
	$ 9, 9 $	$8$	$9$	$( 1,11,18, 6,10,16, 4, 8,14)( 2,12,17, 5, 9,15, 3, 7,13)$

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	$6$	$-2$	$0$	$-3$	$0$	$1$	$0$	$0$	$0$