Galois group: 18T37

• Label: 18T37
• Degree: $18$
• Order: $72$
• Cyclic: no
• Abelian: no
• Solvable: yes
• Primitive: no
• $p$-group: no
• Group: $C_3:S_4$

Group invariants

Abstract group:		$C_3:S_4$
Order:		$72=2^{3} \cdot 3^{2}$
Cyclic:		no
Abelian:		no
Solvable:		yes
Nilpotency class:		not nilpotent

Group action invariants

Degree $n$:		$18$
Transitive number $t$:		$37$
Parity:		$1$
Primitive:		no
$\card{\Aut(F/K)}$:		$2$
Generators:		$(1,6,3,2,5,4)(7,12,10,8,11,9)(13,18,16)(14,17,15)$, $(1,6)(2,5)(3,4)(7,16,8,15)(9,14,10,13)(11,18,12,17)$, $(1,16)(2,15)(3,13)(4,14)(5,18)(6,17)(7,11)(8,12)$

Low degree resolvents

$\card{(G/N)}$	Galois groups for stem field(s)
$2$:	$C_2$
$6$:	$S_3$ x 4
$18$:	$C_3^2:C_2$
$24$:	$S_4$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 3: $S_3$ x 4

Degree 6: $S_4$

Degree 9: $C_3^2:C_2$

Low degree siblings

12T44 x 3, 18T40, 24T79 x 3, 36T23, 36T56

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	$1^{18}$	$1$	$1$	$0$	$()$
2A	$2^{6},1^{6}$	$3$	$2$	$6$	$( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)$
2B	$2^{8},1^{2}$	$18$	$2$	$8$	$( 1, 5)( 2, 6)( 7,16)( 8,15)( 9,14)(10,13)(11,18)(12,17)$
3A	$3^{6}$	$2$	$3$	$12$	$( 1, 5, 3)( 2, 6, 4)( 7,11,10)( 8,12, 9)(13,18,16)(14,17,15)$
3B	$3^{6}$	$8$	$3$	$12$	$( 1,14,11)( 2,13,12)( 3,15, 7)( 4,16, 8)( 5,17,10)( 6,18, 9)$
3C	$3^{6}$	$8$	$3$	$12$	$( 1,15,10)( 2,16, 9)( 3,17,11)( 4,18,12)( 5,14, 7)( 6,13, 8)$
3D	$3^{6}$	$8$	$3$	$12$	$( 1,17, 7)( 2,18, 8)( 3,14,10)( 4,13, 9)( 5,15,11)( 6,16,12)$
4A	$4^{3},2^{3}$	$18$	$4$	$12$	$( 1, 4)( 2, 3)( 5, 6)( 7,13, 8,14)( 9,17,10,18)(11,16,12,15)$
6A	$6^{2},3^{2}$	$6$	$6$	$14$	$( 1, 3, 5)( 2, 4, 6)( 7, 9,11, 8,10,12)(13,15,18,14,16,17)$

Malle's constant $a(G)$:     $1/6$

Character table

		1A	2A	2B	3A	3B	3C	3D	4A	6A
	Size	1	3	18	2	8	8	8	18	6
	2 P	1A	1A	1A	3A	3B	3C	3D	2A	3A
	3 P	1A	2A	2B	1A	1A	1A	1A	4A	2A
	Type
72.43.1a	R	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
72.43.1b	R	$1$	$1$	$-1$	$1$	$1$	$1$	$1$	$-1$	$1$
72.43.2a	R	$2$	$2$	$0$	$-1$	$-1$	$-1$	$2$	$0$	$-1$
72.43.2b	R	$2$	$2$	$0$	$-1$	$-1$	$2$	$-1$	$0$	$-1$
72.43.2c	R	$2$	$2$	$0$	$-1$	$2$	$-1$	$-1$	$0$	$-1$
72.43.2d	R	$2$	$2$	$0$	$2$	$-1$	$-1$	$-1$	$0$	$2$
72.43.3a	R	$3$	$-1$	$1$	$3$	$0$	$0$	$0$	$-1$	$-1$
72.43.3b	R	$3$	$-1$	$-1$	$3$	$0$	$0$	$0$	$1$	$-1$
72.43.6a	R	$6$	$-2$	$0$	$-3$	$0$	$0$	$0$	$0$	$1$

Regular extensions

Data not computed