Galois group: 18T10

• Label: 18T10
• Degree: $18$
• Order: $36$
• Cyclic: no
• Abelian: no
• Solvable: yes
• Primitive: no
• pp-group: no
• Group: $C_3^2 : C_4$

Group action invariants

Degree $n$:		$18$
Transitive number $t$:		$10$
Group:		$C_3^2 : C_4$
Parity:		$-1$
Primitive:		no
$\card{\Aut(F/K)}$:		$2$
Generators:		(1,13,10,15)(2,14,9,16)(3,5,7,18)(4,6,8,17)(11,12), (1,14)(2,13)(3,12)(4,11)(5,9)(6,10)(15,18)(16,17)

Low degree resolvents

$\card{(G/N)}$	Galois groups for stem field(s)
$2$:	$C_2$
$4$:	$C_4$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 3: None

Degree 6: $C_3^2:C_4$ x 2

Degree 9: $C_3^2:C_4$

Low degree siblings

6T10 x 2, 9T9, 12T17 x 2, 36T14

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	$2^{8},1^{2}$	$9$	$2$	$8$	$( 1,10)( 2, 9)( 3, 7)( 4, 8)( 5,18)( 6,17)(13,15)(14,16)$
3A	$3^{6}$	$4$	$3$	$12$	$( 1, 6,16)( 2, 5,15)( 3, 7,12)( 4, 8,11)( 9,13,18)(10,14,17)$
3B	$3^{6}$	$4$	$3$	$12$	$( 1,12,10)( 2,11, 9)( 3,14, 6)( 4,13, 5)( 7,17,16)( 8,18,15)$
4A1	$4^{4},2$	$9$	$4$	$13$	$( 1,13,10,15)( 2,14, 9,16)( 3, 5, 7,18)( 4, 6, 8,17)(11,12)$
4A-1	$4^{4},2$	$9$	$4$	$13$	$( 1,15,10,13)( 2,16, 9,14)( 3,18, 7, 5)( 4,17, 8, 6)(11,12)$

Malle's constant $a(G)$:     $1/8$

Group invariants

Order:		$36=2^{2} \cdot 3^{2}$
Cyclic:		no
Abelian:		no
Solvable:		yes
Nilpotency class:		not nilpotent
Label:		36.9
Character table:

		1A	2A	3A	3B	4A1	4A-1
	Size	1	9	4	4	9	9
	2 P	1A	1A	3A	3B	2A	2A
	3 P	1A	2A	1A	1A	4A-1	4A1
	Type
36.9.1a	R	$1$	$1$	$1$	$1$	$1$	$1$
36.9.1b	R	$1$	$1$	$1$	$1$	$-1$	$-1$
36.9.1c1	C	$1$	$-1$	$1$	$1$	$-i$	$i$
36.9.1c2	C	$1$	$-1$	$1$	$1$	$i$	$-i$
36.9.4a	R	$4$	$0$	$-2$	$1$	$0$	$0$
36.9.4b	R	$4$	$0$	$1$	$-2$	$0$	$0$