Galois group: 36T14

• Label: 36T14
• Degree: $36$
• Order: $36$
• Cyclic: no
• Abelian: no
• Solvable: yes
• Primitive: no
• pp-group: no
• Group: $C_3^2:C_4$

Group invariants

Abstract group:		$C_3^2:C_4$
Order:		$36=2^{2} \cdot 3^{2}$
Cyclic:		no
Abelian:		no
Solvable:		yes
Nilpotency class:		not nilpotent

Group action invariants

Degree $n$:		$36$
Transitive number $t$:		$14$
Parity:		$-1$
Primitive:		no
$\card{\Aut(F/K)}$:		$36$
Generators:		$(1,16,6,30)(2,15,5,29)(3,13,8,32)(4,14,7,31)(9,27,21,19)(10,28,22,20)(11,26,24,18)(12,25,23,17)(33,35,34,36)$, $(1,8,28,21)(2,7,27,22)(3,5,26,23)(4,6,25,24)(9,31,18,33)(10,32,17,34)(11,29,20,36)(12,30,19,35)(13,16,14,15)$

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 4: $C_4$

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

Degree 9: $C_3^2:C_4$

Degree 12: $(C_3\times C_3):C_4$ x 2

Degree 18: $C_3^2 : C_4$

Low degree siblings

6T10 x 2, 9T9, 12T17 x 2, 18T10

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	$1^{36}$	$1$	$1$	$0$	$()$
2A	$2^{18}$	$9$	$2$	$18$	$( 1,13)( 2,14)( 3,15)( 4,16)( 5,11)( 6,12)( 7, 9)( 8,10)(17,35)(18,36)(19,33)(20,34)(21,30)(22,29)(23,32)(24,31)(25,26)(27,28)$
3A	$3^{12}$	$4$	$3$	$24$	$( 1,32,12)( 2,31,11)( 3,29,10)( 4,30, 9)( 5,24,14)( 6,23,13)( 7,21,16)( 8,22,15)(17,36,26)(18,35,25)(19,34,28)(20,33,27)$
3B	$3^{12}$	$4$	$3$	$24$	$( 1,20,24)( 2,19,23)( 3,18,21)( 4,17,22)( 5,12,27)( 6,11,28)( 7,10,25)( 8, 9,26)(13,31,34)(14,32,33)(15,30,36)(16,29,35)$
4A1	$4^{9}$	$9$	$4$	$27$	$( 1,18,13,36)( 2,17,14,35)( 3,19,15,33)( 4,20,16,34)( 5,21,11,30)( 6,22,12,29)( 7,23, 9,32)( 8,24,10,31)(25,28,26,27)$
4A-1	$4^{9}$	$9$	$4$	$27$	$( 1,36,13,18)( 2,35,14,17)( 3,33,15,19)( 4,34,16,20)( 5,30,11,21)( 6,29,12,22)( 7,32, 9,23)( 8,31,10,24)(25,27,26,28)$

Malle's constant $a(G)$:     $1/18$

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	$4$	$0$	$1$	$-2$	$0$	$0$

Regular extensions

Data not computed