Galois group: 12T17

• Label: 12T17
• Degree: $12$
• Order: $36$
• Cyclic: no
• Abelian: no
• Solvable: yes
• Primitive: no
• $p$-group: no
• Group: $(C_3\times C_3):C_4$

Group invariants

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

Group action invariants

Degree $n$:		$12$
Transitive number $t$:		$17$
CHM label:		$[3^{2}]4$
Parity:		$-1$
Primitive:		no
$\card{\Aut(F/K)}$:		$6$
Generators:		$(2,10,6)(4,8,12)$, $(1,4,7,10)(2,5,8,11)(3,6,9,12)$

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: $C_3^2:C_4$

Low degree siblings

6T10 x 2, 9T9, 12T17, 18T10, 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	$1^{12}$	$1$	$1$	$0$	$()$
2A	$2^{6}$	$9$	$2$	$6$	$( 1, 3)( 2, 8)( 4,10)( 5, 7)( 6,12)( 9,11)$
3A	$3^{2},1^{6}$	$4$	$3$	$4$	$( 1, 9, 5)( 3, 7,11)$
3B	$3^{4}$	$4$	$3$	$8$	$( 1, 9, 5)( 2, 6,10)( 3, 7,11)( 4,12, 8)$
4A1	$4^{3}$	$9$	$4$	$9$	$( 1, 8, 3, 2)( 4,11,10, 9)( 5,12, 7, 6)$
4A-1	$4^{3}$	$9$	$4$	$9$	$( 1, 2, 3, 8)( 4, 9,10,11)( 5, 6, 7,12)$

Malle's constant $a(G)$:     $1/4$

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