Galois group: 12T4

• Label: 12T4
• Degree: $12$
• Order: $12$
• Cyclic: no
• Abelian: no
• Solvable: yes
• Primitive: no
• $p$-group: no
• Group: $A_4$

Group action invariants

Degree $n$:		$12$
Transitive number $t$:		$4$
Group:		$A_4$
CHM label:		$A_{4}(12)$
Parity:		$1$
Primitive:		no
$\card{\Aut(F/K)}$:		$12$
Generators:		(1,9,5)(2,4,3)(6,8,7)(10,12,11), (1,11,6)(2,9,7)(3,10,5)(4,8,12)

Low degree resolvents

|G/N|	Galois groups for stem field(s)
$3$:	$C_3$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 3: $C_3$

Degree 4: $A_4$

Degree 6: $A_4$

Low degree siblings

4T4, 6T4

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 $	$1$	$1$	$()$
	$ 3, 3, 3, 3 $	$4$	$3$	$( 1, 2,12)( 3, 7,11)( 4, 5, 6)( 8, 9,10)$
	$ 3, 3, 3, 3 $	$4$	$3$	$( 1, 3, 8)( 2,10, 6)( 4, 9,11)( 5, 7,12)$
	$ 2, 2, 2, 2, 2, 2 $	$3$	$2$	$( 1, 4)( 2,11)( 3, 6)( 5, 8)( 7,10)( 9,12)$

Group invariants

Order:		$12=2^{2} \cdot 3$
Cyclic:		no
Abelian:		no
Solvable:		yes
Nilpotency class:		not nilpotent
Label:		12.3
Character table:

		1A	2A	3A1	3A-1
	Size	1	3	4	4
	2 P	1A	1A	3A-1	3A1
	3 P	1A	2A	1A	1A
	Type
12.3.1a	R	$1$	$1$	$1$	$1$
12.3.1b1	C	$1$	$1$	${\zeta }_3^{-1}$	${\zeta }_3$
12.3.1b2	C	$1$	$1$	${\zeta }_3$	${\zeta }_3^{-1}$
12.3.3a	R	$3$	$-1$	$0$	$0$