Galois group: 4T4

• Label: 4T4
• Degree: $4$
• Order: $12$
• Cyclic: no
• Abelian: no
• Solvable: yes
• Primitive: yes
• $p$-group: no
• Group: $A_4$

Group action invariants

Degree $n$:		$4$
Transitive number $t$:		$4$
Group:		$A_4$
CHM label:		$A4$
Parity:		$1$
Primitive:		yes
$\card{\Aut(F/K)}$:		$1$
Generators:		(2,3,4), (1,3,4)

Low degree resolvents

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

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Low degree siblings

6T4, 12T4

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	$1^{4}$	$1$	$1$	$0$	$()$
2A	$2^{2}$	$3$	$2$	$2$	$(1,2)(3,4)$
3A1	$3,1$	$4$	$3$	$2$	$(1,2,3)$
3A-1	$3,1$	$4$	$3$	$2$	$(1,3,2)$

Malle's constant $a(G)$:     $1/2$

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	$3$	$-1$	$0$	$0$