Galois group: 12T30

• Label: 12T30
• Degree: $12$
• Order: $48$
• Cyclic: no
• Abelian: no
• Solvable: yes
• Primitive: no
• $p$-group: no
• Group: $A_4:C_4$

Group invariants

Abstract group:		$A_4:C_4$
Order:		$48=2^{4} \cdot 3$
Cyclic:		no
Abelian:		no
Solvable:		yes
Nilpotency class:		not nilpotent

Group action invariants

Degree $n$:		$12$
Transitive number $t$:		$30$
CHM label:		$1/2[1/4.4^{3}]S(3)$
Parity:		$-1$
Primitive:		no
$\card{\Aut(F/K)}$:		$4$
Generators:		$(3,9)(6,12)$, $(1,8,7,2)(3,6,9,12)(4,11,10,5)$, $(1,5,9)(2,6,10)(3,7,11)(4,8,12)$

Low degree resolvents

$\card{(G/N)}$	Galois groups for stem field(s)
$2$:	$C_2$
$4$:	$C_4$
$6$:	$S_3$
$12$:	$C_3 : C_4$
$24$:	$S_4$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 3: $S_3$

Degree 4: None

Degree 6: $S_3$

Low degree siblings

12T27, 16T62, 24T51, 24T57

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	$2^{6}$	$1$	$2$	$6$	$( 1, 7)( 2, 8)( 3, 9)( 4,10)( 5,11)( 6,12)$
2B	$2^{4},1^{4}$	$3$	$2$	$4$	$( 1, 7)( 3, 9)( 4,10)( 6,12)$
2C	$2^{2},1^{8}$	$3$	$2$	$2$	$( 1, 7)( 4,10)$
3A	$3^{4}$	$8$	$3$	$8$	$( 1, 5, 9)( 2, 6,10)( 3, 7,11)( 4, 8,12)$
4A1	$4^{3}$	$6$	$4$	$9$	$( 1, 8, 7, 2)( 3, 6, 9,12)( 4,11,10, 5)$
4A-1	$4^{3}$	$6$	$4$	$9$	$( 1, 2, 7, 8)( 3,12, 9, 6)( 4, 5,10,11)$
4B1	$4,2^{4}$	$6$	$4$	$7$	$( 1, 4, 7,10)( 2, 9)( 3, 8)( 5,12)( 6,11)$
4B-1	$4,2^{4}$	$6$	$4$	$7$	$( 1,10, 7, 4)( 2, 9)( 3, 8)( 5,12)( 6,11)$
6A	$6^{2}$	$8$	$6$	$10$	$( 1, 3, 5, 7, 9,11)( 2, 4, 6, 8,10,12)$

Malle's constant $a(G)$:     $1/2$

Character table

		1A	2A	2B	2C	3A	4A1	4A-1	4B1	4B-1	6A
	Size	1	1	3	3	8	6	6	6	6	8
	2 P	1A	1A	1A	1A	3A	2A	2A	2C	2C	3A
	3 P	1A	2A	2B	2C	1A	4A-1	4A1	4B-1	4B1	2A
	Type
48.30.1a	R	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
48.30.1b	R	$1$	$1$	$1$	$1$	$1$	$-1$	$-1$	$-1$	$-1$	$1$
48.30.1c1	C	$1$	$-1$	$1$	$-1$	$1$	$-i$	$i$	$i$	$-i$	$-1$
48.30.1c2	C	$1$	$-1$	$1$	$-1$	$1$	$i$	$-i$	$-i$	$i$	$-1$
48.30.2a	R	$2$	$2$	$2$	$2$	$-1$	$0$	$0$	$0$	$0$	$-1$
48.30.2b	S	$2$	$-2$	$2$	$-2$	$-1$	$0$	$0$	$0$	$0$	$1$
48.30.3a	R	$3$	$3$	$-1$	$-1$	$0$	$1$	$1$	$-1$	$-1$	$0$
48.30.3b	R	$3$	$3$	$-1$	$-1$	$0$	$-1$	$-1$	$1$	$1$	$0$
48.30.3c1	C	$3$	$-3$	$-1$	$1$	$0$	$-i$	$i$	$-i$	$i$	$0$
48.30.3c2	C	$3$	$-3$	$-1$	$1$	$0$	$i$	$-i$	$i$	$-i$	$0$

Regular extensions

Data not computed