Galois group: 12T43

• Label: 12T43
• Degree: $12$
• Order: $72$
• Cyclic: no
• Abelian: no
• Solvable: yes
• Primitive: no
• pp-group: no
• Group: $S_3\times A_4$

Group invariants

Abstract group:		$S_3\times A_4$
Order:		$72=2^{3} \cdot 3^{2}$
Cyclic:		no
Abelian:		no
Solvable:		yes
Nilpotency class:		not nilpotent

Group action invariants

Degree $n$:		$12$
Transitive number $t$:		$43$
CHM label:		$A(4)[x]S(3)$
Parity:		$1$
Primitive:		no
$\card{\Aut(F/K)}$:		$1$
Generators:		$(1,7,10)(2,5,11)(3,6,9)$, $(2,8,11)(3,6,12)(4,7,10)$, $(1,5)(2,10)(4,8)(7,11)$, $(1,5,9)(2,6,10)(3,7,11)(4,8,12)$

Low degree resolvents

$\card{(G/N)}$	Galois groups for stem field(s)
$2$:	$C_2$
$3$:	$C_3$
$6$:	$S_3$, $C_6$
$12$:	$A_4$
$18$:	$S_3\times C_3$
$24$:	$A_4\times C_2$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 3: $S_3$

Degree 4: $A_4$

Degree 6: None

Low degree siblings

18T31, 18T32, 24T78, 24T83, 36T21, 36T50, 36T51

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	$2^{6}$	$3$	$2$	$6$	$( 1, 7)( 2, 8)( 3, 9)( 4,10)( 5,11)( 6,12)$
2B	$2^{4},1^{4}$	$3$	$2$	$4$	$( 1, 5)( 2,10)( 4, 8)( 7,11)$
2C	$2^{6}$	$9$	$2$	$6$	$( 1,10)( 2, 9)( 3, 8)( 4, 7)( 5, 6)(11,12)$
3A	$3^{4}$	$2$	$3$	$8$	$( 1, 5, 9)( 2, 6,10)( 3, 7,11)( 4, 8,12)$
3B1	$3^{3},1^{3}$	$4$	$3$	$6$	$( 1, 7, 4)( 3,12, 9)( 5,11, 8)$
3B-1	$3^{3},1^{3}$	$4$	$3$	$6$	$( 1, 4, 7)( 3, 9,12)( 5, 8,11)$
3C1	$3^{4}$	$8$	$3$	$8$	$( 1, 5, 9)( 2, 3, 4)( 6, 7, 8)(10,11,12)$
3C-1	$3^{4}$	$8$	$3$	$8$	$( 1, 5, 9)( 2,12, 7)( 3,10, 8)( 4,11, 6)$
6A	$6^{2}$	$6$	$6$	$10$	$( 1, 3, 5, 7, 9,11)( 2, 4, 6, 8,10,12)$
6B1	$6,3,2,1$	$12$	$6$	$8$	$( 1, 8, 7, 5, 4,11)( 2,10)( 3, 9,12)$
6B-1	$6,3,2,1$	$12$	$6$	$8$	$( 1,11, 4, 5, 7, 8)( 2,10)( 3,12, 9)$

Malle's constant $a(G)$:     $1/4$

Character table

		1A	2A	2B	2C	3A	3B1	3B-1	3C1	3C-1	6A	6B1	6B-1
	Size	1	3	3	9	2	4	4	8	8	6	12	12
	2 P	1A	1A	1A	1A	3A	3B-1	3B1	3C-1	3C1	3A	3B1	3B-1
	3 P	1A	2A	2B	2C	1A	1A	1A	1A	1A	2A	2B	2B
	Type
72.44.1a	R	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
72.44.1b	R	$1$	$1$	$-1$	$-1$	$1$	$1$	$1$	$1$	$1$	$1$	$-1$	$-1$
72.44.1c1	C	$1$	$1$	$1$	$1$	$1$	${\zeta }_3^{-1}$	${\zeta }_3$	${\zeta }_3$	${\zeta }_3^{-1}$	$1$	${\zeta }_3$	${\zeta }_3^{-1}$
72.44.1c2	C	$1$	$1$	$1$	$1$	$1$	${\zeta }_3$	${\zeta }_3^{-1}$	${\zeta }_3^{-1}$	${\zeta }_3$	$1$	${\zeta }_3^{-1}$	${\zeta }_3$
72.44.1d1	C	$1$	$1$	$-1$	$-1$	$1$	${\zeta }_3^{-1}$	${\zeta }_3$	${\zeta }_3$	${\zeta }_3^{-1}$	$1$	$-{\zeta }_3$	$-{\zeta }_3^{-1}$
72.44.1d2	C	$1$	$1$	$-1$	$-1$	$1$	${\zeta }_3$	${\zeta }_3^{-1}$	${\zeta }_3^{-1}$	${\zeta }_3$	$1$	$-{\zeta }_3^{-1}$	$-{\zeta }_3$
72.44.2a	R	$2$	$2$	$0$	$0$	$-1$	$2$	$2$	$-1$	$-1$	$-1$	$0$	$0$
72.44.2b1	C	$2$	$2$	$0$	$0$	$-1$	$2{\zeta }_3^{-1}$	$2{\zeta }_3$	$-{\zeta }_3$	$-{\zeta }_3^{-1}$	$-1$	$0$	$0$
72.44.2b2	C	$2$	$2$	$0$	$0$	$-1$	$2{\zeta }_3$	$2{\zeta }_3^{-1}$	$-{\zeta }_3^{-1}$	$-{\zeta }_3$	$-1$	$0$	$0$
72.44.3a	R	$3$	$-1$	$3$	$-1$	$3$	$0$	$0$	$0$	$0$	$-1$	$0$	$0$
72.44.3b	R	$3$	$-1$	$-3$	$1$	$3$	$0$	$0$	$0$	$0$	$-1$	$0$	$0$
72.44.6a	R	$6$	$-2$	$0$	$0$	$-3$	$0$	$0$	$0$	$0$	$1$	$0$	$0$

Regular extensions

Data not computed