Galois group: 6T6

• Label: 6T6
• Degree: $6$
• Order: $24$
• Cyclic: no
• Abelian: no
• Solvable: yes
• Primitive: no
• $p$-group: no
• Group: $A_4\times C_2$

Group action invariants

Degree $n$:		$6$
Transitive number $t$:		$6$
Group:		$A_4\times C_2$
CHM label:		$2A_{4}(6) = [2^{3}]3 = 2 wr 3$
Parity:		$-1$
Primitive:		no
$\card{\Aut(F/K)}$:		$2$
Generators:		(1,3,5)(2,4,6), (3,6)

Low degree resolvents

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

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 3: $C_3$

Low degree siblings

8T13, 12T6, 12T7, 24T9

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	$1^{6}$	$1$	$1$	$0$	$()$
2A	$2^{3}$	$1$	$2$	$3$	$(1,4)(2,5)(3,6)$
2B	$2^{2},1^{2}$	$3$	$2$	$2$	$(1,4)(2,5)$
2C	$2,1^{4}$	$3$	$2$	$1$	$(3,6)$
3A1	$3^{2}$	$4$	$3$	$4$	$(1,3,5)(2,4,6)$
3A-1	$3^{2}$	$4$	$3$	$4$	$(1,5,3)(2,6,4)$
6A1	$6$	$4$	$6$	$5$	$(1,2,3,4,5,6)$
6A-1	$6$	$4$	$6$	$5$	$(1,6,5,4,3,2)$

Malle's constant $a(G)$:     $1$

Group invariants

Order:		$24=2^{3} \cdot 3$
Cyclic:		no
Abelian:		no
Solvable:		yes
Nilpotency class:		not nilpotent
Label:		24.13
Character table:

		1A	2A	2B	2C	3A1	3A-1	6A1	6A-1
	Size	1	1	3	3	4	4	4	4
	2 P	1A	1A	1A	1A	3A-1	3A1	3A1	3A-1
	3 P	1A	2A	2B	2C	1A	1A	2A	2A
	Type
24.13.1a	R	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
24.13.1b	R	$1$	$-1$	$1$	$-1$	$1$	$1$	$-1$	$-1$
24.13.1c1	C	$1$	$1$	$1$	$1$	${\zeta }_3^{-1}$	${\zeta }_3$	${\zeta }_3$	${\zeta }_3^{-1}$
24.13.1c2	C	$1$	$1$	$1$	$1$	${\zeta }_3$	${\zeta }_3^{-1}$	${\zeta }_3^{-1}$	${\zeta }_3$
24.13.1d1	C	$1$	$-1$	$1$	$-1$	${\zeta }_3^{-1}$	${\zeta }_3$	$-{\zeta }_3$	$-{\zeta }_3^{-1}$
24.13.1d2	C	$1$	$-1$	$1$	$-1$	${\zeta }_3$	${\zeta }_3^{-1}$	$-{\zeta }_3^{-1}$	$-{\zeta }_3$
24.13.3a	R	$3$	$3$	$-1$	$-1$	$0$	$0$	$0$	$0$
24.13.3b	R	$3$	$-3$	$-1$	$1$	$0$	$0$	$0$	$0$