Galois group: 6T7

• Label: 6T7
• Degree: $6$
• Order: $24$
• Cyclic: no
• Abelian: no
• Solvable: yes
• Primitive: no
• $p$-group: no
• Group: $S_4$

Group action invariants

Degree $n$:		$6$
Transitive number $t$:		$7$
Group:		$S_4$
CHM label:		$S_{4}(6d) = [2^{2}]S(3)$
Parity:		$1$
Primitive:		no
$\card{\Aut(F/K)}$:		$2$
Generators:		(1,4)(2,5), (1,5)(2,4), (1,3,5)(2,4,6)

Low degree resolvents

$\card{(G/N)}$	Galois groups for stem field(s)
$2$:	$C_2$
$6$:	$S_3$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 3: $S_3$

Low degree siblings

4T5, 6T8, 8T14, 12T8, 12T9, 24T10

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	$2^{2},1^{2}$	$3$	$2$	$2$	$(1,4)(2,5)$
2B	$2^{2},1^{2}$	$6$	$2$	$2$	$(1,2)(4,5)$
3A	$3^{2}$	$8$	$3$	$4$	$(1,3,2)(4,6,5)$
4A	$4,2$	$6$	$4$	$4$	$(1,4)(2,6,5,3)$

Malle's constant $a(G)$:     $1/2$

Group invariants

Order:		$24=2^{3} \cdot 3$
Cyclic:		no
Abelian:		no
Solvable:		yes
Nilpotency class:		not nilpotent
Label:		24.12
Character table:

		1A	2A	2B	3A	4A
	Size	1	3	6	8	6
	2 P	1A	1A	1A	3A	2A
	3 P	1A	2A	2B	1A	4A
	Type
24.12.1a	R	$1$	$1$	$1$	$1$	$1$
24.12.1b	R	$1$	$1$	$-1$	$1$	$-1$
24.12.2a	R	$2$	$2$	$0$	$-1$	$0$
24.12.3a	R	$3$	$-1$	$1$	$0$	$-1$
24.12.3b	R	$3$	$-1$	$-1$	$0$	$1$