Galois group: 8T14

• Label: 8T14
• Degree: $8$
• Order: $24$
• Cyclic: no
• Abelian: no
• Solvable: yes
• Primitive: no
• $p$-group: no
• Group: $S_4$

Group action invariants

Degree $n$:		$8$
Transitive number $t$:		$14$
Group:		$S_4$
CHM label:		$S(4)[1/2]2=1/2(S_{4}[x]2)$
Parity:		$1$
Primitive:		no
$\card{\Aut(F/K)}$:		$2$
Generators:		(1,3)(2,8)(4,6)(5,7), (1,2,3)(5,6,7), (1,4)(2,6)(3,7)(5,8)

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: $C_2$

Degree 4: $S_4$

Low degree siblings

4T5, 6T7, 6T8, 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	$1^{8}$	$1$	$1$	$0$	$()$
2A	$2^{4}$	$3$	$2$	$4$	$(1,8)(2,3)(4,5)(6,7)$
2B	$2^{4}$	$6$	$2$	$4$	$(1,5)(2,7)(3,6)(4,8)$
3A	$3^{2},1^{2}$	$8$	$3$	$4$	$(1,3,2)(5,7,6)$
4A	$4^{2}$	$6$	$4$	$6$	$(1,4,3,6)(2,5,8,7)$

Malle's constant $a(G)$:     $1/4$

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