Galois group: 24T10

• Label: 24T10
• Degree: $24$
• Order: $24$
• Cyclic: no
• Abelian: no
• Solvable: yes
• Primitive: no
• $p$-group: no
• Group: $S_4$

Group action invariants

Degree $n$:		$24$
Transitive number $t$:		$10$
Group:		$S_4$
Parity:		$1$
Primitive:		no
$\card{\Aut(F/K)}$:		$24$
Generators:		(1,3)(2,4)(5,9)(6,10)(7,11)(8,12)(13,17)(14,18)(15,19)(16,20)(21,22)(23,24), (1,5,12,24)(2,6,11,23)(3,7,14,20)(4,8,13,19)(9,16,21,17)(10,15,22,18)

Low degree resolvents

|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 3: $S_3$

Degree 4: $S_4$

Degree 6: $S_3$, $S_4$, $S_4$

Degree 8: $S_4$

Degree 12: $S_4$, $S_4$

Low degree siblings

4T5, 6T7, 6T8, 8T14, 12T8, 12T9

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	Representative
	$1^{24}$	$1$	$1$	$()$
	$2^{12}$	$6$	$2$	$( 1, 2)( 3,21)( 4,22)( 5, 7)( 6, 8)( 9,19)(10,20)(11,15)(12,16)(13,24)(14,23) (17,18)$
	$3^{8}$	$8$	$3$	$( 1, 4,21)( 2, 3,22)( 5,11,19)( 6,12,20)( 7, 9,15)( 8,10,16)(13,23,18) (14,24,17)$
	$4^{6}$	$6$	$4$	$( 1, 5,12,24)( 2, 6,11,23)( 3, 7,14,20)( 4, 8,13,19)( 9,16,21,17)(10,15,22,18)$
	$2^{12}$	$3$	$2$	$( 1,12)( 2,11)( 3,14)( 4,13)( 5,24)( 6,23)( 7,20)( 8,19)( 9,21)(10,22)(15,18) (16,17)$

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