Galois group: 24T10

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

Group invariants

Abstract group:		$S_4$
Order:		$24=2^{3} \cdot 3$
Cyclic:		no
Abelian:		no
Solvable:		yes
Nilpotency class:		not nilpotent

Group action invariants

Degree $n$:		$24$
Transitive number $t$:		$10$
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

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

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 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	Index	Representative
1A	$1^{24}$	$1$	$1$	$0$	$()$
2A	$2^{12}$	$3$	$2$	$12$	$( 1,15)( 2,16)( 3,19)( 4,20)( 5,10)( 6, 9)( 7,13)( 8,14)(11,17)(12,18)(21,23)(22,24)$
2B	$2^{12}$	$6$	$2$	$12$	$( 1,19)( 2,20)( 3,15)( 4,16)( 5, 6)( 7,17)( 8,18)( 9,10)(11,13)(12,14)(21,24)(22,23)$
3A	$3^{8}$	$8$	$3$	$16$	$( 1,20, 9)( 2,19,10)( 3,24,11)( 4,23,12)( 5,17, 8)( 6,18, 7)(13,21,15)(14,22,16)$
4A	$4^{6}$	$6$	$4$	$18$	$( 1, 8,15,14)( 2, 7,16,13)( 3,18,19,12)( 4,17,20,11)( 5,23,10,21)( 6,24, 9,22)$

Malle's constant $a(G)$:     $1/12$

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

Regular extensions

Data not computed