Galois group: 24T202

• Label: 24T202
• Degree: $24$
• Order: $120$
• Cyclic: no
• Abelian: no
• Solvable: no
• Primitive: no
• $p$-group: no
• Group: $S_5$

Group invariants

Abstract group:		$S_5$
Order:		$120=2^{3} \cdot 3 \cdot 5$
Cyclic:		no
Abelian:		no
Solvable:		no
Nilpotency class:		not nilpotent

Group action invariants

Degree $n$:		$24$
Transitive number $t$:		$202$
Parity:		$1$
Primitive:		no
$\card{\Aut(F/K)}$:		$4$
Generators:		$(1,3)(2,4)(5,9)(6,10)(7,13)(8,14)(11,19)(12,20)(15,17)(16,18)(21,23)(22,24)$, $(3,17,11,7,5)(4,18,12,8,6)(9,14,22,20,15)(10,13,21,19,16)$

Low degree resolvents

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

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 3: None

Degree 4: None

Degree 6: $\PGL(2,5)$

Degree 8: None

Degree 12: $S_5$

Low degree siblings

5T5, 6T14, 10T12, 10T13, 12T74, 15T10, 20T30, 20T32, 20T35, 30T22, 30T25, 30T27, 40T62

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	$2^{12}$	$10$	$2$	$12$	$( 1,18)( 2,17)( 3,10)( 4, 9)( 5,15)( 6,16)( 7,22)( 8,21)(11,13)(12,14)(19,24)(20,23)$
2B	$2^{12}$	$15$	$2$	$12$	$( 1,15)( 2,16)( 3, 5)( 4, 6)( 7, 8)( 9,10)(11,23)(12,24)(13,21)(14,22)(17,18)(19,20)$
3A	$3^{8}$	$20$	$3$	$16$	$( 1,13,10)( 2,14, 9)( 3,18,11)( 4,17,12)( 5, 7,24)( 6, 8,23)(15,22,19)(16,21,20)$
4A	$4^{6}$	$30$	$4$	$18$	$( 1, 3,15, 5)( 2, 4,16, 6)( 7, 9, 8,10)(11,13,23,21)(12,14,24,22)(17,19,18,20)$
5A	$5^{4},1^{4}$	$24$	$5$	$16$	$( 1,13,15, 9,19)( 2,14,16,10,20)( 5,12, 8,17,24)( 6,11, 7,18,23)$
6A	$6^{4}$	$20$	$6$	$20$	$( 1, 3,13,18,10,11)( 2, 4,14,17, 9,12)( 5,19, 7,15,24,22)( 6,20, 8,16,23,21)$

Malle's constant $a(G)$:     $1/12$

Character table

		1A	2A	2B	3A	4A	5A	6A
	Size	1	10	15	20	30	24	20
	2 P	1A	1A	1A	3A	2B	5A	3A
	3 P	1A	2A	2B	1A	4A	5A	2A
	5 P	1A	2A	2B	3A	4A	1A	6A
	Type
120.34.1a	R	$1$	$1$	$1$	$1$	$1$	$1$	$1$
120.34.1b	R	$1$	$-1$	$1$	$1$	$-1$	$1$	$-1$
120.34.4a	R	$4$	$2$	$0$	$1$	$0$	$-1$	$-1$
120.34.4b	R	$4$	$-2$	$0$	$1$	$0$	$-1$	$1$
120.34.5a	R	$5$	$-1$	$1$	$-1$	$1$	$0$	$-1$
120.34.5b	R	$5$	$1$	$1$	$-1$	$-1$	$0$	$1$
120.34.6a	R	$6$	$0$	$-2$	$0$	$0$	$1$	$0$

Regular extensions

Data not computed