Galois group: 24T202

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

Group action invariants

Degree $n$:		$24$
Transitive number $t$:		$202$
Group:		$S_5$
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

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

Group invariants

Order:		$120=2^{3} \cdot 3 \cdot 5$
Cyclic:		no
Abelian:		no
Solvable:		no
Nilpotency class:		not nilpotent
Label:		120.34
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	$6$	$0$	$-2$	$0$	$0$	$1$	$0$