Galois group: 20T30

• Label: 20T30
• Degree: $20$
• Order: $120$
• Cyclic: no
• Abelian: no
• Solvable: no
• Primitive: no
• $p$-group: no
• Group: $S_5$

Group action invariants

Degree $n$:		$20$
Transitive number $t$:		$30$
Group:		$S_5$
Parity:		$-1$
Primitive:		no
$\card{\Aut(F/K)}$:		$2$
Generators:		(1,3)(2,4)(5,13)(6,14)(15,17)(16,18)(19,20), (1,6,10,13,17)(2,5,9,14,18)(3,8,12,15,20)(4,7,11,16,19)

Low degree resolvents

|G/N|	Galois groups for stem field(s)
$2$:	$C_2$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 4: None

Degree 5: $S_5$

Degree 10: $S_5$

Low degree siblings

5T5, 6T14, 10T12, 10T13, 12T74, 15T10, 20T32, 20T35, 24T202, 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
	$1^{20}$	$1$	$1$	$()$
	$2^{7},1^{6}$	$10$	$2$	$( 5,17)( 6,18)( 7, 9)( 8,10)(11,12)(13,15)(14,16)$
	$3^{6},1^{2}$	$20$	$3$	$( 3, 7, 9)( 4, 8,10)( 5,11,18)( 6,12,17)(13,15,19)(14,16,20)$
	$2^{10}$	$15$	$2$	$( 1, 2)( 3,13)( 4,14)( 5, 6)( 7,19)( 8,20)( 9,15)(10,16)(11,17)(12,18)$
	$6^{2},3^{2},2$	$20$	$6$	$( 1, 2)( 3,13, 9,19, 7,15)( 4,14,10,20, 8,16)( 5,11,18)( 6,12,17)$
	$4^{5}$	$30$	$4$	$( 1, 3, 7, 9)( 2, 4, 8,10)( 5,11,16,19)( 6,12,15,20)(13,17,14,18)$
	$5^{4}$	$24$	$5$	$( 1, 4, 6,12,15)( 2, 3, 5,11,16)( 7,17,14,10,19)( 8,18,13, 9,20)$

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	$6$	$0$	$-2$	$0$	$0$	$1$	$0$