Galois group: 30T27

• Label: 30T27
• Degree: $30$
• Order: $120$
• Cyclic: no
• Abelian: no
• Solvable: no
• Primitive: no
• $p$-group: no
• Group: $S_5$

Group action invariants

Degree $n$:		$30$
Transitive number $t$:		$27$
Group:		$S_5$
Parity:		$1$
Primitive:		no
$\card{\Aut(F/K)}$:		$2$
Generators:		(1,17,7,4)(2,18,8,3)(5,6)(9,30,12,15)(10,29,11,16)(13,22,24,19)(14,21,23,20)(25,27,26,28), (1,11,13,28)(2,12,14,27)(3,29,5,23)(4,30,6,24)(7,25,15,21)(8,26,16,22)(9,19,10,20)(17,18)

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 3: None

Degree 5: $S_5$

Degree 6: None

Degree 10: $S_5$

Degree 15: $S_5$

Low degree siblings

5T5, 6T14, 10T12, 10T13, 12T74, 15T10, 20T30, 20T32, 20T35, 24T202, 30T22, 30T25, 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
	$1^{30}$	$1$	$1$	$()$
	$2^{12},1^{6}$	$10$	$2$	$( 3,11)( 4,12)( 5, 7)( 6, 8)(13,25)(14,26)(15,23)(16,24)(17,19)(18,20)(27,30) (28,29)$
	$2^{14},1^{2}$	$15$	$2$	$( 3,20)( 4,19)( 5, 7)( 6, 8)( 9,22)(10,21)(11,18)(12,17)(13,27)(14,28)(15,16) (23,24)(25,30)(26,29)$
	$4^{7},2$	$30$	$4$	$( 1, 2)( 3,25,20,30)( 4,26,19,29)( 5, 9, 7,22)( 6,10, 8,21)(11,27,18,13) (12,28,17,14)(15,23,16,24)$
	$5^{6}$	$24$	$5$	$( 1, 3, 9, 6,14)( 2, 4,10, 5,13)( 7,15,22,17,26)( 8,16,21,18,25) (11,19,27,29,24)(12,20,28,30,23)$
	$3^{10}$	$20$	$3$	$( 1, 3,28)( 2, 4,27)( 5,15,21)( 6,16,22)( 7,13,29)( 8,14,30)( 9,17,20) (10,18,19)(11,25,23)(12,26,24)$
	$6^{4},3^{2}$	$20$	$6$	$( 1, 4,24,21,16,12)( 2, 3,23,22,15,11)( 5,17,30,28,20, 8)( 6,18,29,27,19, 7) ( 9,13,25)(10,14,26)$

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