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

$\card{(G/N)}$	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	Index	Representative
1A	$1^{30}$	$1$	$1$	$0$	$()$
2A	$2^{12},1^{6}$	$10$	$2$	$12$	$( 1,15)( 2,16)( 3, 5)( 4, 6)( 7,13)( 8,14)( 9,20)(10,19)(11,25)(12,26)(21,28)(22,27)$
2B	$2^{14},1^{2}$	$15$	$2$	$14$	$( 1, 5)( 2, 6)( 3,10)( 4, 9)(11,20)(12,19)(13,14)(15,26)(16,25)(17,22)(18,21)(23,27)(24,28)(29,30)$
3A	$3^{10}$	$20$	$3$	$20$	$( 1,15,23)( 2,16,24)( 3,25, 7)( 4,26, 8)( 5,13,11)( 6,14,12)( 9,20,18)(10,19,17)(21,28,29)(22,27,30)$
4A	$4^{7},2$	$30$	$4$	$22$	$( 1,12, 5,19)( 2,11, 6,20)( 3,28,10,24)( 4,27, 9,23)( 7, 8)(13,29,14,30)(15,21,26,18)(16,22,25,17)$
5A	$5^{6}$	$24$	$5$	$24$	$( 1,30,19, 4,25)( 2,29,20, 3,26)( 5,24, 8,17,11)( 6,23, 7,18,12)( 9,21,27,15,14)(10,22,28,16,13)$
6A	$6^{4},3^{2}$	$20$	$6$	$24$	$( 1, 9,14,15,20, 8)( 2,10,13,16,19, 7)( 3,27,12, 5,22,26)( 4,28,11, 6,21,25)(17,29,23)(18,30,24)$

Malle's constant $a(G)$:     $1/12$

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