Galois group: 30T25

• Label: 30T25
• 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 $t$:		$25$
Group:		$S_5$
Parity:		$-1$
Primitive:		no
$\card{\Aut(F/K)}$:		$6$
Generators:		(1,14,17,12)(2,13,18,11)(3,24,8,10)(4,23,7,9)(5,25)(6,26)(15,30,19,21)(16,29,20,22)(27,28), (1,9,13,3,5)(2,10,14,4,6)(7,22,25,15,18)(8,21,26,16,17)(11,28,23,20,30)(12,27,24,19,29)

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

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