Galois group: 30T22

• Label: 30T22
• 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 $t$:		$22$
Group:		$S_5$
Parity:		$-1$
Primitive:		no
$\card{\Aut(F/K)}$:		$2$
Generators:		(1,7,5,2,8,6)(3,9,11,17,21,20)(4,10,12,18,22,19)(13,26,23,30,27,16)(14,25,24,29,28,15), (1,29,26,4)(2,30,25,3)(5,22,23,14)(6,21,24,13)(7,27,12,16)(8,28,11,15)(9,17,10,18)

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: $\PGL(2,5)$

Degree 10: None

Degree 15: $S_5$

Low degree siblings

5T5, 6T14, 10T12, 10T13, 12T74, 15T10, 20T30, 20T32, 20T35, 24T202, 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	Index	Representative
1A	$1^{30}$	$1$	$1$	$0$	$()$
2A	$2^{15}$	$10$	$2$	$15$	$( 1, 9)( 2,10)( 3,11)( 4,12)( 5,26)( 6,25)( 7,14)( 8,13)(15,20)(16,19)(17,24)(18,23)(21,22)(27,28)(29,30)$
2B	$2^{14},1^{2}$	$15$	$2$	$14$	$( 1, 4)( 2, 3)( 5, 6)( 7,18)( 8,17)( 9,13)(10,14)(11,23)(12,24)(15,21)(16,22)(19,30)(20,29)(27,28)$
3A	$3^{10}$	$20$	$3$	$20$	$( 1,18,25)( 2,17,26)( 3,12,21)( 4,11,22)( 5,24,10)( 6,23, 9)( 7,30,13)( 8,29,14)(15,19,28)(16,20,27)$
4A	$4^{7},1^{2}$	$30$	$4$	$21$	$( 1,20, 4,29)( 2,19, 3,30)( 5,27, 6,28)( 7,15,18,21)( 8,16,17,22)( 9,11,13,23)(10,12,14,24)$
5A	$5^{6}$	$24$	$5$	$24$	$( 1,17,13,28,12)( 2,18,14,27,11)( 3,15,20, 8, 6)( 4,16,19, 7, 5)( 9,29,25,21,23)(10,30,26,22,24)$
6A	$6^{5}$	$20$	$6$	$25$	$( 1,19,23, 9,16,18)( 2,20,24,10,15,17)( 3, 5, 8,11,26,13)( 4, 6, 7,12,25,14)(21,28,30,22,27,29)$

Malle's constant $a(G)$:     $1/14$

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