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

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

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