Galois group: 30T9

• Label: 30T9
• Degree: $30$
• Order: $60$
• Cyclic: no
• Abelian: no
• Solvable: no
• Primitive: no
• $p$-group: no
• Group: $A_5$

Group action invariants

Degree $n$:		$30$
Transitive number $t$:		$9$
Group:		$A_5$
Parity:		$1$
Primitive:		no
$\card{\Aut(F/K)}$:		$2$
Generators:		(1,12,6,7,3)(2,11,5,8,4)(9,21,30,27,17)(10,22,29,28,18)(13,25,15,20,23)(14,26,16,19,24), (1,13,5)(2,14,6)(3,19,9)(4,20,10)(7,18,15)(8,17,16)(11,28,21)(12,27,22)(23,29,25)(24,30,26)

Low degree resolvents

none

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 3: None

Degree 5: $A_5$

Degree 6: $\PSL(2,5)$

Degree 10: $A_{5}$

Degree 15: $A_5$

Low degree siblings

5T4, 6T12, 10T7, 12T33, 15T5, 20T15

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	$2^{14},1^{2}$	$15$	$2$	$14$	$( 1,14)( 2,13)( 3,15)( 4,16)( 5, 6)( 7, 9)( 8,10)(11,21)(12,22)(17,20)(18,19)(23,30)(24,29)(25,26)$
3A	$3^{10}$	$20$	$3$	$20$	$( 1,30,10)( 2,29, 9)( 3,23,11)( 4,24,12)( 5,27,26)( 6,28,25)( 7,16,17)( 8,15,18)(13,20,22)(14,19,21)$
5A1	$5^{6}$	$12$	$5$	$24$	$( 1, 3, 7, 6,12)( 2, 4, 8, 5,11)( 9,17,27,30,21)(10,18,28,29,22)(13,23,20,15,25)(14,24,19,16,26)$
5A2	$5^{6}$	$12$	$5$	$24$	$( 1,11, 9,16,27)( 2,12,10,15,28)( 3,17, 5,23,13)( 4,18, 6,24,14)( 7,25,22,30,19)( 8,26,21,29,20)$

Malle's constant $a(G)$:     $1/14$

Group invariants

Order:		$60=2^{2} \cdot 3 \cdot 5$
Cyclic:		no
Abelian:		no
Solvable:		no
Nilpotency class:		not nilpotent
Label:		60.5
Character table:

		1A	2A	3A	5A1	5A2
	Size	1	15	20	12	12
	2 P	1A	1A	3A	5A2	5A1
	3 P	1A	2A	1A	5A2	5A1
	5 P	1A	2A	3A	1A	1A
	Type
60.5.1a	R	$1$	$1$	$1$	$1$	$1$
60.5.3a1	R	$3$	$-1$	$0$	$-{\zeta }_5^{-1}-{\zeta }_5$	$-{\zeta }_5^{-2}-{\zeta }_5^2$
60.5.3a2	R	$3$	$-1$	$0$	$-{\zeta }_5^{-2}-{\zeta }_5^2$	$-{\zeta }_5^{-1}-{\zeta }_5$
60.5.4a	R	$4$	$0$	$1$	$-1$	$-1$
60.5.5a	R	$5$	$1$	$-1$	$0$	$0$