Galois group: 30T27

• Label: 30T27
• Degree: $30$
• Order: $120$
• Cyclic: no
• Abelian: no
• Solvable: no
• Primitive: no
• $p$-group: no
• Group: $S_5$

Group invariants

Abstract group:		$S_5$
Order:		$120=2^{3} \cdot 3 \cdot 5$
Cyclic:		no
Abelian:		no
Solvable:		no
Nilpotency class:		not nilpotent

Group action invariants

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

Malle's constant $a(G)$:     $1/12$

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

Regular extensions

Data not computed