Galois group: 30T25

• Label: 30T25
• 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 $t$:		$25$
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	$2^{15}$	$10$	$2$	$15$	$( 1,15)( 2,16)( 3,22)( 4,21)( 5,27)( 6,28)( 7,20)( 8,19)( 9,14)(10,13)(11,12)(17,29)(18,30)(23,24)(25,26)$
2B	$2^{12},1^{6}$	$15$	$2$	$12$	$( 1,20)( 2,19)( 3,26)( 4,25)( 5,11)( 6,12)( 9,16)(10,15)(17,23)(18,24)(21,28)(22,27)$
3A	$3^{10}$	$20$	$3$	$20$	$( 1, 3,28)( 2, 4,27)( 5,16,21)( 6,15,22)( 7,14,29)( 8,13,30)( 9,17,20)(10,18,19)(11,26,23)(12,25,24)$
4A	$4^{6},2^{3}$	$30$	$4$	$21$	$( 1, 4,20,25)( 2, 3,19,26)( 5,15,11,10)( 6,16,12, 9)( 7,13)( 8,14)(17,27,23,22)(18,28,24,21)(29,30)$
5A	$5^{6}$	$24$	$5$	$24$	$( 1,17,13,28,11)( 2,18,14,27,12)( 3,16,20, 8, 5)( 4,15,19, 7, 6)( 9,30,26,21,23)(10,29,25,22,24)$
6A	$6^{5}$	$20$	$6$	$25$	$( 1, 6, 3,15,28,22)( 2, 5, 4,16,27,21)( 7,17,14,20,29, 9)( 8,18,13,19,30,10)(11,24,26,12,23,25)$

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