Galois group: 30T30

• Label: 30T30
• Degree: $30$
• Order: $120$
• Cyclic: no
• Abelian: no
• Solvable: no
• Primitive: no
• pp-group: no
• Group: $C_2\times A_5$

Group invariants

Abstract group:		$C_2\times A_5$
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 $t$:		$30$
Parity:		$-1$
Primitive:		no
$\card{\Aut(F/K)}$:		$6$
Generators:		$(1,21,26,20,3,2,22,25,19,4)(5,27,29,7,11,6,28,30,8,12)(9,17,16,13,24,10,18,15,14,23)$, $(1,25,5,2,26,6)(3,18,8,4,17,7)(9,19,30,10,20,29)(11,27,13,12,28,14)(15,24,22,16,23,21)$

Low degree resolvents

$\card{(G/N)}$	Galois groups for stem field(s)
$2$:	$C_2$
$60$:	$A_5$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 3: None

Degree 5: $A_5$

Degree 6: None

Degree 10: $A_5\times C_2$

Degree 15: $A_5$

Low degree siblings

10T11, 12T75, 12T76, 20T31, 20T36, 24T203, 30T29, 40T61

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	$2^{15}$	$1$	$2$	$15$	$( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22)(23,24)(25,26)(27,28)(29,30)$
2B	$2^{15}$	$15$	$2$	$15$	$( 1, 2)( 3, 9)( 4,10)( 5,25)( 6,26)( 7,19)( 8,20)(11,12)(13,27)(14,28)(15,16)(17,30)(18,29)(21,23)(22,24)$
2C	$2^{12},1^{6}$	$15$	$2$	$12$	$( 1,29)( 2,30)( 3, 5)( 4, 6)( 7,14)( 8,13)( 9,12)(10,11)(15,19)(16,20)(17,22)(18,21)$
3A	$3^{10}$	$20$	$3$	$20$	$( 1,11,15)( 2,12,16)( 3,29,26)( 4,30,25)( 5,22,13)( 6,21,14)( 7, 9,27)( 8,10,28)(17,19,23)(18,20,24)$
5A1	$5^{6}$	$12$	$5$	$24$	$( 1,29,22,23,17)( 2,30,21,24,18)( 3,11,10, 5,26)( 4,12, 9, 6,25)( 7,16,20,14,27)( 8,15,19,13,28)$
5A2	$5^{6}$	$12$	$5$	$24$	$( 1,22,17,29,23)( 2,21,18,30,24)( 3,10,26,11, 5)( 4, 9,25,12, 6)( 7,20,27,16,14)( 8,19,28,15,13)$
6A	$6^{5}$	$20$	$6$	$25$	$( 1,16,11, 2,15,12)( 3,25,29, 4,26,30)( 5,14,22, 6,13,21)( 7,28, 9, 8,27,10)(17,24,19,18,23,20)$
10A1	$10^{3}$	$12$	$10$	$27$	$( 1,24,29,18,22, 2,23,30,17,21)( 3, 6,11,25,10, 4, 5,12,26, 9)( 7,13,16,28,20, 8,14,15,27,19)$
10A3	$10^{3}$	$12$	$10$	$27$	$( 1,18,23,21,29, 2,17,24,22,30)( 3,25, 5, 9,11, 4,26, 6,10,12)( 7,28,14,19,16, 8,27,13,20,15)$

Malle's constant $a(G)$:     $1/12$

Character table

		1A	2A	2B	2C	3A	5A1	5A2	6A	10A1	10A3
	Size	1	1	15	15	20	12	12	20	12	12
	2 P	1A	1A	1A	1A	3A	5A2	5A1	3A	5A1	5A2
	3 P	1A	2A	2B	2C	1A	5A2	5A1	2A	10A3	10A1
	5 P	1A	2A	2B	2C	3A	1A	1A	6A	2A	2A
	Type
120.35.1a	R	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
120.35.1b	R	$1$	$-1$	$-1$	$1$	$1$	$1$	$1$	$-1$	$-1$	$-1$
120.35.3a1	R	$3$	$3$	$-1$	$-1$	$0$	$-{\zeta }_5^{-1}-{\zeta }_5$	$-{\zeta }_5^{-2}-{\zeta }_5^2$	$0$	$-{\zeta }_5^{-2}-{\zeta }_5^2$	$-{\zeta }_5^{-1}-{\zeta }_5$
120.35.3a2	R	$3$	$3$	$-1$	$-1$	$0$	$-{\zeta }_5^{-2}-{\zeta }_5^2$	$-{\zeta }_5^{-1}-{\zeta }_5$	$0$	$-{\zeta }_5^{-1}-{\zeta }_5$	$-{\zeta }_5^{-2}-{\zeta }_5^2$
120.35.3b1	R	$3$	$-3$	$1$	$-1$	$0$	$-{\zeta }_5^{-1}-{\zeta }_5$	$-{\zeta }_5^{-2}-{\zeta }_5^2$	$0$	${\zeta }_5^{-2}+{\zeta }_5^2$	${\zeta }_5^{-1}+{\zeta }_5$
120.35.3b2	R	$3$	$-3$	$1$	$-1$	$0$	$-{\zeta }_5^{-2}-{\zeta }_5^2$	$-{\zeta }_5^{-1}-{\zeta }_5$	$0$	${\zeta }_5^{-1}+{\zeta }_5$	${\zeta }_5^{-2}+{\zeta }_5^2$
120.35.4a	R	$4$	$4$	$0$	$0$	$1$	$-1$	$-1$	$1$	$-1$	$-1$
120.35.4b	R	$4$	$-4$	$0$	$0$	$1$	$-1$	$-1$	$-1$	$1$	$1$
120.35.5a	R	$5$	$5$	$1$	$1$	$-1$	$0$	$0$	$-1$	$0$	$0$
120.35.5b	R	$5$	$-5$	$-1$	$1$	$-1$	$0$	$0$	$1$	$0$	$0$

Regular extensions

Data not computed