Galois group: 30T32

• Label: 30T32
• Degree: $30$
• Order: $120$
• Cyclic: no
• Abelian: no
• Solvable: yes
• Primitive: no
• pp-group: no
• Group: $S_3\times F_5$

Group invariants

Abstract group:		$S_3\times F_5$
Order:		$120=2^{3} \cdot 3 \cdot 5$
Cyclic:		no
Abelian:		no
Solvable:		yes
Nilpotency class:		not nilpotent

Group action invariants

Degree $n$:		$30$
Transitive number $t$:		$32$
Parity:		$-1$
Primitive:		no
$\card{\Aut(F/K)}$:		$6$
Generators:		$(1,22,12)(2,21,11)(3,5,20,28,24,26,10,18,14,16,29,7)(4,6,19,27,23,25,9,17,13,15,30,8)$, $(1,27,14,9,26,21,7,4,20,15)(2,28,13,10,25,22,8,3,19,16)(5,11,18,23,29,6,12,17,24,30)$

Low degree resolvents

$\card{(G/N)}$	Galois groups for stem field(s)
$2$:	$C_2$ x 3
$4$:	$C_4$ x 2, $C_2^2$
$6$:	$S_3$
$8$:	$C_4\times C_2$
$12$:	$D_{6}$
$20$:	$F_5$
$24$:	$S_3 \times C_4$
$40$:	$F_{5}\times C_2$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 3: $S_3$

Degree 5: $F_5$

Degree 6: $S_3$

Degree 10: $F_{5}\times C_2$

Degree 15: $F_5 \times S_3$

Low degree siblings

15T11, 30T23, 30T24

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	$2^{15}$	$3$	$2$	$15$	$( 1,11)( 2,12)( 3, 4)( 5,25)( 6,26)( 7,17)( 8,18)( 9,10)(13,24)(14,23)(15,16)(19,29)(20,30)(21,22)(27,28)$
2B	$2^{12},1^{6}$	$5$	$2$	$12$	$( 1, 7)( 2, 8)( 3,16)( 4,15)( 5,24)( 6,23)(11,17)(12,18)(13,25)(14,26)(21,27)(22,28)$
2C	$2^{15}$	$15$	$2$	$15$	$( 1, 4)( 2, 3)( 5,30)( 6,29)( 7,27)( 8,28)( 9,26)(10,25)(11,24)(12,23)(13,22)(14,21)(15,20)(16,19)(17,18)$
3A	$3^{10}$	$2$	$3$	$20$	$( 1,22,12)( 2,21,11)( 3,24,14)( 4,23,13)( 5,26,16)( 6,25,15)( 7,28,18)( 8,27,17)( 9,30,19)(10,29,20)$
4A1	$4^{6},1^{6}$	$5$	$4$	$18$	$( 1,14, 7,26)( 2,13, 8,25)( 3,28,16,22)( 4,27,15,21)( 5,12,24,18)( 6,11,23,17)$
4A-1	$4^{6},1^{6}$	$5$	$4$	$18$	$( 1,26, 7,14)( 2,25, 8,13)( 3,22,16,28)( 4,21,15,27)( 5,18,24,12)( 6,17,23,11)$
4B1	$4^{6},2^{3}$	$15$	$4$	$21$	$( 1, 8,20,13)( 2, 7,19,14)( 3,11,28,30)( 4,12,27,29)( 5,15)( 6,16)( 9,24,21,18)(10,23,22,17)(25,26)$
4B-1	$4^{6},2^{3}$	$15$	$4$	$21$	$( 1,13,20, 8)( 2,14,19, 7)( 3,30,28,11)( 4,29,27,12)( 5,15)( 6,16)( 9,18,21,24)(10,17,22,23)(25,26)$
5A	$5^{6}$	$4$	$5$	$24$	$( 1,14,26, 7,20)( 2,13,25, 8,19)( 3,16,28,10,22)( 4,15,27, 9,21)( 5,18,29,12,24)( 6,17,30,11,23)$
6A	$6^{4},3^{2}$	$10$	$6$	$24$	$( 1,18,22, 7,12,28)( 2,17,21, 8,11,27)( 3,26,24,16,14, 5)( 4,25,23,15,13, 6)( 9,19,30)(10,20,29)$
10A	$10^{3}$	$12$	$10$	$27$	$( 1, 6,20,23, 7,11,26,30,14,17)( 2, 5,19,24, 8,12,25,29,13,18)( 3,27,22,15,10, 4,28,21,16, 9)$
12A1	$12^{2},3^{2}$	$10$	$12$	$26$	$( 1,16,18,14,22, 5, 7, 3,12,26,28,24)( 2,15,17,13,21, 6, 8, 4,11,25,27,23)( 9,30,19)(10,29,20)$
12A-1	$12^{2},3^{2}$	$10$	$12$	$26$	$( 1,16,24,20,22, 5,14,10,12,26, 3,29)( 2,15,23,19,21, 6,13, 9,11,25, 4,30)( 7,28,18)( 8,27,17)$
15A	$15^{2}$	$8$	$15$	$28$	$( 1,16,29,14,28,12,26,10,24, 7,22, 5,20, 3,18)( 2,15,30,13,27,11,25, 9,23, 8,21, 6,19, 4,17)$

Malle's constant $a(G)$:     $1/12$

Character table

		1A	2A	2B	2C	3A	4A1	4A-1	4B1	4B-1	5A	6A	10A	12A1	12A-1	15A
	Size	1	3	5	15	2	5	5	15	15	4	10	12	10	10	8
	2 P	1A	1A	1A	1A	3A	2B	2B	2B	2B	5A	3A	5A	6A	6A	15A
	3 P	1A	2A	2B	2C	1A	4A-1	4A1	4B-1	4B1	5A	2B	10A	4A1	4A-1	5A
	5 P	1A	2A	2B	2C	3A	4A1	4A-1	4B1	4B-1	1A	6A	2A	12A1	12A-1	3A
	Type
120.36.1a	R	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
120.36.1b	R	$1$	$-1$	$1$	$-1$	$1$	$-1$	$-1$	$1$	$1$	$1$	$1$	$-1$	$-1$	$-1$	$1$
120.36.1c	R	$1$	$-1$	$1$	$-1$	$1$	$1$	$1$	$-1$	$-1$	$1$	$1$	$-1$	$1$	$1$	$1$
120.36.1d	R	$1$	$1$	$1$	$1$	$1$	$-1$	$-1$	$-1$	$-1$	$1$	$1$	$1$	$-1$	$-1$	$1$
120.36.1e1	C	$1$	$-1$	$-1$	$1$	$1$	$-i$	$i$	$i$	$-i$	$1$	$-1$	$-1$	$i$	$-i$	$1$
120.36.1e2	C	$1$	$-1$	$-1$	$1$	$1$	$i$	$-i$	$-i$	$i$	$1$	$-1$	$-1$	$-i$	$i$	$1$
120.36.1f1	C	$1$	$1$	$-1$	$-1$	$1$	$-i$	$i$	$-i$	$i$	$1$	$-1$	$1$	$i$	$-i$	$1$
120.36.1f2	C	$1$	$1$	$-1$	$-1$	$1$	$i$	$-i$	$i$	$-i$	$1$	$-1$	$1$	$-i$	$i$	$1$
120.36.2a	R	$2$	$0$	$2$	$0$	$-1$	$2$	$2$	$0$	$0$	$2$	$-1$	$0$	$-1$	$-1$	$-1$
120.36.2b	R	$2$	$0$	$2$	$0$	$-1$	$-2$	$-2$	$0$	$0$	$2$	$-1$	$0$	$1$	$1$	$-1$
120.36.2c1	C	$2$	$0$	$-2$	$0$	$-1$	$-2i$	$2i$	$0$	$0$	$2$	$1$	$0$	$-i$	$i$	$-1$
120.36.2c2	C	$2$	$0$	$-2$	$0$	$-1$	$2i$	$-2i$	$0$	$0$	$2$	$1$	$0$	$i$	$-i$	$-1$
120.36.4a	R	$4$	$4$	$0$	$0$	$4$	$0$	$0$	$0$	$0$	$-1$	$0$	$-1$	$0$	$0$	$-1$
120.36.4b	R	$4$	$-4$	$0$	$0$	$4$	$0$	$0$	$0$	$0$	$-1$	$0$	$1$	$0$	$0$	$-1$
120.36.8a	R	$8$	$0$	$0$	$0$	$-4$	$0$	$0$	$0$	$0$	$-2$	$0$	$0$	$0$	$0$	$1$

Regular extensions

Data not computed