Galois group: 30T3

• Label: 30T3
• Degree: $30$
• Order: $30$
• Cyclic: no
• Abelian: no
• Solvable: yes
• Primitive: no
• $p$-group: no
• Group: $D_{15}$

Group invariants

Abstract group:		$D_{15}$
Order:		$30=2 \cdot 3 \cdot 5$
Cyclic:		no
Abelian:		no
Solvable:		yes
Nilpotency class:		not nilpotent

Group action invariants

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

Low degree resolvents

$\card{(G/N)}$	Galois groups for stem field(s)
$2$:	$C_2$
$6$:	$S_3$
$10$:	$D_{5}$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 3: $S_3$

Degree 5: $D_{5}$

Degree 6: $S_3$

Degree 10: $D_5$

Degree 15: $D_{15}$

Low degree siblings

15T2

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	$2^{15}$	$15$	$2$	$15$	$( 1,27)( 2,28)( 3,26)( 4,25)( 5,24)( 6,23)( 7,22)( 8,21)( 9,20)(10,19)(11,18)(12,17)(13,16)(14,15)(29,30)$
3A	$3^{10}$	$2$	$3$	$20$	$( 1,12,22)( 2,11,21)( 3,13,23)( 4,14,24)( 5,15,25)( 6,16,26)( 7,17,27)( 8,18,28)( 9,19,30)(10,20,29)$
5A1	$5^{6}$	$2$	$5$	$24$	$( 1, 8,14,20,26)( 2, 7,13,19,25)( 3, 9,15,21,27)( 4,10,16,22,28)( 5,11,17,23,30)( 6,12,18,24,29)$
5A2	$5^{6}$	$2$	$5$	$24$	$( 1,14,26, 8,20)( 2,13,25, 7,19)( 3,15,27, 9,21)( 4,16,28,10,22)( 5,17,30,11,23)( 6,18,29,12,24)$
15A1	$15^{2}$	$2$	$15$	$28$	$( 1, 4, 6, 8,10,12,14,16,18,20,22,24,26,28,29)( 2, 3, 5, 7, 9,11,13,15,17,19,21,23,25,27,30)$
15A2	$15^{2}$	$2$	$15$	$28$	$( 1, 6,10,14,18,22,26,29, 4, 8,12,16,20,24,28)( 2, 5, 9,13,17,21,25,30, 3, 7,11,15,19,23,27)$
15A4	$15^{2}$	$2$	$15$	$28$	$( 1,10,18,26, 4,12,20,28, 6,14,22,29, 8,16,24)( 2, 9,17,25, 3,11,19,27, 5,13,21,30, 7,15,23)$
15A7	$15^{2}$	$2$	$15$	$28$	$( 1,16,29,14,28,12,26,10,24, 8,22, 6,20, 4,18)( 2,15,30,13,27,11,25, 9,23, 7,21, 5,19, 3,17)$

Malle's constant $a(G)$:     $1/15$

Character table

		1A	2A	3A	5A1	5A2	15A1	15A2	15A4	15A7
	Size	1	15	2	2	2	2	2	2	2
	2 P	1A	1A	3A	5A2	5A1	15A2	15A4	15A7	15A1
	3 P	1A	2A	1A	5A2	5A1	5A1	5A2	5A1	5A2
	5 P	1A	2A	3A	1A	1A	3A	3A	3A	3A
	Type
30.3.1a	R	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
30.3.1b	R	$1$	$-1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
30.3.2a	R	$2$	$0$	$-1$	$2$	$2$	$-1$	$-1$	$-1$	$-1$
30.3.2b1	R	$2$	$0$	$2$	${\zeta }_5^{-2}+{\zeta }_5^2$	${\zeta }_5^{-1}+{\zeta }_5$	${\zeta }_5^{-1}+{\zeta }_5$	${\zeta }_5^{-2}+{\zeta }_5^2$	${\zeta }_5^{-1}+{\zeta }_5$	${\zeta }_5^{-2}+{\zeta }_5^2$
30.3.2b2	R	$2$	$0$	$2$	${\zeta }_5^{-1}+{\zeta }_5$	${\zeta }_5^{-2}+{\zeta }_5^2$	${\zeta }_5^{-2}+{\zeta }_5^2$	${\zeta }_5^{-1}+{\zeta }_5$	${\zeta }_5^{-2}+{\zeta }_5^2$	${\zeta }_5^{-1}+{\zeta }_5$
30.3.2c1	R	$2$	$0$	$-1$	${\zeta }_{15}^{-6}+{\zeta }_{15}^6$	${\zeta }_{15}^{-3}+{\zeta }_{15}^3$	${\zeta }_{15}^{-7}+{\zeta }_{15}^7$	${\zeta }_{15}^{-1}+{\zeta }_{15}$	${\zeta }_{15}^{-2}+{\zeta }_{15}^2$	${\zeta }_{15}^{-4}+{\zeta }_{15}^4$
30.3.2c2	R	$2$	$0$	$-1$	${\zeta }_{15}^{-6}+{\zeta }_{15}^6$	${\zeta }_{15}^{-3}+{\zeta }_{15}^3$	${\zeta }_{15}^{-2}+{\zeta }_{15}^2$	${\zeta }_{15}^{-4}+{\zeta }_{15}^4$	${\zeta }_{15}^{-7}+{\zeta }_{15}^7$	${\zeta }_{15}^{-1}+{\zeta }_{15}$
30.3.2c3	R	$2$	$0$	$-1$	${\zeta }_{15}^{-3}+{\zeta }_{15}^3$	${\zeta }_{15}^{-6}+{\zeta }_{15}^6$	${\zeta }_{15}^{-4}+{\zeta }_{15}^4$	${\zeta }_{15}^{-7}+{\zeta }_{15}^7$	${\zeta }_{15}^{-1}+{\zeta }_{15}$	${\zeta }_{15}^{-2}+{\zeta }_{15}^2$
30.3.2c4	R	$2$	$0$	$-1$	${\zeta }_{15}^{-3}+{\zeta }_{15}^3$	${\zeta }_{15}^{-6}+{\zeta }_{15}^6$	${\zeta }_{15}^{-1}+{\zeta }_{15}$	${\zeta }_{15}^{-2}+{\zeta }_{15}^2$	${\zeta }_{15}^{-4}+{\zeta }_{15}^4$	${\zeta }_{15}^{-7}+{\zeta }_{15}^7$

Regular extensions

Data not computed