Galois group: 15T2

• Label: 15T2
• Degree: $15$
• Order: $30$
• Cyclic: no
• Abelian: no
• Solvable: yes
• Primitive: no
• $p$-group: no
• Group: $D_{15}$

Group action invariants

Degree $n$:		$15$
Transitive number $t$:		$2$
Group:		$D_{15}$
CHM label:		$D(15)$
Parity:		$-1$
Primitive:		no
$\card{\Aut(F/K)}$:		$1$
Generators:		(1,14)(2,13)(3,12)(4,11)(5,10)(6,9)(7,8), (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15)

Low degree resolvents

|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 3: $S_3$

Degree 5: $D_{5}$

Low degree siblings

30T3

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	Representative
	$1^{15}$	$1$	$1$	$()$
	$2^{7},1$	$15$	$2$	$( 2,15)( 3,14)( 4,13)( 5,12)( 6,11)( 7,10)( 8, 9)$
	$15$	$2$	$15$	$( 1, 2, 3, 4, 5, 6, 7, 8, 9,10,11,12,13,14,15)$
	$15$	$2$	$15$	$( 1, 3, 5, 7, 9,11,13,15, 2, 4, 6, 8,10,12,14)$
	$5^{3}$	$2$	$5$	$( 1, 4, 7,10,13)( 2, 5, 8,11,14)( 3, 6, 9,12,15)$
	$15$	$2$	$15$	$( 1, 5, 9,13, 2, 6,10,14, 3, 7,11,15, 4, 8,12)$
	$3^{5}$	$2$	$3$	$( 1, 6,11)( 2, 7,12)( 3, 8,13)( 4, 9,14)( 5,10,15)$
	$5^{3}$	$2$	$5$	$( 1, 7,13, 4,10)( 2, 8,14, 5,11)( 3, 9,15, 6,12)$
	$15$	$2$	$15$	$( 1, 8,15, 7,14, 6,13, 5,12, 4,11, 3,10, 2, 9)$

Group invariants

Order:		$30=2 \cdot 3 \cdot 5$
Cyclic:		no
Abelian:		no
Solvable:		yes
Nilpotency class:		not nilpotent
Label:		30.3
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	$2$	$0$	$-1$	${\zeta }_{15}^{-3}+{\zeta }_{15}^3$	${\zeta }_{15}^{-6}+{\zeta }_{15}^6$	${\zeta }_{15}^{-1}+{\zeta }_{15}$	${\zeta }_{15}^{-2}+{\zeta }_{15}^2$	${\zeta }_{15}^{-4}+{\zeta }_{15}^4$	${\zeta }_{15}^{-7}+{\zeta }_{15}^7$