Galois group: 15T7

• Label: 15T7
• Degree: $15$
• Order: $60$
• Cyclic: no
• Abelian: no
• Solvable: yes
• Primitive: no
• $p$-group: no
• Group: $D_5\times S_3$

Group action invariants

Degree $n$:		$15$
Transitive number $t$:		$7$
Group:		$D_5\times S_3$
CHM label:		$D(5)[x]S(3)$
Parity:		$-1$
Primitive:		no
$\card{\Aut(F/K)}$:		$1$
Generators:		(1,4)(2,8)(3,12)(6,9)(7,13)(11,14), (1,11)(2,7)(4,14)(5,10)(8,13), (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)
$2$:	$C_2$ x 3
$4$:	$C_2^2$
$6$:	$S_3$
$10$:	$D_{5}$
$12$:	$D_{6}$
$20$:	$D_{10}$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 3: $S_3$

Degree 5: $D_{5}$

Low degree siblings

30T8, 30T10, 30T13

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

Group invariants

Order:		$60=2^{2} \cdot 3 \cdot 5$
Cyclic:		no
Abelian:		no
Solvable:		yes
Nilpotency class:		not nilpotent
Label:		60.8
Character table:

		1A	2A	2B	2C	3A	5A1	5A2	6A	10A1	10A3	15A1	15A2
	Size	1	3	5	15	2	2	2	10	6	6	4	4
	2 P	1A	1A	1A	1A	3A	5A2	5A1	3A	5A1	5A2	15A2	15A1
	3 P	1A	2A	2B	2C	1A	5A2	5A1	2B	10A3	10A1	5A1	5A2
	5 P	1A	2A	2B	2C	3A	1A	1A	6A	2A	2A	3A	3A
	Type
60.8.1a	R	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
60.8.1b	R	$1$	$-1$	$-1$	$1$	$1$	$1$	$1$	$-1$	$-1$	$-1$	$1$	$1$
60.8.1c	R	$1$	$-1$	$1$	$-1$	$1$	$1$	$1$	$1$	$-1$	$-1$	$1$	$1$
60.8.1d	R	$1$	$1$	$-1$	$-1$	$1$	$1$	$1$	$-1$	$1$	$1$	$1$	$1$
60.8.2a	R	$2$	$0$	$2$	$0$	$-1$	$2$	$2$	$-1$	$0$	$0$	$-1$	$-1$
60.8.2b	R	$2$	$0$	$-2$	$0$	$-1$	$2$	$2$	$1$	$0$	$0$	$-1$	$-1$
60.8.2c1	R	$2$	$2$	$0$	$0$	$2$	${\zeta }_5^{-2}+{\zeta }_5^2$	${\zeta }_5^{-1}+{\zeta }_5$	$0$	${\zeta }_5^{-1}+{\zeta }_5$	${\zeta }_5^{-2}+{\zeta }_5^2$	${\zeta }_5^{-1}+{\zeta }_5$	${\zeta }_5^{-2}+{\zeta }_5^2$
60.8.2c2	R	$2$	$2$	$0$	$0$	$2$	${\zeta }_5^{-1}+{\zeta }_5$	${\zeta }_5^{-2}+{\zeta }_5^2$	$0$	${\zeta }_5^{-2}+{\zeta }_5^2$	${\zeta }_5^{-1}+{\zeta }_5$	${\zeta }_5^{-2}+{\zeta }_5^2$	${\zeta }_5^{-1}+{\zeta }_5$
60.8.2d1	R	$2$	$-2$	$0$	$0$	$2$	${\zeta }_5^{-2}+{\zeta }_5^2$	${\zeta }_5^{-1}+{\zeta }_5$	$0$	$-{\zeta }_5^{-1}-{\zeta }_5$	$-{\zeta }_5^{-2}-{\zeta }_5^2$	${\zeta }_5^{-1}+{\zeta }_5$	${\zeta }_5^{-2}+{\zeta }_5^2$
60.8.2d2	R	$2$	$-2$	$0$	$0$	$2$	${\zeta }_5^{-1}+{\zeta }_5$	${\zeta }_5^{-2}+{\zeta }_5^2$	$0$	$-{\zeta }_5^{-2}-{\zeta }_5^2$	$-{\zeta }_5^{-1}-{\zeta }_5$	${\zeta }_5^{-2}+{\zeta }_5^2$	${\zeta }_5^{-1}+{\zeta }_5$
60.8.4a1	R	$4$	$0$	$0$	$0$	$-2$	$2{\zeta }_5^{-2}+2{\zeta }_5^2$	$2{\zeta }_5^{-1}+2{\zeta }_5$	$0$	$0$	$0$	$-{\zeta }_5^{-1}-{\zeta }_5$	$-{\zeta }_5^{-2}-{\zeta }_5^2$
60.8.4a2	R	$4$	$0$	$0$	$0$	$-2$	$2{\zeta }_5^{-1}+2{\zeta }_5$	$2{\zeta }_5^{-2}+2{\zeta }_5^2$	$0$	$0$	$0$	$-{\zeta }_5^{-2}-{\zeta }_5^2$	$-{\zeta }_5^{-1}-{\zeta }_5$