Galois group: 10T5

• Label: 10T5
• Degree: $10$
• Order: $40$
• Cyclic: no
• Abelian: no
• Solvable: yes
• Primitive: no
• $p$-group: no
• Group: $F_{5}\times C_2$

Group action invariants

Degree $n$:		$10$
Transitive number $t$:		$5$
Group:		$F_{5}\times C_2$
CHM label:		$F(5)[x]2$
Parity:		$-1$
Primitive:		no
$\card{\Aut(F/K)}$:		$2$
Generators:		(1,7,9,3)(2,4,8,6), (1,2,3,4,5,6,7,8,9,10)

Low degree resolvents

|G/N|	Galois groups for stem field(s)
$2$:	$C_2$ x 3
$4$:	$C_4$ x 2, $C_2^2$
$8$:	$C_4\times C_2$
$20$:	$F_5$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 5: $F_5$

Low degree siblings

10T5, 20T9, 20T13, 40T14

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, 1, 1, 1, 1, 1, 1, 1, 1, 1 $	$1$	$1$	$()$
	$ 4, 4, 1, 1 $	$5$	$4$	$( 2, 4,10, 8)( 3, 7, 9, 5)$
	$ 4, 4, 1, 1 $	$5$	$4$	$( 2, 8,10, 4)( 3, 5, 9, 7)$
	$ 2, 2, 2, 2, 1, 1 $	$5$	$2$	$( 2,10)( 3, 9)( 4, 8)( 5, 7)$
	$ 2, 2, 2, 2, 2 $	$5$	$2$	$( 1, 2)( 3,10)( 4, 9)( 5, 8)( 6, 7)$
	$ 10 $	$4$	$10$	$( 1, 2, 3, 4, 5, 6, 7, 8, 9,10)$
	$ 4, 4, 2 $	$5$	$4$	$( 1, 2, 5, 4)( 3, 8)( 6, 7,10, 9)$
	$ 4, 4, 2 $	$5$	$4$	$( 1, 2, 9, 8)( 3, 6, 7, 4)( 5,10)$
	$ 5, 5 $	$4$	$5$	$( 1, 3, 5, 7, 9)( 2, 4, 6, 8,10)$
	$ 2, 2, 2, 2, 2 $	$1$	$2$	$( 1, 6)( 2, 7)( 3, 8)( 4, 9)( 5,10)$

Group invariants

Order:		$40=2^{3} \cdot 5$
Cyclic:		no
Abelian:		no
Solvable:		yes
Nilpotency class:		not nilpotent
Label:		40.12
Character table:

		1A	2A	2B	2C	4A1	4A-1	4B1	4B-1	5A	10A
	Size	1	1	5	5	5	5	5	5	4	4
	2 P	1A	1A	1A	1A	2C	2C	2C	2C	5A	5A
	5 P	1A	2A	2B	2C	4A-1	4B-1	4B1	4A1	1A	2A
	Type
40.12.1a	R	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
40.12.1b	R	$1$	$-1$	$-1$	$1$	$-1$	$-1$	$1$	$1$	$1$	$-1$
40.12.1c	R	$1$	$-1$	$-1$	$1$	$1$	$1$	$-1$	$-1$	$1$	$-1$
40.12.1d	R	$1$	$1$	$1$	$1$	$-1$	$-1$	$-1$	$-1$	$1$	$1$
40.12.1e1	C	$1$	$-1$	$1$	$-1$	$-i$	$i$	$i$	$-i$	$1$	$-1$
40.12.1e2	C	$1$	$-1$	$1$	$-1$	$i$	$-i$	$-i$	$i$	$1$	$-1$
40.12.1f1	C	$1$	$1$	$-1$	$-1$	$-i$	$i$	$-i$	$i$	$1$	$1$
40.12.1f2	C	$1$	$1$	$-1$	$-1$	$i$	$-i$	$i$	$-i$	$1$	$1$
40.12.4a	R	$4$	$4$	$0$	$0$	$0$	$0$	$0$	$0$	$-1$	$-1$
40.12.4b	R	$4$	$-4$	$0$	$0$	$0$	$0$	$0$	$0$	$-1$	$1$