Galois group: 10T8

• Label: 10T8
• Degree: $10$
• Order: $80$
• Cyclic: no
• Abelian: no
• Solvable: yes
• Primitive: no
• $p$-group: no
• Group: $C_2^4 : C_5$

Group action invariants

Degree $n$:		$10$
Transitive number $t$:		$8$
Group:		$C_2^4 : C_5$
CHM label:		$[2^{4}]5$
Parity:		$1$
Primitive:		no
$\card{\Aut(F/K)}$:		$2$
Generators:		(1,3,5,7,9)(2,4,6,8,10), (2,7)(5,10)

Low degree resolvents

|G/N|	Galois groups for stem field(s)
$5$:	$C_5$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 5: $C_5$

Low degree siblings

10T8 x 2, 16T178, 20T17 x 6, 20T23, 40T57 x 3

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 $	$1$	$1$	$()$
	$ 2, 2, 1, 1, 1, 1, 1, 1 $	$5$	$2$	$( 4, 9)( 5,10)$
	$ 2, 2, 1, 1, 1, 1, 1, 1 $	$5$	$2$	$( 3, 8)( 5,10)$
	$ 2, 2, 2, 2, 1, 1 $	$5$	$2$	$( 2, 7)( 3, 8)( 4, 9)( 5,10)$
	$ 5, 5 $	$16$	$5$	$( 1, 2, 3, 4, 5)( 6, 7, 8, 9,10)$
	$ 5, 5 $	$16$	$5$	$( 1, 3, 5, 2, 4)( 6, 8,10, 7, 9)$
	$ 5, 5 $	$16$	$5$	$( 1, 4, 2, 5, 3)( 6, 9, 7,10, 8)$
	$ 5, 5 $	$16$	$5$	$( 1, 5, 4, 3, 2)( 6,10, 9, 8, 7)$

Group invariants

Order:		$80=2^{4} \cdot 5$
Cyclic:		no
Abelian:		no
Solvable:		yes
Nilpotency class:		not nilpotent
Label:		80.49
Character table:

		1A	2A	2B	2C	5A1	5A-1	5A2	5A-2
	Size	1	5	5	5	16	16	16	16
	2 P	1A	1A	1A	1A	5A-1	5A1	5A2	5A-2
	5 P	1A	2C	2A	2B	1A	1A	1A	1A
	Type
80.49.1a	R	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
80.49.1b1	C	$1$	$1$	$1$	$1$	${\zeta }_5^{-2}$	${\zeta }_5^2$	${\zeta }_5$	${\zeta }_5^{-1}$
80.49.1b2	C	$1$	$1$	$1$	$1$	${\zeta }_5^2$	${\zeta }_5^{-2}$	${\zeta }_5^{-1}$	${\zeta }_5$
80.49.1b3	C	$1$	$1$	$1$	$1$	${\zeta }_5^{-1}$	${\zeta }_5$	${\zeta }_5^{-2}$	${\zeta }_5^2$
80.49.1b4	C	$1$	$1$	$1$	$1$	${\zeta }_5$	${\zeta }_5^{-1}$	${\zeta }_5^2$	${\zeta }_5^{-2}$
80.49.5a	R	$5$	$-3$	$1$	$1$	$0$	$0$	$0$	$0$
80.49.5b	R	$5$	$1$	$-3$	$1$	$0$	$0$	$0$	$0$
80.49.5c	R	$5$	$1$	$1$	$-3$	$0$	$0$	$0$	$0$