Galois group: 8T1

• Label: 8T1
• Degree: $8$
• Order: $8$
• Cyclic: yes
• Abelian: yes
• Solvable: yes
• Primitive: no
• $p$-group: yes
• Group: $C_8$

Group action invariants

Degree $n$:		$8$
Transitive number $t$:		$1$
Group:		$C_8$
CHM label:		$C(8)=8$
Parity:		$-1$
Primitive:		no
$\card{\Aut(F/K)}$:		$8$
Generators:		(1,2,3,4,5,6,7,8)

Low degree resolvents

|G/N|	Galois groups for stem field(s)
$2$:	$C_2$
$4$:	$C_4$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 4: $C_4$

Low degree siblings

There are no siblings 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 $	$1$	$1$	$()$
	$ 8 $	$1$	$8$	$(1,2,3,4,5,6,7,8)$
	$ 4, 4 $	$1$	$4$	$(1,3,5,7)(2,4,6,8)$
	$ 8 $	$1$	$8$	$(1,4,7,2,5,8,3,6)$
	$ 2, 2, 2, 2 $	$1$	$2$	$(1,5)(2,6)(3,7)(4,8)$
	$ 8 $	$1$	$8$	$(1,6,3,8,5,2,7,4)$
	$ 4, 4 $	$1$	$4$	$(1,7,5,3)(2,8,6,4)$
	$ 8 $	$1$	$8$	$(1,8,7,6,5,4,3,2)$

Group invariants

Order:		$8=2^{3}$
Cyclic:		yes
Abelian:		yes
Solvable:		yes
Nilpotency class:		$1$
Label:		8.1
Character table:

		1A	2A	4A1	4A-1	8A1	8A-1	8A3	8A-3
	Size	1	1	1	1	1	1	1	1
	2 P	1A	1A	2A	2A	4A-1	4A-1	4A1	4A1
	Type
8.1.1a	R	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
8.1.1b	R	$1$	$1$	$1$	$1$	$-1$	$-1$	$-1$	$-1$
8.1.1c1	C	$1$	$1$	$-1$	$-1$	$-i$	$i$	$i$	$-i$
8.1.1c2	C	$1$	$1$	$-1$	$-1$	$i$	$-i$	$-i$	$i$
8.1.1d1	C	$1$	$-1$	$-{\zeta }_8^2$	${\zeta }_8^2$	${\zeta }_8^3$	$-{\zeta }_8$	${\zeta }_8$	$-{\zeta }_8^3$
8.1.1d2	C	$1$	$-1$	${\zeta }_8^2$	$-{\zeta }_8^2$	$-{\zeta }_8$	${\zeta }_8^3$	$-{\zeta }_8^3$	${\zeta }_8$
8.1.1d3	C	$1$	$-1$	$-{\zeta }_8^2$	${\zeta }_8^2$	$-{\zeta }_8^3$	${\zeta }_8$	$-{\zeta }_8$	${\zeta }_8^3$
8.1.1d4	C	$1$	$-1$	${\zeta }_8^2$	$-{\zeta }_8^2$	${\zeta }_8$	$-{\zeta }_8^3$	${\zeta }_8^3$	$-{\zeta }_8$