Galois group: 8T1

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

Group invariants

Abstract group:		$C_8$
Order:		$8=2^{3}$
Cyclic:		yes
Abelian:		yes
Solvable:		yes
Nilpotency class:		$1$

Group action invariants

Degree $n$:		$8$
Transitive number $t$:		$1$
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

$\card{(G/N)}$	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	Index	Representative
1A	$1^{8}$	$1$	$1$	$0$	$()$
2A	$2^{4}$	$1$	$2$	$4$	$(1,5)(2,6)(3,7)(4,8)$
4A1	$4^{2}$	$1$	$4$	$6$	$(1,3,5,7)(2,4,6,8)$
4A-1	$4^{2}$	$1$	$4$	$6$	$(1,7,5,3)(2,8,6,4)$
8A1	$8$	$1$	$8$	$7$	$(1,2,3,4,5,6,7,8)$
8A-1	$8$	$1$	$8$	$7$	$(1,8,7,6,5,4,3,2)$
8A3	$8$	$1$	$8$	$7$	$(1,4,7,2,5,8,3,6)$
8A-3	$8$	$1$	$8$	$7$	$(1,6,3,8,5,2,7,4)$

Malle's constant $a(G)$:     $1/4$

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	4A1	4A-1	4A-1	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	$1$	$-1$	$-{\zeta }_8^2$	${\zeta }_8^2$	${\zeta }_8^3$	$-{\zeta }_8^3$	$-{\zeta }_8$	${\zeta }_8$
8.1.1d2	C	$1$	$-1$	${\zeta }_8^2$	$-{\zeta }_8^2$	$-{\zeta }_8$	${\zeta }_8$	${\zeta }_8^3$	$-{\zeta }_8^3$
8.1.1d3	C	$1$	$-1$	$-{\zeta }_8^2$	${\zeta }_8^2$	$-{\zeta }_8^3$	${\zeta }_8^3$	${\zeta }_8$	$-{\zeta }_8$
8.1.1d4	C	$1$	$-1$	${\zeta }_8^2$	$-{\zeta }_8^2$	${\zeta }_8$	$-{\zeta }_8$	$-{\zeta }_8^3$	${\zeta }_8^3$

Regular extensions

$f_{ 1 } =$

$t^{2} x^{8} + \left(4 t^{4} + 4\right) x^{6} + \left(2 t^{6} + 8 t^{4} + 2 t^{2} + 8\right) x^{4} + \left(4 t^{6} + 4 t^{4} + 4 t^{2} + 4\right) x^{2} + \left(t^{6} + t^{2}\right)$