Galois group: 8T3

• Label: 8T3
• Degree: $8$
• Order: $8$
• Cyclic: no
• Abelian: yes
• Solvable: yes
• Primitive: no
• $p$-group: yes
• Group: $C_2^3$

Group action invariants

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

Low degree resolvents

$\card{(G/N)}$	Galois groups for stem field(s)
$2$:	$C_2$ x 7
$4$:	$C_2^2$ x 7

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$ x 7

Degree 4: $C_2^2$ x 7

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	$2^{4}$	$1$	$2$	$4$	$(1,6)(2,5)(3,4)(7,8)$
2B	$2^{4}$	$1$	$2$	$4$	$(1,8)(2,3)(4,5)(6,7)$
2C	$2^{4}$	$1$	$2$	$4$	$(1,2)(3,8)(4,7)(5,6)$
2D	$2^{4}$	$1$	$2$	$4$	$(1,5)(2,6)(3,7)(4,8)$
2E	$2^{4}$	$1$	$2$	$4$	$(1,7)(2,4)(3,5)(6,8)$
2F	$2^{4}$	$1$	$2$	$4$	$(1,4)(2,7)(3,6)(5,8)$
2G	$2^{4}$	$1$	$2$	$4$	$(1,3)(2,8)(4,6)(5,7)$

Malle's constant $a(G)$:     $1/4$

Group invariants

Order:		$8=2^{3}$
Cyclic:		no
Abelian:		yes
Solvable:		yes
Nilpotency class:		$1$
Label:		8.5
Character table:

		1A	2A	2B	2C	2D	2E	2F	2G
	Size	1	1	1	1	1	1	1	1
	2 P	1A	1A	1A	1A	1A	1A	1A	1A
	Type
8.5.1a	R	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
8.5.1b	R	$1$	$-1$	$-1$	$-1$	$-1$	$1$	$1$	$1$
8.5.1c	R	$1$	$-1$	$-1$	$1$	$1$	$-1$	$-1$	$1$
8.5.1d	R	$1$	$-1$	$1$	$-1$	$1$	$1$	$-1$	$-1$
8.5.1e	R	$1$	$-1$	$1$	$1$	$-1$	$-1$	$1$	$-1$
8.5.1f	R	$1$	$1$	$-1$	$-1$	$1$	$-1$	$1$	$-1$
8.5.1g	R	$1$	$1$	$-1$	$1$	$-1$	$1$	$-1$	$-1$
8.5.1h	R	$1$	$1$	$1$	$-1$	$-1$	$-1$	$-1$	$1$