Galois group: 8T6

• Label: 8T6
• Degree: $8$
• Order: $16$
• Cyclic: no
• Abelian: no
• Solvable: yes
• Primitive: no
• $p$-group: yes
• Group: $D_{8}$

Group invariants

Abstract group:		$D_{8}$
Order:		$16=2^{4}$
Cyclic:		no
Abelian:		no
Solvable:		yes
Nilpotency class:		$3$

Group action invariants

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

Low degree resolvents

$\card{(G/N)}$	Galois groups for stem field(s)
$2$:	$C_2$ x 3
$4$:	$C_2^2$
$8$:	$D_{4}$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 4: $D_{4}$

Low degree siblings

8T6, 16T13

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	Index	Representative
1A	$1^{8}$	$1$	$1$	$0$	$()$
2A	$2^{4}$	$1$	$2$	$4$	$(1,5)(2,6)(3,7)(4,8)$
2B	$2^{3},1^{2}$	$4$	$2$	$3$	$(2,8)(3,7)(4,6)$
2C	$2^{4}$	$4$	$2$	$4$	$(1,2)(3,8)(4,7)(5,6)$
4A	$4^{2}$	$2$	$4$	$6$	$(1,3,5,7)(2,4,6,8)$
8A1	$8$	$2$	$8$	$7$	$(1,2,3,4,5,6,7,8)$
8A3	$8$	$2$	$8$	$7$	$(1,6,3,8,5,2,7,4)$

Malle's constant $a(G)$:     $1/3$

Character table

		1A	2A	2B	2C	4A	8A1	8A3
	Size	1	1	4	4	2	2	2
	2 P	1A	1A	1A	1A	2A	4A	4A
	Type
16.7.1a	R	$1$	$1$	$1$	$1$	$1$	$1$	$1$
16.7.1b	R	$1$	$1$	$-1$	$-1$	$1$	$1$	$1$
16.7.1c	R	$1$	$1$	$-1$	$1$	$1$	$-1$	$-1$
16.7.1d	R	$1$	$1$	$1$	$-1$	$1$	$-1$	$-1$
16.7.2a	R	$2$	$2$	$0$	$0$	$-2$	$0$	$0$
16.7.2b1	R	$2$	$-2$	$0$	$0$	$0$	$-{\zeta }_8^{-1}-{\zeta }_8$	${\zeta }_8^{-1}+{\zeta }_8$
16.7.2b2	R	$2$	$-2$	$0$	$0$	$0$	${\zeta }_8^{-1}+{\zeta }_8$	$-{\zeta }_8^{-1}-{\zeta }_8$

Regular extensions

$f_{ 1 } =$

$x^{8} + 5 x^{7} - t x^{6} + 14 x^{5} + \left(2 t + 35\right) x^{4} + 7 x^{3} - t x^{2} - 2 x + 4$