Galois group: 8T5

• Label: 8T5
• Degree: $8$
• Order: $8$
• Cyclic: no
• Abelian: no
• Solvable: yes
• Primitive: no
• $p$-group: yes
• Group: $Q_8$

Group invariants

Abstract group:		$Q_8$
Order:		$8=2^{3}$
Cyclic:		no
Abelian:		no
Solvable:		yes
Nilpotency class:		$2$

Group action invariants

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

Low degree resolvents

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

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$ x 3

Degree 4: $C_2^2$

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	$2^{4}$	$1$	$2$	$4$	$(1,3)(2,8)(4,6)(5,7)$
4A	$4^{2}$	$2$	$4$	$6$	$(1,2,3,8)(4,5,6,7)$
4B	$4^{2}$	$2$	$4$	$6$	$(1,7,3,5)(2,6,8,4)$
4C	$4^{2}$	$2$	$4$	$6$	$(1,4,3,6)(2,7,8,5)$

Malle's constant $a(G)$:     $1/4$

Character table

		1A	2A	4A	4B	4C
	Size	1	1	2	2	2
	2 P	1A	1A	2A	2A	2A
	Type
8.4.1a	R	$1$	$1$	$1$	$1$	$1$
8.4.1b	R	$1$	$1$	$-1$	$-1$	$1$
8.4.1c	R	$1$	$1$	$-1$	$1$	$-1$
8.4.1d	R	$1$	$1$	$1$	$-1$	$-1$
8.4.2a	S	$2$	$-2$	$0$	$0$	$0$

Regular extensions

$f_{ 1 } =$

$x^{8} + \left(t^{3} + 8 t\right) x^{7} + \left(t^{4} + 4 t^{2} - 28\right) x^{6} + \left(-7 t^{3} - 56 t\right) x^{5} + \left(-2 t^{4} - 8 t^{2} + 70\right) x^{4} + \left(7 t^{3} + 56 t\right) x^{3} + \left(t^{4} + 4 t^{2} - 28\right) x^{2} + \left(-t^{3} - 8 t\right) x + 1$