Galois group: 8T7

• Label: 8T7
• Degree: $8$
• Order: $16$
• Cyclic: no
• Abelian: no
• Solvable: yes
• Primitive: no
• $p$-group: yes
• Group: $C_8:C_2$

Group invariants

Abstract group:		$C_8:C_2$
Order:		$16=2^{4}$
Cyclic:		no
Abelian:		no
Solvable:		yes
Nilpotency class:		$2$

Group action invariants

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

Low degree resolvents

$\card{(G/N)}$	Galois groups for stem field(s)
$2$:	$C_2$ x 3
$4$:	$C_4$ x 2, $C_2^2$
$8$:	$C_4\times C_2$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 4: $C_4$

Low degree siblings

16T6

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	$2^{2},1^{4}$	$2$	$2$	$2$	$(2,6)(4,8)$
4A1	$4^{2}$	$1$	$4$	$6$	$(1,3,5,7)(2,4,6,8)$
4A-1	$4^{2}$	$1$	$4$	$6$	$(1,7,5,3)(2,8,6,4)$
4B	$4^{2}$	$2$	$4$	$6$	$(1,3,5,7)(2,8,6,4)$
8A1	$8$	$2$	$8$	$7$	$(1,2,3,4,5,6,7,8)$
8A-1	$8$	$2$	$8$	$7$	$(1,4,7,2,5,8,3,6)$
8B1	$8$	$2$	$8$	$7$	$(1,2,7,8,5,6,3,4)$
8B-1	$8$	$2$	$8$	$7$	$(1,4,3,6,5,8,7,2)$

Malle's constant $a(G)$:     $1/2$

Character table

		1A	2A	2B	4A1	4A-1	4B	8A1	8A-1	8B1	8B-1
	Size	1	1	2	1	1	2	2	2	2	2
	2 P	1A	1A	1A	2A	2A	2A	4A1	4A-1	4A-1	4A1
	Type
16.6.1a	R	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
16.6.1b	R	$1$	$1$	$-1$	$1$	$1$	$-1$	$-1$	$1$	$-1$	$1$
16.6.1c	R	$1$	$1$	$-1$	$1$	$1$	$-1$	$1$	$-1$	$1$	$-1$
16.6.1d	R	$1$	$1$	$1$	$1$	$1$	$1$	$-1$	$-1$	$-1$	$-1$
16.6.1e1	C	$1$	$1$	$-1$	$-1$	$-1$	$1$	$-i$	$-i$	$i$	$i$
16.6.1e2	C	$1$	$1$	$-1$	$-1$	$-1$	$1$	$i$	$i$	$-i$	$-i$
16.6.1f1	C	$1$	$1$	$1$	$-1$	$-1$	$-1$	$-i$	$i$	$i$	$-i$
16.6.1f2	C	$1$	$1$	$1$	$-1$	$-1$	$-1$	$i$	$-i$	$-i$	$i$
16.6.2a1	C	$2$	$-2$	$0$	$-2i$	$2i$	$0$	$0$	$0$	$0$	$0$
16.6.2a2	C	$2$	$-2$	$0$	$2i$	$-2i$	$0$	$0$	$0$	$0$	$0$

Regular extensions

$f_{ 1 } =$

$t x^{8} + \left(4 t^{2} + 4\right) x^{6} + \left(2 t^{3} + 8 t^{2} + 2 t + 8\right) x^{4} + \left(4 t^{3} + 4 t^{2} + 4 t + 4\right) x^{2} + \left(t^{3} + t\right)$