Galois group: 8T2

• Label: 8T2
• Degree: $8$
• Order: $8$
• Cyclic: no
• Abelian: yes
• Solvable: yes
• Primitive: no
• $p$-group: yes
• Group: $C_4\times C_2$

Group invariants

Abstract group:		$C_4\times C_2$
Order:		$8=2^{3}$
Cyclic:		no
Abelian:		yes
Solvable:		yes
Nilpotency class:		$1$

Group action invariants

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

Low degree resolvents

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

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$ x 3

Degree 4: $C_4$ x 2, $C_2^2$

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

Malle's constant $a(G)$:     $1/4$

Character table

		1A	2A	2B	2C	4A1	4A-1	4B1	4B-1
	Size	1	1	1	1	1	1	1	1
	2 P	1A	1A	1A	1A	2C	2C	2C	2C
	Type
8.2.1a	R	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
8.2.1b	R	$1$	$-1$	$1$	$-1$	$-1$	$-1$	$1$	$1$
8.2.1c	R	$1$	$-1$	$1$	$-1$	$1$	$1$	$-1$	$-1$
8.2.1d	R	$1$	$1$	$1$	$1$	$-1$	$-1$	$-1$	$-1$
8.2.1e1	C	$1$	$-1$	$-1$	$1$	$-i$	$i$	$-i$	$i$
8.2.1e2	C	$1$	$-1$	$-1$	$1$	$i$	$-i$	$i$	$-i$
8.2.1f1	C	$1$	$1$	$-1$	$-1$	$-i$	$i$	$i$	$-i$
8.2.1f2	C	$1$	$1$	$-1$	$-1$	$i$	$-i$	$-i$	$i$

Regular extensions

$f_{ 1 } =$

$t^{2} x^{8} + \left(2 t^{3} + 16 t\right) x^{6} + \left(t^{4} + 14 t^{2} + 64\right) x^{4} + \left(2 t^{3} + 48 t\right) x^{2} + t^{2}$