Galois group: 8T10

• Label: 8T10
• Degree: $8$
• Order: $16$
• Cyclic: no
• Abelian: no
• Solvable: yes
• Primitive: no
• $p$-group: yes
• Group: $C_2^2:C_4$

Group invariants

Abstract group:		$C_2^2:C_4$
Order:		$16=2^{4}$
Cyclic:		no
Abelian:		no
Solvable:		yes
Nilpotency class:		$2$

Group action invariants

Degree $n$:		$8$
Transitive number $t$:		$10$
CHM label:		$[2^{2}]4$
Parity:		$1$
Primitive:		no
$\card{\Aut(F/K)}$:		$4$
Generators:		$(1,2,3,8)(4,5,6,7)$, $(1,5)(3,7)$

Low degree resolvents

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

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 4: $C_4$, $D_{4}$ x 2

Low degree siblings

8T10, 16T10

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

Malle's constant $a(G)$:     $1/2$

Character table

		1A	2A	2B	2C	2D	2E	4A1	4A-1	4B1	4B-1
	Size	1	1	1	1	2	2	2	2	2	2
	2 P	1A	1A	1A	1A	1A	1A	2B	2B	2C	2C
	Type
16.3.1a	R	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
16.3.1b	R	$1$	$1$	$1$	$1$	$-1$	$-1$	$-1$	$-1$	$1$	$1$
16.3.1c	R	$1$	$1$	$1$	$1$	$-1$	$-1$	$1$	$1$	$-1$	$-1$
16.3.1d	R	$1$	$1$	$1$	$1$	$1$	$1$	$-1$	$-1$	$-1$	$-1$
16.3.1e1	C	$1$	$1$	$-1$	$-1$	$-1$	$1$	$-i$	$i$	$i$	$-i$
16.3.1e2	C	$1$	$1$	$-1$	$-1$	$-1$	$1$	$i$	$-i$	$-i$	$i$
16.3.1f1	C	$1$	$1$	$-1$	$-1$	$1$	$-1$	$-i$	$i$	$-i$	$i$
16.3.1f2	C	$1$	$1$	$-1$	$-1$	$1$	$-1$	$i$	$-i$	$i$	$-i$
16.3.2a	R	$2$	$-2$	$-2$	$2$	$0$	$0$	$0$	$0$	$0$	$0$
16.3.2b	R	$2$	$-2$	$2$	$-2$	$0$	$0$	$0$	$0$	$0$	$0$

Regular extensions

$f_{ 1 } =$

$x^{8} - t^{2} x^{6} + \left(-t^{2} + 2\right) x^{4} - t^{2} x^{2} + 1$