Galois group: 8T9

• Label: 8T9
• Degree: $8$
• Order: $16$
• Cyclic: no
• Abelian: no
• Solvable: yes
• Primitive: no
• $p$-group: yes
• Group: $D_4\times C_2$

Group invariants

Abstract group:		$D_4\times C_2$
Order:		$16=2^{4}$
Cyclic:		no
Abelian:		no
Solvable:		yes
Nilpotency class:		$2$

Group action invariants

Degree $n$:		$8$
Transitive number $t$:		$9$
CHM label:		$E(8):2=D(4)[x]2$
Parity:		$1$
Primitive:		no
$\card{\Aut(F/K)}$:		$4$
Generators:		$(1,3)(2,8)(4,6)(5,7)$, $(4,5)(6,7)$, $(1,8)(2,3)(4,5)(6,7)$, $(1,5)(2,6)(3,7)(4,8)$

Low degree resolvents

$\card{(G/N)}$	Galois groups for stem field(s)
$2$:	$C_2$ x 7
$4$:	$C_2^2$ x 7
$8$:	$D_{4}$ x 2, $C_2^3$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$ x 3

Degree 4: $C_2^2$, $D_{4}$ x 2

Low degree siblings

8T9 x 3, 16T9

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

Malle's constant $a(G)$:     $1/2$

Character table

		1A	2A	2B	2C	2D	2E	2F	2G	4A	4B
	Size	1	1	1	1	2	2	2	2	2	2
	2 P	1A	1A	1A	1A	1A	1A	1A	1A	2C	2C
	Type
16.11.1a	R	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
16.11.1b	R	$1$	$-1$	$1$	$-1$	$-1$	$-1$	$1$	$1$	$1$	$-1$
16.11.1c	R	$1$	$-1$	$1$	$-1$	$-1$	$1$	$-1$	$1$	$-1$	$1$
16.11.1d	R	$1$	$-1$	$1$	$-1$	$1$	$-1$	$1$	$-1$	$-1$	$1$
16.11.1e	R	$1$	$-1$	$1$	$-1$	$1$	$1$	$-1$	$-1$	$1$	$-1$
16.11.1f	R	$1$	$1$	$1$	$1$	$-1$	$-1$	$-1$	$-1$	$1$	$1$
16.11.1g	R	$1$	$1$	$1$	$1$	$-1$	$1$	$1$	$-1$	$-1$	$-1$
16.11.1h	R	$1$	$1$	$1$	$1$	$1$	$-1$	$-1$	$1$	$-1$	$-1$
16.11.2a	R	$2$	$-2$	$-2$	$2$	$0$	$0$	$0$	$0$	$0$	$0$
16.11.2b	R	$2$	$2$	$-2$	$-2$	$0$	$0$	$0$	$0$	$0$	$0$

Regular extensions

$f_{ 1 } =$

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