Galois group: 8T15

• Label: 8T15
• Degree: $8$
• Order: $32$
• Cyclic: no
• Abelian: no
• Solvable: yes
• Primitive: no
• $p$-group: yes
• Group: $Z_8 : Z_8^\times$

Group invariants

Abstract group:		$Z_8 : Z_8^\times$
Order:		$32=2^{5}$
Cyclic:		no
Abelian:		no
Solvable:		yes
Nilpotency class:		$3$

Group action invariants

Degree $n$:		$8$
Transitive number $t$:		$15$
CHM label:		$[1/4.cD(4)^{2}]2$
Parity:		$-1$
Primitive:		no
$\card{\Aut(F/K)}$:		$2$
Generators:		$(1,2,3,4,5,6,7,8)$, $(1,5)(3,7)$, $(1,6)(2,5)(3,4)(7,8)$

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, $C_2^3$
$16$:	$D_4\times C_2$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 4: $D_{4}$

Low degree siblings

8T15, 16T35, 16T38 x 2, 16T45, 32T21

Siblings are shown with degree $\leq 47$

A number field with this Galois group has exactly one arithmetically equivalent field.

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

Malle's constant $a(G)$:     $1/2$

Character table

		1A	2A	2B	2C	2D	2E	4A	4B	4C	8A	8B
	Size	1	1	2	4	4	4	2	2	4	4	4
	2 P	1A	1A	1A	1A	1A	1A	2A	2A	2A	4B	4B
	Type
32.43.1a	R	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
32.43.1b	R	$1$	$1$	$-1$	$-1$	$-1$	$1$	$-1$	$1$	$1$	$1$	$-1$
32.43.1c	R	$1$	$1$	$-1$	$-1$	$1$	$1$	$-1$	$1$	$-1$	$-1$	$1$
32.43.1d	R	$1$	$1$	$-1$	$1$	$-1$	$-1$	$-1$	$1$	$1$	$-1$	$1$
32.43.1e	R	$1$	$1$	$-1$	$1$	$1$	$-1$	$-1$	$1$	$-1$	$1$	$-1$
32.43.1f	R	$1$	$1$	$1$	$-1$	$-1$	$-1$	$1$	$1$	$-1$	$1$	$1$
32.43.1g	R	$1$	$1$	$1$	$-1$	$1$	$-1$	$1$	$1$	$1$	$-1$	$-1$
32.43.1h	R	$1$	$1$	$1$	$1$	$-1$	$1$	$1$	$1$	$-1$	$-1$	$-1$
32.43.2a	R	$2$	$2$	$-2$	$0$	$0$	$0$	$2$	$-2$	$0$	$0$	$0$
32.43.2b	R	$2$	$2$	$2$	$0$	$0$	$0$	$-2$	$-2$	$0$	$0$	$0$
32.43.4a	R	$4$	$-4$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$

Regular extensions

$f_{ 1 } =$

$x^{8} + 3 t x^{6} + 3 t^{2} x^{4} + t x^{2} + \left(-t^{2} + 1\right)$