Galois group: 8T20

• Label: 8T20
• Degree: $8$
• Order: $32$
• Cyclic: no
• Abelian: no
• Solvable: yes
• Primitive: no
• $p$-group: yes
• Group: $C_2^3: C_4$

Group invariants

Abstract group:		$C_2^3: C_4$
Order:		$32=2^{5}$
Cyclic:		no
Abelian:		no
Solvable:		yes
Nilpotency class:		$3$

Group action invariants

Degree $n$:		$8$
Transitive number $t$:		$20$
CHM label:		$[2^{3}]4$
Parity:		$1$
Primitive:		no
$\card{\Aut(F/K)}$:		$2$
Generators:		$(1,2,3,8)(4,5,6,7)$, $(2,6)(3,7)$

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, $C_4\times C_2$
$16$:	$C_2^2:C_4$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 4: $C_4$

Low degree siblings

8T19 x 2, 8T21, 16T33 x 2, 16T52, 16T53, 32T19

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	$2^{2},1^{4}$	$2$	$2$	$2$	$(1,5)(3,7)$
2C	$2^{4}$	$2$	$2$	$4$	$(1,3)(2,8)(4,6)(5,7)$
2D	$2^{4}$	$2$	$2$	$4$	$(1,3)(2,4)(5,7)(6,8)$
2E	$2^{2},1^{4}$	$4$	$2$	$2$	$(1,5)(2,6)$
4A	$4^{2}$	$4$	$4$	$6$	$(1,3,5,7)(2,8,6,4)$
4B1	$4^{2}$	$4$	$4$	$6$	$(1,2,3,8)(4,5,6,7)$
4B-1	$4^{2}$	$4$	$4$	$6$	$(1,8,3,2)(4,7,6,5)$
4C1	$4^{2}$	$4$	$4$	$6$	$(1,8,3,6)(2,5,4,7)$
4C-1	$4^{2}$	$4$	$4$	$6$	$(1,6,7,4)(2,3,8,5)$

Malle's constant $a(G)$:     $1/2$

Character table

		1A	2A	2B	2C	2D	2E	4A	4B1	4B-1	4C1	4C-1
	Size	1	1	2	2	2	4	4	4	4	4	4
	2 P	1A	1A	1A	1A	1A	1A	2A	2C	2C	2D	2D
	Type
32.6.1a	R	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
32.6.1b	R	$1$	$1$	$1$	$1$	$1$	$-1$	$-1$	$-1$	$1$	$-1$	$1$
32.6.1c	R	$1$	$1$	$1$	$1$	$1$	$-1$	$1$	$1$	$-1$	$-1$	$-1$
32.6.1d	R	$1$	$1$	$1$	$1$	$1$	$1$	$-1$	$-1$	$-1$	$1$	$-1$
32.6.1e1	C	$1$	$1$	$1$	$-1$	$-1$	$-1$	$-i$	$i$	$i$	$1$	$-i$
32.6.1e2	C	$1$	$1$	$1$	$-1$	$-1$	$-1$	$i$	$-i$	$-i$	$1$	$i$
32.6.1f1	C	$1$	$1$	$1$	$-1$	$-1$	$1$	$-i$	$i$	$-i$	$-1$	$i$
32.6.1f2	C	$1$	$1$	$1$	$-1$	$-1$	$1$	$i$	$-i$	$i$	$-1$	$-i$
32.6.2a	R	$2$	$2$	$-2$	$-2$	$2$	$0$	$0$	$0$	$0$	$0$	$0$
32.6.2b	R	$2$	$2$	$-2$	$2$	$-2$	$0$	$0$	$0$	$0$	$0$	$0$
32.6.4a	R	$4$	$-4$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$

Regular extensions

$f_{ 1 } =$

$3 x^{8} + \left(t^{2} - 5\right) x^{6} + \left(-3 t^{2} - 75\right) x^{4} + \left(-t^{2} + 125\right) x^{2} + 3 t^{2}$