Galois group: 8T30

• Label: 8T30
• Degree: $8$
• Order: $64$
• Cyclic: no
• Abelian: no
• Solvable: yes
• Primitive: no
• pp-group: yes
• Group: $(((C_4 \times C_2): C_2):C_2):C_2$

Group invariants

Abstract group:		$(((C_4 \times C_2): C_2):C_2):C_2$
Order:		$64=2^{6}$
Cyclic:		no
Abelian:		no
Solvable:		yes
Nilpotency class:		$4$

Group action invariants

Degree $n$:		$8$
Transitive number $t$:		$30$
CHM label:		$1/2[2^{4}]cD(4)$
Parity:		$-1$
Primitive:		no
$\card{\Aut(F/K)}$:		$2$
Generators:		$(1,2,3,8)(4,5,6,7)$, $(2,6)(3,7)$, $(1,3)(4,8)(5,7)$

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, $C_4\times C_2$
$16$:	$C_2^2:C_4$
$32$:	$C_2^3 : C_4$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 4: $D_{4}$

Low degree siblings

8T30 x 3, 16T143 x 2, 16T167 x 2, 16T168 x 2, 16T169 x 2, 32T157 x 2, 32T177, 32T178

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	$2^{2},1^{4}$	$2$	$2$	$2$	$(2,6)(4,8)$
2C	$2^{4}$	$4$	$2$	$4$	$(1,3)(2,8)(4,6)(5,7)$
2D	$2^{2},1^{4}$	$4$	$2$	$2$	$(2,6)(3,7)$
2E	$2^{3},1^{2}$	$8$	$2$	$3$	$(1,3)(2,6)(5,7)$
4A	$4,1^{4}$	$4$	$4$	$3$	$(2,8,6,4)$
4B	$4^{2}$	$4$	$4$	$6$	$(1,7,5,3)(2,4,6,8)$
4C	$4,2^{2}$	$4$	$4$	$5$	$(1,5)(2,4,6,8)(3,7)$
4D1	$4^{2}$	$8$	$4$	$6$	$(1,2,3,8)(4,5,6,7)$
4D-1	$4^{2}$	$8$	$4$	$6$	$(1,8,3,2)(4,7,6,5)$
4E1	$4,2^{2}$	$8$	$4$	$5$	$(1,8)(2,3,6,7)(4,5)$
4E-1	$4,2^{2}$	$8$	$4$	$5$	$(1,6,5,2)(3,8)(4,7)$

Malle's constant $a(G)$:     $1/2$

Character table

		1A	2A	2B	2C	2D	2E	4A	4B	4C	4D1	4D-1	4E1	4E-1
	Size	1	1	2	4	4	8	4	4	4	8	8	8	8
	2 P	1A	1A	1A	1A	1A	1A	2B	2A	2B	2C	2C	2D	2D
	Type
64.34.1a	R	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
64.34.1b	R	$1$	$1$	$1$	$1$	$1$	$-1$	$-1$	$1$	$-1$	$-1$	$-1$	$1$	$1$
64.34.1c	R	$1$	$1$	$1$	$1$	$1$	$-1$	$-1$	$1$	$-1$	$1$	$1$	$-1$	$-1$
64.34.1d	R	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$-1$	$-1$	$-1$	$-1$
64.34.1e1	C	$1$	$1$	$1$	$-1$	$-1$	$-1$	$1$	$1$	$1$	$-i$	$i$	$-i$	$i$
64.34.1e2	C	$1$	$1$	$1$	$-1$	$-1$	$-1$	$1$	$1$	$1$	$i$	$-i$	$i$	$-i$
64.34.1f1	C	$1$	$1$	$1$	$-1$	$-1$	$1$	$-1$	$1$	$-1$	$-i$	$i$	$i$	$-i$
64.34.1f2	C	$1$	$1$	$1$	$-1$	$-1$	$1$	$-1$	$1$	$-1$	$i$	$-i$	$-i$	$i$
64.34.2a	R	$2$	$2$	$2$	$-2$	$2$	$0$	$0$	$-2$	$0$	$0$	$0$	$0$	$0$
64.34.2b	R	$2$	$2$	$2$	$2$	$-2$	$0$	$0$	$-2$	$0$	$0$	$0$	$0$	$0$
64.34.4a	R	$4$	$4$	$-4$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$
64.34.4b	R	$4$	$-4$	$0$	$0$	$0$	$0$	$-2$	$0$	$2$	$0$	$0$	$0$	$0$
64.34.4c	R	$4$	$-4$	$0$	$0$	$0$	$0$	$2$	$0$	$-2$	$0$	$0$	$0$	$0$

Regular extensions

$f_{ 1 } =$

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