Galois group: 16T1076

• Label: 16T1076
• Degree: $16$
• Order: $896$
• Cyclic: no
• Abelian: no
• Solvable: yes
• Primitive: no
• $p$-group: no
• Group: $C_2^4.F_8$

Group invariants

Abstract group:		$C_2^4.F_8$
Order:		$896=2^{7} \cdot 7$
Cyclic:		no
Abelian:		no
Solvable:		yes
Nilpotency class:		not nilpotent

Group action invariants

Degree $n$:		$16$
Transitive number $t$:		$1076$
Parity:		$1$
Primitive:		no
$\card{\Aut(F/K)}$:		$2$
Generators:		$(1,3,10,14,6,12,8,2,4,9,13,5,11,7)(15,16)$, $(1,16,12,9,6,13,8,2,15,11,10,5,14,7)(3,4)$

Low degree resolvents

$\card{(G/N)}$	Galois groups for stem field(s)
$7$:	$C_7$
$56$:	$C_2^3:C_7$
$448$:	56T?

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 4: None

Degree 8: $C_2^3:C_7$

Low degree siblings

There are no siblings 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	$1^{16}$	$1$	$1$	$0$	$()$
2A	$2^{8}$	$1$	$2$	$8$	$( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)$
2B	$2^{4},1^{8}$	$14$	$2$	$4$	$( 1, 2)( 5, 6)(11,12)(15,16)$
4A1	$4^{2},2^{4}$	$56$	$4$	$10$	$( 1,11, 2,12)( 3,10)( 4, 9)( 5,16, 6,15)( 7,13)( 8,14)$
4A-1	$4^{2},2^{4}$	$56$	$4$	$10$	$( 1,12, 2,11)( 3,10)( 4, 9)( 5,15, 6,16)( 7,13)( 8,14)$
7A1	$7^{2},1^{2}$	$64$	$7$	$12$	$( 1,11, 4,16,13, 9, 7)( 2,12, 3,15,14,10, 8)$
7A-1	$7^{2},1^{2}$	$64$	$7$	$12$	$( 1, 7, 9,13,16, 4,11)( 2, 8,10,14,15, 3,12)$
7A2	$7^{2},1^{2}$	$64$	$7$	$12$	$( 1, 4,13, 7,11,16, 9)( 2, 3,14, 8,12,15,10)$
7A-2	$7^{2},1^{2}$	$64$	$7$	$12$	$( 1, 9,16,11, 7,13, 4)( 2,10,15,12, 8,14, 3)$
7A3	$7^{2},1^{2}$	$64$	$7$	$12$	$( 1,16, 7, 4, 9,11,13)( 2,15, 8, 3,10,12,14)$
7A-3	$7^{2},1^{2}$	$64$	$7$	$12$	$( 1,13,11, 9, 4, 7,16)( 2,14,12,10, 3, 8,15)$
14A1	$14,2$	$64$	$14$	$14$	$( 1,14,11,10, 4, 8,16, 2,13,12, 9, 3, 7,15)( 5, 6)$
14A-1	$14,2$	$64$	$14$	$14$	$( 1,15, 7, 3, 9,12,13, 2,16, 8, 4,10,11,14)( 5, 6)$
14A3	$14,2$	$64$	$14$	$14$	$( 1,10,16,12, 7,14, 4, 2, 9,15,11, 8,13, 3)( 5, 6)$
14A-3	$14,2$	$64$	$14$	$14$	$( 1, 3,13, 8,11,15, 9, 2, 4,14, 7,12,16,10)( 5, 6)$
14A5	$14,2$	$64$	$14$	$14$	$( 1, 8, 9,14,16, 3,11, 2, 7,10,13,15, 4,12)( 5, 6)$
14A-5	$14,2$	$64$	$14$	$14$	$( 1,12, 4,15,13,10, 7, 2,11, 3,16,14, 9, 8)( 5, 6)$

Malle's constant $a(G)$:     $1/4$

Character table

		1A	2A	2B	4A1	4A-1	7A1	7A-1	7A2	7A-2	7A3	7A-3	14A1	14A-1	14A3	14A-3	14A5	14A-5
	Size	1	1	14	56	56	64	64	64	64	64	64	64	64	64	64	64	64
	2 P	1A	1A	1A	2B	2B	7A2	7A-2	7A-3	7A3	7A-1	7A1	7A1	7A-1	7A3	7A-3	7A-2	7A2
	7 P	1A	2A	2B	4A-1	4A1	7A3	7A-3	7A-1	7A1	7A2	7A-2	14A3	14A-3	14A-5	14A5	14A1	14A-1
	Type
896.564.1a	R	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
896.564.1b1	C	$1$	$1$	$1$	$1$	$1$	${\zeta }_7^{-3}$	${\zeta }_7$	${\zeta }_7^{-2}$	${\zeta }_7^{-1}$	${\zeta }_7^2$	${\zeta }_7^3$	${\zeta }_7^{-2}$	${\zeta }_7^3$	${\zeta }_7^{-3}$	${\zeta }_7^{-1}$	${\zeta }_7$	${\zeta }_7^2$
896.564.1b2	C	$1$	$1$	$1$	$1$	$1$	${\zeta }_7^3$	${\zeta }_7^{-1}$	${\zeta }_7^2$	${\zeta }_7$	${\zeta }_7^{-2}$	${\zeta }_7^{-3}$	${\zeta }_7^2$	${\zeta }_7^{-3}$	${\zeta }_7^3$	${\zeta }_7$	${\zeta }_7^{-1}$	${\zeta }_7^{-2}$
896.564.1b3	C	$1$	$1$	$1$	$1$	$1$	${\zeta }_7^{-2}$	${\zeta }_7^3$	${\zeta }_7$	${\zeta }_7^{-3}$	${\zeta }_7^{-1}$	${\zeta }_7^2$	${\zeta }_7$	${\zeta }_7^2$	${\zeta }_7^{-2}$	${\zeta }_7^{-3}$	${\zeta }_7^3$	${\zeta }_7^{-1}$
896.564.1b4	C	$1$	$1$	$1$	$1$	$1$	${\zeta }_7^2$	${\zeta }_7^{-3}$	${\zeta }_7^{-1}$	${\zeta }_7^3$	${\zeta }_7$	${\zeta }_7^{-2}$	${\zeta }_7^{-1}$	${\zeta }_7^{-2}$	${\zeta }_7^2$	${\zeta }_7^3$	${\zeta }_7^{-3}$	${\zeta }_7$
896.564.1b5	C	$1$	$1$	$1$	$1$	$1$	${\zeta }_7^{-1}$	${\zeta }_7^{-2}$	${\zeta }_7^{-3}$	${\zeta }_7^2$	${\zeta }_7^3$	${\zeta }_7$	${\zeta }_7^{-3}$	${\zeta }_7$	${\zeta }_7^{-1}$	${\zeta }_7^2$	${\zeta }_7^{-2}$	${\zeta }_7^3$
896.564.1b6	C	$1$	$1$	$1$	$1$	$1$	${\zeta }_7$	${\zeta }_7^2$	${\zeta }_7^3$	${\zeta }_7^{-2}$	${\zeta }_7^{-3}$	${\zeta }_7^{-1}$	${\zeta }_7^3$	${\zeta }_7^{-1}$	${\zeta }_7$	${\zeta }_7^{-2}$	${\zeta }_7^2$	${\zeta }_7^{-3}$
896.564.7a	R	$7$	$7$	$7$	$-1$	$-1$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$
896.564.8a	R	$8$	$-8$	$0$	$0$	$0$	$1$	$1$	$1$	$1$	$1$	$1$	$-1$	$-1$	$-1$	$-1$	$-1$	$-1$
896.564.8b1	C	$8$	$-8$	$0$	$0$	$0$	${\zeta }_7^{-3}$	${\zeta }_7$	${\zeta }_7^{-2}$	${\zeta }_7^{-1}$	${\zeta }_7^2$	${\zeta }_7^3$	$-{\zeta }_7^{-2}$	$-{\zeta }_7^3$	$-{\zeta }_7^{-3}$	$-{\zeta }_7^{-1}$	$-{\zeta }_7$	$-{\zeta }_7^2$
896.564.8b2	C	$8$	$-8$	$0$	$0$	$0$	${\zeta }_7^3$	${\zeta }_7^{-1}$	${\zeta }_7^2$	${\zeta }_7$	${\zeta }_7^{-2}$	${\zeta }_7^{-3}$	$-{\zeta }_7^2$	$-{\zeta }_7^{-3}$	$-{\zeta }_7^3$	$-{\zeta }_7$	$-{\zeta }_7^{-1}$	$-{\zeta }_7^{-2}$
896.564.8b3	C	$8$	$-8$	$0$	$0$	$0$	${\zeta }_7^{-2}$	${\zeta }_7^3$	${\zeta }_7$	${\zeta }_7^{-3}$	${\zeta }_7^{-1}$	${\zeta }_7^2$	$-{\zeta }_7$	$-{\zeta }_7^2$	$-{\zeta }_7^{-2}$	$-{\zeta }_7^{-3}$	$-{\zeta }_7^3$	$-{\zeta }_7^{-1}$
896.564.8b4	C	$8$	$-8$	$0$	$0$	$0$	${\zeta }_7^2$	${\zeta }_7^{-3}$	${\zeta }_7^{-1}$	${\zeta }_7^3$	${\zeta }_7$	${\zeta }_7^{-2}$	$-{\zeta }_7^{-1}$	$-{\zeta }_7^{-2}$	$-{\zeta }_7^2$	$-{\zeta }_7^3$	$-{\zeta }_7^{-3}$	$-{\zeta }_7$
896.564.8b5	C	$8$	$-8$	$0$	$0$	$0$	${\zeta }_7^{-1}$	${\zeta }_7^{-2}$	${\zeta }_7^{-3}$	${\zeta }_7^2$	${\zeta }_7^3$	${\zeta }_7$	$-{\zeta }_7^{-3}$	$-{\zeta }_7$	$-{\zeta }_7^{-1}$	$-{\zeta }_7^2$	$-{\zeta }_7^{-2}$	$-{\zeta }_7^3$
896.564.8b6	C	$8$	$-8$	$0$	$0$	$0$	${\zeta }_7$	${\zeta }_7^2$	${\zeta }_7^3$	${\zeta }_7^{-2}$	${\zeta }_7^{-3}$	${\zeta }_7^{-1}$	$-{\zeta }_7^3$	$-{\zeta }_7^{-1}$	$-{\zeta }_7$	$-{\zeta }_7^{-2}$	$-{\zeta }_7^2$	$-{\zeta }_7^{-3}$
896.564.14a1	C	$14$	$14$	$-2$	$-2i$	$2i$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$
896.564.14a2	C	$14$	$14$	$-2$	$2i$	$-2i$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$

Regular extensions

Data not computed