Galois group: 16T14

• Label: 16T14
• Degree: $16$
• Order: $16$
• Cyclic: no
• Abelian: no
• Solvable: yes
• Primitive: no
• $p$-group: yes
• Group: $Q_{16}$

Group invariants

Abstract group:		$Q_{16}$
Order:		$16=2^{4}$
Cyclic:		no
Abelian:		no
Solvable:		yes
Nilpotency class:		$3$

Group action invariants

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

Low degree resolvents

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

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$ x 3

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

Degree 8: $D_4$

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	$2^{8}$	$1$	$2$	$8$	$( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)$
4A	$4^{4}$	$2$	$4$	$12$	$( 1, 5, 2, 6)( 3,15, 4,16)( 7,11, 8,12)( 9,13,10,14)$
4B	$4^{4}$	$4$	$4$	$12$	$( 1,13, 2,14)( 3,12, 4,11)( 5, 9, 6,10)( 7,15, 8,16)$
4C	$4^{4}$	$4$	$4$	$12$	$( 1, 8, 2, 7)( 3,14, 4,13)( 5,11, 6,12)( 9,15,10,16)$
8A1	$8^{2}$	$2$	$8$	$14$	$( 1, 4, 5,16, 2, 3, 6,15)( 7, 9,11,13, 8,10,12,14)$
8A3	$8^{2}$	$2$	$8$	$14$	$( 1, 3, 5,15, 2, 4, 6,16)( 7,10,11,14, 8, 9,12,13)$

Malle's constant $a(G)$:     $1/8$

Character table

		1A	2A	4A	4B	4C	8A1	8A3
	Size	1	1	2	4	4	2	2
	2 P	1A	1A	2A	2A	2A	4A	4A
	Type
16.9.1a	R	$1$	$1$	$1$	$1$	$1$	$1$	$1$
16.9.1b	R	$1$	$1$	$1$	$-1$	$-1$	$1$	$1$
16.9.1c	R	$1$	$1$	$1$	$-1$	$1$	$-1$	$-1$
16.9.1d	R	$1$	$1$	$1$	$1$	$-1$	$-1$	$-1$
16.9.2a	R	$2$	$2$	$-2$	$0$	$0$	$0$	$0$
16.9.2b1	S	$2$	$-2$	$0$	$0$	$0$	$-{\zeta }_8^{-1}-{\zeta }_8$	${\zeta }_8^{-1}+{\zeta }_8$
16.9.2b2	S	$2$	$-2$	$0$	$0$	$0$	${\zeta }_8^{-1}+{\zeta }_8$	$-{\zeta }_8^{-1}-{\zeta }_8$

Regular extensions

Data not computed