Galois group: 16T47

• Label: 16T47
• Degree: $16$
• Order: $32$
• Cyclic: no
• Abelian: no
• Solvable: yes
• Primitive: no
• $p$-group: yes
• Group: $D_8:C_2$

Group invariants

Abstract group:		$D_8:C_2$
Order:		$32=2^{5}$
Cyclic:		no
Abelian:		no
Solvable:		yes
Nilpotency class:		$3$

Group action invariants

Degree $n$:		$16$
Transitive number $t$:		$47$
Parity:		$1$
Primitive:		no
$\card{\Aut(F/K)}$:		$8$
Generators:		$(1,13)(2,14)(3,12)(4,11)(5,9)(6,10)(7,15)(8,16)$, $(1,8,6,4,2,7,5,3)(9,16,14,12,10,15,13,11)$, $(1,10,2,9)(3,16,4,15)(5,13,6,14)(7,12,8,11)$

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: $C_2$ x 3

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

Degree 8: $D_4$

Low degree siblings

16T44 x 2, 32T26

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	$1^{16}$	$1$	$1$	$0$	$()$
2A	$2^{8}$	$1$	$2$	$8$	$( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)$
2B	$2^{4},1^{8}$	$2$	$2$	$4$	$(1,2)(3,4)(5,6)(7,8)$
2C	$2^{8}$	$4$	$2$	$8$	$( 1,10)( 2, 9)( 3,16)( 4,15)( 5,13)( 6,14)( 7,12)( 8,11)$
2D	$2^{8}$	$4$	$2$	$8$	$( 1,11)( 2,12)( 3,10)( 4, 9)( 5,16)( 6,15)( 7,13)( 8,14)$
4A1	$4^{4}$	$1$	$4$	$12$	$( 1, 5, 2, 6)( 3, 7, 4, 8)( 9,14,10,13)(11,16,12,15)$
4A-1	$4^{4}$	$1$	$4$	$12$	$( 1, 6, 2, 5)( 3, 8, 4, 7)( 9,13,10,14)(11,15,12,16)$
4B	$4^{4}$	$2$	$4$	$12$	$( 1, 6, 2, 5)( 3, 8, 4, 7)( 9,14,10,13)(11,16,12,15)$
4C	$4^{4}$	$4$	$4$	$12$	$( 1, 9, 2,10)( 3,15, 4,16)( 5,14, 6,13)( 7,11, 8,12)$
4D	$4^{4}$	$4$	$4$	$12$	$( 1,12, 2,11)( 3, 9, 4,10)( 5,15, 6,16)( 7,14, 8,13)$
8A1	$8^{2}$	$2$	$8$	$14$	$( 1, 7, 6, 3, 2, 8, 5, 4)( 9,15,14,11,10,16,13,12)$
8A-1	$8^{2}$	$2$	$8$	$14$	$( 1, 4, 5, 8, 2, 3, 6, 7)( 9,12,13,16,10,11,14,15)$
8B1	$8^{2}$	$2$	$8$	$14$	$( 1, 8, 6, 4, 2, 7, 5, 3)( 9,15,14,11,10,16,13,12)$
8B3	$8^{2}$	$2$	$8$	$14$	$( 1, 4, 5, 8, 2, 3, 6, 7)( 9,11,13,15,10,12,14,16)$

Malle's constant $a(G)$:     $1/4$

Character table

		1A	2A	2B	2C	2D	4A1	4A-1	4B	4C	4D	8A1	8A-1	8B1	8B3
	Size	1	1	2	4	4	1	1	2	4	4	2	2	2	2
	2 P	1A	1A	1A	1A	1A	2A	2A	2A	2A	2A	4B	4B	4B	4B
	Type
32.42.1a	R	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
32.42.1b	R	$1$	$1$	$-1$	$-1$	$-1$	$-1$	$-1$	$1$	$1$	$1$	$-1$	$1$	$-1$	$1$
32.42.1c	R	$1$	$1$	$-1$	$-1$	$1$	$-1$	$-1$	$1$	$1$	$-1$	$1$	$-1$	$1$	$-1$
32.42.1d	R	$1$	$1$	$-1$	$1$	$-1$	$-1$	$-1$	$1$	$-1$	$1$	$1$	$-1$	$1$	$-1$
32.42.1e	R	$1$	$1$	$-1$	$1$	$1$	$-1$	$-1$	$1$	$-1$	$-1$	$-1$	$1$	$-1$	$1$
32.42.1f	R	$1$	$1$	$1$	$-1$	$-1$	$1$	$1$	$1$	$-1$	$-1$	$1$	$1$	$1$	$1$
32.42.1g	R	$1$	$1$	$1$	$-1$	$1$	$1$	$1$	$1$	$-1$	$1$	$-1$	$-1$	$-1$	$-1$
32.42.1h	R	$1$	$1$	$1$	$1$	$-1$	$1$	$1$	$1$	$1$	$-1$	$-1$	$-1$	$-1$	$-1$
32.42.2a	R	$2$	$2$	$-2$	$0$	$0$	$2$	$2$	$-2$	$0$	$0$	$0$	$0$	$0$	$0$
32.42.2b	R	$2$	$2$	$2$	$0$	$0$	$-2$	$-2$	$-2$	$0$	$0$	$0$	$0$	$0$	$0$
32.42.2c1	C	$2$	$-2$	$0$	$0$	$0$	$-2{\zeta }_8^2$	$2{\zeta }_8^2$	$0$	$0$	$0$	$-{\zeta }_8-{\zeta }_8^3$	$-{\zeta }_8^{-1}-{\zeta }_8$	${\zeta }_8+{\zeta }_8^3$	${\zeta }_8^{-1}+{\zeta }_8$
32.42.2c2	C	$2$	$-2$	$0$	$0$	$0$	$2{\zeta }_8^2$	$-2{\zeta }_8^2$	$0$	$0$	$0$	${\zeta }_8+{\zeta }_8^3$	$-{\zeta }_8^{-1}-{\zeta }_8$	$-{\zeta }_8-{\zeta }_8^3$	${\zeta }_8^{-1}+{\zeta }_8$
32.42.2c3	C	$2$	$-2$	$0$	$0$	$0$	$-2{\zeta }_8^2$	$2{\zeta }_8^2$	$0$	$0$	$0$	${\zeta }_8+{\zeta }_8^3$	${\zeta }_8^{-1}+{\zeta }_8$	$-{\zeta }_8-{\zeta }_8^3$	$-{\zeta }_8^{-1}-{\zeta }_8$
32.42.2c4	C	$2$	$-2$	$0$	$0$	$0$	$2{\zeta }_8^2$	$-2{\zeta }_8^2$	$0$	$0$	$0$	$-{\zeta }_8-{\zeta }_8^3$	${\zeta }_8^{-1}+{\zeta }_8$	${\zeta }_8+{\zeta }_8^3$	$-{\zeta }_8^{-1}-{\zeta }_8$

Regular extensions

Data not computed