Galois group: 16T38

• Label: 16T38
• 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 $t$:		$38$
Parity:		$1$
Primitive:		no
$\card{\Aut(F/K)}$:		$4$
Generators:		$(1,9)(2,10)(3,4)(5,13)(6,14)(7,8)(11,12)(15,16)$, $(1,4,6,8,10,11,13,15)(2,3,5,7,9,12,14,16)$, $(1,12)(2,11)(3,10)(4,9)(5,8)(6,7)(13,16)(14,15)$

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: $D_4\times C_2$, $Z_8 : Z_8^\times$ x 2

Low degree siblings

8T15 x 2, 16T35, 16T38, 16T45, 32T21

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

Malle's constant $a(G)$:     $1/6$

Character table

		1A	2A	2B	2C	2D	2E	4A	4B	4C	8A	8B
	Size	1	1	2	4	4	4	2	2	4	4	4
	2 P	1A	1A	1A	1A	1A	1A	2A	2A	2A	4B	4B
	Type
32.43.1a	R	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
32.43.1b	R	$1$	$1$	$-1$	$-1$	$-1$	$1$	$-1$	$1$	$1$	$1$	$-1$
32.43.1c	R	$1$	$1$	$-1$	$-1$	$1$	$1$	$-1$	$1$	$-1$	$-1$	$1$
32.43.1d	R	$1$	$1$	$-1$	$1$	$-1$	$-1$	$-1$	$1$	$1$	$-1$	$1$
32.43.1e	R	$1$	$1$	$-1$	$1$	$1$	$-1$	$-1$	$1$	$-1$	$1$	$-1$
32.43.1f	R	$1$	$1$	$1$	$-1$	$-1$	$-1$	$1$	$1$	$-1$	$1$	$1$
32.43.1g	R	$1$	$1$	$1$	$-1$	$1$	$-1$	$1$	$1$	$1$	$-1$	$-1$
32.43.1h	R	$1$	$1$	$1$	$1$	$-1$	$1$	$1$	$1$	$-1$	$-1$	$-1$
32.43.2a	R	$2$	$2$	$-2$	$0$	$0$	$0$	$2$	$-2$	$0$	$0$	$0$
32.43.2b	R	$2$	$2$	$2$	$0$	$0$	$0$	$-2$	$-2$	$0$	$0$	$0$
32.43.4a	R	$4$	$-4$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$

Regular extensions

Data not computed