Galois group: 16T137

• Label: 16T137
• Degree: $16$
• Order: $64$
• Cyclic: no
• Abelian: no
• Solvable: yes
• Primitive: no
• $p$-group: yes
• Group: $D_4.D_4$

Group invariants

Abstract group:		$D_4.D_4$
Order:		$64=2^{6}$
Cyclic:		no
Abelian:		no
Solvable:		yes
Nilpotency class:		$3$

Group action invariants

Degree $n$:		$16$
Transitive number $t$:		$137$
Parity:		$1$
Primitive:		no
$\card{\Aut(F/K)}$:		$8$
Generators:		$(1,6,2,5)(3,8,4,7)(9,16,10,15)(11,13,12,14)$, $(9,12,10,11)(13,15,14,16)$, $(1,16)(2,15)(3,9)(4,10)(5,11)(6,12)(7,13)(8,14)$

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 6, $C_2^3$
$16$:	$D_4\times C_2$ x 3
$32$:	$C_2^2 \wr C_2$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$ x 3

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

Degree 8: $D_4\times C_2$

Low degree siblings

16T137, 32T150, 32T344 x 2

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

Malle's constant $a(G)$:     $1/4$

Character table

		1A	2A	2B	2C	2D	4A	4B	4C	4D	4E	4F	4G	4H	4I	8A	8B
	Size	1	1	2	4	4	2	2	4	4	4	4	4	4	8	8	8
	2 P	1A	1A	1A	1A	1A	2A	2A	2B	2A	2B	2A	2A	2A	2A	4A	4B
	Type
64.137.1a	R	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
64.137.1b	R	$1$	$1$	$1$	$-1$	$-1$	$1$	$1$	$-1$	$1$	$-1$	$1$	$-1$	$-1$	$-1$	$1$	$1$
64.137.1c	R	$1$	$1$	$1$	$-1$	$-1$	$1$	$1$	$1$	$1$	$1$	$1$	$-1$	$-1$	$1$	$-1$	$-1$
64.137.1d	R	$1$	$1$	$1$	$-1$	$1$	$1$	$1$	$-1$	$-1$	$-1$	$-1$	$-1$	$1$	$1$	$1$	$-1$
64.137.1e	R	$1$	$1$	$1$	$-1$	$1$	$1$	$1$	$1$	$-1$	$1$	$-1$	$-1$	$1$	$-1$	$-1$	$1$
64.137.1f	R	$1$	$1$	$1$	$1$	$-1$	$1$	$1$	$-1$	$-1$	$-1$	$-1$	$1$	$-1$	$1$	$-1$	$1$
64.137.1g	R	$1$	$1$	$1$	$1$	$-1$	$1$	$1$	$1$	$-1$	$1$	$-1$	$1$	$-1$	$-1$	$1$	$-1$
64.137.1h	R	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$-1$	$1$	$-1$	$1$	$1$	$1$	$-1$	$-1$	$-1$
64.137.2a	R	$2$	$2$	$-2$	$-2$	$0$	$-2$	$2$	$0$	$0$	$0$	$0$	$2$	$0$	$0$	$0$	$0$
64.137.2b	R	$2$	$2$	$-2$	$0$	$-2$	$2$	$-2$	$0$	$0$	$0$	$0$	$0$	$2$	$0$	$0$	$0$
64.137.2c	R	$2$	$2$	$-2$	$0$	$2$	$2$	$-2$	$0$	$0$	$0$	$0$	$0$	$-2$	$0$	$0$	$0$
64.137.2d	R	$2$	$2$	$-2$	$2$	$0$	$-2$	$2$	$0$	$0$	$0$	$0$	$-2$	$0$	$0$	$0$	$0$
64.137.2e	R	$2$	$2$	$2$	$0$	$0$	$-2$	$-2$	$0$	$-2$	$0$	$2$	$0$	$0$	$0$	$0$	$0$
64.137.2f	R	$2$	$2$	$2$	$0$	$0$	$-2$	$-2$	$0$	$2$	$0$	$-2$	$0$	$0$	$0$	$0$	$0$
64.137.4a	S	$4$	$-4$	$0$	$0$	$0$	$0$	$0$	$-2$	$0$	$2$	$0$	$0$	$0$	$0$	$0$	$0$
64.137.4b	S	$4$	$-4$	$0$	$0$	$0$	$0$	$0$	$2$	$0$	$-2$	$0$	$0$	$0$	$0$	$0$	$0$

Regular extensions

Data not computed