Galois group: 16T153

• Label: 16T153
• Degree: $16$
• Order: $64$
• Cyclic: no
• Abelian: no
• Solvable: yes
• Primitive: no
• $p$-group: yes
• Group: $C_2^3.D_4$

Group invariants

Abstract group:		$C_2^3.D_4$
Order:		$64=2^{6}$
Cyclic:		no
Abelian:		no
Solvable:		yes
Nilpotency class:		$4$

Group action invariants

Degree $n$:		$16$
Transitive number $t$:		$153$
Parity:		$1$
Primitive:		no
$\card{\Aut(F/K)}$:		$4$
Generators:		$(1,16,7,13)(2,15,8,14)(3,9,6,12)(4,10,5,11)$, $(5,6)(7,8)(9,13,10,14)(11,15,12,16)$

Low degree resolvents

$\card{(G/N)}$	Galois groups for stem field(s)
$2$:	$C_2$ x 3
$4$:	$C_4$ x 2, $C_2^2$
$8$:	$D_{4}$ x 2, $C_4\times C_2$
$16$:	$C_2^2:C_4$
$32$:	$C_2^3 : C_4 $

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 4: $C_4$, $D_{4}$ x 2

Degree 8: $C_2^2:C_4$

Low degree siblings

16T140, 16T148, 16T160, 32T153, 32T154, 32T164

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	$2^{4},1^{8}$	$2$	$2$	$4$	$( 9,10)(11,12)(13,14)(15,16)$
2C	$2^{8}$	$4$	$2$	$8$	$( 1, 5)( 2, 6)( 3, 7)( 4, 8)( 9,13)(10,14)(11,16)(12,15)$
2D	$2^{8}$	$4$	$2$	$8$	$( 1, 7)( 2, 8)( 3, 6)( 4, 5)( 9,12)(10,11)(13,16)(14,15)$
4A	$4^{4}$	$4$	$4$	$12$	$( 1, 4, 2, 3)( 5, 7, 6, 8)( 9,15,10,16)(11,14,12,13)$
4B1	$4^{4}$	$4$	$4$	$12$	$( 1, 8, 2, 7)( 3, 5, 4, 6)( 9,16,10,15)(11,14,12,13)$
4B-1	$4^{4}$	$4$	$4$	$12$	$( 1, 8, 2, 7)( 3, 5, 4, 6)( 9,15,10,16)(11,13,12,14)$
4C	$4^{2},2^{2},1^{4}$	$8$	$4$	$8$	$( 5, 6)( 7, 8)( 9,13,10,14)(11,15,12,16)$
4D1	$4^{4}$	$8$	$4$	$12$	$( 1,16, 7,13)( 2,15, 8,14)( 3, 9, 6,12)( 4,10, 5,11)$
4D-1	$4^{4}$	$8$	$4$	$12$	$( 1,13, 7,16)( 2,14, 8,15)( 3,12, 6, 9)( 4,11, 5,10)$
8A1	$8^{2}$	$8$	$8$	$14$	$( 1,11, 4,14, 2,12, 3,13)( 5,15, 7,10, 6,16, 8, 9)$
8A-1	$8^{2}$	$8$	$8$	$14$	$( 1,10, 4,15, 2, 9, 3,16)( 5,14, 7,11, 6,13, 8,12)$

Malle's constant $a(G)$:     $1/4$

Character table

		1A	2A	2B	2C	2D	4A	4B1	4B-1	4C	4D1	4D-1	8A1	8A-1
	Size	1	1	2	4	4	4	4	4	8	8	8	8	8
	2 P	1A	1A	1A	1A	1A	2A	2A	2A	2B	2D	2D	4A	4A
	Type
64.33.1a	R	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
64.33.1b	R	$1$	$1$	$1$	$1$	$1$	$1$	$-1$	$-1$	$-1$	$-1$	$-1$	$1$	$1$
64.33.1c	R	$1$	$1$	$1$	$1$	$1$	$1$	$-1$	$-1$	$-1$	$1$	$1$	$-1$	$-1$
64.33.1d	R	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$-1$	$-1$	$-1$	$-1$
64.33.1e1	C	$1$	$1$	$1$	$1$	$-1$	$-1$	$-1$	$-1$	$1$	$-i$	$i$	$-i$	$i$
64.33.1e2	C	$1$	$1$	$1$	$1$	$-1$	$-1$	$-1$	$-1$	$1$	$i$	$-i$	$i$	$-i$
64.33.1f1	C	$1$	$1$	$1$	$1$	$-1$	$-1$	$1$	$1$	$-1$	$-i$	$i$	$i$	$-i$
64.33.1f2	C	$1$	$1$	$1$	$1$	$-1$	$-1$	$1$	$1$	$-1$	$i$	$-i$	$-i$	$i$
64.33.2a	R	$2$	$2$	$2$	$-2$	$-2$	$2$	$0$	$0$	$0$	$0$	$0$	$0$	$0$
64.33.2b	R	$2$	$2$	$2$	$-2$	$2$	$-2$	$0$	$0$	$0$	$0$	$0$	$0$	$0$
64.33.4a	R	$4$	$4$	$-4$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$
64.33.4b1	C	$4$	$-4$	$0$	$0$	$0$	$0$	$-2i$	$2i$	$0$	$0$	$0$	$0$	$0$
64.33.4b2	C	$4$	$-4$	$0$	$0$	$0$	$0$	$2i$	$-2i$	$0$	$0$	$0$	$0$	$0$

Regular extensions

Data not computed