Galois group: 16T133

• Label: 16T133
• Degree: $16$
• Order: $64$
• Cyclic: no
• Abelian: no
• Solvable: yes
• Primitive: no
• $p$-group: yes
• Group: $C_8.D_4$

Group invariants

Abstract group:		$C_8.D_4$
Order:		$64=2^{6}$
Cyclic:		no
Abelian:		no
Solvable:		yes
Nilpotency class:		$3$

Group action invariants

Degree $n$:		$16$
Transitive number $t$:		$133$
Parity:		$1$
Primitive:		no
$\card{\Aut(F/K)}$:		$2$
Generators:		$(1,15,6,11,2,16,5,12)(3,13,8,9,4,14,7,10)$, $(1,10,5,13,2,9,6,14)(3,12,7,15,4,11,8,16)$, $(1,4,5,8,2,3,6,7)(9,11,14,16,10,12,13,15)$

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 4, $C_2^3$
$16$:	$D_4\times C_2$ x 2, $Q_8:C_2$
$32$:	16T34

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$ x 3

Degree 4: $C_2^2$

Degree 8: $Q_8:C_2$

Low degree siblings

16T133, 32T143 x 2, 32T144, 32T357

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	$2^{8}$	$4$	$2$	$8$	$( 1,16)( 2,15)( 3,10)( 4, 9)( 5,12)( 6,11)( 7,13)( 8,14)$
2D	$2^{8}$	$8$	$2$	$8$	$( 1, 7)( 2, 8)( 3, 5)( 4, 6)( 9,15)(10,16)(11,13)(12,14)$
2E	$2^{6},1^{4}$	$8$	$2$	$6$	$( 1, 5)( 2, 6)( 7, 8)( 9,13)(10,14)(11,12)$
4A	$4^{4}$	$2$	$4$	$12$	$( 1, 6, 2, 5)( 3, 8, 4, 7)( 9,13,10,14)(11,15,12,16)$
4B	$4^{4}$	$2$	$4$	$12$	$( 1, 5, 2, 6)( 3, 7, 4, 8)( 9,13,10,14)(11,15,12,16)$
4C	$4^{4}$	$4$	$4$	$12$	$( 1,11, 2,12)( 3,14, 4,13)( 5,16, 6,15)( 7,10, 8, 9)$
8A1	$8^{2}$	$2$	$8$	$14$	$( 1, 8, 6, 4, 2, 7, 5, 3)( 9,15,13,12,10,16,14,11)$
8A3	$8^{2}$	$2$	$8$	$14$	$( 1, 7, 6, 3, 2, 8, 5, 4)( 9,16,13,11,10,15,14,12)$
8B	$8^{2}$	$4$	$8$	$14$	$( 1, 3, 5, 7, 2, 4, 6, 8)( 9,12,14,15,10,11,13,16)$
8C1	$8^{2}$	$4$	$8$	$14$	$( 1,14, 6, 9, 2,13, 5,10)( 3,16, 8,11, 4,15, 7,12)$
8C-1	$8^{2}$	$4$	$8$	$14$	$( 1, 9, 5,14, 2,10, 6,13)( 3,11, 7,16, 4,12, 8,15)$
8D	$8^{2}$	$8$	$8$	$14$	$( 1,10, 5,14, 2, 9, 6,13)( 3,16, 7,11, 4,15, 8,12)$
8E	$8^{2}$	$8$	$8$	$14$	$( 1,16, 5,11, 2,15, 6,12)( 3,14, 7, 9, 4,13, 8,10)$

Malle's constant $a(G)$:     $1/4$

Character table

		1A	2A	2B	2C	2D	2E	4A	4B	4C	8A1	8A3	8B	8C1	8C-1	8D	8E
	Size	1	1	2	4	8	8	2	2	4	2	2	4	4	4	8	8
	2 P	1A	1A	1A	1A	1A	1A	2A	2A	2A	4A	4A	4A	4A	4A	4B	4B
	Type
64.153.1a	R	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
64.153.1b	R	$1$	$1$	$1$	$-1$	$-1$	$-1$	$1$	$1$	$-1$	$1$	$1$	$1$	$-1$	$-1$	$1$	$1$
64.153.1c	R	$1$	$1$	$1$	$-1$	$-1$	$1$	$1$	$1$	$-1$	$-1$	$-1$	$-1$	$1$	$1$	$1$	$-1$
64.153.1d	R	$1$	$1$	$1$	$-1$	$1$	$-1$	$1$	$1$	$-1$	$-1$	$-1$	$-1$	$1$	$1$	$-1$	$1$
64.153.1e	R	$1$	$1$	$1$	$-1$	$1$	$1$	$1$	$1$	$-1$	$1$	$1$	$1$	$-1$	$-1$	$-1$	$-1$
64.153.1f	R	$1$	$1$	$1$	$1$	$-1$	$-1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$-1$	$-1$
64.153.1g	R	$1$	$1$	$1$	$1$	$-1$	$1$	$1$	$1$	$1$	$-1$	$-1$	$-1$	$-1$	$-1$	$-1$	$1$
64.153.1h	R	$1$	$1$	$1$	$1$	$1$	$-1$	$1$	$1$	$1$	$-1$	$-1$	$-1$	$-1$	$-1$	$1$	$-1$
64.153.2a	R	$2$	$2$	$-2$	$-2$	$0$	$0$	$-2$	$2$	$2$	$0$	$0$	$0$	$0$	$0$	$0$	$0$
64.153.2b	R	$2$	$2$	$-2$	$0$	$0$	$0$	$2$	$-2$	$0$	$-2$	$-2$	$2$	$0$	$0$	$0$	$0$
64.153.2c	R	$2$	$2$	$-2$	$0$	$0$	$0$	$2$	$-2$	$0$	$2$	$2$	$-2$	$0$	$0$	$0$	$0$
64.153.2d	R	$2$	$2$	$-2$	$2$	$0$	$0$	$-2$	$2$	$-2$	$0$	$0$	$0$	$0$	$0$	$0$	$0$
64.153.2e1	C	$2$	$2$	$2$	$0$	$0$	$0$	$-2$	$-2$	$0$	$0$	$0$	$0$	$-2i$	$2i$	$0$	$0$
64.153.2e2	C	$2$	$2$	$2$	$0$	$0$	$0$	$-2$	$-2$	$0$	$0$	$0$	$0$	$2i$	$-2i$	$0$	$0$
64.153.4a1	R	$4$	$-4$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$-2{\zeta }_8^{-1}-2{\zeta }_8$	$2{\zeta }_8^{-1}+2{\zeta }_8$	$0$	$0$	$0$	$0$	$0$
64.153.4a2	R	$4$	$-4$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$2{\zeta }_8^{-1}+2{\zeta }_8$	$-2{\zeta }_8^{-1}-2{\zeta }_8$	$0$	$0$	$0$	$0$	$0$

Regular extensions

Data not computed