Galois group: 16T132

• Label: 16T132
• 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 $t$:		$132$
Parity:		$1$
Primitive:		no
$\card{\Aut(F/K)}$:		$2$
Generators:		$(1,15,5,12,2,16,6,11)(3,13,7,10,4,14,8,9)$, $(1,6)(2,5)(3,4)(9,14)(10,13)(15,16)$, $(1,4,2,3)(5,7,6,8)(9,11,10,12)(13,16,14,15)$

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 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

32T141, 32T142, 32T358

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

Malle's constant $a(G)$:     $1/4$

Character table

		1A	2A	2B	2C	2D	4A	4B	4C	4D	8A1	8A-1	8B	8C1	8C-1	8D	8E
	Size	1	1	2	4	8	2	2	4	8	2	2	4	4	4	8	8
	2 P	1A	1A	1A	1A	1A	2A	2A	2A	2A	4A	4A	4A	4A	4A	4B	4B
	Type
64.152.1a	R	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
64.152.1b	R	$1$	$1$	$1$	$-1$	$-1$	$1$	$1$	$-1$	$-1$	$1$	$1$	$1$	$-1$	$-1$	$1$	$1$
64.152.1c	R	$1$	$1$	$1$	$-1$	$-1$	$1$	$1$	$-1$	$1$	$-1$	$-1$	$-1$	$1$	$1$	$1$	$-1$
64.152.1d	R	$1$	$1$	$1$	$-1$	$1$	$1$	$1$	$-1$	$-1$	$-1$	$-1$	$-1$	$1$	$1$	$-1$	$1$
64.152.1e	R	$1$	$1$	$1$	$-1$	$1$	$1$	$1$	$-1$	$1$	$1$	$1$	$1$	$-1$	$-1$	$-1$	$-1$
64.152.1f	R	$1$	$1$	$1$	$1$	$-1$	$1$	$1$	$1$	$-1$	$1$	$1$	$1$	$1$	$1$	$-1$	$-1$
64.152.1g	R	$1$	$1$	$1$	$1$	$-1$	$1$	$1$	$1$	$1$	$-1$	$-1$	$-1$	$-1$	$-1$	$-1$	$1$
64.152.1h	R	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$-1$	$-1$	$-1$	$-1$	$-1$	$-1$	$1$	$-1$
64.152.2a	R	$2$	$2$	$-2$	$-2$	$0$	$-2$	$2$	$2$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$
64.152.2b	R	$2$	$2$	$-2$	$0$	$0$	$2$	$-2$	$0$	$0$	$-2$	$-2$	$2$	$0$	$0$	$0$	$0$
64.152.2c	R	$2$	$2$	$-2$	$0$	$0$	$2$	$-2$	$0$	$0$	$2$	$2$	$-2$	$0$	$0$	$0$	$0$
64.152.2d	R	$2$	$2$	$-2$	$2$	$0$	$-2$	$2$	$-2$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$
64.152.2e1	C	$2$	$2$	$2$	$0$	$0$	$-2$	$-2$	$0$	$0$	$0$	$0$	$0$	$-2i$	$2i$	$0$	$0$
64.152.2e2	C	$2$	$2$	$2$	$0$	$0$	$-2$	$-2$	$0$	$0$	$0$	$0$	$0$	$2i$	$-2i$	$0$	$0$
64.152.4a1	C	$4$	$-4$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$-2{\zeta }_8-2{\zeta }_8^3$	$2{\zeta }_8+2{\zeta }_8^3$	$0$	$0$	$0$	$0$	$0$
64.152.4a2	C	$4$	$-4$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$2{\zeta }_8+2{\zeta }_8^3$	$-2{\zeta }_8-2{\zeta }_8^3$	$0$	$0$	$0$	$0$	$0$

Regular extensions

Data not computed