Galois group: 16T11

• Label: 16T11
• Degree: $16$
• Order: $16$
• Cyclic: no
• Abelian: no
• Solvable: yes
• Primitive: no
• $p$-group: yes
• Group: $Q_8 : C_2$

Group invariants

Abstract group:		$Q_8 : C_2$
Order:		$16=2^{4}$
Cyclic:		no
Abelian:		no
Solvable:		yes
Nilpotency class:		$2$

Group action invariants

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

Low degree resolvents

$\card{(G/N)}$	Galois groups for stem field(s)
$2$:	$C_2$ x 7
$4$:	$C_2^2$ x 7
$8$:	$C_2^3$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$ x 7

Degree 4: $C_2^2$ x 7

Degree 8: $C_2^3$, $Q_8:C_2$ x 3

Low degree siblings

8T11 x 3

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

Malle's constant $a(G)$:     $1/8$

Character table

		1A	2A	2B	2C	2D	4A1	4A-1	4B	4C	4D
	Size	1	1	2	2	2	1	1	2	2	2
	2 P	1A	1A	1A	1A	1A	2A	2A	2A	2A	2A
	Type
16.13.1a	R	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
16.13.1b	R	$1$	$1$	$-1$	$-1$	$-1$	$-1$	$-1$	$1$	$1$	$1$
16.13.1c	R	$1$	$1$	$-1$	$-1$	$1$	$1$	$1$	$-1$	$1$	$-1$
16.13.1d	R	$1$	$1$	$-1$	$1$	$-1$	$1$	$1$	$-1$	$-1$	$1$
16.13.1e	R	$1$	$1$	$-1$	$1$	$1$	$-1$	$-1$	$1$	$-1$	$-1$
16.13.1f	R	$1$	$1$	$1$	$-1$	$-1$	$1$	$1$	$1$	$-1$	$-1$
16.13.1g	R	$1$	$1$	$1$	$-1$	$1$	$-1$	$-1$	$-1$	$-1$	$1$
16.13.1h	R	$1$	$1$	$1$	$1$	$-1$	$-1$	$-1$	$-1$	$1$	$-1$
16.13.2a1	C	$2$	$-2$	$0$	$0$	$0$	$-2i$	$2i$	$0$	$0$	$0$
16.13.2a2	C	$2$	$-2$	$0$	$0$	$0$	$2i$	$-2i$	$0$	$0$	$0$

Regular extensions

Data not computed