Galois group: 16T7

• Label: 16T7
• Degree: $16$
• Order: $16$
• Cyclic: no
• Abelian: no
• Solvable: yes
• Primitive: no
• $p$-group: yes
• Group: $Q_8\times C_2$

Group invariants

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

Group action invariants

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

Low degree resolvents

$\card{(G/N)}$	Galois groups for stem field(s)
$2$:	$C_2$ x 7
$4$:	$C_2^2$ x 7
$8$:	$C_2^3$, $Q_8$ x 2

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$ x 7

Degree 4: $C_2^2$ x 7

Degree 8: $C_2^3$, $Q_8$ x 2

Low degree siblings

There are no siblings 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	$2^{8}$	$1$	$2$	$8$	$( 1, 6)( 2, 5)( 3,15)( 4,16)( 7,12)( 8,11)( 9,14)(10,13)$
2B	$2^{8}$	$1$	$2$	$8$	$( 1, 5)( 2, 6)( 3,16)( 4,15)( 7,11)( 8,12)( 9,13)(10,14)$
2C	$2^{8}$	$1$	$2$	$8$	$( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)$
4A	$4^{4}$	$2$	$4$	$12$	$( 1,14, 2,13)( 3, 8, 4, 7)( 5,10, 6, 9)(11,16,12,15)$
4B	$4^{4}$	$2$	$4$	$12$	$( 1,15, 2,16)( 3, 5, 4, 6)( 7,10, 8, 9)(11,14,12,13)$
4C	$4^{4}$	$2$	$4$	$12$	$( 1,11, 2,12)( 3,10, 4, 9)( 5, 7, 6, 8)(13,16,14,15)$
4D	$4^{4}$	$2$	$4$	$12$	$( 1, 9, 2,10)( 3,11, 4,12)( 5,13, 6,14)( 7,15, 8,16)$
4E	$4^{4}$	$2$	$4$	$12$	$( 1, 3, 2, 4)( 5,16, 6,15)( 7,13, 8,14)( 9,12,10,11)$
4F	$4^{4}$	$2$	$4$	$12$	$( 1, 8, 2, 7)( 3,13, 4,14)( 5,12, 6,11)( 9,15,10,16)$

Malle's constant $a(G)$:     $1/8$

Character table

		1A	2A	2B	2C	4A	4B	4C	4D	4E	4F
	Size	1	1	1	1	2	2	2	2	2	2
	2 P	1A	1A	1A	1A	2C	2C	2C	2C	2C	2C
	Type
16.12.1a	R	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
16.12.1b	R	$1$	$1$	$-1$	$-1$	$-1$	$-1$	$-1$	$1$	$1$	$1$
16.12.1c	R	$1$	$1$	$-1$	$-1$	$-1$	$1$	$1$	$-1$	$1$	$-1$
16.12.1d	R	$1$	$1$	$-1$	$-1$	$1$	$-1$	$1$	$1$	$-1$	$-1$
16.12.1e	R	$1$	$1$	$-1$	$-1$	$1$	$1$	$-1$	$-1$	$-1$	$1$
16.12.1f	R	$1$	$1$	$1$	$1$	$-1$	$-1$	$1$	$-1$	$-1$	$1$
16.12.1g	R	$1$	$1$	$1$	$1$	$-1$	$1$	$-1$	$1$	$-1$	$-1$
16.12.1h	R	$1$	$1$	$1$	$1$	$1$	$-1$	$-1$	$-1$	$1$	$-1$
16.12.2a	S	$2$	$-2$	$-2$	$2$	$0$	$0$	$0$	$0$	$0$	$0$
16.12.2b	S	$2$	$-2$	$2$	$-2$	$0$	$0$	$0$	$0$	$0$	$0$

Regular extensions

Data not computed