Galois group: 16T6

• Label: 16T6
• Degree: $16$
• Order: $16$
• Cyclic: no
• Abelian: no
• Solvable: yes
• Primitive: no
• $p$-group: yes
• Group: $C_8: C_2$

Group invariants

Abstract group:		$C_8: C_2$
Order:		$16=2^{4}$
Cyclic:		no
Abelian:		no
Solvable:		yes
Nilpotency class:		$2$

Group action invariants

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

Low degree resolvents

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

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$ x 3

Degree 4: $C_4$ x 2, $C_2^2$

Degree 8: $C_4\times C_2$, $C_8:C_2$

Low degree siblings

8T7

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

Malle's constant $a(G)$:     $1/8$

Character table

		1A	2A	2B	4A1	4A-1	4B	8A1	8A-1	8B1	8B-1
	Size	1	1	2	1	1	2	2	2	2	2
	2 P	1A	1A	1A	2A	2A	2A	4A1	4A1	4A-1	4A-1
	Type
16.6.1a	R	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
16.6.1b	R	$1$	$1$	$-1$	$1$	$1$	$-1$	$-1$	$-1$	$1$	$1$
16.6.1c	R	$1$	$1$	$-1$	$1$	$1$	$-1$	$1$	$1$	$-1$	$-1$
16.6.1d	R	$1$	$1$	$1$	$1$	$1$	$1$	$-1$	$-1$	$-1$	$-1$
16.6.1e1	C	$1$	$1$	$-1$	$-1$	$-1$	$1$	$-i$	$i$	$-i$	$i$
16.6.1e2	C	$1$	$1$	$-1$	$-1$	$-1$	$1$	$i$	$-i$	$i$	$-i$
16.6.1f1	C	$1$	$1$	$1$	$-1$	$-1$	$-1$	$-i$	$i$	$i$	$-i$
16.6.1f2	C	$1$	$1$	$1$	$-1$	$-1$	$-1$	$i$	$-i$	$-i$	$i$
16.6.2a1	C	$2$	$-2$	$0$	$-2i$	$2i$	$0$	$0$	$0$	$0$	$0$
16.6.2a2	C	$2$	$-2$	$0$	$2i$	$-2i$	$0$	$0$	$0$	$0$	$0$

Regular extensions

Data not computed