Galois group: 16T31

• Label: 16T31
• Degree: $16$
• Order: $32$
• Cyclic: no
• Abelian: no
• Solvable: yes
• Primitive: no
• $p$-group: yes
• Group: $C_2^2:Q_8$

Group invariants

Abstract group:		$C_2^2:Q_8$
Order:		$32=2^{5}$
Cyclic:		no
Abelian:		no
Solvable:		yes
Nilpotency class:		$2$

Group action invariants

Degree $n$:		$16$
Transitive number $t$:		$31$
Parity:		$1$
Primitive:		no
$\card{\Aut(F/K)}$:		$8$
Generators:		$(1,8,16,10)(2,7,15,9)(3,14,5,12)(4,13,6,11)$, $(1,6,2,5)(3,16,4,15)(7,11,8,12)(9,13,10,14)$, $(1,2)(3,4)(5,6)(7,9)(8,10)(11,13)(12,14)(15,16)$

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

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$ x 3

Degree 4: $C_2^2$, $D_{4}$ x 2

Degree 8: $Q_8$, $D_4\times C_2$, $Q_8:C_2$

Low degree siblings

16T31, 32T17

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

Malle's constant $a(G)$:     $1/4$

Character table

		1A	2A	2B	2C	2D	2E	4A	4B	4C1	4C-1	4D	4E	4F	4G
	Size	1	1	1	1	2	2	2	2	2	2	4	4	4	4
	2 P	1A	1A	1A	1A	1A	1A	2B	2B	2B	2B	2C	2B	2B	2C
	Type
32.29.1a	R	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
32.29.1b	R	$1$	$1$	$1$	$1$	$-1$	$-1$	$-1$	$-1$	$1$	$1$	$-1$	$1$	$1$	$-1$
32.29.1c	R	$1$	$1$	$1$	$1$	$-1$	$-1$	$-1$	$-1$	$1$	$1$	$1$	$-1$	$-1$	$1$
32.29.1d	R	$1$	$1$	$1$	$1$	$-1$	$-1$	$1$	$1$	$-1$	$-1$	$-1$	$-1$	$1$	$1$
32.29.1e	R	$1$	$1$	$1$	$1$	$-1$	$-1$	$1$	$1$	$-1$	$-1$	$1$	$1$	$-1$	$-1$
32.29.1f	R	$1$	$1$	$1$	$1$	$1$	$1$	$-1$	$-1$	$-1$	$-1$	$-1$	$1$	$-1$	$1$
32.29.1g	R	$1$	$1$	$1$	$1$	$1$	$1$	$-1$	$-1$	$-1$	$-1$	$1$	$-1$	$1$	$-1$
32.29.1h	R	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$-1$	$-1$	$-1$	$-1$
32.29.2a	R	$2$	$-2$	$-2$	$2$	$0$	$0$	$0$	$0$	$-2$	$2$	$0$	$0$	$0$	$0$
32.29.2b	R	$2$	$-2$	$-2$	$2$	$0$	$0$	$0$	$0$	$2$	$-2$	$0$	$0$	$0$	$0$
32.29.2c	S	$2$	$-2$	$2$	$-2$	$-2$	$2$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$
32.29.2d	S	$2$	$-2$	$2$	$-2$	$2$	$-2$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$
32.29.2e1	C	$2$	$2$	$-2$	$-2$	$0$	$0$	$-2i$	$2i$	$0$	$0$	$0$	$0$	$0$	$0$
32.29.2e2	C	$2$	$2$	$-2$	$-2$	$0$	$0$	$2i$	$-2i$	$0$	$0$	$0$	$0$	$0$	$0$

Regular extensions

Data not computed