Galois group: 16T41

• Label: 16T41
• Degree: $16$
• Order: $32$
• Cyclic: no
• Abelian: no
• Solvable: yes
• Primitive: no
• $p$-group: yes
• Group: $\OD_{16}:C_2$

Group invariants

Abstract group:		$\OD_{16}:C_2$
Order:		$32=2^{5}$
Cyclic:		no
Abelian:		no
Solvable:		yes
Nilpotency class:		$3$

Group action invariants

Degree $n$:		$16$
Transitive number $t$:		$41$
Parity:		$1$
Primitive:		no
$\card{\Aut(F/K)}$:		$4$
Generators:		$(3,12)(4,11)(5,13)(6,14)(7,8)(15,16)$, $(1,3,6,7,9,11,14,15)(2,4,5,8,10,12,13,16)$

Low degree resolvents

$\card{(G/N)}$	Galois groups for stem field(s)
$2$:	$C_2$ x 3
$4$:	$C_4$ x 2, $C_2^2$
$8$:	$D_{4}$ x 2, $C_4\times C_2$
$16$:	$C_2^2:C_4$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 4: $C_4$, $D_{4}$ x 2

Degree 8: $C_2^2:C_4$, $(C_8:C_2):C_2$

Low degree siblings

8T16 x 2, 16T36, 16T41, 32T22

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

Malle's constant $a(G)$:     $1/6$

Character table

		1A	2A	2B	2C	2D	4A	4B	8A1	8A-1	8B1	8B-1
	Size	1	1	2	4	4	2	2	4	4	4	4
	2 P	1A	1A	1A	1A	1A	2A	2A	4A	4A	4B	4B
	Type
32.7.1a	R	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
32.7.1b	R	$1$	$1$	$1$	$-1$	$-1$	$1$	$1$	$-1$	$-1$	$1$	$1$
32.7.1c	R	$1$	$1$	$1$	$-1$	$-1$	$1$	$1$	$1$	$1$	$-1$	$-1$
32.7.1d	R	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$-1$	$-1$	$-1$	$-1$
32.7.1e1	C	$1$	$1$	$1$	$-1$	$1$	$-1$	$-1$	$-i$	$i$	$i$	$-i$
32.7.1e2	C	$1$	$1$	$1$	$-1$	$1$	$-1$	$-1$	$i$	$-i$	$-i$	$i$
32.7.1f1	C	$1$	$1$	$1$	$1$	$-1$	$-1$	$-1$	$-i$	$i$	$-i$	$i$
32.7.1f2	C	$1$	$1$	$1$	$1$	$-1$	$-1$	$-1$	$i$	$-i$	$i$	$-i$
32.7.2a	R	$2$	$2$	$-2$	$0$	$0$	$-2$	$2$	$0$	$0$	$0$	$0$
32.7.2b	R	$2$	$2$	$-2$	$0$	$0$	$2$	$-2$	$0$	$0$	$0$	$0$
32.7.4a	R	$4$	$-4$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$

Regular extensions

Data not computed