Galois group: 16T36

• Label: 16T36
• 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 $t$:		$36$
Parity:		$1$
Primitive:		no
$\card{\Aut(F/K)}$:		$8$
Generators:		$(1,3,6,8,10,12,13,15)(2,4,5,7,9,11,14,16)$, $(1,2)(3,11)(4,12)(5,13)(6,14)(7,8)(9,10)(15,16)$

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: $C_2$ x 3

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

Degree 8: $C_4\times C_2$, $(C_8:C_2):C_2$ x 2

Low degree siblings

8T16 x 2, 16T41 x 2, 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	$2^{8}$	$1$	$2$	$8$	$( 1,10)( 2, 9)( 3,12)( 4,11)( 5,14)( 6,13)( 7,16)( 8,15)$
2B	$2^{4},1^{8}$	$2$	$2$	$4$	$( 1,10)( 2, 9)( 5,14)( 6,13)$
2C	$2^{8}$	$4$	$2$	$8$	$( 1, 2)( 3,11)( 4,12)( 5,13)( 6,14)( 7, 8)( 9,10)(15,16)$
2D	$2^{8}$	$4$	$2$	$8$	$( 1, 5)( 2, 6)( 3,16)( 4,15)( 7,12)( 8,11)( 9,13)(10,14)$
4A	$4^{4}$	$2$	$4$	$12$	$( 1, 6,10,13)( 2, 5, 9,14)( 3, 8,12,15)( 4, 7,11,16)$
4B	$4^{4}$	$2$	$4$	$12$	$( 1,13,10, 6)( 2,14, 9, 5)( 3, 8,12,15)( 4, 7,11,16)$
8A1	$8^{2}$	$4$	$8$	$14$	$( 1, 3, 6, 8,10,12,13,15)( 2, 4, 5, 7, 9,11,14,16)$
8A-1	$8^{2}$	$4$	$8$	$14$	$( 1, 8,13, 3,10,15, 6,12)( 2, 7,14, 4, 9,16, 5,11)$
8B1	$8^{2}$	$4$	$8$	$14$	$( 1, 4,13, 7,10,11, 6,16)( 2, 3,14, 8, 9,12, 5,15)$
8B-1	$8^{2}$	$4$	$8$	$14$	$( 1, 7,13,11,10,16, 6, 4)( 2, 8,14,12, 9,15, 5, 3)$

Malle's constant $a(G)$:     $1/4$

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