Galois group: 16T48

• Label: 16T48
• Degree: $16$
• Order: $32$
• Cyclic: no
• Abelian: no
• Solvable: yes
• Primitive: no
• $p$-group: yes
• Group: $C_2\times \SD_{16}$

Group invariants

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

Group action invariants

Degree $n$:		$16$
Transitive number $t$:		$48$
Parity:		$1$
Primitive:		no
$\card{\Aut(F/K)}$:		$4$
Generators:		$(1,10)(2,9)(3,15)(4,16)(5,6)(7,11)(8,12)(13,14)$, $(1,16,14,12,9,7,6,3)(2,15,13,11,10,8,5,4)$, $(1,3,9,12)(2,4,10,11)(5,15,13,8)(6,16,14,7)$

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, $C_2^3$
$16$:	$QD_{16}$ x 2, $D_4\times C_2$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$ x 3

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

Degree 8: $QD_{16}$ x 2, $D_4\times C_2$

Low degree siblings

16T48, 32T27

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

Malle's constant $a(G)$:     $1/6$

Character table

		1A	2A	2B	2C	2D	2E	4A	4B	4C	4D	8A1	8A-1	8B1	8B-1
	Size	1	1	1	1	4	4	2	2	4	4	2	2	2	2
	2 P	1A	1A	1A	1A	1A	1A	2C	2C	2C	2C	4B	4B	4B	4B
	Type
32.40.1a	R	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
32.40.1b	R	$1$	$-1$	$-1$	$1$	$-1$	$1$	$-1$	$1$	$-1$	$1$	$1$	$1$	$-1$	$-1$
32.40.1c	R	$1$	$-1$	$-1$	$1$	$-1$	$1$	$-1$	$1$	$1$	$-1$	$-1$	$-1$	$1$	$1$
32.40.1d	R	$1$	$-1$	$-1$	$1$	$1$	$-1$	$-1$	$1$	$-1$	$1$	$-1$	$-1$	$1$	$1$
32.40.1e	R	$1$	$-1$	$-1$	$1$	$1$	$-1$	$-1$	$1$	$1$	$-1$	$1$	$1$	$-1$	$-1$
32.40.1f	R	$1$	$1$	$1$	$1$	$-1$	$-1$	$1$	$1$	$-1$	$-1$	$1$	$1$	$1$	$1$
32.40.1g	R	$1$	$1$	$1$	$1$	$-1$	$-1$	$1$	$1$	$1$	$1$	$-1$	$-1$	$-1$	$-1$
32.40.1h	R	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$-1$	$-1$	$-1$	$-1$	$-1$	$-1$
32.40.2a	R	$2$	$-2$	$-2$	$2$	$0$	$0$	$2$	$-2$	$0$	$0$	$0$	$0$	$0$	$0$
32.40.2b	R	$2$	$2$	$2$	$2$	$0$	$0$	$-2$	$-2$	$0$	$0$	$0$	$0$	$0$	$0$
32.40.2c1	C	$2$	$-2$	$2$	$-2$	$0$	$0$	$0$	$0$	$0$	$0$	$-{\zeta }_8-{\zeta }_8^3$	${\zeta }_8+{\zeta }_8^3$	${\zeta }_8+{\zeta }_8^3$	$-{\zeta }_8-{\zeta }_8^3$
32.40.2c2	C	$2$	$-2$	$2$	$-2$	$0$	$0$	$0$	$0$	$0$	$0$	${\zeta }_8+{\zeta }_8^3$	$-{\zeta }_8-{\zeta }_8^3$	$-{\zeta }_8-{\zeta }_8^3$	${\zeta }_8+{\zeta }_8^3$
32.40.2d1	C	$2$	$2$	$-2$	$-2$	$0$	$0$	$0$	$0$	$0$	$0$	$-{\zeta }_8-{\zeta }_8^3$	${\zeta }_8+{\zeta }_8^3$	$-{\zeta }_8-{\zeta }_8^3$	${\zeta }_8+{\zeta }_8^3$
32.40.2d2	C	$2$	$2$	$-2$	$-2$	$0$	$0$	$0$	$0$	$0$	$0$	${\zeta }_8+{\zeta }_8^3$	$-{\zeta }_8-{\zeta }_8^3$	${\zeta }_8+{\zeta }_8^3$	$-{\zeta }_8-{\zeta }_8^3$

Regular extensions

Data not computed