Galois group: 16T49

• Label: 16T49
• Degree: $16$
• Order: $32$
• Cyclic: no
• Abelian: no
• Solvable: yes
• Primitive: no
• $p$-group: yes
• Group: $C_8.C_4$

Group invariants

Abstract group:		$C_8.C_4$
Order:		$32=2^{5}$
Cyclic:		no
Abelian:		no
Solvable:		yes
Nilpotency class:		$3$

Group action invariants

Degree $n$:		$16$
Transitive number $t$:		$49$
Parity:		$1$
Primitive:		no
$\card{\Aut(F/K)}$:		$8$
Generators:		$(1,4,6,8,2,3,5,7)(9,12,14,16,10,11,13,15)$, $(1,12,5,16,2,11,6,15)(3,10,8,13,4,9,7,14)$

Low degree resolvents

$\card{(G/N)}$	Galois groups for stem field(s)
$2$:	$C_2$ x 3
$4$:	$C_4$ x 2, $C_2^2$
$8$:	$D_{4}$, $C_4\times C_2$, $Q_8$
$16$:	$C_4:C_4$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$ x 3

Degree 4: $C_2^2$

Degree 8: $Q_8$

Low degree siblings

32T28

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	$2^{8}$	$1$	$2$	$8$	$( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)$
2B	$2^{4},1^{8}$	$2$	$2$	$4$	$( 9,10)(11,12)(13,14)(15,16)$
4A1	$4^{4}$	$1$	$4$	$12$	$( 1, 5, 2, 6)( 3, 8, 4, 7)( 9,14,10,13)(11,15,12,16)$
4A-1	$4^{4}$	$1$	$4$	$12$	$( 1, 6, 2, 5)( 3, 7, 4, 8)( 9,13,10,14)(11,16,12,15)$
4B	$4^{4}$	$2$	$4$	$12$	$( 1, 6, 2, 5)( 3, 7, 4, 8)( 9,14,10,13)(11,15,12,16)$
8A1	$8^{2}$	$2$	$8$	$14$	$( 1, 4, 6, 8, 2, 3, 5, 7)( 9,12,14,16,10,11,13,15)$
8A3	$8^{2}$	$2$	$8$	$14$	$( 1, 3, 6, 7, 2, 4, 5, 8)( 9,11,14,15,10,12,13,16)$
8B1	$8^{2}$	$2$	$8$	$14$	$( 1, 7, 5, 3, 2, 8, 6, 4)( 9,16,13,12,10,15,14,11)$
8B-1	$8^{2}$	$2$	$8$	$14$	$( 1, 8, 5, 4, 2, 7, 6, 3)( 9,15,13,11,10,16,14,12)$
8C1	$8^{2}$	$4$	$8$	$14$	$( 1,12, 5,16, 2,11, 6,15)( 3,10, 8,13, 4, 9, 7,14)$
8C-1	$8^{2}$	$4$	$8$	$14$	$( 1,16, 6,12, 2,15, 5,11)( 3,13, 7,10, 4,14, 8, 9)$
8D1	$8^{2}$	$4$	$8$	$14$	$( 1,14, 5,10, 2,13, 6, 9)( 3,11, 8,15, 4,12, 7,16)$
8D-1	$8^{2}$	$4$	$8$	$14$	$( 1,10, 6,14, 2, 9, 5,13)( 3,15, 7,11, 4,16, 8,12)$

Malle's constant $a(G)$:     $1/4$

Character table

		1A	2A	2B	4A1	4A-1	4B	8A1	8A3	8B1	8B-1	8C1	8C-1	8D1	8D-1
	Size	1	1	2	1	1	2	2	2	2	2	4	4	4	4
	2 P	1A	1A	1A	2A	2A	2A	4B	4B	4B	4B	4A1	4A-1	4A1	4A-1
	Type
32.15.1a	R	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
32.15.1b	R	$1$	$1$	$1$	$1$	$1$	$1$	$-1$	$-1$	$-1$	$-1$	$-1$	$-1$	$1$	$1$
32.15.1c	R	$1$	$1$	$1$	$1$	$1$	$1$	$-1$	$-1$	$-1$	$-1$	$1$	$1$	$-1$	$-1$
32.15.1d	R	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$-1$	$-1$	$-1$	$-1$
32.15.1e1	C	$1$	$1$	$-1$	$-1$	$-1$	$1$	$-1$	$-1$	$1$	$1$	$-i$	$i$	$i$	$-i$
32.15.1e2	C	$1$	$1$	$-1$	$-1$	$-1$	$1$	$-1$	$-1$	$1$	$1$	$i$	$-i$	$-i$	$i$
32.15.1f1	C	$1$	$1$	$-1$	$-1$	$-1$	$1$	$1$	$1$	$-1$	$-1$	$-i$	$i$	$-i$	$i$
32.15.1f2	C	$1$	$1$	$-1$	$-1$	$-1$	$1$	$1$	$1$	$-1$	$-1$	$i$	$-i$	$i$	$-i$
32.15.2a	R	$2$	$2$	$-2$	$2$	$2$	$-2$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$
32.15.2b	S	$2$	$2$	$2$	$-2$	$-2$	$-2$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$
32.15.2c1	C	$2$	$-2$	$0$	$-2{\zeta }_8^2$	$2{\zeta }_8^2$	$0$	$-{\zeta }_8^{-1}-{\zeta }_8$	${\zeta }_8^{-1}+{\zeta }_8$	${\zeta }_8+{\zeta }_8^3$	$-{\zeta }_8-{\zeta }_8^3$	$0$	$0$	$0$	$0$
32.15.2c2	C	$2$	$-2$	$0$	$2{\zeta }_8^2$	$-2{\zeta }_8^2$	$0$	$-{\zeta }_8^{-1}-{\zeta }_8$	${\zeta }_8^{-1}+{\zeta }_8$	$-{\zeta }_8-{\zeta }_8^3$	${\zeta }_8+{\zeta }_8^3$	$0$	$0$	$0$	$0$
32.15.2c3	C	$2$	$-2$	$0$	$-2{\zeta }_8^2$	$2{\zeta }_8^2$	$0$	${\zeta }_8^{-1}+{\zeta }_8$	$-{\zeta }_8^{-1}-{\zeta }_8$	$-{\zeta }_8-{\zeta }_8^3$	${\zeta }_8+{\zeta }_8^3$	$0$	$0$	$0$	$0$
32.15.2c4	C	$2$	$-2$	$0$	$2{\zeta }_8^2$	$-2{\zeta }_8^2$	$0$	${\zeta }_8^{-1}+{\zeta }_8$	$-{\zeta }_8^{-1}-{\zeta }_8$	${\zeta }_8+{\zeta }_8^3$	$-{\zeta }_8-{\zeta }_8^3$	$0$	$0$	$0$	$0$

Regular extensions

Data not computed