Galois group: 16T30

• Label: 16T30
• Degree: $16$
• Order: $32$
• Cyclic: no
• Abelian: no
• Solvable: yes
• Primitive: no
• pp-group: yes
• Group: $C_4^2:C_2$

Group invariants

Abstract group:		$C_4^2:C_2$
Order:		$32=2^{5}$
Cyclic:		no
Abelian:		no
Solvable:		yes
Nilpotency class:		$2$

Group action invariants

Degree $n$:		$16$
Transitive number $t$:		$30$
Parity:		$1$
Primitive:		no
$\card{\Aut(F/K)}$:		$4$
Generators:		$(1,14,16,12)(2,13,15,11)(3,8,5,10)(4,7,6,9)$, $(1,5)(2,6)(3,16)(4,15)(7,14)(8,13)(9,12)(10,11)$, $(1,4,2,3)(5,16,6,15)(7,13,8,14)(9,11,10,12)$

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

Low degree siblings

16T30, 32T16

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

Malle's constant $a(G)$:     $1/6$

Character table

		1A	2A	2B	2C	2D	2E	4A	4B	4C1	4C-1	4D1	4D-1	4E	4F
	Size	1	1	1	1	4	4	2	2	2	2	2	2	4	4
	2 P	1A	1A	1A	1A	1A	1A	2A	2A	2B	2B	2C	2C	2A	2A
	Type
32.31.1a	R	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
32.31.1b	R	$1$	$1$	$1$	$1$	$-1$	$-1$	$-1$	$-1$	$1$	$1$	$-1$	$-1$	$1$	$1$
32.31.1c	R	$1$	$1$	$1$	$1$	$-1$	$-1$	$1$	$1$	$1$	$1$	$1$	$1$	$-1$	$-1$
32.31.1d	R	$1$	$1$	$1$	$1$	$-1$	$1$	$-1$	$1$	$-1$	$-1$	$1$	$-1$	$-1$	$1$
32.31.1e	R	$1$	$1$	$1$	$1$	$-1$	$1$	$1$	$-1$	$-1$	$-1$	$-1$	$1$	$1$	$-1$
32.31.1f	R	$1$	$1$	$1$	$1$	$1$	$-1$	$-1$	$1$	$-1$	$-1$	$1$	$-1$	$1$	$-1$
32.31.1g	R	$1$	$1$	$1$	$1$	$1$	$-1$	$1$	$-1$	$-1$	$-1$	$-1$	$1$	$-1$	$1$
32.31.1h	R	$1$	$1$	$1$	$1$	$1$	$1$	$-1$	$-1$	$1$	$1$	$-1$	$-1$	$-1$	$-1$
32.31.2a	R	$2$	$2$	$-2$	$-2$	$0$	$0$	$0$	$0$	$-2$	$2$	$0$	$0$	$0$	$0$
32.31.2b	R	$2$	$2$	$-2$	$-2$	$0$	$0$	$0$	$0$	$2$	$-2$	$0$	$0$	$0$	$0$
32.31.2c1	C	$2$	$-2$	$-2$	$2$	$0$	$0$	$0$	$-2i$	$0$	$0$	$2i$	$0$	$0$	$0$
32.31.2c2	C	$2$	$-2$	$-2$	$2$	$0$	$0$	$0$	$2i$	$0$	$0$	$-2i$	$0$	$0$	$0$
32.31.2d1	C	$2$	$-2$	$2$	$-2$	$0$	$0$	$-2i$	$0$	$0$	$0$	$0$	$2i$	$0$	$0$
32.31.2d2	C	$2$	$-2$	$2$	$-2$	$0$	$0$	$2i$	$0$	$0$	$0$	$0$	$-2i$	$0$	$0$

Regular extensions

Data not computed