Galois group: 16T176

• Label: 16T176
• Degree: $16$
• Order: $64$
• Cyclic: no
• Abelian: no
• Solvable: yes
• Primitive: no
• $p$-group: yes
• Group: $C_4^2:C_4$

Group invariants

Abstract group:		$C_4^2:C_4$
Order:		$64=2^{6}$
Cyclic:		no
Abelian:		no
Solvable:		yes
Nilpotency class:		$4$

Group action invariants

Degree $n$:		$16$
Transitive number $t$:		$176$
Parity:		$1$
Primitive:		no
$\card{\Aut(F/K)}$:		$2$
Generators:		$(1,6,2,5)(7,14,11,10)(8,13,12,9)(15,16)$, $(1,7,5,11)(2,8,6,12)(3,14,16,9)(4,13,15,10)$

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, $C_4\times C_2$
$16$:	$C_2^2:C_4$
$32$:	$C_2^3 : C_4 $

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 4: $D_{4}$

Degree 8: $C_2^3 : C_4 $

Low degree siblings

16T154, 16T174 x 2, 16T176, 32T166, 32T179 x 2

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	$2^{8}$	$2$	$2$	$8$	$( 1, 5)( 2, 6)( 3,16)( 4,15)( 7,11)( 8,12)( 9,14)(10,13)$
2C	$2^{6},1^{4}$	$4$	$2$	$6$	$( 3, 4)( 7,12)( 8,11)( 9,14)(10,13)(15,16)$
2D	$2^{8}$	$4$	$2$	$8$	$( 1, 4)( 2, 3)( 5,15)( 6,16)( 7, 9)( 8,10)(11,14)(12,13)$
4A	$4^{4}$	$4$	$4$	$12$	$( 1, 4, 2, 3)( 5,15, 6,16)( 7,13, 8,14)( 9,11,10,12)$
4B1	$4^{4}$	$4$	$4$	$12$	$( 1, 7, 5,11)( 2, 8, 6,12)( 3,14,16, 9)( 4,13,15,10)$
4B-1	$4^{4}$	$4$	$4$	$12$	$( 1, 8, 5,12)( 2, 7, 6,11)( 3,13,16,10)( 4,14,15, 9)$
4C	$4^{4}$	$8$	$4$	$12$	$( 1, 7, 2, 8)( 3,13, 4,14)( 5,11, 6,12)( 9,16,10,15)$
4D1	$4^{3},2,1^{2}$	$8$	$4$	$10$	$( 3,15, 4,16)( 5, 6)( 7,10,12,13)( 8, 9,11,14)$
4D-1	$4^{3},2,1^{2}$	$8$	$4$	$10$	$( 3,16, 4,15)( 5, 6)( 7,13,12,10)( 8,14,11, 9)$
4E1	$4^{4}$	$8$	$4$	$12$	$( 1, 7, 4, 9)( 2, 8, 3,10)( 5,12,15,13)( 6,11,16,14)$
4E-1	$4^{4}$	$8$	$4$	$12$	$( 1, 7,15,14)( 2, 8,16,13)( 3, 9, 6,11)( 4,10, 5,12)$

Malle's constant $a(G)$:     $1/6$

Character table

		1A	2A	2B	2C	2D	4A	4B1	4B-1	4C	4D1	4D-1	4E1	4E-1
	Size	1	1	2	4	4	4	4	4	8	8	8	8	8
	2 P	1A	1A	1A	1A	1A	2A	2B	2B	2A	2C	2C	2D	2D
	Type
64.35.1a	R	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
64.35.1b	R	$1$	$1$	$1$	$1$	$1$	$1$	$-1$	$-1$	$-1$	$-1$	$1$	$-1$	$1$
64.35.1c	R	$1$	$1$	$1$	$1$	$1$	$1$	$-1$	$-1$	$1$	$-1$	$-1$	$1$	$-1$
64.35.1d	R	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$-1$	$1$	$-1$	$-1$	$-1$
64.35.1e1	C	$1$	$1$	$1$	$-1$	$-1$	$1$	$-1$	$-1$	$-i$	$1$	$i$	$i$	$-i$
64.35.1e2	C	$1$	$1$	$1$	$-1$	$-1$	$1$	$-1$	$-1$	$i$	$1$	$-i$	$-i$	$i$
64.35.1f1	C	$1$	$1$	$1$	$-1$	$-1$	$1$	$1$	$1$	$-i$	$-1$	$-i$	$i$	$i$
64.35.1f2	C	$1$	$1$	$1$	$-1$	$-1$	$1$	$1$	$1$	$i$	$-1$	$i$	$-i$	$-i$
64.35.2a	R	$2$	$2$	$2$	$-2$	$2$	$-2$	$0$	$0$	$0$	$0$	$0$	$0$	$0$
64.35.2b	R	$2$	$2$	$2$	$2$	$-2$	$-2$	$0$	$0$	$0$	$0$	$0$	$0$	$0$
64.35.4a	R	$4$	$4$	$-4$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$
64.35.4b1	C	$4$	$-4$	$0$	$0$	$0$	$0$	$-2i$	$2i$	$0$	$0$	$0$	$0$	$0$
64.35.4b2	C	$4$	$-4$	$0$	$0$	$0$	$0$	$2i$	$-2i$	$0$	$0$	$0$	$0$	$0$

Regular extensions

Data not computed