Galois group: 16T53

• Label: 16T53
• Degree: $16$
• Order: $32$
• Cyclic: no
• Abelian: no
• Solvable: yes
• Primitive: no
• $p$-group: yes
• Group: $C_2^3:C_4$

Group invariants

Abstract group:		$C_2^3:C_4$
Order:		$32=2^{5}$
Cyclic:		no
Abelian:		no
Solvable:		yes
Nilpotency class:		$3$

Group action invariants

Degree $n$:		$16$
Transitive number $t$:		$53$
Parity:		$1$
Primitive:		no
$\card{\Aut(F/K)}$:		$8$
Generators:		$(1,4,5,15)(2,3,6,16)(7,9,12,13)(8,10,11,14)$, $(1,2)(3,11)(4,12)(5,13)(6,14)(7,8)(9,10)(15,16)$

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: $C_4\times C_2$, $C_2^3: C_4$, $C_2^3: C_4$

Low degree siblings

8T19 x 2, 8T20, 8T21, 16T33 x 2, 16T52, 32T19

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

Malle's constant $a(G)$:     $1/4$

Character table

		1A	2A	2B	2C	2D	2E	4A	4B1	4B-1	4C1	4C-1
	Size	1	1	2	2	2	4	4	4	4	4	4
	2 P	1A	1A	1A	1A	1A	1A	2A	2C	2C	2D	2D
	Type
32.6.1a	R	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
32.6.1b	R	$1$	$1$	$1$	$1$	$1$	$-1$	$-1$	$-1$	$1$	$-1$	$1$
32.6.1c	R	$1$	$1$	$1$	$1$	$1$	$-1$	$1$	$1$	$-1$	$-1$	$-1$
32.6.1d	R	$1$	$1$	$1$	$1$	$1$	$1$	$-1$	$-1$	$-1$	$1$	$-1$
32.6.1e1	C	$1$	$1$	$1$	$-1$	$-1$	$-1$	$-i$	$i$	$i$	$1$	$-i$
32.6.1e2	C	$1$	$1$	$1$	$-1$	$-1$	$-1$	$i$	$-i$	$-i$	$1$	$i$
32.6.1f1	C	$1$	$1$	$1$	$-1$	$-1$	$1$	$-i$	$i$	$-i$	$-1$	$i$
32.6.1f2	C	$1$	$1$	$1$	$-1$	$-1$	$1$	$i$	$-i$	$i$	$-1$	$-i$
32.6.2a	R	$2$	$2$	$-2$	$-2$	$2$	$0$	$0$	$0$	$0$	$0$	$0$
32.6.2b	R	$2$	$2$	$-2$	$2$	$-2$	$0$	$0$	$0$	$0$	$0$	$0$
32.6.4a	R	$4$	$-4$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$

Regular extensions

Data not computed