Galois group: 16T54

• Label: 16T54
• Degree: $16$
• Order: $32$
• Cyclic: no
• Abelian: no
• Solvable: yes
• Primitive: no
• $p$-group: yes
• Group: $C_2^2.D_4$

Group invariants

Abstract group:		$C_2^2.D_4$
Order:		$32=2^{5}$
Cyclic:		no
Abelian:		no
Solvable:		yes
Nilpotency class:		$2$

Group action invariants

Degree $n$:		$16$
Transitive number $t$:		$54$
Parity:		$1$
Primitive:		no
$\card{\Aut(F/K)}$:		$8$
Generators:		$(1,15,10,7)(2,16,9,8)(3,13,12,6)(4,14,11,5)$, $(1,7,2,8)(3,14,4,13)(5,11,6,12)(9,16,10,15)$, $(1,4)(2,3)(5,8)(6,7)(9,12)(10,11)(13,15)(14,16)$

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: $D_4$, $Q_8:C_2$ x 2

Low degree siblings

16T37, 16T54, 32T23

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

Malle's constant $a(G)$:     $1/4$

Character table

		1A	2A	2B	2C	2D	2E	2F	4A1	4A-1	4B1	4B-1	4C	4D	4E
	Size	1	1	1	1	2	2	4	2	2	2	2	4	4	4
	2 P	1A	1A	1A	1A	1A	1A	1A	2A	2A	2A	2A	2B	2A	2C
	Type
32.30.1a	R	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
32.30.1b	R	$1$	$1$	$1$	$1$	$-1$	$-1$	$-1$	$-1$	$1$	$1$	$-1$	$1$	$1$	$-1$
32.30.1c	R	$1$	$1$	$1$	$1$	$-1$	$-1$	$-1$	$1$	$-1$	$-1$	$1$	$-1$	$1$	$1$
32.30.1d	R	$1$	$1$	$1$	$1$	$-1$	$-1$	$1$	$-1$	$1$	$1$	$-1$	$-1$	$-1$	$1$
32.30.1e	R	$1$	$1$	$1$	$1$	$-1$	$-1$	$1$	$1$	$-1$	$-1$	$1$	$1$	$-1$	$-1$
32.30.1f	R	$1$	$1$	$1$	$1$	$1$	$1$	$-1$	$-1$	$-1$	$-1$	$-1$	$1$	$-1$	$1$
32.30.1g	R	$1$	$1$	$1$	$1$	$1$	$1$	$-1$	$1$	$1$	$1$	$1$	$-1$	$-1$	$-1$
32.30.1h	R	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$-1$	$-1$	$-1$	$-1$	$-1$	$1$	$-1$
32.30.2a	R	$2$	$2$	$-2$	$-2$	$-2$	$2$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$
32.30.2b	R	$2$	$2$	$-2$	$-2$	$2$	$-2$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$
32.30.2c1	C	$2$	$-2$	$-2$	$2$	$0$	$0$	$0$	$-2i$	$0$	$0$	$2i$	$0$	$0$	$0$
32.30.2c2	C	$2$	$-2$	$-2$	$2$	$0$	$0$	$0$	$2i$	$0$	$0$	$-2i$	$0$	$0$	$0$
32.30.2d1	C	$2$	$-2$	$2$	$-2$	$0$	$0$	$0$	$0$	$-2i$	$2i$	$0$	$0$	$0$	$0$
32.30.2d2	C	$2$	$-2$	$2$	$-2$	$0$	$0$	$0$	$0$	$2i$	$-2i$	$0$	$0$	$0$	$0$

Regular extensions

Data not computed