Galois group: 32T16

• Label: 32T16
• Degree: $32$
• 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 $n$:		$32$
Transitive number $t$:		$16$
Parity:		$1$
Primitive:		no
$\card{\Aut(F/K)}$:		$32$
Generators:		$(1,22,31,25)(2,21,32,26)(3,23,30,28)(4,24,29,27)(5,17,10,14)(6,18,9,13)(7,19,12,16)(8,20,11,15)$, $(1,8)(2,7)(3,5)(4,6)(9,29)(10,30)(11,31)(12,32)(13,21)(14,22)(15,23)(16,24)(17,25)(18,26)(19,27)(20,28)$, $(1,16,30,18)(2,15,29,17)(3,13,31,19)(4,14,32,20)(5,26,11,24)(6,25,12,23)(7,28,9,22)(8,27,10,21)$

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 7

Degree 4: $C_2^2$ x 7, $D_{4}$ x 4

Degree 8: $C_2^3$, $D_4$ x 2, $D_4\times C_2$ x 4, $Q_8:C_2$ x 6

Degree 16: $D_4\times C_2$, $Q_8 : C_2$ x 2, 16T30 x 2

Low degree siblings

16T30 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	$1^{32}$	$1$	$1$	$0$	$()$
2A	$2^{16}$	$1$	$2$	$16$	$( 1, 3)( 2, 4)( 5, 8)( 6, 7)( 9,12)(10,11)(13,16)(14,15)(17,20)(18,19)(21,24)(22,23)(25,28)(26,27)(29,32)(30,31)$
2B	$2^{16}$	$1$	$2$	$16$	$( 1,30)( 2,29)( 3,31)( 4,32)( 5,11)( 6,12)( 7, 9)( 8,10)(13,19)(14,20)(15,17)(16,18)(21,27)(22,28)(23,25)(24,26)$
2C	$2^{16}$	$1$	$2$	$16$	$( 1,31)( 2,32)( 3,30)( 4,29)( 5,10)( 6, 9)( 7,12)( 8,11)(13,18)(14,17)(15,20)(16,19)(21,26)(22,25)(23,28)(24,27)$
2D	$2^{16}$	$4$	$2$	$16$	$( 1, 8)( 2, 7)( 3, 5)( 4, 6)( 9,29)(10,30)(11,31)(12,32)(13,21)(14,22)(15,23)(16,24)(17,25)(18,26)(19,27)(20,28)$
2E	$2^{16}$	$4$	$2$	$16$	$( 1, 4)( 2, 3)( 5, 6)( 7, 8)( 9,10)(11,12)(13,20)(14,19)(15,18)(16,17)(21,25)(22,26)(23,27)(24,28)(29,31)(30,32)$
4A	$4^{8}$	$2$	$4$	$24$	$( 1, 7, 3, 6)( 2, 8, 4, 5)( 9,31,12,30)(10,32,11,29)(13,25,16,28)(14,26,15,27)(17,21,20,24)(18,22,19,23)$
4B	$4^{8}$	$2$	$4$	$24$	$( 1, 9, 3,12)( 2,10, 4,11)( 5,29, 8,32)( 6,30, 7,31)(13,23,16,22)(14,24,15,21)(17,27,20,26)(18,28,19,25)$
4C1	$4^{8}$	$2$	$4$	$24$	$( 1,16,30,18)( 2,15,29,17)( 3,13,31,19)( 4,14,32,20)( 5,26,11,24)( 6,25,12,23)( 7,28, 9,22)( 8,27,10,21)$
4C-1	$4^{8}$	$2$	$4$	$24$	$( 1,18,30,16)( 2,17,29,15)( 3,19,31,13)( 4,20,32,14)( 5,24,11,26)( 6,23,12,25)( 7,22, 9,28)( 8,21,10,27)$
4D1	$4^{8}$	$2$	$4$	$24$	$( 1,22,31,25)( 2,21,32,26)( 3,23,30,28)( 4,24,29,27)( 5,17,10,14)( 6,18, 9,13)( 7,19,12,16)( 8,20,11,15)$
4D-1	$4^{8}$	$2$	$4$	$24$	$( 1,25,31,22)( 2,26,32,21)( 3,28,30,23)( 4,27,29,24)( 5,14,10,17)( 6,13, 9,18)( 7,16,12,19)( 8,15,11,20)$
4E	$4^{8}$	$4$	$4$	$24$	$( 1,27, 3,26)( 2,28, 4,25)( 5,13, 8,16)( 6,14, 7,15)( 9,17,12,20)(10,18,11,19)(21,31,24,30)(22,32,23,29)$
4F	$4^{8}$	$4$	$4$	$24$	$( 1,20, 3,17)( 2,19, 4,18)( 5,23, 8,22)( 6,24, 7,21)( 9,27,12,26)(10,28,11,25)(13,32,16,29)(14,31,15,30)$

Malle's constant $a(G)$:     $1/16$

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