Galois group: 32T14

• Label: 32T14
• Degree: $32$
• Order: $32$
• Cyclic: no
• Abelian: no
• Solvable: yes
• Primitive: no
• $p$-group: yes
• Group: $C_4\wr C_2$

Group invariants

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

Group action invariants

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

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, $C_2^2$, $D_{4}$ x 4

Degree 8: $C_4\times C_2$, $D_4$ x 2, $C_2^2:C_4$ x 2, $C_4\wr C_2$ x 2

Degree 16: $C_2^2 : C_4$, 16T28, 16T42

Low degree siblings

8T17 x 2, 16T28, 16T42

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	$2^{16}$	$1$	$2$	$16$	$( 1, 4)( 2, 3)( 5, 8)( 6, 7)( 9,11)(10,12)(13,15)(14,16)(17,20)(18,19)(21,23)(22,24)(25,27)(26,28)(29,31)(30,32)$
2B	$2^{16}$	$2$	$2$	$16$	$( 1,25)( 2,26)( 3,28)( 4,27)( 5,13)( 6,14)( 7,16)( 8,15)( 9,19)(10,20)(11,18)(12,17)(21,31)(22,32)(23,29)(24,30)$
2C	$2^{16}$	$4$	$2$	$16$	$( 1, 3)( 2, 4)( 5,22)( 6,21)( 7,23)( 8,24)( 9,12)(10,11)(13,30)(14,29)(15,32)(16,31)(17,18)(19,20)(25,26)(27,28)$
4A1	$4^{8}$	$1$	$4$	$24$	$( 1,10, 4,12)( 2, 9, 3,11)( 5,31, 8,29)( 6,32, 7,30)(13,21,15,23)(14,22,16,24)(17,25,20,27)(18,26,19,28)$
4A-1	$4^{8}$	$1$	$4$	$24$	$( 1,12, 4,10)( 2,11, 3, 9)( 5,29, 8,31)( 6,30, 7,32)(13,23,15,21)(14,24,16,22)(17,27,20,25)(18,28,19,26)$
4B	$4^{8}$	$2$	$4$	$24$	$( 1,20, 4,17)( 2,19, 3,18)( 5,21, 8,23)( 6,22, 7,24)( 9,28,11,26)(10,27,12,25)(13,31,15,29)(14,32,16,30)$
4C1	$4^{8}$	$2$	$4$	$24$	$( 1,22,25,32)( 2,21,26,31)( 3,23,28,29)( 4,24,27,30)( 5,11,13,18)( 6,12,14,17)( 7,10,16,20)( 8, 9,15,19)$
4C-1	$4^{8}$	$2$	$4$	$24$	$( 1,32,25,22)( 2,31,26,21)( 3,29,28,23)( 4,30,27,24)( 5,18,13,11)( 6,17,14,12)( 7,20,16,10)( 8,19,15, 9)$
4D1	$4^{8}$	$2$	$4$	$24$	$( 1,16,27, 6)( 2,15,28, 5)( 3,13,26, 8)( 4,14,25, 7)( 9,23,18,31)(10,24,17,32)(11,21,19,29)(12,22,20,30)$
4D-1	$4^{8}$	$2$	$4$	$24$	$( 1, 6,27,16)( 2, 5,28,15)( 3, 8,26,13)( 4, 7,25,14)( 9,31,18,23)(10,32,17,24)(11,29,19,21)(12,30,20,22)$
4E	$4^{8}$	$4$	$4$	$24$	$( 1,26, 4,28)( 2,25, 3,27)( 5,30, 8,32)( 6,29, 7,31)( 9,20,11,17)(10,19,12,18)(13,22,15,24)(14,21,16,23)$
8A1	$8^{4}$	$4$	$8$	$28$	$( 1, 5,10,31, 4, 8,12,29)( 2, 6, 9,32, 3, 7,11,30)(13,17,21,25,15,20,23,27)(14,18,22,26,16,19,24,28)$
8A-1	$8^{4}$	$4$	$8$	$28$	$( 1,15,12,21, 4,13,10,23)( 2,16,11,22, 3,14, 9,24)( 5,17,29,27, 8,20,31,25)( 6,18,30,28, 7,19,32,26)$

Malle's constant $a(G)$:     $1/16$

Character table

		1A	2A	2B	2C	4A1	4A-1	4B	4C1	4C-1	4D1	4D-1	4E	8A1	8A-1
	Size	1	1	2	4	1	1	2	2	2	2	2	4	4	4
	2 P	1A	1A	1A	1A	2A	2A	2A	2B	2B	2B	2B	2A	4A1	4A-1
	Type
32.11.1a	R	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
32.11.1b	R	$1$	$1$	$1$	$-1$	$1$	$1$	$-1$	$-1$	$-1$	$1$	$-1$	$-1$	$1$	$1$
32.11.1c	R	$1$	$1$	$1$	$-1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$-1$	$-1$	$-1$
32.11.1d	R	$1$	$1$	$1$	$1$	$1$	$1$	$-1$	$-1$	$-1$	$1$	$-1$	$1$	$-1$	$-1$
32.11.1e1	C	$1$	$1$	$-1$	$-1$	$-1$	$-1$	$-i$	$-i$	$i$	$1$	$i$	$1$	$i$	$-i$
32.11.1e2	C	$1$	$1$	$-1$	$-1$	$-1$	$-1$	$i$	$i$	$-i$	$1$	$-i$	$1$	$-i$	$i$
32.11.1f1	C	$1$	$1$	$-1$	$1$	$-1$	$-1$	$-i$	$-i$	$i$	$1$	$i$	$-1$	$-i$	$i$
32.11.1f2	C	$1$	$1$	$-1$	$1$	$-1$	$-1$	$i$	$i$	$-i$	$1$	$-i$	$-1$	$i$	$-i$
32.11.2a	R	$2$	$2$	$-2$	$0$	$2$	$2$	$0$	$0$	$0$	$-2$	$0$	$0$	$0$	$0$
32.11.2b	R	$2$	$2$	$2$	$0$	$-2$	$-2$	$0$	$0$	$0$	$-2$	$0$	$0$	$0$	$0$
32.11.2c1	C	$2$	$-2$	$0$	$0$	$-2i$	$2i$	$-1+i$	$1-i$	$1+i$	$0$	$-1-i$	$0$	$0$	$0$
32.11.2c2	C	$2$	$-2$	$0$	$0$	$2i$	$-2i$	$-1-i$	$1+i$	$1-i$	$0$	$-1+i$	$0$	$0$	$0$
32.11.2d1	C	$2$	$-2$	$0$	$0$	$-2i$	$2i$	$1-i$	$-1+i$	$-1-i$	$0$	$1+i$	$0$	$0$	$0$
32.11.2d2	C	$2$	$-2$	$0$	$0$	$2i$	$-2i$	$1+i$	$-1-i$	$-1+i$	$0$	$1-i$	$0$	$0$	$0$

Regular extensions

Data not computed