LMFDB - Galois group: 32T24

Show commands: Magma

magma: G := TransitiveGroup(32, 24);

Group action invariants

Degree $n$:	$32$	magma: t, n := TransitiveGroupIdentification(G); n;
Transitive number $t$:	$24$	magma: t, n := TransitiveGroupIdentification(G); t;
Group:	$C_2^2\wr C_2$
Parity:	$1$	magma: IsEven(G);
Primitive:	no	magma: IsPrimitive(G);
magma: NilpotencyClass(G);
$\card{\Aut(F/K)}$:	$32$	magma: Order(Centralizer(SymmetricGroup(n), G));
Generators:	(1,6)(2,5)(3,7)(4,8)(9,29)(10,30)(11,32)(12,31)(13,24)(14,23)(15,22)(16,21)(17,28)(18,27)(19,25)(20,26), (1,20)(2,19)(3,17)(4,18)(5,24)(6,23)(7,21)(8,22)(9,27)(10,28)(11,26)(12,25)(13,31)(14,32)(15,29)(16,30), (1,25,7,15)(2,26,8,16)(3,27,6,13)(4,28,5,14)(9,17,31,23)(10,18,32,24)(11,19,30,22)(12,20,29,21)	magma: Generators(G);

Low degree resolvents

|G/N| Galois groups for stem field(s)
$2$: $C_2$ x 7
$4$: $C_2^2$ x 7
$8$: $D_{4}$ x 6, $C_2^3$
$16$: $D_4\times C_2$ x 3

Resolvents shown for degrees $\leq 47$

\|G/N\|	Galois groups for stem field(s)
$2$:	$C_2$ x 7
$4$:	$C_2^2$ x 7
$8$:	$D_{4}$ x 6, $C_2^3$
$16$:	$D_4\times C_2$ x 3

Subfields

Degree 2: $C_2$ x 7
Degree 4: $C_2^2$ x 7, $D_{4}$ x 12
Degree 8: $C_2^3$, $D_4$ x 6, $D_4\times C_2$ x 12, $C_2^2 \wr C_2$ x 8
Degree 16: $D_4\times C_2$ x 3, 16T39 x 6, 16T46

Low degree siblings

8T18 x 8, 16T39 x 6, 16T46
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	Representative
	$ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $	$1$	$1$	$()$
	$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $	$2$	$2$	$( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,21)(14,22)(15,23)(16,24)(17,25) (18,26)(19,28)(20,27)(29,30)(31,32)$
	$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $	$2$	$2$	$( 1, 3)( 2, 4)( 5, 8)( 6, 7)( 9,12)(10,11)(13,18)(14,17)(15,19)(16,20)(21,26) (22,25)(23,28)(24,27)(29,31)(30,32)$
	$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $	$2$	$2$	$( 1, 4)( 2, 3)( 5, 7)( 6, 8)( 9,11)(10,12)(13,26)(14,25)(15,28)(16,27)(17,22) (18,21)(19,23)(20,24)(29,32)(30,31)$
	$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $	$2$	$2$	$( 1, 6)( 2, 5)( 3, 7)( 4, 8)( 9,29)(10,30)(11,32)(12,31)(13,24)(14,23)(15,22) (16,21)(17,28)(18,27)(19,25)(20,26)$
	$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $	$1$	$2$	$( 1, 7)( 2, 8)( 3, 6)( 4, 5)( 9,31)(10,32)(11,30)(12,29)(13,27)(14,28)(15,25) (16,26)(17,23)(18,24)(19,22)(20,21)$
	$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $	$2$	$2$	$( 1, 8)( 2, 7)( 3, 5)( 4, 6)( 9,32)(10,31)(11,29)(12,30)(13,20)(14,19)(15,17) (16,18)(21,27)(22,28)(23,25)(24,26)$
	$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $	$1$	$2$	$( 1,10)( 2, 9)( 3,11)( 4,12)( 5,29)( 6,30)( 7,32)( 8,31)(13,22)(14,21)(15,24) (16,23)(17,26)(18,25)(19,27)(20,28)$
	$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $	$2$	$2$	$( 1,12)( 2,11)( 3, 9)( 4,10)( 5,32)( 6,31)( 7,29)( 8,30)(13,17)(14,18)(15,20) (16,19)(21,25)(22,26)(23,27)(24,28)$
	$ 4, 4, 4, 4, 4, 4, 4, 4 $	$4$	$4$	$( 1,13,10,22)( 2,14, 9,21)( 3,15,11,24)( 4,16,12,23)( 5,26,29,17)( 6,25,30,18) ( 7,27,32,19)( 8,28,31,20)$
	$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $	$4$	$2$	$( 1,14)( 2,13)( 3,16)( 4,15)( 5,25)( 6,26)( 7,28)( 8,27)( 9,22)(10,21)(11,23) (12,24)(17,30)(18,29)(19,31)(20,32)$
	$ 4, 4, 4, 4, 4, 4, 4, 4 $	$4$	$4$	$( 1,15, 7,25)( 2,16, 8,26)( 3,13, 6,27)( 4,14, 5,28)( 9,23,31,17)(10,24,32,18) (11,22,30,19)(12,21,29,20)$
	$ 4, 4, 4, 4, 4, 4, 4, 4 $	$4$	$4$	$( 1,16,32,17)( 2,15,31,18)( 3,14,30,20)( 4,13,29,19)( 5,27,12,22)( 6,28,11,21) ( 7,26,10,23)( 8,25, 9,24)$
	$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $	$1$	$2$	$( 1,32)( 2,31)( 3,30)( 4,29)( 5,12)( 6,11)( 7,10)( 8, 9)(13,19)(14,20)(15,18) (16,17)(21,28)(22,27)(23,26)(24,25)$

magma: ConjugacyClasses(G);

Group invariants

Order:	$32=2^{5}$	magma: Order(G);
Cyclic:	no	magma: IsCyclic(G);
Abelian:	no	magma: IsAbelian(G);
Solvable:	yes	magma: IsSolvable(G);
Nilpotency class:	$2$
Label:	32.27	magma: IdentifyGroup(G);
Character table:

		1A	2A	2B	2C	2D	2E	2F	2G	2H	2I	2J	4A	4B	4C
	Size	1	1	1	1	2	2	2	2	2	2	4	4	4	4
	2 P	1A	1A	1A	1A	1A	1A	1A	1A	1A	1A	1A	2A	2B	2C
	Type
32.27.1a	R	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
32.27.1b	R	$1$	$1$	$1$	$1$	$- 1$	$- 1$	$1$	$1$	$- 1$	$- 1$	$- 1$	$1$	$1$	$- 1$
32.27.1c	R	$1$	$1$	$1$	$1$	$- 1$	$- 1$	$1$	$1$	$- 1$	$- 1$	$1$	$- 1$	$- 1$	$1$
32.27.1d	R	$1$	$1$	$1$	$1$	$- 1$	$1$	$- 1$	$- 1$	$- 1$	$1$	$- 1$	$- 1$	$1$	$1$
32.27.1e	R	$1$	$1$	$1$	$1$	$- 1$	$1$	$- 1$	$- 1$	$- 1$	$1$	$1$	$1$	$- 1$	$- 1$
32.27.1f	R	$1$	$1$	$1$	$1$	$1$	$- 1$	$- 1$	$- 1$	$1$	$- 1$	$- 1$	$1$	$- 1$	$1$
32.27.1g	R	$1$	$1$	$1$	$1$	$1$	$- 1$	$- 1$	$- 1$	$1$	$- 1$	$1$	$- 1$	$1$	$- 1$
32.27.1h	R	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$- 1$	$- 1$	$- 1$	$- 1$
32.27.2a	R	$2$	$- 2$	$- 2$	$2$	$0$	$0$	$- 2$	$2$	$0$	$0$	$0$	$0$	$0$	$0$
32.27.2b	R	$2$	$- 2$	$- 2$	$2$	$0$	$0$	$2$	$- 2$	$0$	$0$	$0$	$0$	$0$	$0$
32.27.2c	R	$2$	$- 2$	$2$	$- 2$	$- 2$	$0$	$0$	$0$	$2$	$0$	$0$	$0$	$0$	$0$
32.27.2d	R	$2$	$- 2$	$2$	$- 2$	$2$	$0$	$0$	$0$	$- 2$	$0$	$0$	$0$	$0$	$0$
32.27.2e	R	$2$	$2$	$- 2$	$- 2$	$0$	$- 2$	$0$	$0$	$0$	$2$	$0$	$0$	$0$	$0$
32.27.2f	R	$2$	$2$	$- 2$	$- 2$	$0$	$2$	$0$	$0$	$0$	$- 2$	$0$	$0$	$0$	$0$

magma: CharacterTable(G);

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Downloads

Learn more

Group action invariants

Low degree resolvents

Subfields

Low degree siblings

Conjugacy classes

Group invariants

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	32T24
Degree	$32$
Order	$32$
Cyclic	no
Abelian	no
Solvable	yes
Primitive	no
$p$-group	yes
Group:	$C_2^2\wr C_2$