LMFDB - Galois group: 32T34

Show commands: Magma

magma: G := TransitiveGroup(32, 34);

Group action invariants

Degree $n$:	$32$	magma: t, n := TransitiveGroupIdentification(G); n;
Transitive number $t$:	$34$	magma: t, n := TransitiveGroupIdentification(G); t;
Group:	$C_2^3\times C_4$
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:	(1,12)(2,11)(3,10)(4,9)(5,30)(6,29)(7,32)(8,31)(13,24)(14,23)(15,22)(16,21)(17,26)(18,25)(19,28)(20,27), (1,18)(2,17)(3,20)(4,19)(5,22)(6,21)(7,24)(8,23)(9,28)(10,27)(11,26)(12,25)(13,32)(14,31)(15,30)(16,29), (1,5,11,29)(2,6,12,30)(3,7,9,31)(4,8,10,32)(13,19,23,27)(14,20,24,28)(15,17,21,25)(16,18,22,26), (1,7,11,31)(2,8,12,32)(3,5,9,29)(4,6,10,30)(13,17,23,25)(14,18,24,26)(15,19,21,27)(16,20,22,28)	magma: Generators(G);

Low degree resolvents

|G/N| Galois groups for stem field(s)
$2$: $C_2$ x 15
$4$: $C_4$ x 8, $C_2^2$ x 35
$8$: $C_4\times C_2$ x 28, $C_2^3$ x 15
$16$: $C_4\times C_2^2$ x 14, $C_2^4$

Resolvents shown for degrees $\leq 47$

\|G/N\|	Galois groups for stem field(s)
$2$:	$C_2$ x 15
$4$:	$C_4$ x 8, $C_2^2$ x 35
$8$:	$C_4\times C_2$ x 28, $C_2^3$ x 15
$16$:	$C_4\times C_2^2$ x 14, $C_2^4$

Subfields

Degree 2: $C_2$ x 15
Degree 4: $C_4$ x 8, $C_2^2$ x 35
Degree 8: $C_4\times C_2$ x 28, $C_2^3$ x 15
Degree 16: $C_4\times C_2^2$ x 14, $C_2^4$

Low degree siblings

There are no siblings 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 $	$1$	$2$	$( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22) (23,24)(25,26)(27,28)(29,30)(31,32)$
	$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $	$1$	$2$	$( 1, 3)( 2, 4)( 5, 7)( 6, 8)( 9,11)(10,12)(13,15)(14,16)(17,19)(18,20)(21,23) (22,24)(25,27)(26,28)(29,31)(30,32)$
	$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $	$1$	$2$	$( 1, 4)( 2, 3)( 5, 8)( 6, 7)( 9,12)(10,11)(13,16)(14,15)(17,20)(18,19)(21,24) (22,23)(25,28)(26,27)(29,32)(30,31)$
	$ 4, 4, 4, 4, 4, 4, 4, 4 $	$1$	$4$	$( 1, 5,11,29)( 2, 6,12,30)( 3, 7, 9,31)( 4, 8,10,32)(13,19,23,27)(14,20,24,28) (15,17,21,25)(16,18,22,26)$
	$ 4, 4, 4, 4, 4, 4, 4, 4 $	$1$	$4$	$( 1, 6,11,30)( 2, 5,12,29)( 3, 8, 9,32)( 4, 7,10,31)(13,20,23,28)(14,19,24,27) (15,18,21,26)(16,17,22,25)$
	$ 4, 4, 4, 4, 4, 4, 4, 4 $	$1$	$4$	$( 1, 7,11,31)( 2, 8,12,32)( 3, 5, 9,29)( 4, 6,10,30)(13,17,23,25)(14,18,24,26) (15,19,21,27)(16,20,22,28)$
	$ 4, 4, 4, 4, 4, 4, 4, 4 $	$1$	$4$	$( 1, 8,11,32)( 2, 7,12,31)( 3, 6, 9,30)( 4, 5,10,29)(13,18,23,26)(14,17,24,25) (15,20,21,28)(16,19,22,27)$
	$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $	$1$	$2$	$( 1, 9)( 2,10)( 3,11)( 4,12)( 5,31)( 6,32)( 7,29)( 8,30)(13,21)(14,22)(15,23) (16,24)(17,27)(18,28)(19,25)(20,26)$
	$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $	$1$	$2$	$( 1,10)( 2, 9)( 3,12)( 4,11)( 5,32)( 6,31)( 7,30)( 8,29)(13,22)(14,21)(15,24) (16,23)(17,28)(18,27)(19,26)(20,25)$
	$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $	$1$	$2$	$( 1,11)( 2,12)( 3, 9)( 4,10)( 5,29)( 6,30)( 7,31)( 8,32)(13,23)(14,24)(15,21) (16,22)(17,25)(18,26)(19,27)(20,28)$
	$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $	$1$	$2$	$( 1,12)( 2,11)( 3,10)( 4, 9)( 5,30)( 6,29)( 7,32)( 8,31)(13,24)(14,23)(15,22) (16,21)(17,26)(18,25)(19,28)(20,27)$
	$ 4, 4, 4, 4, 4, 4, 4, 4 $	$1$	$4$	$( 1,13,11,23)( 2,14,12,24)( 3,15, 9,21)( 4,16,10,22)( 5,19,29,27)( 6,20,30,28) ( 7,17,31,25)( 8,18,32,26)$
	$ 4, 4, 4, 4, 4, 4, 4, 4 $	$1$	$4$	$( 1,14,11,24)( 2,13,12,23)( 3,16, 9,22)( 4,15,10,21)( 5,20,29,28)( 6,19,30,27) ( 7,18,31,26)( 8,17,32,25)$
	$ 4, 4, 4, 4, 4, 4, 4, 4 $	$1$	$4$	$( 1,15,11,21)( 2,16,12,22)( 3,13, 9,23)( 4,14,10,24)( 5,17,29,25)( 6,18,30,26) ( 7,19,31,27)( 8,20,32,28)$
	$ 4, 4, 4, 4, 4, 4, 4, 4 $	$1$	$4$	$( 1,16,11,22)( 2,15,12,21)( 3,14, 9,24)( 4,13,10,23)( 5,18,29,26)( 6,17,30,25) ( 7,20,31,28)( 8,19,32,27)$
	$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $	$1$	$2$	$( 1,17)( 2,18)( 3,19)( 4,20)( 5,21)( 6,22)( 7,23)( 8,24)( 9,27)(10,28)(11,25) (12,26)(13,31)(14,32)(15,29)(16,30)$
	$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $	$1$	$2$	$( 1,18)( 2,17)( 3,20)( 4,19)( 5,22)( 6,21)( 7,24)( 8,23)( 9,28)(10,27)(11,26) (12,25)(13,32)(14,31)(15,30)(16,29)$
	$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $	$1$	$2$	$( 1,19)( 2,20)( 3,17)( 4,18)( 5,23)( 6,24)( 7,21)( 8,22)( 9,25)(10,26)(11,27) (12,28)(13,29)(14,30)(15,31)(16,32)$
	$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $	$1$	$2$	$( 1,20)( 2,19)( 3,18)( 4,17)( 5,24)( 6,23)( 7,22)( 8,21)( 9,26)(10,25)(11,28) (12,27)(13,30)(14,29)(15,32)(16,31)$
	$ 4, 4, 4, 4, 4, 4, 4, 4 $	$1$	$4$	$( 1,21,11,15)( 2,22,12,16)( 3,23, 9,13)( 4,24,10,14)( 5,25,29,17)( 6,26,30,18) ( 7,27,31,19)( 8,28,32,20)$
	$ 4, 4, 4, 4, 4, 4, 4, 4 $	$1$	$4$	$( 1,22,11,16)( 2,21,12,15)( 3,24, 9,14)( 4,23,10,13)( 5,26,29,18)( 6,25,30,17) ( 7,28,31,20)( 8,27,32,19)$
	$ 4, 4, 4, 4, 4, 4, 4, 4 $	$1$	$4$	$( 1,23,11,13)( 2,24,12,14)( 3,21, 9,15)( 4,22,10,16)( 5,27,29,19)( 6,28,30,20) ( 7,25,31,17)( 8,26,32,18)$
	$ 4, 4, 4, 4, 4, 4, 4, 4 $	$1$	$4$	$( 1,24,11,14)( 2,23,12,13)( 3,22, 9,16)( 4,21,10,15)( 5,28,29,20)( 6,27,30,19) ( 7,26,31,18)( 8,25,32,17)$
	$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $	$1$	$2$	$( 1,25)( 2,26)( 3,27)( 4,28)( 5,15)( 6,16)( 7,13)( 8,14)( 9,19)(10,20)(11,17) (12,18)(21,29)(22,30)(23,31)(24,32)$
	$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $	$1$	$2$	$( 1,26)( 2,25)( 3,28)( 4,27)( 5,16)( 6,15)( 7,14)( 8,13)( 9,20)(10,19)(11,18) (12,17)(21,30)(22,29)(23,32)(24,31)$
	$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $	$1$	$2$	$( 1,27)( 2,28)( 3,25)( 4,26)( 5,13)( 6,14)( 7,15)( 8,16)( 9,17)(10,18)(11,19) (12,20)(21,31)(22,32)(23,29)(24,30)$
	$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $	$1$	$2$	$( 1,28)( 2,27)( 3,26)( 4,25)( 5,14)( 6,13)( 7,16)( 8,15)( 9,18)(10,17)(11,20) (12,19)(21,32)(22,31)(23,30)(24,29)$
	$ 4, 4, 4, 4, 4, 4, 4, 4 $	$1$	$4$	$( 1,29,11, 5)( 2,30,12, 6)( 3,31, 9, 7)( 4,32,10, 8)(13,27,23,19)(14,28,24,20) (15,25,21,17)(16,26,22,18)$
	$ 4, 4, 4, 4, 4, 4, 4, 4 $	$1$	$4$	$( 1,30,11, 6)( 2,29,12, 5)( 3,32, 9, 8)( 4,31,10, 7)(13,28,23,20)(14,27,24,19) (15,26,21,18)(16,25,22,17)$
	$ 4, 4, 4, 4, 4, 4, 4, 4 $	$1$	$4$	$( 1,31,11, 7)( 2,32,12, 8)( 3,29, 9, 5)( 4,30,10, 6)(13,25,23,17)(14,26,24,18) (15,27,21,19)(16,28,22,20)$
	$ 4, 4, 4, 4, 4, 4, 4, 4 $	$1$	$4$	$( 1,32,11, 8)( 2,31,12, 7)( 3,30, 9, 6)( 4,29,10, 5)(13,26,23,18)(14,25,24,17) (15,28,21,20)(16,27,22,19)$

magma: ConjugacyClasses(G);

Group invariants

Order:	$32=2^{5}$	magma: Order(G);
Cyclic:	no	magma: IsCyclic(G);
Abelian:	yes	magma: IsAbelian(G);
Solvable:	yes	magma: IsSolvable(G);
Nilpotency class:	$1$
Label:	32.45	magma: IdentifyGroup(G);
Character table:	32 x 32 character table

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

Related objects

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	32T34
Degree	$32$
Order	$32$
Cyclic	no
Abelian	yes
Solvable	yes
Primitive	no
$p$-group	yes
Group:	$C_2^3\times C_4$