LMFDB - Galois group: 32T4401

Show commands: Magma

magma:G := TransitiveGroup(32, 4401);

Group invariants

Abstract group:	$C_2^5.D_4$	magma:IdentifyGroup(G);
Order:	$256=2^{8}$	magma:Order(G);
Cyclic:	no	magma:IsCyclic(G);
Abelian:	no	magma:IsAbelian(G);
Solvable:	yes	magma:IsSolvable(G);
Nilpotency class:	$4$	magma:NilpotencyClass(G);

Group action invariants

Degree $n$:	$32$	magma:t, n := TransitiveGroupIdentification(G); n;
Transitive number $t$:	$4401$	magma:t, n := TransitiveGroupIdentification(G); t;
Parity:	$1$	magma:IsEven(G);
Primitive:	no	magma:IsPrimitive(G);
$\card{\Aut(F/K)}$:	$4$	magma:Order(Centralizer(SymmetricGroup(n), G));
Generators:	$(1,26,3,27)(2,25,4,28)(5,30,7,32)(6,29,8,31)(9,21,11,23)(10,22,12,24)(13,18,15,20)(14,17,16,19)$, $(1,14,3,16)(2,13,4,15)(5,10,7,12)(6,9,8,11)(17,26,18,25)(19,27,20,28)(21,29,22,30)(23,31,24,32)$, $(1,7,3,5)(2,8,4,6)(9,24,10,23)(11,22,12,21)(13,17,14,18)(15,19,16,20)(25,30,28,32)(26,29,27,31)$	magma:Generators(G);

Low degree resolvents

$\card{(G/N)}$ Galois groups for stem field(s)
$2$: $C_2$ x 7
$4$: $C_4$ x 4, $C_2^2$ x 7
$8$: $D_{4}$ x 4, $C_4\times C_2$ x 6, $C_2^3$
$16$: $D_4\times C_2$ x 2, $C_2^2:C_4$ x 4, $C_4\times C_2^2$
$32$: $Z_8 : Z_8^\times$, $C_4\wr C_2$ x 2, $C_2^3 : C_4 $ x 2, $C_2 \times (C_2^2:C_4)$, 16T32
$64$: $((C_8 : C_2):C_2):C_2$ x 2, 16T76, 16T111, 32T264
$128$: 16T227, 16T234, 16T254

Resolvents shown for degrees $\leq 47$

$\card{(G/N)}$	Galois groups for stem field(s)
$2$:	$C_2$ x 7
$4$:	$C_4$ x 4, $C_2^2$ x 7
$8$:	$D_{4}$ x 4, $C_4\times C_2$ x 6, $C_2^3$
$16$:	$D_4\times C_2$ x 2, $C_2^2:C_4$ x 4, $C_4\times C_2^2$
$32$:	$Z_8 : Z_8^\times$, $C_4\wr C_2$ x 2, $C_2^3 : C_4 $ x 2, $C_2 \times (C_2^2:C_4)$, 16T32
$64$:	$((C_8 : C_2):C_2):C_2$ x 2, 16T76, 16T111, 32T264
$128$:	16T227, 16T234, 16T254

Subfields

Degree 2: $C_2$ x 3
Degree 4: $C_2^2$, $D_{4}$ x 2
Degree 8: $D_4\times C_2$, $(((C_4 \times C_2): C_2):C_2):C_2$ x 2
Degree 16: 16T32, 16T250, 16T283

Low degree siblings

32T4016 x 4, 32T4394 x 2, 32T4401, 32T4438 x 8, 32T5321 x 2, 32T5326 x 2, 32T5347 x 4
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	$2^{16}$	$1$	$2$	$16$	$( 1, 4)( 2, 3)( 5, 8)( 6, 7)( 9,12)(10,11)(13,16)(14,15)(17,20)(18,19)(21,24)(22,23)(25,27)(26,28)(29,32)(30,31)$
2B	$2^{16}$	$1$	$2$	$16$	$( 1, 3)( 2, 4)( 5, 7)( 6, 8)( 9,11)(10,12)(13,15)(14,16)(17,19)(18,20)(21,23)(22,24)(25,28)(26,27)(29,31)(30,32)$
2C	$2^{16}$	$1$	$2$	$16$	$( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22)(23,24)(25,26)(27,28)(29,30)(31,32)$
2D	$2^{16}$	$2$	$2$	$16$	$( 1, 7)( 2, 8)( 3, 5)( 4, 6)( 9,15)(10,16)(11,13)(12,14)(17,24)(18,23)(19,22)(20,21)(25,31)(26,32)(27,30)(28,29)$
2E	$2^{16}$	$2$	$2$	$16$	$( 1, 6)( 2, 5)( 3, 8)( 4, 7)( 9,14)(10,13)(11,16)(12,15)(17,21)(18,22)(19,23)(20,24)(25,30)(26,29)(27,31)(28,32)$
2F	$2^{16}$	$2$	$2$	$16$	$( 1, 3)( 2, 4)( 5, 7)( 6, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22)(23,24)(25,28)(26,27)(29,31)(30,32)$
2G	$2^{8},1^{16}$	$2$	$2$	$8$	$( 9,12)(10,11)(13,16)(14,15)(17,20)(18,19)(21,24)(22,23)$
2H	$2^{16}$	$4$	$2$	$16$	$( 1, 5)( 2, 6)( 3, 7)( 4, 8)( 9,16)(10,15)(11,14)(12,13)(17,23)(18,24)(19,21)(20,22)(25,29)(26,30)(27,32)(28,31)$
2I	$2^{8},1^{16}$	$4$	$2$	$8$	$( 9,12)(10,11)(13,16)(14,15)(25,27)(26,28)(29,32)(30,31)$
2J	$2^{16}$	$4$	$2$	$16$	$( 1, 5)( 2, 6)( 3, 7)( 4, 8)( 9,16)(10,15)(11,14)(12,13)(17,22)(18,21)(19,24)(20,23)(25,32)(26,31)(27,29)(28,30)$
2K	$2^{16}$	$4$	$2$	$16$	$( 1, 7)( 2, 8)( 3, 5)( 4, 6)( 9,14)(10,13)(11,16)(12,15)(17,24)(18,23)(19,22)(20,21)(25,30)(26,29)(27,31)(28,32)$
2L	$2^{16}$	$4$	$2$	$16$	$( 1, 3)( 2, 4)( 5, 7)( 6, 8)( 9,10)(11,12)(13,14)(15,16)(17,19)(18,20)(21,23)(22,24)(25,26)(27,28)(29,30)(31,32)$
2M	$2^{12},1^{8}$	$8$	$2$	$12$	$( 5, 7)( 6, 8)( 9,20)(10,19)(11,18)(12,17)(13,21)(14,22)(15,23)(16,24)(29,31)(30,32)$
2N	$2^{16}$	$8$	$2$	$16$	$( 1,30)( 2,29)( 3,32)( 4,31)( 5,27)( 6,28)( 7,26)( 8,25)( 9,12)(10,11)(13,14)(15,16)(17,18)(19,20)(21,24)(22,23)$
4A1	$4^{8}$	$4$	$4$	$24$	$( 1,27, 3,26)( 2,28, 4,25)( 5,32, 7,30)( 6,31, 8,29)( 9,22,11,24)(10,21,12,23)(13,17,15,19)(14,18,16,20)$
4A-1	$4^{8}$	$4$	$4$	$24$	$( 1,26, 3,27)( 2,25, 4,28)( 5,30, 7,32)( 6,29, 8,31)( 9,24,11,22)(10,23,12,21)(13,19,15,17)(14,20,16,18)$
4B1	$4^{8}$	$4$	$4$	$24$	$( 1,29, 2,30)( 3,31, 4,32)( 5,25, 6,26)( 7,28, 8,27)( 9,17,10,18)(11,19,12,20)(13,22,14,21)(15,24,16,23)$
4B-1	$4^{8}$	$4$	$4$	$24$	$( 1,30, 2,29)( 3,32, 4,31)( 5,26, 6,25)( 7,27, 8,28)( 9,18,10,17)(11,20,12,19)(13,21,14,22)(15,23,16,24)$
4C	$4^{8}$	$8$	$4$	$24$	$( 1,30, 3,32)( 2,29, 4,31)( 5,26, 7,27)( 6,25, 8,28)( 9,19,11,17)(10,20,12,18)(13,24,15,22)(14,23,16,21)$
4D	$4^{8}$	$8$	$4$	$24$	$( 1, 5, 3, 7)( 2, 6, 4, 8)( 9,23,11,21)(10,24,12,22)(13,18,15,20)(14,17,16,19)(25,29,28,31)(26,30,27,32)$
4E	$4^{8}$	$8$	$4$	$24$	$( 1,26, 3,27)( 2,25, 4,28)( 5,32, 7,30)( 6,31, 8,29)( 9,16,11,14)(10,15,12,13)(17,21,19,23)(18,22,20,24)$
4F	$4^{8}$	$8$	$4$	$24$	$( 1,26, 2,25)( 3,27, 4,28)( 5,30, 6,29)( 7,32, 8,31)( 9,21,10,22)(11,23,12,24)(13,18,14,17)(15,20,16,19)$
4G1	$4^{8}$	$8$	$4$	$24$	$( 1,22, 3,24)( 2,21, 4,23)( 5,17, 7,19)( 6,18, 8,20)( 9,28,10,27)(11,25,12,26)(13,31,14,32)(15,29,16,30)$
4G-1	$4^{8}$	$8$	$4$	$24$	$( 1,24, 3,22)( 2,23, 4,21)( 5,19, 7,17)( 6,20, 8,18)( 9,27,10,28)(11,26,12,25)(13,32,14,31)(15,30,16,29)$
4H1	$4^{8}$	$8$	$4$	$24$	$( 1,24, 3,22)( 2,23, 4,21)( 5,19, 7,17)( 6,20, 8,18)( 9,25,10,26)(11,28,12,27)(13,29,14,30)(15,31,16,32)$
4H-1	$4^{8}$	$8$	$4$	$24$	$( 1,22, 3,24)( 2,21, 4,23)( 5,17, 7,19)( 6,18, 8,20)( 9,26,10,25)(11,27,12,28)(13,30,14,29)(15,32,16,31)$
4I1	$4^{8}$	$8$	$4$	$24$	$( 1,19, 3,17)( 2,20, 4,18)( 5,24, 7,22)( 6,23, 8,21)( 9,29,10,30)(11,31,12,32)(13,25,14,26)(15,28,16,27)$
4I-1	$4^{8}$	$8$	$4$	$24$	$( 1,17, 3,19)( 2,18, 4,20)( 5,22, 7,24)( 6,21, 8,23)( 9,30,10,29)(11,32,12,31)(13,26,14,25)(15,27,16,28)$
4J1	$4^{8}$	$8$	$4$	$24$	$( 1,17, 3,19)( 2,18, 4,20)( 5,22, 7,24)( 6,21, 8,23)( 9,31,10,32)(11,29,12,30)(13,28,14,27)(15,25,16,26)$
4J-1	$4^{8}$	$8$	$4$	$24$	$( 1,19, 3,17)( 2,20, 4,18)( 5,24, 7,22)( 6,23, 8,21)( 9,32,10,31)(11,30,12,29)(13,27,14,28)(15,26,16,25)$
4K	$4^{8}$	$16$	$4$	$24$	$( 1, 7, 3, 5)( 2, 8, 4, 6)( 9,24,10,23)(11,22,12,21)(13,17,14,18)(15,19,16,20)(25,30,28,32)(26,29,27,31)$
4L	$4^{4},2^{6},1^{4}$	$16$	$4$	$18$	$( 1, 3)( 2, 4)( 9,19,12,18)(10,20,11,17)(13,22,16,23)(14,21,15,24)(25,26)(27,28)(29,32)(30,31)$
8A1	$8^{4}$	$16$	$8$	$28$	$( 1,12,29,20, 2,11,30,19)( 3,10,31,18, 4, 9,32,17)( 5,14,25,21, 6,13,26,22)( 7,16,28,23, 8,15,27,24)$
8A-1	$8^{4}$	$16$	$8$	$28$	$( 1,10,32,18, 2, 9,31,17)( 3,12,30,20, 4,11,29,19)( 5,16,27,23, 6,15,28,24)( 7,14,26,21, 8,13,25,22)$
8B1	$8^{4}$	$16$	$8$	$28$	$( 1,16,31,21, 2,15,32,22)( 3,14,29,23, 4,13,30,24)( 5,10,28,20, 6, 9,27,19)( 7,12,25,18, 8,11,26,17)$
8B-1	$8^{4}$	$16$	$8$	$28$	$( 1,14,30,23, 2,13,29,24)( 3,16,32,21, 4,15,31,22)( 5,12,26,18, 6,11,25,17)( 7,10,27,20, 8, 9,28,19)$

Malle's constant $a(G)$: $1/8$

magma:ConjugacyClasses(G);

Character table

37 x 37 character table

magma:CharacterTable(G);

Regular extensions

Data not computed

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 $\F_{q}$
Belyi maps

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more

Group invariants

Group action invariants

Low degree resolvents

Subfields

Low degree siblings

Conjugacy classes

Character table

Regular extensions

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	32T4401

Degree	$32$
Order	$256$
Cyclic	no
Abelian	no
Solvable	yes
Primitive	no
$p$-group	yes
Group:	$C_2^5.D_4$