LMFDB - Galois group: 32T3215

Show commands: Magma

magma:G := TransitiveGroup(32, 3215);

Group invariants

Abstract group:	$(C_4\times C_8).D_4$	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:	$6$	magma:NilpotencyClass(G);

Group action invariants

Degree $n$:	$32$	magma:t, n := TransitiveGroupIdentification(G); n;
Transitive number $t$:	$3215$	magma:t, n := TransitiveGroupIdentification(G); t;
Parity:	$1$	magma:IsEven(G);
Primitive:	no	magma:IsPrimitive(G);
$\card{\Aut(F/K)}$:	$8$	magma:Order(Centralizer(SymmetricGroup(n), G));
Generators:	$(1,30,8,28,11,22,15,19,3,32,5,25,9,23,14,18)(2,29,7,27,12,21,16,20,4,31,6,26,10,24,13,17)$, $(1,24,12,32)(2,23,11,31)(3,21,10,30)(4,22,9,29)(5,27,7,25)(6,28,8,26)(13,18,14,17)(15,20,16,19)$	magma:Generators(G);

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, $C_4\times C_2$
$16$: $C_2^2:C_4$
$32$: $C_2^3 : C_4 $
$64$: $((C_8 : C_2):C_2):C_2$
$128$: 16T335

Resolvents shown for degrees $\leq 47$

$\card{(G/N)}$	Galois groups for stem field(s)
$2$:	$C_2$ x 3
$4$:	$C_4$ x 2, $C_2^2$
$8$:	$D_{4}$ x 2, $C_4\times C_2$
$16$:	$C_2^2:C_4$
$32$:	$C_2^3 : C_4 $
$64$:	$((C_8 : C_2):C_2):C_2$
$128$:	16T335

Subfields

Degree 2: $C_2$
Degree 4: $C_4$, $D_{4}$ x 2
Degree 8: $C_2^2:C_4$, $((C_8 : C_2):C_2):C_2$, $(((C_4 \times C_2): C_2):C_2):C_2$
Degree 16: 16T158, 16T658 x 2

Low degree siblings

16T658 x 2, 32T3214, 32T3216, 32T7128, 32T7142, 32T7145, 32T7297
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	$2^{16}$	$1$	$2$	$16$	$( 1, 3)( 2, 4)( 5, 8)( 6, 7)( 9,11)(10,12)(13,16)(14,15)(17,20)(18,19)(21,24)(22,23)(25,28)(26,27)(29,31)(30,32)$
2B	$2^{8},1^{16}$	$2$	$2$	$8$	$(17,20)(18,19)(21,24)(22,23)(25,28)(26,27)(29,31)(30,32)$
2C	$2^{12},1^{8}$	$16$	$2$	$12$	$( 1, 6)( 2, 5)( 3, 7)( 4, 8)( 9,13)(10,14)(11,16)(12,15)(21,24)(22,23)(29,31)(30,32)$
2D	$2^{16}$	$16$	$2$	$16$	$( 1, 2)( 3, 4)( 5,13)( 6,14)( 7,15)( 8,16)( 9,12)(10,11)(17,19)(18,20)(21,32)(22,31)(23,29)(24,30)(25,26)(27,28)$
4A	$4^{4},2^{8}$	$4$	$4$	$20$	$( 1, 3)( 2, 4)( 5, 8)( 6, 7)( 9,11)(10,12)(13,16)(14,15)(17,26,20,27)(18,25,19,28)(21,29,24,31)(22,30,23,32)$
4B	$4^{8}$	$4$	$4$	$24$	$( 1, 9, 3,11)( 2,10, 4,12)( 5,15, 8,14)( 6,16, 7,13)(17,27,20,26)(18,28,19,25)(21,31,24,29)(22,32,23,30)$
4C	$4^{4},1^{16}$	$4$	$4$	$12$	$(17,26,20,27)(18,25,19,28)(21,29,24,31)(22,30,23,32)$
4D1	$4^{4},2^{8}$	$8$	$4$	$20$	$( 1,16)( 2,15)( 3,13)( 4,14)( 5,10)( 6, 9)( 7,11)( 8,12)(17,27,20,26)(18,28,19,25)(21,29,24,31)(22,30,23,32)$
4D-1	$4^{4},2^{8}$	$8$	$4$	$20$	$( 1,13)( 2,14)( 3,16)( 4,15)( 5,12)( 6,11)( 7, 9)( 8,10)(17,26,20,27)(18,25,19,28)(21,31,24,29)(22,32,23,30)$
4E	$4^{8}$	$16$	$4$	$24$	$( 1,16, 3,13)( 2,15, 4,14)( 5,12, 8,10)( 6,11, 7, 9)(17,23,20,22)(18,24,19,21)(25,29,28,31)(26,30,27,32)$
4F1	$4^{8}$	$32$	$4$	$24$	$( 1,26, 2,25)( 3,27, 4,28)( 5,29,13,23)( 6,30,14,24)( 7,32,15,21)( 8,31,16,22)( 9,17,12,19)(10,18,11,20)$
4F-1	$4^{8}$	$32$	$4$	$24$	$( 1,25, 2,26)( 3,28, 4,27)( 5,23,13,29)( 6,24,14,30)( 7,21,15,32)( 8,22,16,31)( 9,19,12,17)(10,20,11,18)$
8A1	$8^{4}$	$4$	$8$	$28$	$( 1,14, 9, 5, 3,15,11, 8)( 2,13,10, 6, 4,16,12, 7)(17,31,27,24,20,29,26,21)(18,32,28,23,19,30,25,22)$
8A-1	$8^{4}$	$4$	$8$	$28$	$( 1, 8,11,15, 3, 5, 9,14)( 2, 7,12,16, 4, 6,10,13)(17,21,26,29,20,24,27,31)(18,22,25,30,19,23,28,32)$
8B	$8^{4}$	$8$	$8$	$28$	$( 1,15, 9, 8, 3,14,11, 5)( 2,16,10, 7, 4,13,12, 6)(17,31,27,24,20,29,26,21)(18,32,28,23,19,30,25,22)$
8C1	$8^{2},2^{8}$	$16$	$8$	$22$	$( 1, 2)( 3, 4)( 5,16)( 6,15)( 7,14)( 8,13)( 9,12)(10,11)(17,29,26,24,20,31,27,21)(18,30,25,23,19,32,28,22)$
8C3	$8^{2},2^{8}$	$16$	$8$	$22$	$( 1,14,11, 8, 3,15, 9, 5)( 2,13,12, 7, 4,16,10, 6)(17,25)(18,26)(19,27)(20,28)(21,22)(23,24)(29,32)(30,31)$
16A1	$16^{2}$	$16$	$16$	$30$	$( 1,26,14,21, 9,17, 5,31, 3,27,15,24,11,20, 8,29)( 2,25,13,22,10,18, 6,32, 4,28,16,23,12,19, 7,30)$
16A-1	$16^{2}$	$16$	$16$	$30$	$( 1,18,14,23, 9,25, 5,32, 3,19,15,22,11,28, 8,30)( 2,17,13,24,10,26, 6,31, 4,20,16,21,12,27, 7,29)$
16A3	$16^{2}$	$16$	$16$	$30$	$( 1,24, 5,26,11,31,14,20, 3,21, 8,27, 9,29,15,17)( 2,23, 6,25,12,32,13,19, 4,22, 7,28,10,30,16,18)$
16A-3	$16^{2}$	$16$	$16$	$30$	$( 1,17,15,29, 9,27, 8,21, 3,20,14,31,11,26, 5,24)( 2,18,16,30,10,28, 7,22, 4,19,13,32,12,25, 6,23)$

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

magma:ConjugacyClasses(G);

Character table

		1A	2A	2B	2C	2D	4A	4B	4C	4D1	4D-1	4E	4F1	4F-1	8A1	8A-1	8B	8C1	8C3	16A1	16A-1	16A3	16A-3
	Size	1	1	2	16	16	4	4	4	8	8	16	32	32	4	4	8	16	16	16	16	16	16
	2 P	1A	1A	1A	1A	1A	2B	2A	2B	2B	2B	2A	2D	2D	4B	4B	4B	4C	4C	8A1	8A-1	8A1	8A-1
	Type
256.507.1a	R	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
256.507.1b	R	$1$	$1$	$1$	$- 1$	$1$	$1$	$1$	$1$	$- 1$	$- 1$	$1$	$- 1$	$- 1$	$1$	$1$	$1$	$- 1$	$- 1$	$1$	$1$	$1$	$1$
256.507.1c	R	$1$	$1$	$1$	$- 1$	$1$	$1$	$1$	$1$	$- 1$	$- 1$	$1$	$1$	$1$	$1$	$1$	$1$	$- 1$	$- 1$	$- 1$	$- 1$	$- 1$	$- 1$
256.507.1d	R	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$- 1$	$- 1$	$1$	$1$	$1$	$1$	$1$	$- 1$	$- 1$	$- 1$	$- 1$
256.507.1e1	C	$1$	$1$	$1$	$- 1$	$- 1$	$1$	$1$	$1$	$- 1$	$- 1$	$1$	$- i$	$i$	$- 1$	$- 1$	$- 1$	$1$	$1$	$i$	$- i$	$- i$	$i$
256.507.1e2	C	$1$	$1$	$1$	$- 1$	$- 1$	$1$	$1$	$1$	$- 1$	$- 1$	$1$	$i$	$- i$	$- 1$	$- 1$	$- 1$	$1$	$1$	$- i$	$i$	$i$	$- i$
256.507.1f1	C	$1$	$1$	$1$	$1$	$- 1$	$1$	$1$	$1$	$1$	$1$	$1$	$- i$	$i$	$- 1$	$- 1$	$- 1$	$- 1$	$- 1$	$- i$	$i$	$i$	$- i$
256.507.1f2	C	$1$	$1$	$1$	$1$	$- 1$	$1$	$1$	$1$	$1$	$1$	$1$	$i$	$- i$	$- 1$	$- 1$	$- 1$	$- 1$	$- 1$	$i$	$- i$	$- i$	$i$
256.507.2a	R	$2$	$2$	$2$	$0$	$- 2$	$2$	$2$	$2$	$0$	$0$	$- 2$	$0$	$0$	$2$	$2$	$2$	$0$	$0$	$0$	$0$	$0$	$0$
256.507.2b	R	$2$	$2$	$2$	$0$	$2$	$2$	$2$	$2$	$0$	$0$	$- 2$	$0$	$0$	$- 2$	$- 2$	$- 2$	$0$	$0$	$0$	$0$	$0$	$0$
256.507.4a	R	$4$	$4$	$4$	$0$	$0$	$- 4$	$4$	$- 4$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$
256.507.4b	R	$4$	$4$	$4$	$- 2$	$0$	$0$	$- 4$	$0$	$2$	$2$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$
256.507.4c	R	$4$	$4$	$4$	$2$	$0$	$0$	$- 4$	$0$	$- 2$	$- 2$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$
256.507.4d1	C	$4$	$4$	$- 4$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$- 2$	$- 2$	$2$	$0$	$0$	$- ζ_{8} - ζ_{8}^{3}$	$ζ_{8} + ζ_{8}^{3}$	$- ζ_{8} - ζ_{8}^{3}$	$ζ_{8} + ζ_{8}^{3}$
256.507.4d2	C	$4$	$4$	$- 4$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$- 2$	$- 2$	$2$	$0$	$0$	$ζ_{8} + ζ_{8}^{3}$	$- ζ_{8} - ζ_{8}^{3}$	$ζ_{8} + ζ_{8}^{3}$	$- ζ_{8} - ζ_{8}^{3}$
256.507.4e1	R	$4$	$4$	$- 4$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$2$	$2$	$- 2$	$0$	$0$	$- ζ_{8}^{- 1} - ζ_{8}$	$- ζ_{8}^{- 1} - ζ_{8}$	$ζ_{8}^{- 1} + ζ_{8}$	$ζ_{8}^{- 1} + ζ_{8}$
256.507.4e2	R	$4$	$4$	$- 4$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$2$	$2$	$- 2$	$0$	$0$	$ζ_{8}^{- 1} + ζ_{8}$	$ζ_{8}^{- 1} + ζ_{8}$	$- ζ_{8}^{- 1} - ζ_{8}$	$- ζ_{8}^{- 1} - ζ_{8}$
256.507.4f1	C	$4$	$- 4$	$0$	$0$	$0$	$2$	$0$	$- 2$	$- 2 ζ_{8}^{2}$	$2 ζ_{8}^{2}$	$0$	$0$	$0$	$- 2 ζ_{8} - 2 ζ_{8}^{3}$	$2 ζ_{8} + 2 ζ_{8}^{3}$	$0$	$- ζ_{8}^{- 1} - ζ_{8}$	$ζ_{8}^{- 1} + ζ_{8}$	$0$	$0$	$0$	$0$
256.507.4f2	C	$4$	$- 4$	$0$	$0$	$0$	$2$	$0$	$- 2$	$2 ζ_{8}^{2}$	$- 2 ζ_{8}^{2}$	$0$	$0$	$0$	$2 ζ_{8} + 2 ζ_{8}^{3}$	$- 2 ζ_{8} - 2 ζ_{8}^{3}$	$0$	$- ζ_{8}^{- 1} - ζ_{8}$	$ζ_{8}^{- 1} + ζ_{8}$	$0$	$0$	$0$	$0$
256.507.4f3	C	$4$	$- 4$	$0$	$0$	$0$	$2$	$0$	$- 2$	$- 2 ζ_{8}^{2}$	$2 ζ_{8}^{2}$	$0$	$0$	$0$	$2 ζ_{8} + 2 ζ_{8}^{3}$	$- 2 ζ_{8} - 2 ζ_{8}^{3}$	$0$	$ζ_{8}^{- 1} + ζ_{8}$	$- ζ_{8}^{- 1} - ζ_{8}$	$0$	$0$	$0$	$0$
256.507.4f4	C	$4$	$- 4$	$0$	$0$	$0$	$2$	$0$	$- 2$	$2 ζ_{8}^{2}$	$- 2 ζ_{8}^{2}$	$0$	$0$	$0$	$- 2 ζ_{8} - 2 ζ_{8}^{3}$	$2 ζ_{8} + 2 ζ_{8}^{3}$	$0$	$ζ_{8}^{- 1} + ζ_{8}$	$- ζ_{8}^{- 1} - ζ_{8}$	$0$	$0$	$0$	$0$
256.507.8a	R	$8$	$- 8$	$0$	$0$	$0$	$- 4$	$0$	$4$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$

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	32T3215

Degree	$32$
Order	$256$
Cyclic	no
Abelian	no
Solvable	yes
Primitive	no
$p$-group	yes
Group:	$(C_4\times C_8).D_4$