LMFDB - Galois group: 15T43

Show commands: Magma

magma: G := TransitiveGroup(15, 43);

Group action invariants

Degree $n$:	$15$	magma: t, n := TransitiveGroupIdentification(G); n;
Transitive number $t$:	$43$	magma: t, n := TransitiveGroupIdentification(G); t;
Group:	$C_3^4:D_{10}$
CHM label:	$[3^{4}:2]D(5)$
Parity:	$-1$	magma: IsEven(G);
Primitive:	no	magma: IsPrimitive(G);
magma: NilpotencyClass(G);
$\card{\Aut(F/K)}$:	$1$	magma: Order(Centralizer(SymmetricGroup(n), G));
Generators:	(1,4)(2,8)(3,12)(6,9)(7,13)(11,14), (1,6,11)(4,14,9), (1,11)(2,7)(4,14)(5,10)(8,13), (1,4,7,10,13)(2,5,8,11,14)(3,6,9,12,15)	magma: Generators(G);

Low degree resolvents

|G/N| Galois groups for stem field(s)
$2$: $C_2$ x 3
$4$: $C_2^2$
$10$: $D_{5}$
$20$: $D_{10}$

Resolvents shown for degrees $\leq 47$

\|G/N\|	Galois groups for stem field(s)
$2$:	$C_2$ x 3
$4$:	$C_2^2$
$10$:	$D_{5}$
$20$:	$D_{10}$

Subfields

Degree 3: None
Degree 5: $D_{5}$

Low degree siblings

15T43 x 7, 30T290 x 8, 30T291 x 8, 30T297 x 8, 45T205 x 16, 45T206 x 2
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 $	$1$	$1$	$()$
	$ 3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1 $	$10$	$3$	$( 1,11, 6)( 4, 9,14)$
	$ 3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1 $	$10$	$3$	$( 1,11, 6)( 2, 7,12)$
	$ 3, 3, 3, 1, 1, 1, 1, 1, 1 $	$10$	$3$	$( 1, 6,11)( 2, 7,12)( 4, 9,14)$
	$ 3, 3, 3, 1, 1, 1, 1, 1, 1 $	$10$	$3$	$( 1, 6,11)( 4, 9,14)( 5,10,15)$
	$ 3, 3, 3, 3, 1, 1, 1 $	$10$	$3$	$( 1,11, 6)( 2, 7,12)( 4,14, 9)( 5,10,15)$
	$ 3, 3, 3, 3, 1, 1, 1 $	$10$	$3$	$( 1,11, 6)( 2,12, 7)( 4, 9,14)( 5,10,15)$
	$ 3, 3, 3, 3, 1, 1, 1 $	$10$	$3$	$( 1, 6,11)( 2,12, 7)( 4,14, 9)( 5,10,15)$
	$ 3, 3, 3, 3, 3 $	$10$	$3$	$( 1, 6,11)( 2, 7,12)( 3, 8,13)( 4, 9,14)( 5,15,10)$
	$ 5, 5, 5 $	$162$	$5$	$( 1, 4,12,10, 3)( 2,15, 8, 6, 9)( 5,13,11,14, 7)$
	$ 5, 5, 5 $	$162$	$5$	$( 1,12, 3, 4,10)( 2, 8, 9,15, 6)( 5,11, 7,13,14)$
	$ 2, 2, 2, 2, 2, 2, 1, 1, 1 $	$45$	$2$	$( 2,15)( 3, 4)( 5, 7)( 8, 9)(10,12)(13,14)$
	$ 6, 3, 2, 2, 2 $	$90$	$6$	$( 1,11, 6)( 2,15)( 3, 9, 8,14,13, 4)( 5, 7)(10,12)$
	$ 6, 6, 1, 1, 1 $	$90$	$6$	$( 2,15, 7, 5,12,10)( 3,14,13, 9, 8, 4)$
	$ 6, 3, 2, 2, 2 $	$90$	$6$	$( 1,11, 6)( 2,15, 7, 5,12,10)( 3, 4)( 8, 9)(13,14)$
	$ 6, 6, 3 $	$90$	$6$	$( 1, 6,11)( 2,15, 7, 5,12,10)( 3, 9, 8,14,13, 4)$
	$ 2, 2, 2, 2, 2, 1, 1, 1, 1, 1 $	$81$	$2$	$( 6,11)( 7,12)( 8,13)( 9,14)(10,15)$
	$ 10, 5 $	$162$	$10$	$( 1, 4,12, 5,13, 6,14, 2,15, 3)( 7,10, 8,11, 9)$
	$ 10, 5 $	$162$	$10$	$( 1,12,13, 9, 5,11, 2, 8,14,15)( 3, 4,10, 6, 7)$
	$ 6, 2, 2, 2, 2, 1 $	$90$	$6$	$( 2,15,12, 5, 7,10)( 3, 4)( 6,11)( 8,14)( 9,13)$
	$ 6, 6, 2, 1 $	$90$	$6$	$( 1,11)( 2,15,12, 5, 7,10)( 3, 9,13,14, 8, 4)$
	$ 6, 6, 2, 1 $	$90$	$6$	$( 1, 6)( 2,15,12, 5, 7,10)( 3,14, 8, 9,13, 4)$
	$ 6, 2, 2, 2, 2, 1 $	$90$	$6$	$( 2,15)( 3,14, 8, 9,13, 4)( 5,12)( 6,11)( 7,10)$
	$ 2, 2, 2, 2, 2, 2, 2, 1 $	$45$	$2$	$( 1,11)( 2,15)( 3, 4)( 5,12)( 7,10)( 8,14)( 9,13)$

magma: ConjugacyClasses(G);

Group invariants

Order:	$1620=2^{2} \cdot 3^{4} \cdot 5$	magma: Order(G);
Cyclic:	no	magma: IsCyclic(G);
Abelian:	no	magma: IsAbelian(G);
Solvable:	yes	magma: IsSolvable(G);
Nilpotency class:	not nilpotent
Label:	1620.422	magma: IdentifyGroup(G);
Character table:

		1A	2A	2B	2C	3A	3B	3C	3D	3E	3F	3G	3H	5A1	5A2	6A	6B	6C	6D	6E	6F	6G	6H	10A1	10A3
	Size	1	45	45	81	10	10	10	10	10	10	10	10	162	162	90	90	90	90	90	90	90	90	162	162
	2 P	1A	1A	1A	1A	3F	3A	3H	3D	3C	3B	3G	3E	5A2	5A1	3H	3A	3D	3F	3E	3C	3G	3B	5A1	5A2
	3 P	1A	2A	2B	2C	1A	1A	1A	1A	1A	1A	1A	1A	5A2	5A1	2B	2A	2B	2B	2A	2A	2A	2B	10A3	10A1
	5 P	1A	2A	2B	2C	3F	3A	3H	3D	3C	3B	3G	3E	1A	1A	6H	6A	6D	6F	6E	6C	6G	6B	2C	2C
	Type
1620.422.1a	R	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
1620.422.1b	R	$1$	$- 1$	$- 1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$- 1$	$- 1$	$- 1$	$- 1$	$- 1$	$- 1$	$- 1$	$- 1$	$1$	$1$
1620.422.1c	R	$1$	$- 1$	$1$	$- 1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$- 1$	$1$	$- 1$	$1$	$- 1$	$1$	$- 1$	$1$	$- 1$	$- 1$
1620.422.1d	R	$1$	$1$	$- 1$	$- 1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$- 1$	$1$	$- 1$	$1$	$- 1$	$1$	$- 1$	$- 1$	$- 1$
1620.422.2a1	R	$2$	$0$	$0$	$2$	$2$	$2$	$2$	$2$	$2$	$2$	$2$	$2$	$ζ_{5}^{- 2} + ζ_{5}^{2}$	$ζ_{5}^{- 1} + ζ_{5}$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$ζ_{5}^{- 1} + ζ_{5}$	$ζ_{5}^{- 2} + ζ_{5}^{2}$
1620.422.2a2	R	$2$	$0$	$0$	$2$	$2$	$2$	$2$	$2$	$2$	$2$	$2$	$2$	$ζ_{5}^{- 1} + ζ_{5}$	$ζ_{5}^{- 2} + ζ_{5}^{2}$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$ζ_{5}^{- 2} + ζ_{5}^{2}$	$ζ_{5}^{- 1} + ζ_{5}$
1620.422.2b1	R	$2$	$0$	$0$	$- 2$	$2$	$2$	$2$	$2$	$2$	$2$	$2$	$2$	$ζ_{5}^{- 2} + ζ_{5}^{2}$	$ζ_{5}^{- 1} + ζ_{5}$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$- ζ_{5}^{- 1} - ζ_{5}$	$- ζ_{5}^{- 2} - ζ_{5}^{2}$
1620.422.2b2	R	$2$	$0$	$0$	$- 2$	$2$	$2$	$2$	$2$	$2$	$2$	$2$	$2$	$ζ_{5}^{- 1} + ζ_{5}$	$ζ_{5}^{- 2} + ζ_{5}^{2}$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$- ζ_{5}^{- 2} - ζ_{5}^{2}$	$- ζ_{5}^{- 1} - ζ_{5}$
1620.422.10a	R	$10$	$0$	$2$	$0$	$- 2$	$- 5$	$4$	$- 2$	$- 2$	$1$	$4$	$1$	$0$	$0$	$0$	$- 1$	$0$	$2$	$0$	$- 1$	$0$	$- 1$	$0$	$0$
1620.422.10b	R	$10$	$0$	$2$	$0$	$- 2$	$1$	$4$	$1$	$4$	$- 2$	$- 2$	$- 5$	$0$	$0$	$0$	$- 1$	$0$	$- 1$	$0$	$2$	$0$	$- 1$	$0$	$0$
1620.422.10c	R	$10$	$0$	$2$	$0$	$4$	$- 2$	$- 2$	$- 5$	$4$	$1$	$- 2$	$1$	$0$	$0$	$0$	$2$	$0$	$- 1$	$0$	$- 1$	$0$	$- 1$	$0$	$0$
1620.422.10d	R	$10$	$0$	$2$	$0$	$4$	$1$	$- 2$	$1$	$- 2$	$- 5$	$4$	$- 2$	$0$	$0$	$0$	$- 1$	$0$	$- 1$	$0$	$- 1$	$0$	$2$	$0$	$0$
1620.422.10e	R	$10$	$2$	$0$	$0$	$- 5$	$4$	$- 2$	$- 2$	$1$	$- 2$	$1$	$4$	$0$	$0$	$- 1$	$0$	$2$	$0$	$- 1$	$0$	$- 1$	$0$	$0$	$0$
1620.422.10f	R	$10$	$2$	$0$	$0$	$- 2$	$- 2$	$- 5$	$4$	$1$	$4$	$1$	$- 2$	$0$	$0$	$2$	$0$	$- 1$	$0$	$- 1$	$0$	$- 1$	$0$	$0$	$0$
1620.422.10g	R	$10$	$2$	$0$	$0$	$1$	$- 2$	$1$	$4$	$- 2$	$- 2$	$- 5$	$4$	$0$	$0$	$- 1$	$0$	$- 1$	$0$	$2$	$0$	$- 1$	$0$	$0$	$0$
1620.422.10h	R	$10$	$2$	$0$	$0$	$1$	$4$	$1$	$- 2$	$- 5$	$4$	$- 2$	$- 2$	$0$	$0$	$- 1$	$0$	$- 1$	$0$	$- 1$	$0$	$2$	$0$	$0$	$0$
1620.422.10i	R	$10$	$- 2$	$0$	$0$	$- 5$	$4$	$- 2$	$- 2$	$1$	$- 2$	$1$	$4$	$0$	$0$	$1$	$0$	$- 2$	$0$	$1$	$0$	$1$	$0$	$0$	$0$
1620.422.10j	R	$10$	$- 2$	$0$	$0$	$- 2$	$- 2$	$- 5$	$4$	$1$	$4$	$1$	$- 2$	$0$	$0$	$- 2$	$0$	$1$	$0$	$1$	$0$	$1$	$0$	$0$	$0$
1620.422.10k	R	$10$	$- 2$	$0$	$0$	$1$	$- 2$	$1$	$4$	$- 2$	$- 2$	$- 5$	$4$	$0$	$0$	$1$	$0$	$1$	$0$	$- 2$	$0$	$1$	$0$	$0$	$0$
1620.422.10l	R	$10$	$- 2$	$0$	$0$	$1$	$4$	$1$	$- 2$	$- 5$	$4$	$- 2$	$- 2$	$0$	$0$	$1$	$0$	$1$	$0$	$1$	$0$	$- 2$	$0$	$0$	$0$
1620.422.10m	R	$10$	$0$	$- 2$	$0$	$- 2$	$- 5$	$4$	$- 2$	$- 2$	$1$	$4$	$1$	$0$	$0$	$0$	$1$	$0$	$- 2$	$0$	$1$	$0$	$1$	$0$	$0$
1620.422.10n	R	$10$	$0$	$- 2$	$0$	$- 2$	$1$	$4$	$1$	$4$	$- 2$	$- 2$	$- 5$	$0$	$0$	$0$	$1$	$0$	$1$	$0$	$- 2$	$0$	$1$	$0$	$0$
1620.422.10o	R	$10$	$0$	$- 2$	$0$	$4$	$- 2$	$- 2$	$- 5$	$4$	$1$	$- 2$	$1$	$0$	$0$	$0$	$- 2$	$0$	$1$	$0$	$1$	$0$	$1$	$0$	$0$
1620.422.10p	R	$10$	$0$	$- 2$	$0$	$4$	$1$	$- 2$	$1$	$- 2$	$- 5$	$4$	$- 2$	$0$	$0$	$0$	$1$	$0$	$1$	$0$	$1$	$0$	$- 2$	$0$	$0$

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

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	15T43
Degree	$15$
Order	$1620$
Cyclic	no
Abelian	no
Solvable	yes
Primitive	no
$p$-group	no
Group:	$C_3^4:D_{10}$