LMFDB - Galois group: 30T15

Show commands: Magma

magma: G := TransitiveGroup(30, 15);

Group action invariants

Degree $n$:	$30$	magma: t, n := TransitiveGroupIdentification(G); n;
Transitive number $t$:	$15$	magma: t, n := TransitiveGroupIdentification(G); t;
Group:	$S_3\times C_{15}$
Parity:	$-1$	magma: IsEven(G);
Primitive:	no	magma: IsPrimitive(G);
Nilpotency class:	$-1$ (not nilpotent)	magma: NilpotencyClass(G);
$\card{\Aut(F/K)}$:	$15$	magma: Order(Centralizer(SymmetricGroup(n), G));
Generators:	(1,2,3)(4,5,6)(7,8,9)(10,11,12)(13,14,15)(16,18,17)(19,21,20)(22,24,23)(25,27,26)(28,30,29), (1,30,13,26,11,22,7,19,5,18,3,29,15,25,10,24,9,21,4,17,2,28,14,27,12,23,8,20,6,16)	magma: Generators(G);

Low degree resolvents

|G/N| Galois groups for stem field(s)
$2$: $C_2$
$3$: $C_3$
$5$: $C_5$
$6$: $S_3$, $C_6$
$10$: $C_{10}$
$15$: $C_{15}$
$18$: $S_3\times C_3$
$30$: $S_3 \times C_5$, $C_{30}$

Resolvents shown for degrees $\leq 47$

\|G/N\|	Galois groups for stem field(s)
$2$:	$C_2$
$3$:	$C_3$
$5$:	$C_5$
$6$:	$S_3$, $C_6$
$10$:	$C_{10}$
$15$:	$C_{15}$
$18$:	$S_3\times C_3$
$30$:	$S_3 \times C_5$, $C_{30}$

Subfields

Degree 2: $C_2$
Degree 3: None
Degree 5: $C_5$
Degree 6: $S_3\times C_3$
Degree 10: $C_{10}$
Degree 15: None

Low degree siblings

45T3
Siblings are shown with degree $\leq 47$

A number field with this Galois group has no arithmetically equivalent fields.

Conjugacy classes

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

magma: ConjugacyClasses(G);

Group invariants

Order:	$90=2 \cdot 3^{2} \cdot 5$	magma: Order(G);
Cyclic:	no	magma: IsCyclic(G);
Abelian:	no	magma: IsAbelian(G);
Solvable:	yes	magma: IsSolvable(G);
Label:	90.6	magma: IdentifyGroup(G);

Character table: not available.

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	30T15
Degree	$30$
Order	$90$
Cyclic	no
Abelian	no
Solvable	yes
Primitive	no
$p$-group	no
Group:	$S_3\times C_{15}$