LMFDB - Abstract group 3420.a: $\PSL(2,19)$

Show commands: Gap / Magma / Oscar / SageMath

comment:Construction of abstract group

magma:G := PSL(2,19);

gap:G := PSL(2,19);

sage:G = PSL(2,19)

sage_gap:G = gap.new('Group( (2,4,6,9,12,15,18,5,8)(7,10,13,16,19,20,17,11,14), (1,2,3)(4,5,7)(6,8,11)(9,10,12)(14,16,19)(15,17,18) )')

oscar:G = @permutation_group(20, (2,4,6,9,12,15,18,5,8)(7,10,13,16,19,20,17,11,14), (1,2,3)(4,5,7)(6,8,11)(9,10,12)(14,16,19)(15,17,18))

Group information

Description:	$\PSL(2,19)$
Order:	$3420$$\medspace = 2^{2} \cdot 3^{2} \cdot 5 \cdot 19 $	comment:Order of the group magma:Order(G); gap:Order(G); sage:G.order() sage_gap:G.Order() oscar:order(G)
Exponent:	$1710$$\medspace = 2 \cdot 3^{2} \cdot 5 \cdot 19 $	comment:Exponent of the group magma:Exponent(G); gap:Exponent(G); sage:G.exponent() sage_gap:G.Exponent() oscar:exponent(G)
Automorphism group:	$\PGL(2,19)$, of order $6840$$\medspace = 2^{3} \cdot 3^{2} \cdot 5 \cdot 19 $	comment:Automorphism group gap:AutomorphismGroup(G); magma:AutomorphismGroup(G); sage:libgap(G).AutomorphismGroup() sage_gap:G.AutomorphismGroup() oscar:automorphism_group(G)
Composition factors:	$\PSL(2,19)$	comment:Composition factors of the group magma:CompositionFactors(G); gap:CompositionSeries(G); sage:G.composition_series() sage_gap:G.CompositionSeries() oscar:composition_series(G)
Derived length:	$0$	comment:Derived length of the group magma:DerivedLength(G); gap:DerivedLength(G); sage:libgap(G).DerivedLength() sage_gap:G.DerivedLength() oscar:derived_length(G)

This group is nonabelian, simple (hence nonsolvable, perfect, quasisimple, and almost simple), and an A-group.

comment:Determine if the group G is abelian

magma:IsAbelian(G);

gap:IsAbelian(G);

sage:G.is_abelian()

sage_gap:G.IsAbelian()

oscar:is_abelian(G)

comment:Determine if the group G is cyclic

magma:IsCyclic(G);

gap:IsCyclic(G);

sage:G.is_cyclic()

sage_gap:G.IsCyclic()

oscar:is_cyclic(G)

comment:Determine if the group G is nilpotent

magma:IsNilpotent(G);

gap:IsNilpotentGroup(G);

sage:G.is_nilpotent()

sage_gap:G.IsNilpotentGroup()

oscar:is_nilpotent(G)

comment:Determine if the group G is solvable

magma:IsSolvable(G);

gap:IsSolvableGroup(G);

sage:G.is_solvable()

sage_gap:G.IsSolvableGroup()

oscar:is_solvable(G)

comment:Determine if the group G is supersolvable

gap:IsSupersolvableGroup(G);

sage:G.is_supersolvable()

sage_gap:G.IsSupersolvableGroup()

oscar:is_supersolvable(G)

comment:Determine if the group G is simple

magma:IsSimple(G);

gap:IsSimpleGroup(G);

sage:G.is_simple()

sage_gap:G.IsSimpleGroup()

oscar:is_simple(G)

Group statistics

comment:Compute statistics for the group G

magma:// Magma code to output the first two rows of the group statistics table element_orders := [Order(g) : g in G]; orders := Set(element_orders); printf "Orders: %o\n", orders; printf "Elements: %o %o\n", [#[x : x in element_orders | x eq n] : n in orders], Order(G); cc_orders := [cc[1] : cc in ConjugacyClasses(G)]; printf "Conjugacy classes: %o %o\n", [#[x : x in cc_orders | x eq n] : n in orders], #cc_orders;

gap:# Gap code to output the first two rows of the group statistics table element_orders := List(Elements(G), g -> Order(g)); orders := Set(element_orders); Print("Orders: ", orders, "\n"); element_counts := List(orders, n -> Length(Filtered(element_orders, x -> x = n))); Print("Elements: ", element_counts, " ", Size(G), "\n"); cc_orders := List(ConjugacyClasses(G), cc -> Order(Representative(cc))); cc_counts := List(orders, n -> Length(Filtered(cc_orders, x -> x = n))); Print("Conjugacy classes: ", cc_counts, " ", Length(ConjugacyClasses(G)), "\n");

sage:# Sage code to output the first two rows of the group statistics table element_orders = [g.order() for g in G] orders = sorted(list(set(element_orders))) print("Orders:", orders) print("Elements:", [element_orders.count(n) for n in orders], G.order()) cc_orders = [cc[0].order() for cc in G.conjugacy_classes()] print("Conjugacy classes:", [cc_orders.count(n) for n in orders], len(cc_orders))

sage_gap:# Sage code (using the GAP interface) to output the first two rows of the group statistics table element_orders = [g.Order() for g in G.Elements()] orders = sorted(list(set(element_orders))) print("Orders:", orders) print("Elements:", [element_orders.count(n) for n in orders], G.Order()) cc_orders = [cc.Representative().Order() for cc in G.ConjugacyClasses()] print("Conjugacy classes:", [cc_orders.count(n) for n in orders], len(cc_orders))

oscar:# Oscar code to output the first two rows of the group statistics table element_orders = [order(g) for g in elements(G)] orders = sort(unique(element_orders)) println("Orders: ", orders) element_counts = [count(==(n), element_orders) for n in orders] println("Elements: ", element_counts, " ", order(G)) ccs = conjugacy_classes(G) cc_orders = [order(representative(cc)) for cc in ccs] cc_counts = [count(==(n), cc_orders) for n in orders] println("Conjugacy classes: ", cc_counts, " ", length(ccs))

Order	1	2	3	5	9	10	19
Elements	1	171	380	684	1140	684	360	3420
Conjugacy classes	1	1	1	2	3	2	2	12
Divisions	1	1	1	1	1	1	1	7
Autjugacy classes	1	1	1	2	3	2	1	11

comment:Compute statistics about the characters of G

magma:// Outputs [<d_1,c_1>, <d_2,c_2>, ...] where c_i is the number of irr. complex chars. of G with degree d_i CharacterDegrees(G);

gap:# Outputs [[d_1,c_1], [d_2,c_2], ...] where c_i is the number of irr. complex chars. of G with degree d_i CharacterDegrees(G);

sage:# Outputs [[d_1,c_1], [d_2,c_2], ...] where c_i is the number of irr. complex chars. of G with degree d_i character_degrees = [c[0] for c in G.character_table()] [[n, character_degrees.count(n)] for n in set(character_degrees)]

sage_gap:# Outputs [[d_1,c_1], [d_2,c_2], ...] where c_i is the number of irr. complex chars. of G with degree d_i G.CharacterDegrees()

oscar:# Outputs an MSet containing the absolutely irreducible degrees of G and their multiplicities. character_degrees(G)

Dimension	1	9	18	19	20	36	60
Irr. complex chars.	1	2	4	1	4	0	0	12
Irr. rational chars.	1	0	1	1	1	2	1	7

Minimal presentations

Permutation degree:	$20$
Transitive degree:	$20$
Rank:	$2$
Inequivalent generating pairs:	$1570$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	9	18	18
Arbitrary	9	18	18

Constructions

Show commands: Gap / Magma / Oscar / SageMath

Groups of Lie type:	$\PSL(2,19)$, $\PSU(2,19)$, $\Omega(3,19)$, $\POmega(3,19)$, $\PSigmaL(2,19)$
magma:G := PSL(2,19); gap:G := PSL(2,19); sage:G = PSL(2,19) oscar:G = matrix_group([matrix(GF(19), [[2, 0], [0, 10]]), matrix(GF(19), [[18, 1], [18, 0]])]) magma:G := PSU(2,19); gap:G := PSU(2,19); sage:G = PSU(2,19) magma:G := Omega(3,19); gap:G := Omega(3,19); magma:G := POmega(3,19); gap:G := POmega(3,19); magma:G := PSigmaL(2,19); gap:G := PSigmaL(2,19);
Permutation group:	Degree $20$ $\langle(2,4,6,9,12,15,18,5,8)(7,10,13,16,19,20,17,11,14), (1,2,3)(4,5,7)(6,8,11)(9,10,12)(14,16,19)(15,17,18)\rangle$ (2,4,6,9,12,15,18,5,8)(7,10,13,16,19,20,17,11,14), (1,2,3)(4,5,7)(6,8,11)(9,10,12)(14,16,19)(15,17,18)
comment:Define the group as a permutation group magma:G := PermutationGroup< 20 \| (2,4,6,9,12,15,18,5,8)(7,10,13,16,19,20,17,11,14), (1,2,3)(4,5,7)(6,8,11)(9,10,12)(14,16,19)(15,17,18) >; gap:G := Group( (2,4,6,9,12,15,18,5,8)(7,10,13,16,19,20,17,11,14), (1,2,3)(4,5,7)(6,8,11)(9,10,12)(14,16,19)(15,17,18) ); sage:G = PermutationGroup(['(2,4,6,9,12,15,18,5,8)(7,10,13,16,19,20,17,11,14)', '(1,2,3)(4,5,7)(6,8,11)(9,10,12)(14,16,19)(15,17,18)']) sage_gap:G = gap.new('Group( (2,4,6,9,12,15,18,5,8)(7,10,13,16,19,20,17,11,14), (1,2,3)(4,5,7)(6,8,11)(9,10,12)(14,16,19)(15,17,18) )') oscar:G = @permutation_group(20, (2,4,6,9,12,15,18,5,8)(7,10,13,16,19,20,17,11,14), (1,2,3)(4,5,7)(6,8,11)(9,10,12)(14,16,19)(15,17,18))
Transitive group:	20T272	more information
magma:G := TransitiveGroup(20, 272); gap:G := TransitiveGroup(20, 272); sage:G = TransitiveGroup(20, 272) sage_gap:G = libgap.TransitiveGroup(20, 272) oscar:G = transitive_group(20, 272)
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	not isomorphic to a non-trivial semidirect product
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product

Elements of the group are displayed as equivalence classes (represented by square brackets) of matrices in $\SL(2,19)$.

Homology

Abelianization:	$C_1 $	comment:The abelianization of the group magma:quo< G \| CommutatorSubgroup(G) >; gap:FactorGroup(G, DerivedSubgroup(G)); sage:G.quotient(G.commutator()) sage_gap:G.FactorGroup(G.DerivedSubgroup()) oscar:quo(G, derived_subgroup(G)[1])
Schur multiplier:	$C_{2}$	comment:The Schur multiplier of the group gap:AbelianInvariantsMultiplier(G); sage:G.homology(2) sage_gap:G.AbelianInvariantsMultiplier()
Commutator length:	$1$	comment:The commutator length of the group gap:CommutatorLength(G); sage_gap:G.CommutatorLength()

Subgroups

comment:List of subgroups of the group

magma:Subgroups(G);

gap:AllSubgroups(G);

sage:G.subgroups()

sage_gap:G.AllSubgroups()

oscar:subgroups(G)

There are 2912 subgroups in 19 conjugacy classes, 2 normal, and all normal subgroups are characteristic.

Characteristic subgroups are shown in this color.

Special subgroups

Center:	$Z \simeq$ $C_1$	$G/Z \simeq$ $\PSL(2,19)$	comment:Center of the group magma:Center(G); gap:Center(G); sage:G.center() sage_gap:G.Center() oscar:center(G)
Commutator:	$G' \simeq$ $\PSL(2,19)$	$G/G' \simeq$ $C_1$	comment:Commutator subgroup of the group G magma:CommutatorSubgroup(G); gap:DerivedSubgroup(G); sage:G.commutator() sage_gap:G.DerivedSubgroup() oscar:derived_subgroup(G)
Frattini:	$\Phi \simeq$ $C_1$	$G/\Phi \simeq$ $\PSL(2,19)$	comment:Frattini subgroup of the group G magma:FrattiniSubgroup(G); gap:FrattiniSubgroup(G); sage:G.frattini_subgroup() sage_gap:G.FrattiniSubgroup() oscar:frattini_subgroup(G)
Fitting:	$\operatorname{Fit} \simeq$ $C_1$	$G/\operatorname{Fit} \simeq$ $\PSL(2,19)$	comment:Fitting subgroup of the group G magma:FittingSubgroup(G); gap:FittingSubgroup(G); sage:G.fitting_subgroup() sage_gap:G.FittingSubgroup() oscar:fitting_subgroup(G)
Radical:	$R \simeq$ $C_1$	$G/R \simeq$ $\PSL(2,19)$	comment:Radical of the group G magma:Radical(G); gap:SolvableRadical(G); sage_gap:G.SolvableRadical() oscar:solvable_radical(G)
Socle:	$\operatorname{soc} \simeq$ $\PSL(2,19)$	$G/\operatorname{soc} \simeq$ $C_1$	comment:Socle of the group G magma:Socle(G); gap:Socle(G); sage:G.socle() sage_gap:G.Socle() oscar:socle(G)
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^2$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_9$
5-Sylow subgroup:	$P_{ 5 } \simeq$ $C_5$
19-Sylow subgroup:	$P_{ 19 } \simeq$ $C_{19}$

Subgroup diagram and profile

For the default diagram, subgroups are sorted vertically by the number of prime divisors (counted with multiplicity) in their orders.
To see subgroups sorted vertically by order instead, check this box.

Subgroup information

Click on a subgroup in the diagram to see information about it.

See a full page version of the diagram

Series

Derived series

$\PSL(2,19)$

comment:Derived series of the group G

magma:DerivedSeries(G);

gap:DerivedSeriesOfGroup(G);

sage:G.derived_series()

sage_gap:G.DerivedSeriesOfGroup()

oscar:derived_series(G)

Chief series

$\PSL(2,19)$

$\rhd$

$C_1$

comment:Chief series of the group G

magma:ChiefSeries(G);

gap:ChiefSeries(G);

sage:libgap(G).ChiefSeries()

sage_gap:G.ChiefSeries()

oscar:chief_series(G)

Lower central series

$\PSL(2,19)$

comment:The lower central series of the group G

magma:LowerCentralSeries(G);

gap:LowerCentralSeriesOfGroup(G);

sage:G.lower_central_series()

sage_gap:G.LowerCentralSeriesOfGroup()

oscar:lower_central_series(G)

Upper central series

$C_1$

comment:The upper central series of the group G

magma:UpperCentralSeries(G);

gap:UpperCentralSeriesOfGroup(G);

sage:G.upper_central_series()

sage_gap:G.UpperCentralSeriesOfGroup()

oscar:upper_central_series(G)

Supergroups

This group is a maximal subgroup of 4 larger groups in the database.

This group is a maximal quotient of 1 larger groups in the database.

Character theory

comment:Character table

magma:CharacterTable(G); // Output not guaranteed to exactly match the LMFDB table

gap:CharacterTable(G); # Output not guaranteed to exactly match the LMFDB table

sage:G.character_table() # Output not guaranteed to exactly match the LMFDB table

sage_gap:G.CharacterTable() # Output not guaranteed to exactly match the LMFDB table

oscar:character_table(G) # Output not guaranteed to exactly match the LMFDB table

Complex character table

		1A	2A	3A	5A1	5A2	9A1	9A2	9A4	10A1	10A3	19A1	19A-1
	Size	1	171	380	342	342	380	380	380	342	342	180	180
	2 P	1A	1A	3A	5A2	5A1	9A2	9A4	9A1	5A1	5A2	19A-1	19A1
	3 P	1A	2A	1A	5A2	5A1	3A	3A	3A	10A3	10A1	19A-1	19A1
	5 P	1A	2A	3A	1A	1A	9A4	9A1	9A2	2A	2A	19A1	19A-1
	19 P	1A	2A	3A	5A1	5A2	9A1	9A2	9A4	10A1	10A3	1A	1A
	Type
3420.a.1a	R	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
3420.a.9a1	C	$9$	$1$	$0$	$- 1$	$- 1$	$0$	$0$	$0$	$1$	$1$	$- ζ_{19}^{- 8} - ζ_{19}^{- 3} - ζ_{19}^{- 2} - 1 - ζ_{19} - ζ_{19}^{4} - ζ_{19}^{5} - ζ_{19}^{6} - ζ_{19}^{7} - ζ_{19}^{9}$	$ζ_{19}^{- 8} + ζ_{19}^{- 3} + ζ_{19}^{- 2} + ζ_{19} + ζ_{19}^{4} + ζ_{19}^{5} + ζ_{19}^{6} + ζ_{19}^{7} + ζ_{19}^{9}$
3420.a.9a2	C	$9$	$1$	$0$	$- 1$	$- 1$	$0$	$0$	$0$	$1$	$1$	$ζ_{19}^{- 8} + ζ_{19}^{- 3} + ζ_{19}^{- 2} + ζ_{19} + ζ_{19}^{4} + ζ_{19}^{5} + ζ_{19}^{6} + ζ_{19}^{7} + ζ_{19}^{9}$	$- ζ_{19}^{- 8} - ζ_{19}^{- 3} - ζ_{19}^{- 2} - 1 - ζ_{19} - ζ_{19}^{4} - ζ_{19}^{5} - ζ_{19}^{6} - ζ_{19}^{7} - ζ_{19}^{9}$
3420.a.18a1	R	$18$	$2$	$0$	$- ζ_{5}^{- 1} - ζ_{5}$	$- ζ_{5}^{- 2} - ζ_{5}^{2}$	$0$	$0$	$0$	$ζ_{5}^{- 2} + ζ_{5}^{2}$	$ζ_{5}^{- 1} + ζ_{5}$	$- 1$	$- 1$
3420.a.18a2	R	$18$	$2$	$0$	$- ζ_{5}^{- 2} - ζ_{5}^{2}$	$- ζ_{5}^{- 1} - ζ_{5}$	$0$	$0$	$0$	$ζ_{5}^{- 1} + ζ_{5}$	$ζ_{5}^{- 2} + ζ_{5}^{2}$	$- 1$	$- 1$
3420.a.18b1	R	$18$	$- 2$	$0$	$- ζ_{5}^{- 1} - ζ_{5}$	$- ζ_{5}^{- 2} - ζ_{5}^{2}$	$0$	$0$	$0$	$- ζ_{5}^{- 2} - ζ_{5}^{2}$	$- ζ_{5}^{- 1} - ζ_{5}$	$- 1$	$- 1$
3420.a.18b2	R	$18$	$- 2$	$0$	$- ζ_{5}^{- 2} - ζ_{5}^{2}$	$- ζ_{5}^{- 1} - ζ_{5}$	$0$	$0$	$0$	$- ζ_{5}^{- 1} - ζ_{5}$	$- ζ_{5}^{- 2} - ζ_{5}^{2}$	$- 1$	$- 1$
3420.a.19a	R	$19$	$- 1$	$1$	$- 1$	$- 1$	$1$	$1$	$1$	$- 1$	$- 1$	$0$	$0$
3420.a.20a	R	$20$	$0$	$2$	$0$	$0$	$- 1$	$- 1$	$- 1$	$0$	$0$	$1$	$1$
3420.a.20b1	R	$20$	$0$	$- 1$	$0$	$0$	$ζ_{9}^{- 4} + ζ_{9}^{4}$	$ζ_{9}^{- 1} + ζ_{9}$	$ζ_{9}^{- 2} + ζ_{9}^{2}$	$0$	$0$	$1$	$1$
3420.a.20b2	R	$20$	$0$	$- 1$	$0$	$0$	$ζ_{9}^{- 2} + ζ_{9}^{2}$	$ζ_{9}^{- 4} + ζ_{9}^{4}$	$ζ_{9}^{- 1} + ζ_{9}$	$0$	$0$	$1$	$1$
3420.a.20b3	R	$20$	$0$	$- 1$	$0$	$0$	$ζ_{9}^{- 1} + ζ_{9}$	$ζ_{9}^{- 2} + ζ_{9}^{2}$	$ζ_{9}^{- 4} + ζ_{9}^{4}$	$0$	$0$	$1$	$1$

Rational character table

		1A	2A	3A	5A	9A	10A	19A
	Size	1	171	380	684	1140	684	360
	2 P	1A	1A	3A	5A	9A	5A	19A
	3 P	1A	2A	1A	5A	3A	10A	19A
	5 P	1A	2A	3A	1A	9A	2A	19A
	19 P	1A	2A	3A	5A	9A	10A	1A
3420.a.1a		$1$	$1$	$1$	$1$	$1$	$1$	$1$
3420.a.9a		$18$	$2$	$0$	$- 2$	$0$	$2$	$- 1$
3420.a.18a		$36$	$4$	$0$	$1$	$0$	$- 1$	$- 2$
3420.a.18b		$36$	$- 4$	$0$	$1$	$0$	$1$	$- 2$
3420.a.19a		$19$	$- 1$	$1$	$- 1$	$1$	$- 1$	$0$
3420.a.20a		$20$	$0$	$2$	$0$	$- 1$	$0$	$1$
3420.a.20b		$60$	$0$	$- 3$	$0$	$0$	$0$	$3$

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$ over number fields
Genus 2 curves
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Label	3420.a
Order	\( 2^{2} \cdot 3^{2} \cdot 5 \cdot 19 \)
Exponent	\( 2 \cdot 3^{2} \cdot 5 \cdot 19 \)

Simple	yes
$\card{G^{\mathrm{ab}}}$	\( 1 \)
$\card{Z(G)}$	\( 1 \)
$\card{\Aut(G)}$	\( 2^{3} \cdot 3^{2} \cdot 5 \cdot 19 \)
$\card{\mathrm{Out}(G)}$	\( 2 \)
Perm deg.	$20$
Trans deg.	$20$
Rank	$2$

Introduction

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Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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Group information

Group statistics

Minimal presentations

Minimal degrees of faithful linear representations

Constructions

Homology

Subgroups

Special subgroups

Subgroup diagram and profile

Classes of subgroups up to conjugation

Classes of subgroups up to automorphism

Normal subgroups

Normal subgroups up to automorphism

Subgroup information

Classes of subgroups up to conjugation

Classes of subgroups up to automorphism

Normal subgroups (quotient in parentheses)

Normal subgroups up to automorphism (quotient in parentheses)

Series

Supergroups

Character theory

Complex character table

Rational character table

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.