LMFDB - Abstract group 2048.cqm: $(C_4\times D_4^2).D

comment:Define the group as a permutation group

magma:G := PermutationGroup< 20 | (2,6)(3,8)(11,16)(13,18), (1,17)(2,6)(3,8)(4,9)(7,12)(11,16)(13,18)(14,19), (2,3)(6,8)(11,13)(16,18), (1,8,14,2)(3,19,6,17)(4,13,12,16)(7,11,9,18), (5,10,20,15), (1,12,19,9)(4,14,7,17)(6,8)(16,18), (1,19)(2,3)(4,7)(6,8)(9,12)(11,13)(14,17)(16,18), (2,8)(3,6)(4,7)(9,12)(11,16)(13,18), (2,6)(3,8)(4,9)(7,12) >;

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

sage:G = PermutationGroup(['(2,6)(3,8)(11,16)(13,18)', '(1,17)(2,6)(3,8)(4,9)(7,12)(11,16)(13,18)(14,19)', '(2,3)(6,8)(11,13)(16,18)', '(1,8,14,2)(3,19,6,17)(4,13,12,16)(7,11,9,18)', '(5,10,20,15)', '(1,12,19,9)(4,14,7,17)(6,8)(16,18)', '(1,19)(2,3)(4,7)(6,8)(9,12)(11,13)(14,17)(16,18)', '(2,8)(3,6)(4,7)(9,12)(11,16)(13,18)', '(2,6)(3,8)(4,9)(7,12)'])

sage_gap:# This uses Sage's interface to GAP, as Sage (currently) has no native support for PC groups G = gap.new('PcGroupCode(7070840823869233431361319832255775671574661617292883210933703467771887917,2048)'); a = G.1; b = G.3; c = G.4; d = G.6; e = G.8; f = G.9; g = G.11;

Group information

Description:	$(C_4\times D_4^2).D_4$
Order:	$2048$$\medspace = 2^{11} $	comment:Order of the group magma:Order(G); gap:Order(G); sage:G.order() sage_gap:G.Order()
Exponent:	$8$$\medspace = 2^{3} $	comment:Exponent of the group magma:Exponent(G); gap:Exponent(G); sage:G.exponent() sage_gap:G.Exponent()
Automorphism group:	$C_5^4.(C_4\times D_4)$, of order $524288$$\medspace = 2^{19} $	comment:Automorphism group gap:AutomorphismGroup(G); magma:AutomorphismGroup(G); sage_gap:G.AutomorphismGroup()
Composition factors:	$C_2$ x 11	comment:Composition factors of the group magma:CompositionFactors(G); gap:CompositionSeries(G); sage:G.composition_series() sage_gap:G.CompositionSeries()
Nilpotency class:	$4$	comment:Nilpotency class of the group magma:NilpotencyClass(G); gap:NilpotencyClassOfGroup(G); sage_gap:G.NilpotencyClassOfGroup()
Derived length:	$3$	comment:Derived length of the group magma:DerivedLength(G); gap:DerivedLength(G); sage_gap:G.DerivedLength()

This group is nonabelian and a $p$-group (hence nilpotent, solvable, supersolvable, monomial, elementary, and hyperelementary).

comment:Determine if the group G is abelian

magma:IsAbelian(G);

gap:IsAbelian(G);

sage:G.is_abelian()

sage_gap:G.IsAbelian()

comment:Determine if the group G is cyclic

magma:IsCyclic(G);

gap:IsCyclic(G);

sage:G.is_cyclic()

sage_gap:G.IsCyclic()

comment:Determine if the group G is nilpotent

magma:IsNilpotent(G);

gap:IsNilpotentGroup(G);

sage:G.is_nilpotent()

sage_gap:G.IsNilpotentGroup()

comment:Determine if the group G is solvable

magma:IsSolvable(G);

gap:IsSolvableGroup(G);

sage:G.is_solvable()

sage_gap:G.IsSolvableGroup()

comment:Determine if the group G is supersolvable

gap:IsSupersolvableGroup(G);

sage:G.is_supersolvable()

sage_gap:G.IsSupersolvableGroup()

comment:Determine if the group G is simple

magma:IsSimple(G);

gap:IsSimpleGroup(G);

sage_gap:G.IsSimpleGroup()

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))

Order	1	2	4	8
Elements	1	255	1536	256	2048
Conjugacy classes	1	49	154	8	212
Divisions	1	49	103	6	159
Autjugacy classes	1	21	37	2	61

Minimal presentations

Permutation degree:	not computed
Transitive degree:	not computed
Rank:	$5$
Inequivalent generating 5-tuples:	not computed

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	none	not computed	none
Arbitrary	not computed	not computed	not computed

Constructions

Show commands: Gap / Magma / SageMath

Presentation:	${\langle a, b, c, d, e, f, g \mid a^{4}=c^{4}=d^{4}=e^{2}=f^{4}=g^{2}=[a,b]= \!\cdots\! \rangle}$ < a, b, c, d, e, f, g \| a^4,c^4,d^4,e^2,f^4,g^2,[a,b],[a,c],[a,d],[a,e],[a,f],[a,g],[b,f],[b,g],[c,g],[d,e],[d,f],[e,g],[f,g], f^2gb^-2, c^3f^2gb^-1c^-1b, dfb^-1d^-1b, ef^2gb^-1e^-1b, dfc^-1d^-1c, ef^2gc^-1e^-1c, d^2f^3c^-1f^-1c, f^2gd^-1g^-1d, f^3e^-1f^-1e >
comment:Define the group with the given generators and relations magma:G := PCGroup([11, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 22, 12696, 17185, 124, 4779, 2414, 445, 192, 11116, 5583, 36637, 18344, 10337, 4804, 679, 294, 5873]); a,b,c,d,e,f,g := Explode([G.1, G.3, G.4, G.6, G.8, G.9, G.11]); AssignNames(~G, ["a", "a2", "b", "c", "c2", "d", "d2", "e", "f", "f2", "g"]); gap:G := PcGroupCode(7070840823869233431361319832255775671574661617292883210933703467771887917,2048); a := G.1; b := G.3; c := G.4; d := G.6; e := G.8; f := G.9; g := G.11; sage:# This uses Sage's interface to GAP, as Sage (currently) has no native support for PC groups G = gap.new('PcGroupCode(7070840823869233431361319832255775671574661617292883210933703467771887917,2048)'); a = G.1; b = G.3; c = G.4; d = G.6; e = G.8; f = G.9; g = G.11; sage_gap:# This uses Sage's interface to GAP, as Sage (currently) has no native support for PC groups G = gap.new('PcGroupCode(7070840823869233431361319832255775671574661617292883210933703467771887917,2048)'); a = G.1; b = G.3; c = G.4; d = G.6; e = G.8; f = G.9; g = G.11;
Permutation group:	Degree $20$ $\langle(2,6)(3,8)(11,16)(13,18), (1,17)(2,6)(3,8)(4,9)(7,12)(11,16)(13,18)(14,19) \!\cdots\! \rangle$ (2,6)(3,8)(11,16)(13,18), (1,17)(2,6)(3,8)(4,9)(7,12)(11,16)(13,18)(14,19), (2,3)(6,8)(11,13)(16,18), (1,8,14,2)(3,19,6,17)(4,13,12,16)(7,11,9,18), (5,10,20,15), (1,12,19,9)(4,14,7,17)(6,8)(16,18), (1,19)(2,3)(4,7)(6,8)(9,12)(11,13)(14,17)(16,18), (2,8)(3,6)(4,7)(9,12)(11,16)(13,18), (2,6)(3,8)(4,9)(7,12)
comment:Define the group as a permutation group magma:G := PermutationGroup< 20 \| (2,6)(3,8)(11,16)(13,18), (1,17)(2,6)(3,8)(4,9)(7,12)(11,16)(13,18)(14,19), (2,3)(6,8)(11,13)(16,18), (1,8,14,2)(3,19,6,17)(4,13,12,16)(7,11,9,18), (5,10,20,15), (1,12,19,9)(4,14,7,17)(6,8)(16,18), (1,19)(2,3)(4,7)(6,8)(9,12)(11,13)(14,17)(16,18), (2,8)(3,6)(4,7)(9,12)(11,16)(13,18), (2,6)(3,8)(4,9)(7,12) >; gap:G := Group( (2,6)(3,8)(11,16)(13,18), (1,17)(2,6)(3,8)(4,9)(7,12)(11,16)(13,18)(14,19), (2,3)(6,8)(11,13)(16,18), (1,8,14,2)(3,19,6,17)(4,13,12,16)(7,11,9,18), (5,10,20,15), (1,12,19,9)(4,14,7,17)(6,8)(16,18), (1,19)(2,3)(4,7)(6,8)(9,12)(11,13)(14,17)(16,18), (2,8)(3,6)(4,7)(9,12)(11,16)(13,18), (2,6)(3,8)(4,9)(7,12) ); sage:G = PermutationGroup(['(2,6)(3,8)(11,16)(13,18)', '(1,17)(2,6)(3,8)(4,9)(7,12)(11,16)(13,18)(14,19)', '(2,3)(6,8)(11,13)(16,18)', '(1,8,14,2)(3,19,6,17)(4,13,12,16)(7,11,9,18)', '(5,10,20,15)', '(1,12,19,9)(4,14,7,17)(6,8)(16,18)', '(1,19)(2,3)(4,7)(6,8)(9,12)(11,13)(14,17)(16,18)', '(2,8)(3,6)(4,7)(9,12)(11,16)(13,18)', '(2,6)(3,8)(4,9)(7,12)'])
Direct product:	not computed
Semidirect product:	not computed
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Possibly split product:	$(D_4^2:D_4)$ . $C_4$ (16)	$(D_4^2:C_4)$ . $D_4$ (4)	$(C_4\times D_4^2)$ . $D_4$ (4)	$(C_2^5.D_4)$ . $D_4$	all 338
Aut. group:	$\Aut(C_{20}:Q_8)$	$\Aut((C_2^2\times C_4):C_{20})$

Elements of the group are displayed as permutations of degree 20.

Homology

Abelianization:	$C_{2}^{4} \times C_{4} $	comment:The abelianization of the group magma:quo< G \| CommutatorSubgroup(G) >; gap:FactorGroup(G, DerivedSubgroup(G)); sage:G.quotient(G.commutator())
Schur multiplier:	$C_{2}^{12}$	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()

There are 1744 normal subgroups (202 characteristic).

Characteristic subgroups are shown in this color. Normal (but not characteristic) subgroups are shown in this color.

Special subgroups

Center:	a subgroup isomorphic to $C_2\times C_4$	comment:Center of the group magma:Center(G); gap:Center(G); sage:G.center() sage_gap:G.Center()
Commutator:	a subgroup isomorphic to $C_2^2\times D_4$	comment:Commutator subgroup of the group G magma:CommutatorSubgroup(G); gap:DerivedSubgroup(G); sage:G.commutator() sage_gap:G.DerivedSubgroup()
Frattini:	a subgroup isomorphic to $D_4\times C_2^3$	comment:Frattini subgroup of the group G magma:FrattiniSubgroup(G); gap:FrattiniSubgroup(G); sage:G.frattini_subgroup() sage_gap:G.FrattiniSubgroup()
Fitting:	not computed	comment:Fitting subgroup of the group G magma:FittingSubgroup(G); gap:FittingSubgroup(G); sage:G.fitting_subgroup() sage_gap:G.FittingSubgroup()
Radical:	not computed	comment:Radical of the group G magma:Radical(G); gap:SolvableRadical(G); sage_gap:G.SolvableRadical()
Socle:	not computed	comment:Socle of the group G magma:Socle(G); gap:Socle(G); sage:G.socle() sage_gap:G.Socle()
2-Sylow subgroup:	$P_{ 2 } \simeq$ $(C_4\times D_4^2).D_4$

Subgroup diagram and profile

Series

Derived series

not computed

comment:Derived series of the group GF

magma:DerivedSeries(G);

gap:DerivedSeriesOfGroup(G);

sage:G.derived_series()

sage_gap:G.DerivedSeriesOfGroup()

Chief series

not computed

comment:Chief series of the group G

magma:ChiefSeries(G);

gap:ChiefSeries(G);

sage_gap:G.ChiefSeries()

Lower central series

not computed

comment:The lower central series of the group G

magma:LowerCentralSeries(G);

gap:LowerCentralSeriesOfGroup(G);

sage:G.lower_central_series()

sage_gap:G.LowerCentralSeriesOfGroup()

Upper central series

not computed

comment:The upper central series of the group G

magma:UpperCentralSeries(G);

gap:UpperCentralSeriesOfGroup(G);

sage:G.upper_central_series()

sage_gap:G.UpperCentralSeriesOfGroup()

Supergroups

This group is a maximal subgroup of 2 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

Complex character table

The $212 \times 212$ character table is not available for this group.

Rational character table

The $159 \times 159$ rational character table is not available for this group.

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

Label	2048.cqm
Order	\( 2^{11} \)
Exponent	\( 2^{3} \)

Nilpotent	yes
Solvable	yes
$\card{G^{\mathrm{ab}}}$	\( 2^{6} \)
$\card{Z(G)}$	8
$\card{\Aut(G)}$	\( 2^{19} \)
$\card{\mathrm{Out}(G)}$	\( 2^{11} \)
Perm deg.	not computed
Trans deg.	not computed
Rank	$5$

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more

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

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.