LMFDB - Abstract group 139968.ct: $C_6^4.(C_3^2\times D

comment:Define the group as a permutation group

magma:G := PermutationGroup< 28 | (1,3,6)(2,5,8,4,7,9)(10,12)(11,13,14,16,15,17)(18,20,23,21,24,26,19,22,25), (1,2,4)(3,5,7)(6,9,8)(10,11)(14,15)(18,19,21)(20,22,24)(23,26,25)(27,28) >;

gap:G := Group( (1,3,6)(2,5,8,4,7,9)(10,12)(11,13,14,16,15,17)(18,20,23,21,24,26,19,22,25), (1,2,4)(3,5,7)(6,9,8)(10,11)(14,15)(18,19,21)(20,22,24)(23,26,25)(27,28) );

sage:G = PermutationGroup(['(1,3,6)(2,5,8,4,7,9)(10,12)(11,13,14,16,15,17)(18,20,23,21,24,26,19,22,25)', '(1,2,4)(3,5,7)(6,9,8)(10,11)(14,15)(18,19,21)(20,22,24)(23,26,25)(27,28)'])

sage_gap:# This uses Sage's interface to GAP, as Sage (currently) has no native support for PC groups G = gap.new('PcGroupCode(698371481729146808123892143331914951026538052259846808520558035776706977485484016460777801089592447687415006485965475245678686620820204607285662550502297445823435256159441066298464134004356253465555598829817960679974523162600082009570879465968321800337683843336801064168072197785728745991082,139968)'); a = G.1; b = G.3; c = G.5; d = G.7; e = G.9; f = G.11; g = G.13;

Group information

Description:	$C_6^4.(C_3^2\times D_6)$
Order:	$139968$$\medspace = 2^{6} \cdot 3^{7} $	comment:Order of the group magma:Order(G); gap:Order(G); sage:G.order() sage_gap:G.Order()
Exponent:	$36$$\medspace = 2^{2} \cdot 3^{2} $	comment:Exponent of the group magma:Exponent(G); gap:Exponent(G); sage:G.exponent() sage_gap:G.Exponent()
Automorphism group:	$(C_2\times C_6^3).C_3^5.C_6^2.C_2^4$, of order $60466176$$\medspace = 2^{10} \cdot 3^{10} $	comment:Automorphism group gap:AutomorphismGroup(G); magma:AutomorphismGroup(G); sage_gap:G.AutomorphismGroup()
Composition factors:	$C_2$ x 6, $C_3$ x 7	comment:Composition factors of the group magma:CompositionFactors(G); gap:CompositionSeries(G); sage:G.composition_series() sage_gap:G.CompositionSeries()
Derived length:	$2$	comment:Derived length of the group magma:DerivedLength(G); gap:DerivedLength(G); sage_gap:G.DerivedLength()

This group is nonabelian and metabelian (hence solvable). Whether it is monomial has not been computed.

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	3	4	6	9	12	18
Elements	1	247	3320	648	46328	20736	16848	51840	139968
Conjugacy classes	1	9	131	2	3821	10	52	18	4044
Divisions	1	9	68	2	1926	5	26	9	2046
Autjugacy classes	1	6	16	1	80	3	3	4	114

Minimal presentations

Permutation degree:	$28$
Transitive degree:	$36$
Rank:	$2$
Inequivalent generating pairs:	not computed

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	6	not computed	not computed
Arbitrary	not computed	not computed	not computed

Constructions

Show commands: Gap / Magma / SageMath

Presentation:	${\langle a, b, c, d, e, f, g \mid c^{9}=d^{6}=e^{6}=f^{6}=g^{2}=[d,e]=[d,f]= \!\cdots\! \rangle}$ < a, b, c, d, e, f, g \| c^9,d^6,e^6,f^6,g^2,[d,e],[d,f],[d,g],[e,f],[e,g],[f,g], c^3a^-6, c^3b^-6, b^3c^5d^5e^3ga^-1b^-1a, b^4c^2de^4f^3ga^-1c^-1a, d^3ef^3a^-1d^-1a, d^5e^4f^3a^-1e^-1a, d^2e^3f^5ga^-1f^-1a, d^3a^-1g^-1a, c^4de^2b^-1c^-1b, d^4e^2gb^-1d^-1b, de^3b^-1e^-1b, d^2ef^2gb^-1f^-1b, d^3f^3gb^-1g^-1b, d^3e^2f^3gc^-1d^-1c, de^5gc^-1e^-1c, d^4e^3fgc^-1f^-1c, d^3e^3f^3gc^-1g^-1c >
comment:Define the group with the given generators and relations magma:G := PCGroup([13, 2, 3, 2, 3, 3, 3, 2, 3, 2, 3, 2, 3, 2, 26, 1418, 3027728, 782628, 106, 563475, 2928448, 510, 7356184, 3661337, 47220, 424363, 251, 3449634, 5159719, 1140080, 832968, 277699, 266, 404359, 269588, 146049, 67438, 22523, 5193404, 4188855, 120076, 780320, 260151, 346, 673929, 84262, 14075, 91308, 30481, 19274122, 9127427, 2285748, 1057963, 352700, 426, 7480523, 2476680, 1347877, 311738, 103959, 164280, 3531787, 2984240, 1574286, 524809]); a,b,c,d,e,f,g := Explode([G.1, G.3, G.5, G.7, G.9, G.11, G.13]); AssignNames(~G, ["a", "a2", "b", "b2", "c", "c3", "d", "d2", "e", "e2", "f", "f2", "g"]); gap:G := PcGroupCode(698371481729146808123892143331914951026538052259846808520558035776706977485484016460777801089592447687415006485965475245678686620820204607285662550502297445823435256159441066298464134004356253465555598829817960679974523162600082009570879465968321800337683843336801064168072197785728745991082,139968); a := G.1; b := G.3; c := G.5; d := G.7; e := G.9; f := G.11; g := G.13; sage:# This uses Sage's interface to GAP, as Sage (currently) has no native support for PC groups G = gap.new('PcGroupCode(698371481729146808123892143331914951026538052259846808520558035776706977485484016460777801089592447687415006485965475245678686620820204607285662550502297445823435256159441066298464134004356253465555598829817960679974523162600082009570879465968321800337683843336801064168072197785728745991082,139968)'); a = G.1; b = G.3; c = G.5; d = G.7; e = G.9; f = G.11; g = G.13; sage_gap:# This uses Sage's interface to GAP, as Sage (currently) has no native support for PC groups G = gap.new('PcGroupCode(698371481729146808123892143331914951026538052259846808520558035776706977485484016460777801089592447687415006485965475245678686620820204607285662550502297445823435256159441066298464134004356253465555598829817960679974523162600082009570879465968321800337683843336801064168072197785728745991082,139968)'); a = G.1; b = G.3; c = G.5; d = G.7; e = G.9; f = G.11; g = G.13;
Permutation group:	Degree $28$ $\langle(1,3,6)(2,5,8,4,7,9)(10,12)(11,13,14,16,15,17)(18,20,23,21,24,26,19,22,25), (1,2,4)(3,5,7)(6,9,8)(10,11)(14,15)(18,19,21)(20,22,24)(23,26,25)(27,28)\rangle$ (1,3,6)(2,5,8,4,7,9)(10,12)(11,13,14,16,15,17)(18,20,23,21,24,26,19,22,25), (1,2,4)(3,5,7)(6,9,8)(10,11)(14,15)(18,19,21)(20,22,24)(23,26,25)(27,28)
comment:Define the group as a permutation group magma:G := PermutationGroup< 28 \| (1,3,6)(2,5,8,4,7,9)(10,12)(11,13,14,16,15,17)(18,20,23,21,24,26,19,22,25), (1,2,4)(3,5,7)(6,9,8)(10,11)(14,15)(18,19,21)(20,22,24)(23,26,25)(27,28) >; gap:G := Group( (1,3,6)(2,5,8,4,7,9)(10,12)(11,13,14,16,15,17)(18,20,23,21,24,26,19,22,25), (1,2,4)(3,5,7)(6,9,8)(10,11)(14,15)(18,19,21)(20,22,24)(23,26,25)(27,28) ); sage:G = PermutationGroup(['(1,3,6)(2,5,8,4,7,9)(10,12)(11,13,14,16,15,17)(18,20,23,21,24,26,19,22,25)', '(1,2,4)(3,5,7)(6,9,8)(10,11)(14,15)(18,19,21)(20,22,24)(23,26,25)(27,28)'])
Transitive group:	36T20901				more information
Direct product:	$C_2$ $\, \times\, $ $(C_6^4.(S_3\times C_3^2))$
Semidirect product:	not computed
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Possibly split product:	$C_6^5$ . $(C_3\times S_3)$ (3)	$(C_3\times C_6^5)$ . $C_6$ (4)	$C_6^5$ . $(C_3\times C_6)$ (2)	$C_3^6$ . $(C_2^4:A_4)$	all 116

Elements of the group are displayed as words in the presentation generators from the presentation above.

Homology

Abelianization:	$C_{6}^{2} \simeq C_{2}^{2} \times C_{3}^{2}$	comment:The abelianization of the group magma:quo< G \| CommutatorSubgroup(G) >; gap:FactorGroup(G, DerivedSubgroup(G)); sage:G.quotient(G.commutator())
Schur multiplier:	$C_{2} \times C_{6}^{2}$	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()

There are 148 normal subgroups (114 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_6$	$G/Z \simeq$ $C_6^4.(C_3\times C_6)$	comment:Center of the group magma:Center(G); gap:Center(G); sage:G.center() sage_gap:G.Center()
Commutator:	$G' \simeq$ $C_3\times C_6^4$	$G/G' \simeq$ $C_6^2$	comment:Commutator subgroup of the group G magma:CommutatorSubgroup(G); gap:DerivedSubgroup(G); sage:G.commutator() sage_gap:G.DerivedSubgroup()
Frattini:	$\Phi \simeq$ $C_3^2\times C_6^2$	$G/\Phi \simeq$ $C_6\times S_3\times A_4$	comment:Frattini subgroup of the group G magma:FrattiniSubgroup(G); gap:FrattiniSubgroup(G); sage:G.frattini_subgroup() sage_gap:G.FrattiniSubgroup()
Fitting:	$\operatorname{Fit} \simeq$ $C_3\times C_6^5$	$G/\operatorname{Fit} \simeq$ $C_6$	comment:Fitting subgroup of the group G magma:FittingSubgroup(G); gap:FittingSubgroup(G); sage:G.fitting_subgroup() sage_gap:G.FittingSubgroup()
Radical:	$R \simeq$ $C_6^4.(C_3^2\times D_6)$	$G/R \simeq$ $C_1$	comment:Radical of the group G magma:Radical(G); gap:SolvableRadical(G); sage_gap:G.SolvableRadical()
Socle:	$\operatorname{soc} \simeq$ $C_2\times C_6^2$	$G/\operatorname{soc} \simeq$ $(C_3^2\times C_6^2):C_6$	comment:Socle of the group G magma:Socle(G); gap:Socle(G); sage:G.socle() sage_gap:G.Socle()
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^3:D_4$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3^2\wr C_3$

Subgroup diagram and profile

Series

Derived series

$C_6^4.(C_3^2\times D_6)$

$\rhd$

$C_3\times C_6^4$

$\rhd$

$C_1$

comment:Derived series of the group GF

magma:DerivedSeries(G);

gap:DerivedSeriesOfGroup(G);

sage:G.derived_series()

sage_gap:G.DerivedSeriesOfGroup()

Chief series

$C_6^4.(C_3^2\times D_6)$

$\rhd$

$C_6^4.(S_3\times C_3^2)$

$\rhd$

$C_6^4.C_3^3$

$\rhd$

$(C_2\times C_6^3).C_3^3$

$\rhd$

$C_3\times C_6^4$

$\rhd$

$C_6^4$

$\rhd$

$C_2\times C_6^3$

$\rhd$

$C_3\times C_6^2$

$\rhd$

$C_3^3$

$\rhd$

$C_3^2$

$\rhd$

$C_3$

$\rhd$

$C_1$

comment:Chief series of the group G

magma:ChiefSeries(G);

gap:ChiefSeries(G);

sage_gap:G.ChiefSeries()

Lower central series

$C_6^4.(C_3^2\times D_6)$

$\rhd$

$C_3\times C_6^4$

$\rhd$

$C_6^4$

$\rhd$

$C_2\times C_6^3$

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

$C_1$

$\lhd$

$C_6$

$\lhd$

$C_3\times C_6$

$\lhd$

$C_3^2\times C_6$

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 14 larger groups in the database.

This group is a maximal quotient of 4 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 $4044 \times 4044$ character table is not available for this group.

Rational character table

The $2046 \times 2046$ 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	139968.ct
Order	\( 2^{6} \cdot 3^{7} \)
Exponent	\( 2^{2} \cdot 3^{2} \)
Nilpotent	no
Solvable	yes
$\card{G^{\mathrm{ab}}}$	\( 2^{2} \cdot 3^{2} \)
$\card{Z(G)}$	\( 2 \cdot 3 \)
$\card{\Aut(G)}$	\( 2^{10} \cdot 3^{10} \)
$\card{\mathrm{Out}(G)}$	\( 2^{5} \cdot 3^{4} \)
Perm deg.	$28$
Trans deg.	$36$
Rank	$2$

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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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

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.