LMFDB - Abstract group 1728.26510: $C_6^3.C

Show commands: Gap / Magma / SageMath

magma:G := SmallGroup(1728, 26510);

gap:G := SmallGroup(1728, 26510);

sage_gap:G = libgap.SmallGroup(1728, 26510)

comment:Define the group as a permutation group

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

Group information

Description:	$C_6^3.C_2^3$
Order:	$1728$$\medspace = 2^{6} \cdot 3^{3} $	comment:Order of the group magma:Order(G); gap:Order(G); sage:G.order() sage_gap:G.Order()
Exponent:	$12$$\medspace = 2^{2} \cdot 3 $	comment:Exponent of the group magma:Exponent(G); gap:Exponent(G); sage:G.exponent() sage_gap:G.Exponent()
Automorphism group:	Group of order $1327104$$\medspace = 2^{14} \cdot 3^{4} $	comment:Automorphism group gap:AutomorphismGroup(G); magma:AutomorphismGroup(G); sage_gap:G.AutomorphismGroup()
Composition factors:	$C_2$ x 6, $C_3$ x 3	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, supersolvable (hence solvable and monomial), and metabelian.

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	12
Elements	1	7	26	632	182	880	1728
Conjugacy classes	1	7	9	32	63	92	204
Divisions	1	7	9	18	63	54	152
Autjugacy classes	1	7	3	7	21	7	46

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:G.CharacterDegrees()

Dimension	1	2	4	8
Irr. complex chars.	32	88	84	0	204
Irr. rational chars.	8	52	76	16	152

Minimal presentations

Permutation degree:	$21$
Transitive degree:	$576$
Rank:	$3$
Inequivalent generating triples:	$672$

Minimal degrees of faithful linear representations

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

Constructions

Show commands: Gap / Magma / SageMath

Presentation:	$\langle a, b, c, d \mid a^{4}=b^{12}=c^{12}=d^{3}=[a,d]=[b,d]=1, b^{a}=b^{5}c^{6}, c^{a}=c^{5}, c^{b}=c^{7}, d^{c}=d^{2} \rangle$ < a, b, c, d \| a^4,b^12,c^12,d^3,[a,d],[b,d], b^5c^6a^-1b^-1a, c^5a^-1c^-1a, c^7b^-1c^-1b, d^2c^-1d^-1*c >
comment:Define the group with the given generators and relations magma:G := PCGroup([9, -2, -2, -2, -2, -3, -2, -2, -3, -3, 18, 8318, 74, 1443, 102, 1444, 12965, 4559, 158, 30246, 186, 27655, 1997]); a,b,c,d := Explode([G.1, G.3, G.6, G.9]); AssignNames(~G, ["a", "a2", "b", "b2", "b4", "c", "c2", "c4", "d"]); gap:G := PcGroupCode(6428661695450091308275312787197701405986896225761024531,1728); a := G.1; b := G.3; c := G.6; d := G.9; sage:# This uses Sage's interface to GAP, as Sage (currently) has no native support for PC groups G = gap.new('PcGroupCode(6428661695450091308275312787197701405986896225761024531,1728)'); a = G.1; b = G.3; c = G.6; d = G.9; sage_gap:# This uses Sage's interface to GAP, as Sage (currently) has no native support for PC groups G = gap.new('PcGroupCode(6428661695450091308275312787197701405986896225761024531,1728)'); a = G.1; b = G.3; c = G.6; d = G.9;
Permutation group:	Degree $21$ $\langle(1,2,3,4)(5,6)(7,8)(10,11)(15,16,17,18)(20,21), (1,2,3,4)(13,14)(15,17) \!\cdots\! \rangle$ (1,2,3,4)(5,6)(7,8)(10,11)(15,16,17,18)(20,21), (1,2,3,4)(13,14)(15,17)(16,18), (2,4)(5,7,6,8)(15,17)(16,18), (5,6)(7,8), (1,3)(2,4), (1,3)(2,4)(15,17)(16,18), (9,10,11), (19,20,21), (12,13,14)
comment:Define the group as a permutation group magma:G := PermutationGroup< 21 \| (1,2,3,4)(5,6)(7,8)(10,11)(15,16,17,18)(20,21), (1,2,3,4)(13,14)(15,17)(16,18), (2,4)(5,7,6,8)(15,17)(16,18), (5,6)(7,8), (1,3)(2,4), (1,3)(2,4)(15,17)(16,18), (9,10,11), (19,20,21), (12,13,14) >; gap:G := Group( (1,2,3,4)(5,6)(7,8)(10,11)(15,16,17,18)(20,21), (1,2,3,4)(13,14)(15,17)(16,18), (2,4)(5,7,6,8)(15,17)(16,18), (5,6)(7,8), (1,3)(2,4), (1,3)(2,4)(15,17)(16,18), (9,10,11), (19,20,21), (12,13,14) ); sage:G = PermutationGroup(['(1,2,3,4)(5,6)(7,8)(10,11)(15,16,17,18)(20,21)', '(1,2,3,4)(13,14)(15,17)(16,18)', '(2,4)(5,7,6,8)(15,17)(16,18)', '(5,6)(7,8)', '(1,3)(2,4)', '(1,3)(2,4)(15,17)(16,18)', '(9,10,11)', '(19,20,21)', '(12,13,14)'])
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	$(C_6^2.D_6)$ $\,\rtimes\,$ $C_4$	$(C_6^2.D_6)$ $\,\rtimes\,$ $C_4$	$(C_6^2.D_6)$ $\,\rtimes\,$ $C_4$	$(C_3^2:C_{12})$ $\,\rtimes\,$ $C_4^2$	all 13
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_6^3$ . $C_2^3$	$(C_6^2.S_3)$ . $Q_8$	$C_6^2$ . $(S_3\times Q_8)$ (2)	$(C_6^2.D_4)$ . $S_3$	all 106

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

Homology

Abelianization:	$C_{2} \times C_{4}^{2} $	comment:The abelianization of the group magma:quo< G \| CommutatorSubgroup(G) >; gap:FactorGroup(G, DerivedSubgroup(G)); sage:G.quotient(G.commutator())
Schur multiplier:	$C_{2}^{2} \times C_{12}$	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 8880 subgroups in 1596 conjugacy classes, 384 normal (88 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2^3$	$G/Z \simeq$ $C_6:S_3^2$	comment:Center of the group magma:Center(G); gap:Center(G); sage:G.center() sage_gap:G.Center()
Commutator:	$G' \simeq$ $C_3^2\times C_6$	$G/G' \simeq$ $C_2\times C_4^2$	comment:Commutator subgroup of the group G magma:CommutatorSubgroup(G); gap:DerivedSubgroup(G); sage:G.commutator() sage_gap:G.DerivedSubgroup()
Frattini:	$\Phi \simeq$ $C_2^3$	$G/\Phi \simeq$ $C_6:S_3^2$	comment:Frattini subgroup of the group G magma:FrattiniSubgroup(G); gap:FrattiniSubgroup(G); sage:G.frattini_subgroup() sage_gap:G.FrattiniSubgroup()
Fitting:	$\operatorname{Fit} \simeq$ $C_6^2\times C_{12}$	$G/\operatorname{Fit} \simeq$ $C_2^2$	comment:Fitting subgroup of the group G magma:FittingSubgroup(G); gap:FittingSubgroup(G); sage:G.fitting_subgroup() sage_gap:G.FittingSubgroup()
Radical:	$R \simeq$ $C_6^3.C_2^3$	$G/R \simeq$ $C_1$	comment:Radical of the group G magma:Radical(G); gap:SolvableRadical(G); sage_gap:G.SolvableRadical()
Socle:	$\operatorname{soc} \simeq$ $C_6^3$	$G/\operatorname{soc} \simeq$ $C_2^3$	comment:Socle of the group G magma:Socle(G); gap:Socle(G); sage:G.socle() sage_gap:G.Socle()
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_4:C_4^2$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3^3$

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

$C_6^3.C_2^3$

$\rhd$

$C_3^2\times C_6$

$\rhd$

$C_1$

comment:Derived series of the group GF

magma:DerivedSeries(G);

gap:DerivedSeriesOfGroup(G);

sage:G.derived_series()

sage_gap:G.DerivedSeriesOfGroup()

Chief series

$C_6^3.C_2^3$

$\rhd$

$C_6^3.C_2^2$

$\rhd$

$C_6^2:C_{12}$

$\rhd$

$C_6.C_6^2$

$\rhd$

$C_3^2:C_{12}$

$\rhd$

$C_3:C_{12}$

$\rhd$

$C_3\times C_6$

$\rhd$

$C_3^2$

$\rhd$

$C_3$

$\rhd$

$C_1$

comment:Chief series of the group G

magma:ChiefSeries(G);

gap:ChiefSeries(G);

sage_gap:G.ChiefSeries()

Lower central series

$C_6^3.C_2^3$

$\rhd$

$C_3^2\times C_6$

$\rhd$

$C_3^3$

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

$C_1$

$\lhd$

$C_2^3$

$\lhd$

$C_2^2\times C_4$

comment:The upper central series of the group G

magma:UpperCentralSeries(G);

gap:UpperCentralSeriesOfGroup(G);

sage:G.upper_central_series()

sage_gap:G.UpperCentralSeriesOfGroup()

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

See the $204 \times 204$ character table (warning: may be slow to load). Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $152 \times 152$ rational character table (warning: may be slow to load).

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	1728.26510
Order	\( 2^{6} \cdot 3^{3} \)
Exponent	\( 2^{2} \cdot 3 \)

Nilpotent	no
Solvable	yes
$\card{G^{\mathrm{ab}}}$	\( 2^{5} \)
$\card{Z(G)}$	\( 2^{3} \)
$\card{\Aut(G)}$	\( 2^{14} \cdot 3^{4} \)
$\card{\mathrm{Out}(G)}$	\( 2^{11} \cdot 3 \)
Perm deg.	$21$
Trans deg.	$576$
Rank	$3$

Introduction

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Groups

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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

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.