LMFDB - Abstract group 3456.lz: $C_6^2:C_2^2:S

comment:Define the group as a permutation group

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

gap:G := Group( (4,7,5)(6,14)(8,13)(9,12)(10,11), (1,3,2)(4,5,7)(6,9,13,11)(8,10,14,12), (1,3)(4,7)(6,12,9,13,10,11)(8,14), (2,3)(5,7)(6,10)(8,11)(9,14)(12,13), (1,2)(5,7), (2,3)(4,7)(6,10)(8,14)(9,11)(12,13), (1,2)(5,7)(6,13)(8,14)(9,11)(10,12), (1,3)(4,7)(6,9)(8,12)(10,14)(11,13), (4,5,7), (1,3,2) );

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

sage_gap:# This uses Sage's interface to GAP, as Sage (currently) has no native support for PC groups G = gap.new('PcGroupCode(622121209740903999378877705988515885765974312899752377305814457840505459093992954745475218086547572009121,3456)'); a = G.1; b = G.2; c = G.4; d = G.6; e = G.7; f = G.8;

Group information

Description:	$C_6^2:C_2^2:S_4$
Order:	$3456$$\medspace = 2^{7} \cdot 3^{3} $	comment:Order of the group magma:Order(G); gap:Order(G); sage:G.order() sage_gap:G.Order()
Exponent:	$12$$\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:	$C_2^2\times C_3:S_3.C_2^6.S_3^3$, of order $995328$$\medspace = 2^{12} \cdot 3^{5} $	comment:Automorphism group gap:AutomorphismGroup(G); magma:AutomorphismGroup(G); sage_gap:G.AutomorphismGroup()
Composition factors:	$C_2$ x 7, $C_3$ x 3	comment:Composition factors of the group magma:CompositionFactors(G); gap:CompositionSeries(G); sage:G.composition_series() sage_gap:G.CompositionSeries()
Derived length:	$4$	comment:Derived length of the group magma:DerivedLength(G); gap:DerivedLength(G); sage_gap:G.DerivedLength()

This group is nonabelian, monomial (hence solvable), and rational.

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	439	296	840	1208	672	3456
Conjugacy classes	1	13	9	8	31	16	78
Divisions	1	13	9	8	31	16	78
Autjugacy classes	1	6	3	4	6	2	22

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	3	4	6	8	12	16
Irr. complex chars.	4	10	12	8	26	10	4	4	78
Irr. rational chars.	4	10	12	8	26	10	4	4	78

Minimal presentations

Permutation degree:	$14$
Transitive degree:	$72$
Rank:	$3$
Inequivalent generating triples:	$11760$

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:	${\langle a, b, c, d, e, f \mid a^{2}=b^{6}=c^{6}=d^{2}=e^{2}=f^{12}=[a,d]= \!\cdots\! \rangle}$ < a, b, c, d, e, f \| a^2,b^6,c^6,d^2,e^2,f^12,[a,d],[a,e],[c,d],[c,f],[d,e], b^5c^5f^4a^-1b^-1a, c^5df^6a^-1c^-1a, def^11a^-1f^-1a, c^5df^6b^-1c^-1b, c^3f^6b^-1d^-1b, c^3df^3b^-1e^-1b, c^3ef^11b^-1f^-1b, ef^6c^-1e^-1c, f^7d^-1f^-1d, f^7e^-1f^-1e >
comment:Define the group with the given generators and relations magma:G := PCGroup([10, 2, 2, 3, 2, 3, 2, 2, 2, 2, 3, 24441, 51, 55322, 74403, 37213, 18503, 113, 2404, 1214, 52935, 1645, 34036, 49796, 10956, 270727, 133937, 26667, 2297, 1187, 237, 259208, 129618, 268, 230409, 115219]); a,b,c,d,e,f := Explode([G.1, G.2, G.4, G.6, G.7, G.8]); AssignNames(~G, ["a", "b", "b2", "c", "c2", "d", "e", "f", "f2", "f4"]); gap:G := PcGroupCode(622121209740903999378877705988515885765974312899752377305814457840505459093992954745475218086547572009121,3456); a := G.1; b := G.2; c := G.4; d := G.6; e := G.7; f := G.8; sage:# This uses Sage's interface to GAP, as Sage (currently) has no native support for PC groups G = gap.new('PcGroupCode(622121209740903999378877705988515885765974312899752377305814457840505459093992954745475218086547572009121,3456)'); a = G.1; b = G.2; c = G.4; d = G.6; e = G.7; f = G.8; sage_gap:# This uses Sage's interface to GAP, as Sage (currently) has no native support for PC groups G = gap.new('PcGroupCode(622121209740903999378877705988515885765974312899752377305814457840505459093992954745475218086547572009121,3456)'); a = G.1; b = G.2; c = G.4; d = G.6; e = G.7; f = G.8;
Permutation group:	Degree $14$ $\langle(4,7,5)(6,14)(8,13)(9,12)(10,11), (1,3,2)(4,5,7)(6,9,13,11)(8,10,14,12) \!\cdots\! \rangle$ (4,7,5)(6,14)(8,13)(9,12)(10,11), (1,3,2)(4,5,7)(6,9,13,11)(8,10,14,12), (1,3)(4,7)(6,12,9,13,10,11)(8,14), (2,3)(5,7)(6,10)(8,11)(9,14)(12,13), (1,2)(5,7), (2,3)(4,7)(6,10)(8,14)(9,11)(12,13), (1,2)(5,7)(6,13)(8,14)(9,11)(10,12), (1,3)(4,7)(6,9)(8,12)(10,14)(11,13), (4,5,7), (1,3,2)
comment:Define the group as a permutation group magma:G := PermutationGroup< 14 \| (4,7,5)(6,14)(8,13)(9,12)(10,11), (1,3,2)(4,5,7)(6,9,13,11)(8,10,14,12), (1,3)(4,7)(6,12,9,13,10,11)(8,14), (2,3)(5,7)(6,10)(8,11)(9,14)(12,13), (1,2)(5,7), (2,3)(4,7)(6,10)(8,14)(9,11)(12,13), (1,2)(5,7)(6,13)(8,14)(9,11)(10,12), (1,3)(4,7)(6,9)(8,12)(10,14)(11,13), (4,5,7), (1,3,2) >; gap:G := Group( (4,7,5)(6,14)(8,13)(9,12)(10,11), (1,3,2)(4,5,7)(6,9,13,11)(8,10,14,12), (1,3)(4,7)(6,12,9,13,10,11)(8,14), (2,3)(5,7)(6,10)(8,11)(9,14)(12,13), (1,2)(5,7), (2,3)(4,7)(6,10)(8,14)(9,11)(12,13), (1,2)(5,7)(6,13)(8,14)(9,11)(10,12), (1,3)(4,7)(6,9)(8,12)(10,14)(11,13), (4,5,7), (1,3,2) ); sage:G = PermutationGroup(['(4,7,5)(6,14)(8,13)(9,12)(10,11)', '(1,3,2)(4,5,7)(6,9,13,11)(8,10,14,12)', '(1,3)(4,7)(6,12,9,13,10,11)(8,14)', '(2,3)(5,7)(6,10)(8,11)(9,14)(12,13)', '(1,2)(5,7)', '(2,3)(4,7)(6,10)(8,14)(9,11)(12,13)', '(1,2)(5,7)(6,13)(8,14)(9,11)(10,12)', '(1,3)(4,7)(6,9)(8,12)(10,14)(11,13)', '(4,5,7)', '(1,3,2)'])
Direct product:	$(C_3:S_3)$ $\, \times\, $ $(C_2^3:S_4)$
Semidirect product:	$(D_4:C_6^2)$ $\,\rtimes\,$ $D_6$	$(C_6^2.C_2^4)$ $\,\rtimes\,$ $S_3$	$(C_6^2:C_2^2)$ $\,\rtimes\,$ $S_4$	$(C_6.C_2^4)$ $\,\rtimes\,$ $S_3^2$	all 17
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_6$ . $(C_2^4:S_3^2)$	$(C_6:S_3)$ . $(C_2^2:S_4)$	$(C_3\times C_6)$ . $(C_2^3:S_4)$	$C_2$ . $(C_6^2:(C_2\times S_4))$	more information

Elements of the group are displayed as permutations of degree 14.

Homology

Abelianization:	$C_{2}^{2} $	comment:The abelianization of the group magma:quo< G \| CommutatorSubgroup(G) >; gap:FactorGroup(G, DerivedSubgroup(G)); sage:G.quotient(G.commutator())
Schur multiplier:	$C_{2}^{3} \times C_{6}$	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 65322 subgroups in 2818 conjugacy classes, 58 normal (15 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2$	$G/Z \simeq$ $C_6^2:(C_2\times S_4)$	comment:Center of the group magma:Center(G); gap:Center(G); sage:G.center() sage_gap:G.Center()
Commutator:	$G' \simeq$ $C_3^2\times Q_8:A_4$	$G/G' \simeq$ $C_2^2$	comment:Commutator subgroup of the group G magma:CommutatorSubgroup(G); gap:DerivedSubgroup(G); sage:G.commutator() sage_gap:G.DerivedSubgroup()
Frattini:	$\Phi \simeq$ $C_2$	$G/\Phi \simeq$ $C_6^2:(C_2\times S_4)$	comment:Frattini subgroup of the group G magma:FrattiniSubgroup(G); gap:FrattiniSubgroup(G); sage:G.frattini_subgroup() sage_gap:G.FrattiniSubgroup()
Fitting:	$\operatorname{Fit} \simeq$ $D_4:C_6^2$	$G/\operatorname{Fit} \simeq$ $D_6$	comment:Fitting subgroup of the group G magma:FittingSubgroup(G); gap:FittingSubgroup(G); sage:G.fitting_subgroup() sage_gap:G.FittingSubgroup()
Radical:	$R \simeq$ $C_6^2:C_2^2:S_4$	$G/R \simeq$ $C_1$	comment:Radical of the group G magma:Radical(G); gap:SolvableRadical(G); sage_gap:G.SolvableRadical()
Socle:	$\operatorname{soc} \simeq$ $C_3\times C_6$	$G/\operatorname{soc} \simeq$ $C_2^3:S_4$	comment:Socle of the group G magma:Socle(G); gap:Socle(G); sage:G.socle() sage_gap:G.Socle()
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^4:D_4$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3^3$

Subgroup diagram and profile

Series

Derived series

$C_6^2:C_2^2:S_4$

$\rhd$

$C_3^2\times Q_8:A_4$

$\rhd$

$D_4:C_2^2$

$\rhd$

$C_2$

$\rhd$

$C_1$

comment:Derived series of the group GF

magma:DerivedSeries(G);

gap:DerivedSeriesOfGroup(G);

sage:G.derived_series()

sage_gap:G.DerivedSeriesOfGroup()

Chief series

$C_6^2:C_2^2:S_4$

$\rhd$

$C_3^2\times C_2^3:S_4$

$\rhd$

$C_3^2\times Q_8:A_4$

$\rhd$

$C_3\times Q_8:A_4$

$\rhd$

$Q_8:A_4$

$\rhd$

$D_4:C_2^2$

$\rhd$

$C_2^3$

$\rhd$

$C_2$

$\rhd$

$C_1$

comment:Chief series of the group G

magma:ChiefSeries(G);

gap:ChiefSeries(G);

sage_gap:G.ChiefSeries()

Lower central series

$C_6^2:C_2^2:S_4$

$\rhd$

$C_3^2\times Q_8:A_4$

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

$C_1$

$\lhd$

$C_2$

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 7 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

Every character has rational values, so the complex character table is the same as the rational character table below.

Rational character table

See the $78 \times 78$ rational character table. Alternatively, you may search for characters of this group with desired properties.

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	3456.lz
Order	\( 2^{7} \cdot 3^{3} \)
Exponent	\( 2^{2} \cdot 3 \)

Nilpotent	no
Solvable	yes
$\card{G^{\mathrm{ab}}}$	\( 2^{2} \)
$\card{Z(G)}$	\( 2 \)
$\card{\Aut(G)}$	\( 2^{12} \cdot 3^{5} \)
$\card{\mathrm{Out}(G)}$	\( 2^{6} \cdot 3^{2} \)
Perm deg.	$14$
Trans deg.	$72$
Rank	$3$

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.