Properties

Label 648.727
Order \( 2^{3} \cdot 3^{4} \)
Exponent \( 2^{2} \cdot 3 \)
Nilpotent no
Solvable yes
$\card{G^{\mathrm{ab}}}$ \( 2^{2} \cdot 3 \)
$\card{Z(G)}$ \( 3 \)
$\card{\Aut(G)}$ \( 2^{6} \cdot 3^{3} \)
$\card{\mathrm{Out}(G)}$ \( 2^{3} \)
Perm deg. $15$
Trans deg. $24$
Rank $2$

Related objects

Downloads

Learn more

Show commands: Gap / Magma / Oscar / SageMath

Copy content comment:Construction of abstract group
 
Copy content magma:G := SmallGroup(648, 727);
 
Copy content gap:G := SmallGroup(648, 727);
 
Copy content sage:G = PermutationGroup(['(2,4,6,8)(3,7,5,9)(10,11)(12,14)(13,15)', '(2,5,6,3)(4,7,8,9)', '(10,12,13)(11,14,15)', '(2,6)(3,5)(4,8)(7,9)', '(1,2,6)(3,7,4)(5,8,9)', '(1,3,5)(2,7,8)(4,9,6)', '(10,13,12)(11,14,15)'])
 
Copy content sage_gap:G = libgap.SmallGroup(648, 727)
 
Copy content oscar:G = small_group(648, 727)
 

Group information

Description:$C_3^4:Q_8$
Order: \(648\)\(\medspace = 2^{3} \cdot 3^{4} \)
Copy content comment:Order of the group
 
Copy content magma:Order(G);
 
Copy content gap:Order(G);
 
Copy content sage:G.order()
 
Copy content sage_gap:G.Order()
 
Copy content oscar:order(G)
 
Exponent: \(12\)\(\medspace = 2^{2} \cdot 3 \)
Copy content comment:Exponent of the group
 
Copy content magma:Exponent(G);
 
Copy content gap:Exponent(G);
 
Copy content sage:G.exponent()
 
Copy content sage_gap:G.Exponent()
 
Copy content oscar:exponent(G)
 
Automorphism group:$C_2\times F_9:D_6$, of order \(1728\)\(\medspace = 2^{6} \cdot 3^{3} \)
Copy content comment:Automorphism group
 
Copy content gap:AutomorphismGroup(G);
 
Copy content magma:AutomorphismGroup(G);
 
Copy content sage:libgap(G).AutomorphismGroup()
 
Copy content sage_gap:G.AutomorphismGroup()
 
Copy content oscar:automorphism_group(G)
 
Composition factors:$C_2$ x 3, $C_3$ x 4
Copy content comment:Composition factors of the group
 
Copy content magma:CompositionFactors(G);
 
Copy content gap:CompositionSeries(G);
 
Copy content sage:G.composition_series()
 
Copy content sage_gap:G.CompositionSeries()
 
Copy content oscar:composition_series(G)
 
Derived length:$3$
Copy content comment:Derived length of the group
 
Copy content magma:DerivedLength(G);
 
Copy content gap:DerivedLength(G);
 
Copy content sage:libgap(G).DerivedLength()
 
Copy content sage_gap:G.DerivedLength()
 
Copy content oscar:derived_length(G)
 

This group is nonabelian and monomial (hence solvable).

Copy content comment:Determine if the group G is abelian
 
Copy content magma:IsAbelian(G);
 
Copy content gap:IsAbelian(G);
 
Copy content sage:G.is_abelian()
 
Copy content sage_gap:G.IsAbelian()
 
Copy content oscar:is_abelian(G)
 
Copy content comment:Determine if the group G is cyclic
 
Copy content magma:IsCyclic(G);
 
Copy content gap:IsCyclic(G);
 
Copy content sage:G.is_cyclic()
 
Copy content sage_gap:G.IsCyclic()
 
Copy content oscar:is_cyclic(G)
 
Copy content comment:Determine if the group G is nilpotent
 
Copy content magma:IsNilpotent(G);
 
Copy content gap:IsNilpotentGroup(G);
 
Copy content sage:G.is_nilpotent()
 
Copy content sage_gap:G.IsNilpotentGroup()
 
Copy content oscar:is_nilpotent(G)
 
Copy content comment:Determine if the group G is solvable
 
Copy content magma:IsSolvable(G);
 
Copy content gap:IsSolvableGroup(G);
 
Copy content sage:G.is_solvable()
 
Copy content sage_gap:G.IsSolvableGroup()
 
Copy content oscar:is_solvable(G)
 
Copy content comment:Determine if the group G is supersolvable
 
Copy content gap:IsSupersolvableGroup(G);
 
Copy content sage:G.is_supersolvable()
 
Copy content sage_gap:G.IsSupersolvableGroup()
 
Copy content oscar:is_supersolvable(G)
 
Copy content comment:Determine if the group G is simple
 
Copy content magma:IsSimple(G);
 
Copy content gap:IsSimpleGroup(G);
 
Copy content sage:G.is_simple()
 
Copy content sage_gap:G.IsSimpleGroup()
 
Copy content oscar:is_simple(G)
 

Group statistics

Copy content comment:Compute statistics for the group G
 
Copy content 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;
 
Copy content 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");
 
Copy content 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))
 
Copy content 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))
 
Copy content 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 4 6 12
Elements 1 9 80 126 72 360 648
Conjugacy classes   1 1 14 3 5 12 36
Divisions 1 1 8 3 3 5 21
Autjugacy classes 1 1 7 2 3 4 18

Copy content comment:Compute statistics about the characters of G
 
Copy content 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);
 
Copy content 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);
 
Copy content 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)]
 
Copy content 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()
 
Copy content oscar:# Outputs an MSet containing the absolutely irreducible degrees of G and their multiplicities. character_degrees(G)
 

Dimension 1 2 4 8 16
Irr. complex chars.   12 15 0 9 0 36
Irr. rational chars. 4 7 4 2 4 21

Minimal presentations

Permutation degree:$15$
Transitive degree:$24$
Rank: $2$
Inequivalent generating pairs: $48$

Minimal degrees of faithful linear representations

Over $\mathbb{C}$ Over $\mathbb{R}$ Over $\mathbb{Q}$
Irreducible 8 16 16
Arbitrary 8 12 12

Constructions

Show commands: Gap / Magma / Oscar / SageMath


Presentation: $\langle a, b, c, d \mid b^{12}=c^{3}=d^{3}=[c,d]=1, a^{6}=b^{6}, b^{a}=b^{11}, c^{a}=c^{2}d, d^{a}=cd, c^{b}=cd, d^{b}=cd^{2} \rangle$ Copy content Toggle raw display
Copy content comment:Define the group with the given generators and relations
 
Copy content magma:G := PCGroup([7, -2, -3, -2, -2, -3, -3, 3, 14, 260, 1388, 58, 1683, 80, 1684, 15125, 3036, 2035, 530, 14118, 10597, 4136, 1791]); a,b,c,d := Explode([G.1, G.3, G.6, G.7]); AssignNames(~G, ["a", "a2", "b", "b2", "b4", "c", "d"]);
 
Copy content gap:G := PcGroupCode(39726475548830774046997208869518121674727710246823,648); a := G.1; b := G.3; c := G.6; d := G.7;
 
Copy content sage:# This uses Sage's interface to GAP, as Sage (currently) has no native support for PC groups G = gap.new('PcGroupCode(39726475548830774046997208869518121674727710246823,648)'); a = G.1; b = G.3; c = G.6; d = G.7;
 
Copy content sage_gap:# This uses Sage's interface to GAP, as Sage (currently) has no native support for PC groups G = gap.new('PcGroupCode(39726475548830774046997208869518121674727710246823,648)'); a = G.1; b = G.3; c = G.6; d = G.7;
 
Permutation group:Degree $15$ $\langle(2,4,6,8)(3,7,5,9)(10,11)(12,14)(13,15), (2,5,6,3)(4,7,8,9), (10,12,13) \!\cdots\! \rangle$ Copy content Toggle raw display
Copy content comment:Define the group as a permutation group
 
Copy content magma:G := PermutationGroup< 15 | (2,4,6,8)(3,7,5,9)(10,11)(12,14)(13,15), (2,5,6,3)(4,7,8,9), (10,12,13)(11,14,15), (2,6)(3,5)(4,8)(7,9), (1,2,6)(3,7,4)(5,8,9), (1,3,5)(2,7,8)(4,9,6), (10,13,12)(11,14,15) >;
 
Copy content gap:G := Group( (2,4,6,8)(3,7,5,9)(10,11)(12,14)(13,15), (2,5,6,3)(4,7,8,9), (10,12,13)(11,14,15), (2,6)(3,5)(4,8)(7,9), (1,2,6)(3,7,4)(5,8,9), (1,3,5)(2,7,8)(4,9,6), (10,13,12)(11,14,15) );
 
Copy content sage:G = PermutationGroup(['(2,4,6,8)(3,7,5,9)(10,11)(12,14)(13,15)', '(2,5,6,3)(4,7,8,9)', '(10,12,13)(11,14,15)', '(2,6)(3,5)(4,8)(7,9)', '(1,2,6)(3,7,4)(5,8,9)', '(1,3,5)(2,7,8)(4,9,6)', '(10,13,12)(11,14,15)'])
 
Copy content sage_gap:G = gap.new('Group( (2,4,6,8)(3,7,5,9)(10,11)(12,14)(13,15), (2,5,6,3)(4,7,8,9), (10,12,13)(11,14,15), (2,6)(3,5)(4,8)(7,9), (1,2,6)(3,7,4)(5,8,9), (1,3,5)(2,7,8)(4,9,6), (10,13,12)(11,14,15) )')
 
Copy content oscar:G = @permutation_group(15, (2,4,6,8)(3,7,5,9)(10,11)(12,14)(13,15), (2,5,6,3)(4,7,8,9), (10,12,13)(11,14,15), (2,6)(3,5)(4,8)(7,9), (1,2,6)(3,7,4)(5,8,9), (1,3,5)(2,7,8)(4,9,6), (10,13,12)(11,14,15))
 
Transitive group: 24T1542 36T1181 more information
Copy content magma:G := TransitiveGroup(24, 1542);
 
Copy content gap:G := TransitiveGroup(24, 1542);
 
Copy content sage:G = TransitiveGroup(24, 1542)
 
Copy content sage_gap:G = libgap.TransitiveGroup(24, 1542)
 
Copy content oscar:G = transitive_group(24, 1542)
 
Copy content magma:G := TransitiveGroup(36, 1181);
 
Copy content gap:G := TransitiveGroup(36, 1181);
 
Copy content sage:G = TransitiveGroup(36, 1181)
 
Copy content sage_gap:G = libgap.TransitiveGroup(36, 1181)
 
Copy content oscar:G = transitive_group(36, 1181)
 
Direct product: $C_3$ $\, \times\, $ $(C_3^3:Q_8)$
Semidirect product: $C_3^4$ $\,\rtimes\,$ $Q_8$ $C_3^2$ $\,\rtimes\,$ $\PSU(3,2)$ $C_3^3$ $\,\rtimes\,$ $(C_3:Q_8)$ $C_3^3$ $\,\rtimes\,$ $(C_3\times Q_8)$ all 6
Trans. wreath product: not isomorphic to a non-trivial transitive wreath product
Non-split product: $(C_3\wr C_4)$ . $C_2$ (2) $(C_3^2:C_6)$ . $D_6$ $(C_3^3:C_4)$ . $C_6$ (2) $(C_3^2:C_{12})$ . $S_3$ all 10

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

Homology

Abelianization: $C_{2} \times C_{6} \simeq C_{2}^{2} \times C_{3}$
Copy content comment:The abelianization of the group
 
Copy content magma:quo< G | CommutatorSubgroup(G) >;
 
Copy content gap:FactorGroup(G, DerivedSubgroup(G));
 
Copy content sage:G.quotient(G.commutator())
 
Copy content sage_gap:G.FactorGroup(G.DerivedSubgroup())
 
Copy content oscar:quo(G, derived_subgroup(G)[1])
 
Schur multiplier: $C_{3}$
Copy content comment:The Schur multiplier of the group
 
Copy content gap:AbelianInvariantsMultiplier(G);
 
Copy content sage:G.homology(2)
 
Copy content sage_gap:G.AbelianInvariantsMultiplier()
 
Commutator length: $1$
Copy content comment:The commutator length of the group
 
Copy content gap:CommutatorLength(G);
 
Copy content sage_gap:G.CommutatorLength()
 

Subgroups

Copy content comment:List of subgroups of the group
 
Copy content magma:Subgroups(G);
 
Copy content gap:AllSubgroups(G);
 
Copy content sage:G.subgroups()
 
Copy content sage_gap:G.AllSubgroups()
 
Copy content oscar:subgroups(G)
 

There are 644 subgroups in 88 conjugacy classes, 22 normal (18 characteristic).

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

Special subgroups

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

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.
Sorry, your browser does not support the subgroup diagram.
See a full page version of the diagram
See a full page version of the diagram
See a full page version of the diagram
See a full page version of the diagram

Subgroup information

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

Series

Derived series $C_3^4:Q_8$ $\rhd$ $C_3^2:C_6$ $\rhd$ $C_3^2$ $\rhd$ $C_1$
Copy content comment:Derived series of the group G
 
Copy content magma:DerivedSeries(G);
 
Copy content gap:DerivedSeriesOfGroup(G);
 
Copy content sage:G.derived_series()
 
Copy content sage_gap:G.DerivedSeriesOfGroup()
 
Copy content oscar:derived_series(G)
 
Chief series $C_3^4:Q_8$ $\rhd$ $C_3^3:C_{12}$ $\rhd$ $C_3^2:C_{12}$ $\rhd$ $C_3^2:C_6$ $\rhd$ $C_3^3$ $\rhd$ $C_3^2$ $\rhd$ $C_1$
Copy content comment:Chief series of the group G
 
Copy content magma:ChiefSeries(G);
 
Copy content gap:ChiefSeries(G);
 
Copy content sage:libgap(G).ChiefSeries()
 
Copy content sage_gap:G.ChiefSeries()
 
Copy content oscar:chief_series(G)
 
Lower central series $C_3^4:Q_8$ $\rhd$ $C_3^2:C_6$ $\rhd$ $C_3^3$
Copy content comment:The lower central series of the group G
 
Copy content magma:LowerCentralSeries(G);
 
Copy content gap:LowerCentralSeriesOfGroup(G);
 
Copy content sage:G.lower_central_series()
 
Copy content sage_gap:G.LowerCentralSeriesOfGroup()
 
Copy content oscar:lower_central_series(G)
 
Upper central series $C_1$ $\lhd$ $C_3$
Copy content comment:The upper central series of the group G
 
Copy content magma:UpperCentralSeries(G);
 
Copy content gap:UpperCentralSeriesOfGroup(G);
 
Copy content sage:G.upper_central_series()
 
Copy content sage_gap:G.UpperCentralSeriesOfGroup()
 
Copy content oscar:upper_central_series(G)
 

Supergroups

This group is a maximal subgroup of 21 larger groups in the database.

This group is a maximal quotient of 13 larger groups in the database.

Character theory

Copy content comment:Character table
 
Copy content magma:CharacterTable(G); // Output not guaranteed to exactly match the LMFDB table
 
Copy content gap:CharacterTable(G); # Output not guaranteed to exactly match the LMFDB table
 
Copy content sage:G.character_table() # Output not guaranteed to exactly match the LMFDB table
 
Copy content sage_gap:G.CharacterTable() # Output not guaranteed to exactly match the LMFDB table
 
Copy content oscar:character_table(G) # Output not guaranteed to exactly match the LMFDB table
 

Complex character table

See the $36 \times 36$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $21 \times 21$ rational character table.