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Properties

Label 264.30
Order \( 2^{3} \cdot 3 \cdot 11 \)
Exponent \( 2^{2} \cdot 3 \cdot 11 \)
Nilpotent yes
Solvable yes
$\card{G^{\mathrm{ab}}}$ \( 2^{2} \cdot 3 \cdot 11 \)
$\card{Z(G)}$ \( 2 \cdot 3 \cdot 11 \)
$\card{\Aut(G)}$ \( 2^{5} \cdot 3 \cdot 5 \)
$\card{\mathrm{Out}(G)}$ \( 2^{3} \cdot 3 \cdot 5 \)
Perm deg. $22$
Trans deg. $264$
Rank $2$

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Show commands: Gap / Magma / SageMath

Copy content magma:G := SmallGroup(264, 30);
 
Copy content gap:G := SmallGroup(264, 30);
 
Copy content sage_gap:G = libgap.SmallGroup(264, 30)
 
Copy content comment:Define the group as a permutation group
 
Copy content sage:G = PermutationGroup(['(1,6,2,5)(3,8,4,7)', '(1,4,2,3)(5,7,6,8)', '(9,11,10)', '(12,22,21,20,19,18,17,16,15,14,13)', '(1,2)(3,4)(5,6)(7,8)'])
 

Group information

Description:$Q_8\times C_{33}$
Order: \(264\)\(\medspace = 2^{3} \cdot 3 \cdot 11 \)
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()
 
Exponent: \(132\)\(\medspace = 2^{2} \cdot 3 \cdot 11 \)
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()
 
Automorphism group:$C_2\times C_{10}\times S_4$, of order \(480\)\(\medspace = 2^{5} \cdot 3 \cdot 5 \)
Copy content comment:Automorphism group
 
Copy content gap:AutomorphismGroup(G);
 
Copy content magma:AutomorphismGroup(G);
 
Copy content sage_gap:G.AutomorphismGroup()
 
Composition factors:$C_2$ x 3, $C_3$, $C_{11}$
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()
 
Nilpotency class:$2$
Copy content comment:Nilpotency class of the group
 
Copy content magma:NilpotencyClass(G);
 
Copy content gap:NilpotencyClassOfGroup(G);
 
Copy content sage_gap:G.NilpotencyClassOfGroup()
 
Derived length:$2$
Copy content comment:Derived length of the group
 
Copy content magma:DerivedLength(G);
 
Copy content gap:DerivedLength(G);
 
Copy content sage_gap:G.DerivedLength()
 

This group is nonabelian, elementary for $p = 2$ (hence nilpotent, solvable, supersolvable, monomial, and hyperelementary), and metacyclic (hence metabelian).

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 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 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 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 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 comment:Determine if the group G is simple
 
Copy content magma:IsSimple(G);
 
Copy content gap:IsSimpleGroup(G);
 
Copy content sage_gap:G.IsSimpleGroup()
 

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))
 

Order 1 2 3 4 6 11 12 22 33 44 66 132
Elements 1 1 2 6 2 10 12 10 20 60 20 120 264
Conjugacy classes   1 1 2 3 2 10 6 10 20 30 20 60 165
Divisions 1 1 1 3 1 1 3 1 1 3 1 3 20
Autjugacy classes 1 1 1 1 1 1 1 1 1 1 1 1 12

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:G.CharacterDegrees()
 

Dimension 1 2 4 10 20 40
Irr. complex chars.   132 33 0 0 0 0 165
Irr. rational chars. 4 5 1 4 5 1 20

Minimal presentations

Permutation degree:$22$
Transitive degree:$264$
Rank: $2$
Inequivalent generating pairs: $48$

Minimal degrees of faithful linear representations

Over $\mathbb{C}$ Over $\mathbb{R}$ Over $\mathbb{Q}$
Irreducible 2 4 40
Arbitrary 2 4 14

Constructions

Show commands: Gap / Magma / SageMath


Presentation: $\langle a, b \mid b^{132}=1, a^{2}=b^{66}, b^{a}=b^{67} \rangle$ Copy content Toggle raw display
Copy content comment:Define the group with the given generators and relations
 
Copy content magma:G := PCGroup([5, -2, -2, -2, -3, -11, 660, 1341, 26, 42, 78]); a,b := Explode([G.1, G.2]); AssignNames(~G, ["a", "b", "b2", "b4", "b12"]);
 
Copy content gap:G := PcGroupCode(390001890992131100019,264); a := G.1; b := G.2;
 
Copy content sage:# This uses Sage's interface to GAP, as Sage (currently) has no native support for PC groups G = gap.new('PcGroupCode(390001890992131100019,264)'); a = G.1; b = G.2;
 
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(390001890992131100019,264)'); a = G.1; b = G.2;
 
Permutation group:Degree $22$ $\langle(1,6,2,5)(3,8,4,7), (1,4,2,3)(5,7,6,8), (9,11,10), (12,22,21,20,19,18,17,16,15,14,13), (1,2)(3,4)(5,6)(7,8)\rangle$ Copy content Toggle raw display
Copy content comment:Define the group as a permutation group
 
Copy content magma:G := PermutationGroup< 22 | (1,6,2,5)(3,8,4,7), (1,4,2,3)(5,7,6,8), (9,11,10), (12,22,21,20,19,18,17,16,15,14,13), (1,2)(3,4)(5,6)(7,8) >;
 
Copy content gap:G := Group( (1,6,2,5)(3,8,4,7), (1,4,2,3)(5,7,6,8), (9,11,10), (12,22,21,20,19,18,17,16,15,14,13), (1,2)(3,4)(5,6)(7,8) );
 
Copy content sage:G = PermutationGroup(['(1,6,2,5)(3,8,4,7)', '(1,4,2,3)(5,7,6,8)', '(9,11,10)', '(12,22,21,20,19,18,17,16,15,14,13)', '(1,2)(3,4)(5,6)(7,8)'])
 
Matrix group:$\left\langle \left(\begin{array}{rr} 39 & 19 \\ 24 & 28 \end{array}\right), \left(\begin{array}{rr} 0 & 58 \\ 29 & 0 \end{array}\right) \right\rangle \subseteq \GL_{2}(\F_{67})$
Copy content comment:Define the group as a matrix group with coefficients in GLFp
 
Copy content magma:G := MatrixGroup< 2, GF(67) | [[39, 19, 24, 28], [0, 58, 29, 0]] >;
 
Copy content gap:G := Group([[[ Z(67)^58, Z(67)^10 ], [ Z(67)^42, Z(67)^25 ]], [[ 0*Z(67), Z(67)^45 ], [ Z(67)^44, 0*Z(67) ]]]);
 
Copy content sage:MS = MatrixSpace(GF(67), 2, 2) G = MatrixGroup([MS([[39, 19], [24, 28]]), MS([[0, 58], [29, 0]])])
 
Direct product: $C_3$ $\, \times\, $ $C_{11}$ $\, \times\, $ $Q_8$
Semidirect product: not computed
Trans. wreath product: not isomorphic to a non-trivial transitive wreath product
Non-split product: $C_{44}$ . $C_6$ (3) $C_4$ . $C_{66}$ (3) $C_{132}$ . $C_2$ (3) $C_{12}$ . $C_{22}$ (3) all 8

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

Homology

Abelianization: $C_{2} \times C_{66} \simeq C_{2}^{2} \times C_{3} \times C_{11}$
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())
 
Schur multiplier: $C_1$
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()
 

There are 24 subgroups, all normal (12 characteristic).

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

Special subgroups

Center: $Z \simeq$ $C_{66}$ $G/Z \simeq$ $C_2^2$
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()
 
Commutator: $G' \simeq$ $C_2$ $G/G' \simeq$ $C_2\times C_{66}$
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()
 
Frattini: $\Phi \simeq$ $C_2$ $G/\Phi \simeq$ $C_2\times C_{66}$
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()
 
Fitting: $\operatorname{Fit} \simeq$ $Q_8\times C_{33}$ $G/\operatorname{Fit} \simeq$ $C_1$
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()
 
Radical: $R \simeq$ $Q_8\times C_{33}$ $G/R \simeq$ $C_1$
Copy content comment:Radical of the group G
 
Copy content magma:Radical(G);
 
Copy content gap:SolvableRadical(G);
 
Copy content sage_gap:G.SolvableRadical()
 
Socle: $\operatorname{soc} \simeq$ $C_{66}$ $G/\operatorname{soc} \simeq$ $C_2^2$
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()
 
2-Sylow subgroup: $P_{ 2 } \simeq$ $Q_8$
3-Sylow subgroup: $P_{ 3 } \simeq$ $C_3$
11-Sylow subgroup: $P_{ 11 } \simeq$ $C_{11}$

Subgroup diagram and profile

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Subgroup information

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Series

Derived series $Q_8\times C_{33}$ $\rhd$ $C_2$ $\rhd$ $C_1$
Copy content comment:Derived series of the group GF
 
Copy content magma:DerivedSeries(G);
 
Copy content gap:DerivedSeriesOfGroup(G);
 
Copy content sage:G.derived_series()
 
Copy content sage_gap:G.DerivedSeriesOfGroup()
 
Chief series $Q_8\times C_{33}$ $\rhd$ $C_{132}$ $\rhd$ $C_{66}$ $\rhd$ $C_{33}$ $\rhd$ $C_{11}$ $\rhd$ $C_1$
Copy content comment:Chief series of the group G
 
Copy content magma:ChiefSeries(G);
 
Copy content gap:ChiefSeries(G);
 
Copy content sage_gap:G.ChiefSeries()
 
Lower central series $Q_8\times C_{33}$ $\rhd$ $C_2$ $\rhd$ $C_1$
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()
 
Upper central series $C_1$ $\lhd$ $C_{66}$ $\lhd$ $Q_8\times C_{33}$
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()
 

Supergroups

This group is a maximal subgroup of 34 larger groups in the database.

This group is a maximal quotient of 16 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
 

Complex character table

See the $165 \times 165$ 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 $20 \times 20$ rational character table.