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Properties

Label 324.9
Order \( 2^{2} \cdot 3^{4} \)
Exponent \( 2^{2} \cdot 3^{2} \)
Nilpotent no
Solvable yes
$\card{G^{\mathrm{ab}}}$ \( 2^{2} \cdot 3^{2} \)
$\card{Z(G)}$ \( 2 \cdot 3 \)
$\card{\Aut(G)}$ \( 2^{2} \cdot 3^{4} \)
$\card{\mathrm{Out}(G)}$ \( 2 \cdot 3 \)
Perm deg. $22$
Trans deg. $36$
Rank $2$

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Show commands: Gap / Magma / SageMath

Copy content magma:G := SmallGroup(324, 9);
 
Copy content gap:G := SmallGroup(324, 9);
 
Copy content sage_gap:G = libgap.SmallGroup(324, 9)
 
Copy content comment:Define the group as a permutation group
 
Copy content sage:G = PermutationGroup(['(2,4)(3,7)(5,6)(8,9)(19,20,21,22)', '(2,5,9)(4,6,8)(10,11,13,12,14,16,15,17,18)', '(19,21)(20,22)', '(10,12,15)(11,14,17)(13,16,18)', '(1,2,6,7,9,8,3,5,4)', '(1,3,7)(2,5,9)(4,8,6)'])
 

Group information

Description:$C_9:C_{36}$
Order: \(324\)\(\medspace = 2^{2} \cdot 3^{4} \)
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()
 
Exponent: \(36\)\(\medspace = 2^{2} \cdot 3^{2} \)
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:$C_9:C_6^2$, of order \(324\)\(\medspace = 2^{2} \cdot 3^{4} \)
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 2, $C_3$ 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()
 
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 and metacyclic (hence solvable, supersolvable, monomial, and 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 9 12 18 36
Elements 1 1 8 18 8 72 36 72 108 324
Conjugacy classes   1 1 5 2 5 15 4 15 12 60
Divisions 1 1 3 1 3 4 1 4 1 19
Autjugacy classes 1 1 5 1 5 7 2 7 2 31

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 6 12
Irr. complex chars.   36 18 0 6 0 60
Irr. rational chars. 2 5 3 4 5 19

Minimal presentations

Permutation degree:$22$
Transitive degree:$36$
Rank: $2$
Inequivalent generating pairs: $144$

Minimal degrees of faithful linear representations

Over $\mathbb{C}$ Over $\mathbb{R}$ Over $\mathbb{Q}$
Irreducible 6 12 12
Arbitrary 6 8 12

Constructions

Show commands: Gap / Magma / SageMath


Presentation: $\langle a, b \mid a^{36}=b^{9}=1, b^{a}=b^{2} \rangle$ Copy content Toggle raw display
Copy content comment:Define the group with the given generators and relations
 
Copy content magma:G := PCGroup([6, -2, -2, -3, -3, -3, -3, 12, 31, 68, 2164, 2170, 1906, 118, 7781]); a,b := Explode([G.1, G.5]); AssignNames(~G, ["a", "a2", "a4", "a12", "b", "b3"]);
 
Copy content gap:G := PcGroupCode(50384474091699982234338767,324); a := G.1; b := G.5;
 
Copy content sage:# This uses Sage's interface to GAP, as Sage (currently) has no native support for PC groups G = gap.new('PcGroupCode(50384474091699982234338767,324)'); a = G.1; b = G.5;
 
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(50384474091699982234338767,324)'); a = G.1; b = G.5;
 
Permutation group:Degree $22$ $\langle(2,4)(3,7)(5,6)(8,9)(19,20,21,22), (2,5,9)(4,6,8)(10,11,13,12,14,16,15,17,18) \!\cdots\! \rangle$ Copy content Toggle raw display
Copy content comment:Define the group as a permutation group
 
Copy content magma:G := PermutationGroup< 22 | (2,4)(3,7)(5,6)(8,9)(19,20,21,22), (2,5,9)(4,6,8)(10,11,13,12,14,16,15,17,18), (19,21)(20,22), (10,12,15)(11,14,17)(13,16,18), (1,2,6,7,9,8,3,5,4), (1,3,7)(2,5,9)(4,8,6) >;
 
Copy content gap:G := Group( (2,4)(3,7)(5,6)(8,9)(19,20,21,22), (2,5,9)(4,6,8)(10,11,13,12,14,16,15,17,18), (19,21)(20,22), (10,12,15)(11,14,17)(13,16,18), (1,2,6,7,9,8,3,5,4), (1,3,7)(2,5,9)(4,8,6) );
 
Copy content sage:G = PermutationGroup(['(2,4)(3,7)(5,6)(8,9)(19,20,21,22)', '(2,5,9)(4,6,8)(10,11,13,12,14,16,15,17,18)', '(19,21)(20,22)', '(10,12,15)(11,14,17)(13,16,18)', '(1,2,6,7,9,8,3,5,4)', '(1,3,7)(2,5,9)(4,8,6)'])
 
Matrix group:$\left\langle \left(\begin{array}{rr} 80 & 95 \\ 27 & 82 \end{array}\right), \left(\begin{array}{rr} 1 & 54 \\ 54 & 1 \end{array}\right), \left(\begin{array}{rr} 37 & 0 \\ 0 & 73 \end{array}\right), \left(\begin{array}{rr} 1 & 36 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 1 & 12 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 49 & 0 \\ 0 & 25 \end{array}\right) \right\rangle \subseteq \GL_{2}(\Z/108\Z)$
Copy content comment:Define the group as a matrix group with coefficients in GLZN
 
Copy content magma:G := MatrixGroup< 2, Integers(108) | [[80, 95, 27, 82], [1, 54, 54, 1], [37, 0, 0, 73], [1, 36, 0, 1], [1, 12, 0, 1], [49, 0, 0, 25]] >;
 
Copy content gap:G := Group([[[ZmodnZObj(80,108), ZmodnZObj(95,108)], [ZmodnZObj(27,108), ZmodnZObj(82,108)]],[[ZmodnZObj(1,108), ZmodnZObj(54,108)], [ZmodnZObj(54,108), ZmodnZObj(1,108)]],[[ZmodnZObj(37,108), ZmodnZObj(0,108)], [ZmodnZObj(0,108), ZmodnZObj(73,108)]],[[ZmodnZObj(1,108), ZmodnZObj(36,108)], [ZmodnZObj(0,108), ZmodnZObj(1,108)]],[[ZmodnZObj(1,108), ZmodnZObj(12,108)], [ZmodnZObj(0,108), ZmodnZObj(1,108)]],[[ZmodnZObj(49,108), ZmodnZObj(0,108)], [ZmodnZObj(0,108), ZmodnZObj(25,108)]]]);
 
Copy content sage:MS = MatrixSpace(Integers(108), 2, 2) G = MatrixGroup([MS([[80, 95], [27, 82]]), MS([[1, 54], [54, 1]]), MS([[37, 0], [0, 73]]), MS([[1, 36], [0, 1]]), MS([[1, 12], [0, 1]]), MS([[49, 0], [0, 25]])])
 
Transitive group: 36T473 more information
Direct product: not isomorphic to a non-trivial direct product
Semidirect product: $C_9$ $\,\rtimes\,$ $C_{36}$ $(C_9:C_9)$ $\,\rtimes\,$ $C_4$ $(C_9:C_4)$ $\,\rtimes\,$ $C_9$ more information
Trans. wreath product: not isomorphic to a non-trivial transitive wreath product
Non-split product: $C_{18}$ . $C_{18}$ $C_6$ . $(S_3\times C_9)$ $C_6$ . $(C_9:C_6)$ $(C_3\times C_{18})$ . $S_3$ all 14

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

Homology

Abelianization: $C_{36} \simeq C_{4} \times C_{9}$
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 85 subgroups in 35 conjugacy classes, 19 normal, and all normal subgroups are characteristic.

Characteristic subgroups are shown in this color.

Special subgroups

Center: $Z \simeq$ $C_6$ $G/Z \simeq$ $C_9:C_6$
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: $G' \simeq$ $C_9$ $G/G' \simeq$ $C_{36}$
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: $\Phi \simeq$ $C_3\times C_6$ $G/\Phi \simeq$ $C_3\times S_3$
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: $\operatorname{Fit} \simeq$ $C_9:C_{18}$ $G/\operatorname{Fit} \simeq$ $C_2$
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: $R \simeq$ $C_9:C_{36}$ $G/R \simeq$ $C_1$
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: $\operatorname{soc} \simeq$ $C_3\times C_6$ $G/\operatorname{soc} \simeq$ $C_3\times S_3$
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: $P_{ 2 } \simeq$ $C_4$
3-Sylow subgroup: $P_{ 3 } \simeq$ $C_9:C_9$

Subgroup diagram and profile

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Subgroup information

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Series

Derived series $C_9:C_{36}$ $\rhd$ $C_9$ $\rhd$ $C_1$
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 $C_9:C_{36}$ $\rhd$ $C_9:C_{18}$ $\rhd$ $C_9:C_9$ $\rhd$ $C_3\times C_9$ $\rhd$ $C_9$ $\rhd$ $C_3$ $\rhd$ $C_1$
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 $C_9:C_{36}$ $\rhd$ $C_9$
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 $C_1$ $\lhd$ $C_6$
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 28 larger groups in the database.

This group is a maximal quotient of 15 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 $60 \times 60$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $19 \times 19$ rational character table.