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Properties

Label 243.61
Order \( 3^{5} \)
Exponent \( 3^{2} \)
Abelian yes
$\card{\Aut(G)}$ \( 2^{6} \cdot 3^{10} \cdot 13 \)
Perm deg. $18$
Trans deg. $243$
Rank $4$

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Show commands: Gap / Magma / Oscar / SageMath

Copy content comment:Construction of abstract group
 
Copy content magma:G := SmallGroup(243, 61);
 
Copy content gap:G := SmallGroup(243, 61);
 
Copy content sage:G = PermutationGroup(['(10,18,15,12,17,14,11,16,13)', '(1,3,2)', '(4,6,5)', '(7,9,8)', '(10,12,11)(13,15,14)(16,18,17)'])
 
Copy content sage_gap:G = libgap.SmallGroup(243, 61)
 
Copy content oscar:G = small_group(243, 61)
 

Group information

Description:$C_3^3\times C_9$
Order: \(243\)\(\medspace = 3^{5} \)
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: \(9\)\(\medspace = 3^{2} \)
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()
 
Copy content oscar:exponent(G)
 
Automorphism group:$C_3^4.(C_2\times C_3^3:\GL(3,3))$, of order \(49128768\)\(\medspace = 2^{6} \cdot 3^{10} \cdot 13 \)
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:$C_3$ x 5
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)
 
Nilpotency class:$1$
Copy content comment:Nilpotency class of the group
 
Copy content magma:NilpotencyClass(G);
 
Copy content gap:if IsNilpotentGroup(G) then NilpotencyClassOfGroup(G); fi;
 
Copy content sage:libgap(G).NilpotencyClassOfGroup() if G.is_nilpotent() else -1
 
Copy content sage_gap:G.NilpotencyClassOfGroup() if G.IsNilpotentGroup() else -1
 
Copy content oscar:if is_nilpotent(G) nilpotency_class(G) end
 
Derived length:$1$
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 abelian (hence nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group) and a $p$-group (hence elementary and hyperelementary).

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 3 9
Elements 1 80 162 243
Conjugacy classes   1 80 162 243
Divisions 1 40 27 68
Autjugacy classes 1 2 1 4

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 6
Irr. complex chars.   243 0 0 243
Irr. rational chars. 1 40 27 68

Minimal presentations

Permutation degree:$18$
Transitive degree:$243$
Rank: $4$
Inequivalent generating quadruples: $40$

Minimal degrees of faithful linear representations

Over $\mathbb{C}$ Over $\mathbb{R}$ Over $\mathbb{Q}$
Irreducible none none none
Arbitrary 4 not computed not computed

Constructions

Show commands: Gap / Magma / Oscar / SageMath


Presentation:Abelian group $\langle a, b, c, d \mid a^{3}=b^{3}=c^{3}=d^{9}=1 \rangle$ Copy content Toggle raw display
Copy content comment:Define the group with the given generators and relations
 
Copy content magma:G := PCGroup([5, -3, 3, 3, 3, -3, 78]); a,b,c,d := Explode([G.1, G.2, G.3, G.4]); AssignNames(~G, ["a", "b", "c", "d", "d3"]);
 
Copy content gap:G := PcGroupCode(16392,243); a := G.1; b := G.2; c := G.3; d := G.4;
 
Copy content sage:# This uses Sage's interface to GAP, as Sage (currently) has no native support for PC groups G = gap.new('PcGroupCode(16392,243)'); a = G.1; b = G.2; c = G.3; d = G.4;
 
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(16392,243)'); a = G.1; b = G.2; c = G.3; d = G.4;
 
Permutation group:Degree $18$ $\langle(10,18,15,12,17,14,11,16,13), (1,3,2), (4,6,5), (7,9,8), (10,12,11)(13,15,14)(16,18,17)\rangle$ Copy content Toggle raw display
Copy content comment:Define the group as a permutation group
 
Copy content magma:G := PermutationGroup< 18 | (10,18,15,12,17,14,11,16,13), (1,3,2), (4,6,5), (7,9,8), (10,12,11)(13,15,14)(16,18,17) >;
 
Copy content gap:G := Group( (10,18,15,12,17,14,11,16,13), (1,3,2), (4,6,5), (7,9,8), (10,12,11)(13,15,14)(16,18,17) );
 
Copy content sage:G = PermutationGroup(['(10,18,15,12,17,14,11,16,13)', '(1,3,2)', '(4,6,5)', '(7,9,8)', '(10,12,11)(13,15,14)(16,18,17)'])
 
Copy content sage_gap:G = gap.new('Group( (10,18,15,12,17,14,11,16,13), (1,3,2), (4,6,5), (7,9,8), (10,12,11)(13,15,14)(16,18,17) )')
 
Copy content oscar:G = @permutation_group(18, (10,18,15,12,17,14,11,16,13), (1,3,2), (4,6,5), (7,9,8), (10,12,11)(13,15,14)(16,18,17))
 
Matrix group:$\left\langle \left(\begin{array}{rr} 10 & 0 \\ 0 & 10 \end{array}\right), \left(\begin{array}{rr} 4 & 0 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 19 & 18 \\ 9 & 10 \end{array}\right), \left(\begin{array}{rr} 10 & 0 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 1 & 9 \\ 0 & 1 \end{array}\right) \right\rangle \subseteq \GL_{2}(\Z/27\Z)$
Copy content comment:Define the group as a matrix group with coefficients in GLZq
 
Copy content magma:G := MatrixGroup< 2, Integers(27) | [[10, 0, 0, 10], [4, 0, 0, 1], [19, 18, 9, 10], [10, 0, 0, 1], [1, 9, 0, 1]] >;
 
Copy content gap:G := Group([[[ZmodnZObj(10,27), ZmodnZObj(0,27)], [ZmodnZObj(0,27), ZmodnZObj(10,27)]],[[ZmodnZObj(4,27), ZmodnZObj(0,27)], [ZmodnZObj(0,27), ZmodnZObj(1,27)]],[[ZmodnZObj(19,27), ZmodnZObj(18,27)], [ZmodnZObj(9,27), ZmodnZObj(10,27)]],[[ZmodnZObj(10,27), ZmodnZObj(0,27)], [ZmodnZObj(0,27), ZmodnZObj(1,27)]],[[ZmodnZObj(1,27), ZmodnZObj(9,27)], [ZmodnZObj(0,27), ZmodnZObj(1,27)]]]);
 
Copy content sage:MS = MatrixSpace(Integers(27), 2, 2) G = MatrixGroup([MS([[10, 0], [0, 10]]), MS([[4, 0], [0, 1]]), MS([[19, 18], [9, 10]]), MS([[10, 0], [0, 1]]), MS([[1, 9], [0, 1]])])
 
Copy content sage_gap:G = gap.new('Group([[[ZmodnZObj(10,27), ZmodnZObj(0,27)], [ZmodnZObj(0,27), ZmodnZObj(10,27)]],[[ZmodnZObj(4,27), ZmodnZObj(0,27)], [ZmodnZObj(0,27), ZmodnZObj(1,27)]],[[ZmodnZObj(19,27), ZmodnZObj(18,27)], [ZmodnZObj(9,27), ZmodnZObj(10,27)]],[[ZmodnZObj(10,27), ZmodnZObj(0,27)], [ZmodnZObj(0,27), ZmodnZObj(1,27)]],[[ZmodnZObj(1,27), ZmodnZObj(9,27)], [ZmodnZObj(0,27), ZmodnZObj(1,27)]]])')
 
Copy content oscar:G = matrix_group([matrix(residue_ring(ZZ, 27)[1][[10, 0], [0, 10]]), matrix(residue_ring(ZZ, 27)[1][[4, 0], [0, 1]]), matrix(residue_ring(ZZ, 27)[1][[19, 18], [9, 10]]), matrix(residue_ring(ZZ, 27)[1][[10, 0], [0, 1]]), matrix(residue_ring(ZZ, 27)[1][[1, 9], [0, 1]])])
 
$\left\langle \left(\begin{array}{rr} 19 & 0 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 1 & 18 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 37 & 24 \\ 18 & 1 \end{array}\right), \left(\begin{array}{rr} 19 & 45 \\ 9 & 46 \end{array}\right), \left(\begin{array}{rr} 19 & 0 \\ 0 & 19 \end{array}\right) \right\rangle \subseteq \GL_{2}(\Z/54\Z)$
Copy content comment:Define the group as a matrix group with coefficients in GLZN
 
Copy content magma:G := MatrixGroup< 2, Integers(54) | [[19, 0, 0, 1], [1, 18, 0, 1], [37, 24, 18, 1], [19, 45, 9, 46], [19, 0, 0, 19]] >;
 
Copy content gap:G := Group([[[ZmodnZObj(19,54), ZmodnZObj(0,54)], [ZmodnZObj(0,54), ZmodnZObj(1,54)]],[[ZmodnZObj(1,54), ZmodnZObj(18,54)], [ZmodnZObj(0,54), ZmodnZObj(1,54)]],[[ZmodnZObj(37,54), ZmodnZObj(24,54)], [ZmodnZObj(18,54), ZmodnZObj(1,54)]],[[ZmodnZObj(19,54), ZmodnZObj(45,54)], [ZmodnZObj(9,54), ZmodnZObj(46,54)]],[[ZmodnZObj(19,54), ZmodnZObj(0,54)], [ZmodnZObj(0,54), ZmodnZObj(19,54)]]]);
 
Copy content sage:MS = MatrixSpace(Integers(54), 2, 2) G = MatrixGroup([MS([[19, 0], [0, 1]]), MS([[1, 18], [0, 1]]), MS([[37, 24], [18, 1]]), MS([[19, 45], [9, 46]]), MS([[19, 0], [0, 19]])])
 
Copy content sage_gap:G = gap.new('Group([[[ZmodnZObj(19,54), ZmodnZObj(0,54)], [ZmodnZObj(0,54), ZmodnZObj(1,54)]],[[ZmodnZObj(1,54), ZmodnZObj(18,54)], [ZmodnZObj(0,54), ZmodnZObj(1,54)]],[[ZmodnZObj(37,54), ZmodnZObj(24,54)], [ZmodnZObj(18,54), ZmodnZObj(1,54)]],[[ZmodnZObj(19,54), ZmodnZObj(45,54)], [ZmodnZObj(9,54), ZmodnZObj(46,54)]],[[ZmodnZObj(19,54), ZmodnZObj(0,54)], [ZmodnZObj(0,54), ZmodnZObj(19,54)]]])')
 
Copy content oscar:G = matrix_group([matrix(residue_ring(ZZ, 54)[1][[19, 0], [0, 1]]), matrix(residue_ring(ZZ, 54)[1][[1, 18], [0, 1]]), matrix(residue_ring(ZZ, 54)[1][[37, 24], [18, 1]]), matrix(residue_ring(ZZ, 54)[1][[19, 45], [9, 46]]), matrix(residue_ring(ZZ, 54)[1][[19, 0], [0, 19]])])
 
Direct product: $C_3$ ${}^3$ $\, \times\, $ $C_9$
Semidirect product: not isomorphic to a non-trivial semidirect product
Trans. wreath product: not isomorphic to a non-trivial transitive wreath product
Non-split product: $C_3^4$ . $C_3$ $C_3$ . $C_3^4$ $C_3^3$ . $C_3^2$ $C_3^2$ . $C_3^3$ more information

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

Homology

Primary decomposition: $C_{3}^{3} \times C_{9}$
Copy content comment:The primary decomposition of the group
 
Copy content magma:PrimaryInvariants(G);
 
Copy content gap:AbelianInvariants(G);
 
Copy content sage_gap:G.AbelianInvariants()
 
Copy content oscar:abelian_invariants(G)
 
Schur multiplier: $C_{3}^{6}$
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: $0$
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 396 subgroups, all normal (4 characteristic).

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

Special subgroups

Center: $Z \simeq$ $C_3^3\times C_9$ $G/Z \simeq$ $C_1$
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: $G' \simeq$ $C_1$ $G/G' \simeq$ $C_3^3\times C_9$
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: $\Phi \simeq$ $C_3$ $G/\Phi \simeq$ $C_3^4$
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: $\operatorname{Fit} \simeq$ $C_3^3\times C_9$ $G/\operatorname{Fit} \simeq$ $C_1$
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()
 
Copy content oscar:fitting_subgroup(G)
 
Radical: $R \simeq$ $C_3^3\times C_9$ $G/R \simeq$ $C_1$
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: $\operatorname{soc} \simeq$ $C_3^4$ $G/\operatorname{soc} \simeq$ $C_3$
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()
 
Copy content oscar:socle(G)
 
3-Sylow subgroup: $P_{ 3 } \simeq$ $C_3^3\times C_9$

Subgroup diagram and profile

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Subgroup information

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Series

Derived series $C_3^3\times C_9$ $\rhd$ $C_1$
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 $C_3^3\times C_9$ $\rhd$ $C_3^2\times C_9$ $\rhd$ $C_3\times C_9$ $\rhd$ $C_9$ $\rhd$ $C_3$ $\rhd$ $C_1$
Copy content comment:Chief series of the group G
 
Copy content magma:ChiefSeries(G);
 
Copy content gap:ChiefSeries(G);
 
Copy content sage:libgap(G).ChiefSeries()
 
Copy content sage_gap:G.ChiefSeries()
 
Copy content oscar:chief_series(G)
 
Lower central series $C_3^3\times C_9$ $\rhd$ $C_1$
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()
 
Copy content oscar:lower_central_series(G)
 
Upper central series $C_1$ $\lhd$ $C_3^3\times C_9$
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 63 larger groups in the database.

This group is a maximal quotient of 22 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 $243 \times 243$ 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 $68 \times 68$ rational character table.