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Properties

Label 162.13
Order \( 2 \cdot 3^{4} \)
Exponent \( 2 \cdot 3^{2} \)
Nilpotent no
Solvable yes
$\card{G^{\mathrm{ab}}}$ \( 2 \cdot 3 \)
$\card{Z(G)}$ \( 1 \)
$\card{\Aut(G)}$ \( 2^{2} \cdot 3^{5} \)
$\card{\mathrm{Out}(G)}$ \( 2 \cdot 3 \)
Perm deg. $27$
Trans deg. $27$
Rank $2$

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Show commands: Gap / Magma / Oscar / SageMath

Copy content comment:Construction of abstract group
 
Copy content magma:G := SmallGroup(162, 13);
 
Copy content gap:G := SmallGroup(162, 13);
 
Copy content sage:G = PermutationGroup(['(2,3)(4,7)(5,9)(6,8)(10,21)(11,20)(12,19)(13,27)(14,26)(15,25)(16,24)(17,23)(18,22)', '(4,5,6)(7,9,8)(10,13,17)(11,14,18)(12,15,16)(19,25,24)(20,26,22)(21,27,23)', '(1,21,12,3,20,11,2,19,10)(4,24,15,6,23,14,5,22,13)(7,27,18,9,26,17,8,25,16)', '(1,7,4)(2,8,5)(3,9,6)(10,16,13)(11,17,14)(12,18,15)(19,25,22)(20,26,23)(21,27,24)', '(1,3,2)(4,6,5)(7,9,8)(10,12,11)(13,15,14)(16,18,17)(19,21,20)(22,24,23)(25,27,26)'])
 
Copy content sage_gap:G = libgap.SmallGroup(162, 13)
 
Copy content oscar:G = small_group(162, 13)
 

Group information

Description:$\He_3.S_3$
Order: \(162\)\(\medspace = 2 \cdot 3^{4} \)
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()
 
Copy content oscar:order(G)
 
Exponent: \(18\)\(\medspace = 2 \cdot 3^{2} \)
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()
 
Copy content oscar:exponent(G)
 
Automorphism group:$C_3^3.S_3^2$, of order \(972\)\(\medspace = 2^{2} \cdot 3^{5} \)
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:$C_2$, $C_3$ 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:$2$
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, supersolvable (hence solvable and 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 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 6 9
Elements 1 27 26 54 54 162
Conjugacy classes   1 1 4 2 5 13
Divisions 1 1 3 1 2 8
Autjugacy classes 1 1 3 1 2 8

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 6 18
Irr. complex chars.   6 3 0 4 0 13
Irr. rational chars. 2 3 1 1 1 8

Minimal presentations

Permutation degree:$27$
Transitive degree:$27$
Rank: $2$
Inequivalent generating pairs: $12$

Minimal degrees of faithful linear representations

Over $\mathbb{C}$ Over $\mathbb{R}$ Over $\mathbb{Q}$
Irreducible 6 6 18
Arbitrary 6 6 18

Constructions

Show commands: Gap / Magma / Oscar / SageMath


Presentation: $\langle a, b, c \mid a^{6}=b^{3}=c^{9}=[b,c]=1, b^{a}=b^{2}c^{6}, c^{a}=b^{2}c^{8} \rangle$ Copy content Toggle raw display
Copy content comment:Define the group with the given generators and relations
 
Copy content magma:G := PCGroup([5, -2, -3, -3, 3, -3, 10, 1802, 862, 3123, 848, 78, 2704]); a,b,c := Explode([G.1, G.3, G.4]); AssignNames(~G, ["a", "a2", "b", "c", "c3"]);
 
Copy content gap:G := PcGroupCode(180175900447401868591,162); a := G.1; b := G.3; c := 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(180175900447401868591,162)'); a = G.1; b = G.3; c = 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(180175900447401868591,162)'); a = G.1; b = G.3; c = G.4;
 
Permutation group:Degree $27$ $\langle(2,3)(4,7)(5,9)(6,8)(10,21)(11,20)(12,19)(13,27)(14,26)(15,25)(16,24)(17,23) \!\cdots\! \rangle$ Copy content Toggle raw display
Copy content comment:Define the group as a permutation group
 
Copy content magma:G := PermutationGroup< 27 | (2,3)(4,7)(5,9)(6,8)(10,21)(11,20)(12,19)(13,27)(14,26)(15,25)(16,24)(17,23)(18,22), (4,5,6)(7,9,8)(10,13,17)(11,14,18)(12,15,16)(19,25,24)(20,26,22)(21,27,23), (1,21,12,3,20,11,2,19,10)(4,24,15,6,23,14,5,22,13)(7,27,18,9,26,17,8,25,16), (1,7,4)(2,8,5)(3,9,6)(10,16,13)(11,17,14)(12,18,15)(19,25,22)(20,26,23)(21,27,24), (1,3,2)(4,6,5)(7,9,8)(10,12,11)(13,15,14)(16,18,17)(19,21,20)(22,24,23)(25,27,26) >;
 
Copy content gap:G := Group( (2,3)(4,7)(5,9)(6,8)(10,21)(11,20)(12,19)(13,27)(14,26)(15,25)(16,24)(17,23)(18,22), (4,5,6)(7,9,8)(10,13,17)(11,14,18)(12,15,16)(19,25,24)(20,26,22)(21,27,23), (1,21,12,3,20,11,2,19,10)(4,24,15,6,23,14,5,22,13)(7,27,18,9,26,17,8,25,16), (1,7,4)(2,8,5)(3,9,6)(10,16,13)(11,17,14)(12,18,15)(19,25,22)(20,26,23)(21,27,24), (1,3,2)(4,6,5)(7,9,8)(10,12,11)(13,15,14)(16,18,17)(19,21,20)(22,24,23)(25,27,26) );
 
Copy content sage:G = PermutationGroup(['(2,3)(4,7)(5,9)(6,8)(10,21)(11,20)(12,19)(13,27)(14,26)(15,25)(16,24)(17,23)(18,22)', '(4,5,6)(7,9,8)(10,13,17)(11,14,18)(12,15,16)(19,25,24)(20,26,22)(21,27,23)', '(1,21,12,3,20,11,2,19,10)(4,24,15,6,23,14,5,22,13)(7,27,18,9,26,17,8,25,16)', '(1,7,4)(2,8,5)(3,9,6)(10,16,13)(11,17,14)(12,18,15)(19,25,22)(20,26,23)(21,27,24)', '(1,3,2)(4,6,5)(7,9,8)(10,12,11)(13,15,14)(16,18,17)(19,21,20)(22,24,23)(25,27,26)'])
 
Copy content sage_gap:G = gap.new('Group( (2,3)(4,7)(5,9)(6,8)(10,21)(11,20)(12,19)(13,27)(14,26)(15,25)(16,24)(17,23)(18,22), (4,5,6)(7,9,8)(10,13,17)(11,14,18)(12,15,16)(19,25,24)(20,26,22)(21,27,23), (1,21,12,3,20,11,2,19,10)(4,24,15,6,23,14,5,22,13)(7,27,18,9,26,17,8,25,16), (1,7,4)(2,8,5)(3,9,6)(10,16,13)(11,17,14)(12,18,15)(19,25,22)(20,26,23)(21,27,24), (1,3,2)(4,6,5)(7,9,8)(10,12,11)(13,15,14)(16,18,17)(19,21,20)(22,24,23)(25,27,26) )')
 
Copy content oscar:G = @permutation_group(27, (2,3)(4,7)(5,9)(6,8)(10,21)(11,20)(12,19)(13,27)(14,26)(15,25)(16,24)(17,23)(18,22), (4,5,6)(7,9,8)(10,13,17)(11,14,18)(12,15,16)(19,25,24)(20,26,22)(21,27,23), (1,21,12,3,20,11,2,19,10)(4,24,15,6,23,14,5,22,13)(7,27,18,9,26,17,8,25,16), (1,7,4)(2,8,5)(3,9,6)(10,16,13)(11,17,14)(12,18,15)(19,25,22)(20,26,23)(21,27,24), (1,3,2)(4,6,5)(7,9,8)(10,12,11)(13,15,14)(16,18,17)(19,21,20)(22,24,23)(25,27,26))
 
Transitive group: 27T41 27T72 more information
Copy content magma:G := TransitiveGroup(27, 41);
 
Copy content gap:G := TransitiveGroup(27, 41);
 
Copy content sage:G = TransitiveGroup(27, 41)
 
Copy content sage_gap:G = libgap.TransitiveGroup(27, 41)
 
Copy content oscar:G = transitive_group(27, 41)
 
Copy content magma:G := TransitiveGroup(27, 72);
 
Copy content gap:G := TransitiveGroup(27, 72);
 
Copy content sage:G = TransitiveGroup(27, 72)
 
Copy content sage_gap:G = libgap.TransitiveGroup(27, 72)
 
Copy content oscar:G = transitive_group(27, 72)
 
Direct product: not isomorphic to a non-trivial direct product
Semidirect product: $(C_3:D_9)$ $\,\rtimes\,$ $C_3$ $(C_3\times C_9)$ $\,\rtimes\,$ $C_6$ $(C_3.\He_3)$ $\,\rtimes\,$ $C_2$ more information
Trans. wreath product: not isomorphic to a non-trivial transitive wreath product
Non-split product: $\He_3$ . $S_3$ $C_3^2$ . $(C_3\times S_3)$ $C_3$ . $(C_3^2:C_6)$ more information

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

Homology

Abelianization: $C_{6} \simeq C_{2} \times C_{3}$
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 148 subgroups in 22 conjugacy classes, 8 normal, and all normal subgroups are characteristic.

Characteristic subgroups are shown in this color.

Special subgroups

Center: $Z \simeq$ $C_1$ $G/Z \simeq$ $\He_3.S_3$
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: $G' \simeq$ $C_3\times C_9$ $G/G' \simeq$ $C_6$
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: $\Phi \simeq$ $C_3^2$ $G/\Phi \simeq$ $C_3\times S_3$
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()
 
Copy content oscar:frattini_subgroup(G)
 
Fitting: $\operatorname{Fit} \simeq$ $C_3.\He_3$ $G/\operatorname{Fit} \simeq$ $C_2$
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()
 
Copy content oscar:fitting_subgroup(G)
 
Radical: $R \simeq$ $\He_3.S_3$ $G/R \simeq$ $C_1$
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: $\operatorname{soc} \simeq$ $C_3$ $G/\operatorname{soc} \simeq$ $C_3^2:C_6$
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: $P_{ 2 } \simeq$ $C_2$
3-Sylow subgroup: $P_{ 3 } \simeq$ $C_3.\He_3$

Subgroup diagram and profile

For the default diagram, subgroups are sorted vertically by the number of prime divisors (counted with multiplicity) in their orders.
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Subgroup information

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Series

Derived series $\He_3.S_3$ $\rhd$ $C_3\times C_9$ $\rhd$ $C_1$
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 $\He_3.S_3$ $\rhd$ $C_3.\He_3$ $\rhd$ $C_3\times C_9$ $\rhd$ $C_3^2$ $\rhd$ $C_3$ $\rhd$ $C_1$
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 $\He_3.S_3$ $\rhd$ $C_3\times C_9$
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()
 
Copy content oscar:lower_central_series(G)
 
Upper central series $C_1$
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 17 larger groups in the database.

This group is a maximal quotient of 20 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

1A 2A 3A 3B 3C1 3C-1 6A1 6A-1 9A1 9A2 9A4 9B1 9B-1
Size 1 27 2 6 9 9 27 27 6 6 6 18 18
2 P 1A 1A 3A 3B 3C-1 3C1 3C1 3C-1 9A2 9A4 9A1 9B-1 9B1
3 P 1A 2A 1A 1A 1A 1A 2A 2A 3A 3A 3A 3A 3A
Type
162.13.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1
162.13.1b R 1 −1 1 1 1 1 −1 −1 1 1 1 1 1
162.13.1c1 C 1 1 1 1 ζ3−1 ζ3 ζ3 ζ3−1 1 1 1 ζ3−1 ζ3
162.13.1c2 C 1 1 1 1 ζ3 ζ3−1 ζ3−1 ζ3 1 1 1 ζ3 ζ3−1
162.13.1d1 C 1 −1 1 1 ζ3−1 ζ3 −ζ3 −ζ3−1 1 1 1 ζ3−1 ζ3
162.13.1d2 C 1 −1 1 1 ζ3 ζ3−1 −ζ3−1 −ζ3 1 1 1 ζ3 ζ3−1
162.13.2a R 2 0 2 2 2 2 0 0 −1 −1 −1 −1 −1
162.13.2b1 C 2 0 2 2 2ζ3−1 2ζ3 0 0 −1 −1 −1 −ζ3−1 −ζ3
162.13.2b2 C 2 0 2 2 2ζ3 2ζ3−1 0 0 −1 −1 −1 −ζ3 −ζ3−1
162.13.6a R 6 0 6 −3 0 0 0 0 0 0 0 0 0
162.13.6b1 R 6 0 −3 0 0 0 0 0 2ζ9−4−ζ9+ζ92+ζ94 −ζ9−4+2ζ9−2ζ92+ζ94 −ζ9−4−ζ9+ζ92−2ζ94 0 0
162.13.6b2 R 6 0 −3 0 0 0 0 0 −ζ9−4+2ζ9−2ζ92+ζ94 −ζ9−4−ζ9+ζ92−2ζ94 2ζ9−4−ζ9+ζ92+ζ94 0 0
162.13.6b3 R 6 0 −3 0 0 0 0 0 −ζ9−4−ζ9+ζ92−2ζ94 2ζ9−4−ζ9+ζ92+ζ94 −ζ9−4+2ζ9−2ζ92+ζ94 0 0

Rational character table

1A 2A 3A 3B 3C 6A 9A 9B
Size 1 27 2 6 18 54 18 36
2 P 1A 1A 3A 3B 3C 3C 9A 9B
3 P 1A 2A 1A 1A 1A 2A 3A 3A
162.13.1a 1 1 1 1 1 1 1 1
162.13.1b 1 −1 1 1 1 −1 1 1
162.13.1c 2 2 2 2 −1 −1 2 −1
162.13.1d 2 −2 2 2 −1 1 2 −1
162.13.2a 2 0 2 2 2 0 −1 −1
162.13.2b 4 0 4 4 −2 0 −2 1
162.13.6a 6 0 6 −3 0 0 0 0
162.13.6b 18 0 −9 0 0 0 0 0