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Properties

Label 216.96
Order \( 2^{3} \cdot 3^{3} \)
Exponent \( 2 \cdot 3^{2} \)
Nilpotent no
Solvable yes
$\card{G^{\mathrm{ab}}}$ \( 2 \cdot 3 \)
$\card{Z(G)}$ \( 1 \)
$\card{\Aut(G)}$ \( 2^{3} \cdot 3^{4} \)
$\card{\mathrm{Out}(G)}$ \( 3 \)
Perm deg. $13$
Trans deg. $36$
Rank $2$

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Show commands: Gap / Magma / Oscar / SageMath

Copy content comment:Construction of abstract group
 
Copy content magma:G := SmallGroup(216, 96);
 
Copy content gap:G := SmallGroup(216, 96);
 
Copy content sage:G = PermutationGroup(['(2,4)(3,7)(5,6)(8,9)', '(2,5,9)(4,6,8)(11,13,12)', '(10,11)(12,13)', '(10,12)(11,13)', '(1,2,6,7,9,8,3,5,4)', '(1,3,7)(2,5,9)(4,8,6)'])
 
Copy content sage_gap:G = libgap.SmallGroup(216, 96)
 
Copy content oscar:G = small_group(216, 96)
 

Group information

Description:$D_9:A_4$
Order: \(216\)\(\medspace = 2^{3} \cdot 3^{3} \)
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:$A_4\times C_9:C_6$, of order \(648\)\(\medspace = 2^{3} \cdot 3^{4} \)
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 3
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, monomial (hence solvable), 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 18
Elements 1 39 26 78 54 18 216
Conjugacy classes   1 3 3 3 3 3 16
Divisions 1 3 2 2 2 1 11
Autjugacy classes 1 3 3 3 3 1 14

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 3 4 6 18
Irr. complex chars.   6 3 2 0 5 0 16
Irr. rational chars. 2 3 2 1 2 1 11

Minimal presentations

Permutation degree:$13$
Transitive degree:$36$
Rank: $2$
Inequivalent generating pairs: $24$

Minimal degrees of faithful linear representations

Over $\mathbb{C}$ Over $\mathbb{R}$ Over $\mathbb{Q}$
Irreducible 6 6 18
Arbitrary 6 6 9

Constructions

Show commands: Gap / Magma / Oscar / SageMath


Presentation: $\langle a, b, c \mid a^{6}=b^{2}=c^{18}=[b,c]=1, b^{a}=bc^{9}, c^{a}=bc^{2} \rangle$ Copy content Toggle raw display
Copy content comment:Define the group with the given generators and relations
 
Copy content magma:G := PCGroup([6, -2, -3, -2, 2, -3, -3, 12, 2054, 980, 723, 1953, 69, 1444, 1450, 118, 5189]); a,b,c := Explode([G.1, G.3, G.4]); AssignNames(~G, ["a", "a2", "b", "c", "c2", "c6"]);
 
Copy content gap:G := PcGroupCode(75090667753138104702226396755,216); 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(75090667753138104702226396755,216)'); 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(75090667753138104702226396755,216)'); a = G.1; b = G.3; c = G.4;
 
Permutation group:Degree $13$ $\langle(2,4)(3,7)(5,6)(8,9), (2,5,9)(4,6,8)(11,13,12), (10,11)(12,13), (10,12)(11,13), (1,2,6,7,9,8,3,5,4), (1,3,7)(2,5,9)(4,8,6)\rangle$ Copy content Toggle raw display
Copy content comment:Define the group as a permutation group
 
Copy content magma:G := PermutationGroup< 13 | (2,4)(3,7)(5,6)(8,9), (2,5,9)(4,6,8)(11,13,12), (10,11)(12,13), (10,12)(11,13), (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), (2,5,9)(4,6,8)(11,13,12), (10,11)(12,13), (10,12)(11,13), (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)', '(2,5,9)(4,6,8)(11,13,12)', '(10,11)(12,13)', '(10,12)(11,13)', '(1,2,6,7,9,8,3,5,4)', '(1,3,7)(2,5,9)(4,8,6)'])
 
Copy content sage_gap:G = gap.new('Group( (2,4)(3,7)(5,6)(8,9), (2,5,9)(4,6,8)(11,13,12), (10,11)(12,13), (10,12)(11,13), (1,2,6,7,9,8,3,5,4), (1,3,7)(2,5,9)(4,8,6) )')
 
Copy content oscar:G = @permutation_group(13, (2,4)(3,7)(5,6)(8,9), (2,5,9)(4,6,8)(11,13,12), (10,11)(12,13), (10,12)(11,13), (1,2,6,7,9,8,3,5,4), (1,3,7)(2,5,9)(4,8,6))
 
Matrix group:$\left\langle \left(\begin{array}{rr} 17 & 6 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 19 & 0 \\ 18 & 19 \end{array}\right), \left(\begin{array}{rr} 7 & 13 \\ 3 & 28 \end{array}\right), \left(\begin{array}{rr} 19 & 33 \\ 27 & 4 \end{array}\right), \left(\begin{array}{rr} 1 & 18 \\ 18 & 1 \end{array}\right), \left(\begin{array}{rr} 1 & 12 \\ 0 & 1 \end{array}\right) \right\rangle \subseteq \GL_{2}(\Z/36\Z)$
Copy content comment:Define the group as a matrix group with coefficients in GLZN
 
Copy content magma:G := MatrixGroup< 2, Integers(36) | [[17, 6, 0, 1], [19, 0, 18, 19], [7, 13, 3, 28], [19, 33, 27, 4], [1, 18, 18, 1], [1, 12, 0, 1]] >;
 
Copy content gap:G := Group([[[ZmodnZObj(17,36), ZmodnZObj(6,36)], [ZmodnZObj(0,36), ZmodnZObj(1,36)]],[[ZmodnZObj(19,36), ZmodnZObj(0,36)], [ZmodnZObj(18,36), ZmodnZObj(19,36)]],[[ZmodnZObj(7,36), ZmodnZObj(13,36)], [ZmodnZObj(3,36), ZmodnZObj(28,36)]],[[ZmodnZObj(19,36), ZmodnZObj(33,36)], [ZmodnZObj(27,36), ZmodnZObj(4,36)]],[[ZmodnZObj(1,36), ZmodnZObj(18,36)], [ZmodnZObj(18,36), ZmodnZObj(1,36)]],[[ZmodnZObj(1,36), ZmodnZObj(12,36)], [ZmodnZObj(0,36), ZmodnZObj(1,36)]]]);
 
Copy content sage:MS = MatrixSpace(Integers(36), 2, 2) G = MatrixGroup([MS([[17, 6], [0, 1]]), MS([[19, 0], [18, 19]]), MS([[7, 13], [3, 28]]), MS([[19, 33], [27, 4]]), MS([[1, 18], [18, 1]]), MS([[1, 12], [0, 1]])])
 
Copy content sage_gap:G = gap.new('Group([[[ZmodnZObj(17,36), ZmodnZObj(6,36)], [ZmodnZObj(0,36), ZmodnZObj(1,36)]],[[ZmodnZObj(19,36), ZmodnZObj(0,36)], [ZmodnZObj(18,36), ZmodnZObj(19,36)]],[[ZmodnZObj(7,36), ZmodnZObj(13,36)], [ZmodnZObj(3,36), ZmodnZObj(28,36)]],[[ZmodnZObj(19,36), ZmodnZObj(33,36)], [ZmodnZObj(27,36), ZmodnZObj(4,36)]],[[ZmodnZObj(1,36), ZmodnZObj(18,36)], [ZmodnZObj(18,36), ZmodnZObj(1,36)]],[[ZmodnZObj(1,36), ZmodnZObj(12,36)], [ZmodnZObj(0,36), ZmodnZObj(1,36)]]])')
 
Copy content oscar:G = matrix_group([matrix(residue_ring(ZZ, 36)[1][[17, 6], [0, 1]]), matrix(residue_ring(ZZ, 36)[1][[19, 0], [18, 19]]), matrix(residue_ring(ZZ, 36)[1][[7, 13], [3, 28]]), matrix(residue_ring(ZZ, 36)[1][[19, 33], [27, 4]]), matrix(residue_ring(ZZ, 36)[1][[1, 18], [18, 1]]), matrix(residue_ring(ZZ, 36)[1][[1, 12], [0, 1]])])
 
Transitive group: 36T298 more information
Copy content magma:G := TransitiveGroup(36, 298);
 
Copy content gap:G := TransitiveGroup(36, 298);
 
Copy content sage:G = TransitiveGroup(36, 298)
 
Copy content sage_gap:G = libgap.TransitiveGroup(36, 298)
 
Copy content oscar:G = transitive_group(36, 298)
 
Direct product: not isomorphic to a non-trivial direct product
Semidirect product: $D_9$ $\,\rtimes\,$ $A_4$ $(C_9:A_4)$ $\,\rtimes\,$ $C_2$ $C_9$ $\,\rtimes\,$ $(C_2\times A_4)$ $(C_2\times D_{18})$ $\,\rtimes\,$ $C_3$ all 6
Trans. wreath product: not isomorphic to a non-trivial transitive wreath product
Non-split product: $(C_3\times A_4)$ . $S_3$ $C_3$ . $(S_3\times A_4)$ $(C_2\times C_6)$ . $(C_3\times S_3)$ 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: $C_{2}$
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 258 subgroups in 38 conjugacy classes, 11 normal, and all normal subgroups are characteristic.

Characteristic subgroups are shown in this color.

Special subgroups

Center: $Z \simeq$ $C_1$ $G/Z \simeq$ $D_9:A_4$
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: $G' \simeq$ $C_2\times C_{18}$ $G/G' \simeq$ $C_6$
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: $\Phi \simeq$ $C_3$ $G/\Phi \simeq$ $S_3\times A_4$
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: $\operatorname{Fit} \simeq$ $C_2\times C_{18}$ $G/\operatorname{Fit} \simeq$ $C_6$
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: $R \simeq$ $D_9:A_4$ $G/R \simeq$ $C_1$
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: $\operatorname{soc} \simeq$ $C_2\times C_6$ $G/\operatorname{soc} \simeq$ $C_3\times S_3$
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()
 
Copy content oscar:socle(G)
 
2-Sylow subgroup: $P_{ 2 } \simeq$ $C_2^3$
3-Sylow subgroup: $P_{ 3 } \simeq$ $C_9:C_3$

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 $D_9:A_4$ $\rhd$ $C_2\times C_{18}$ $\rhd$ $C_1$
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 $D_9:A_4$ $\rhd$ $C_9:A_4$ $\rhd$ $C_2\times C_{18}$ $\rhd$ $C_9$ $\rhd$ $C_3$ $\rhd$ $C_1$
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 $D_9:A_4$ $\rhd$ $C_2\times C_{18}$
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 24 larger groups in the database.

This group is a maximal quotient of 26 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 2B 2C 3A 3B1 3B-1 6A 6B1 6B-1 9A 9B1 9B-1 18A1 18A5 18A7
Size 1 3 9 27 2 12 12 6 36 36 6 24 24 6 6 6
2 P 1A 1A 1A 1A 3A 3B-1 3B1 3A 3B1 3B-1 9A 9B-1 9B1 9A 9A 9A
3 P 1A 2A 2B 2C 1A 1A 1A 2A 2B 2B 3A 3A 3A 6A 6A 6A
Type
216.96.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
216.96.1b R 1 1 −1 −1 1 1 1 1 −1 −1 1 1 1 1 1 1
216.96.1c1 C 1 1 1 1 1 ζ3−1 ζ3 1 ζ3 ζ3−1 1 ζ3−1 ζ3 1 1 1
216.96.1c2 C 1 1 1 1 1 ζ3 ζ3−1 1 ζ3−1 ζ3 1 ζ3 ζ3−1 1 1 1
216.96.1d1 C 1 1 −1 −1 1 ζ3−1 ζ3 1 −ζ3 −ζ3−1 1 ζ3−1 ζ3 1 1 1
216.96.1d2 C 1 1 −1 −1 1 ζ3 ζ3−1 1 −ζ3−1 −ζ3 1 ζ3 ζ3−1 1 1 1
216.96.2a R 2 2 0 0 2 2 2 2 0 0 −1 −1 −1 −1 −1 −1
216.96.2b1 C 2 2 0 0 2 2ζ3−1 2ζ3 2 0 0 −1 −ζ3−1 −ζ3 −1 −1 −1
216.96.2b2 C 2 2 0 0 2 2ζ3 2ζ3−1 2 0 0 −1 −ζ3 −ζ3−1 −1 −1 −1
216.96.3a R 3 −1 3 −1 3 0 0 −1 0 0 3 0 0 −1 −1 −1
216.96.3b R 3 −1 −3 1 3 0 0 −1 0 0 3 0 0 −1 −1 −1
216.96.6a R 6 6 0 0 −3 0 0 −3 0 0 0 0 0 0 0 0
216.96.6b R 6 −2 0 0 6 0 0 −2 0 0 −3 0 0 1 1 1
216.96.6c1 R 6 −2 0 0 −3 0 0 1 0 0 0 0 0 2ζ9−4+2ζ94 2ζ9−2+2ζ92 2ζ9−1+2ζ9
216.96.6c2 R 6 −2 0 0 −3 0 0 1 0 0 0 0 0 2ζ9−2+2ζ92 2ζ9−1+2ζ9 2ζ9−4+2ζ94
216.96.6c3 R 6 −2 0 0 −3 0 0 1 0 0 0 0 0 2ζ9−1+2ζ9 2ζ9−4+2ζ94 2ζ9−2+2ζ92

Rational character table

1A 2A 2B 2C 3A 3B 6A 6B 9A 9B 18A
Size 1 3 9 27 2 24 6 72 6 48 18
2 P 1A 1A 1A 1A 3A 3B 3A 3B 9A 9B 9A
3 P 1A 2A 2B 2C 1A 1A 2A 2B 3A 3A 6A
216.96.1a 1 1 1 1 1 1 1 1 1 1 1
216.96.1b 1 1 −1 −1 1 1 1 −1 1 1 1
216.96.1c 2 2 2 2 2 −1 2 −1 2 −1 2
216.96.1d 2 2 −2 −2 2 −1 2 1 2 −1 2
216.96.2a 2 2 0 0 2 2 2 0 −1 −1 −1
216.96.2b 4 4 0 0 4 −2 4 0 −2 1 −2
216.96.3a 3 −1 3 −1 3 0 −1 0 3 0 −1
216.96.3b 3 −1 −3 1 3 0 −1 0 3 0 −1
216.96.6a 6 6 0 0 −3 0 −3 0 0 0 0
216.96.6b 6 −2 0 0 6 0 −2 0 −3 0 1
216.96.6c 18 −6 0 0 −9 0 3 0 0 0 0