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Properties

Label 19494.i
Order \( 2 \cdot 3^{3} \cdot 19^{2} \)
Exponent \( 2 \cdot 3^{2} \cdot 19 \)
Nilpotent no
Solvable yes
$\card{G^{\mathrm{ab}}}$ \( 2 \cdot 3^{2} \)
$\card{Z(G)}$ \( 1 \)
$\card{\Aut(G)}$ \( 2^{2} \cdot 3^{3} \cdot 19^{2} \)
$\card{\mathrm{Out}(G)}$ \( 2 \)
Perm deg. $38$
Trans deg. $38$
Rank $2$

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Show commands: Gap / Magma / SageMath

Copy content comment:Define the group as a permutation group
 
Copy content magma:G := PermutationGroup< 38 | (1,26,18,34,4,33)(2,22,6,25,15,27)(3,37,13,35,7,21)(5,29,8,36,10,28)(9,32,17,38,16,23)(11,24,12,20,19,30)(14,31), (1,2,6,3,10,19,17,9,15)(4,14,16,5,18,13,12,8,11)(20,27,31,36,28,37,34,35,22)(21,33,29,24,32,23,26,25,38) >;
 
Copy content gap:G := Group( (1,26,18,34,4,33)(2,22,6,25,15,27)(3,37,13,35,7,21)(5,29,8,36,10,28)(9,32,17,38,16,23)(11,24,12,20,19,30)(14,31), (1,2,6,3,10,19,17,9,15)(4,14,16,5,18,13,12,8,11)(20,27,31,36,28,37,34,35,22)(21,33,29,24,32,23,26,25,38) );
 
Copy content sage:G = PermutationGroup(['(1,26,18,34,4,33)(2,22,6,25,15,27)(3,37,13,35,7,21)(5,29,8,36,10,28)(9,32,17,38,16,23)(11,24,12,20,19,30)(14,31)', '(1,2,6,3,10,19,17,9,15)(4,14,16,5,18,13,12,8,11)(20,27,31,36,28,37,34,35,22)(21,33,29,24,32,23,26,25,38)'])
 
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(31686707130062717648555099371584944858577313409520693506635509237791,19494)'); a = G.1; b = G.4; c = G.5; d = G.6;
 

Group information

Description:$C_{19}^2:(S_3\times C_9)$
Order: \(19494\)\(\medspace = 2 \cdot 3^{3} \cdot 19^{2} \)
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: \(342\)\(\medspace = 2 \cdot 3^{2} \cdot 19 \)
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:$C_{19}^2:(S_3\times C_{18})$, of order \(38988\)\(\medspace = 2^{2} \cdot 3^{3} \cdot 19^{2} \)
Copy content comment:Automorphism group
 
Copy content gap:AutomorphismGroup(G);
 
Copy content magma:AutomorphismGroup(G);
 
Copy content sage_gap:G.AutomorphismGroup()
 
Composition factors:$C_2$, $C_3$ x 3, $C_{19}$ x 2
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:$3$
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, monomial (hence solvable), and an A-group.

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 6 9 18 19 38 57
Elements 1 57 1520 2166 6498 6498 360 1026 1368 19494
Conjugacy classes   1 1 5 2 12 6 9 2 4 42
Divisions 1 1 3 1 2 1 5 1 1 16
Autjugacy classes 1 1 5 2 12 6 5 1 2 35

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 18 27 36 54 72 108
Irr. complex chars.   18 9 0 0 0 6 4 0 5 0 0 42
Irr. rational chars. 2 3 1 2 1 0 0 1 3 1 2 16

Minimal presentations

Permutation degree:$38$
Transitive degree:$38$
Rank: $2$
Inequivalent generating pairs: $4320$

Minimal degrees of faithful linear representations

Over $\mathbb{C}$ Over $\mathbb{R}$ Over $\mathbb{Q}$
Irreducible 18 36 36
Arbitrary 18 36 36

Constructions

Show commands: Gap / Magma / SageMath


Presentation: ${\langle a, b, c, d \mid a^{18}=b^{3}=c^{19}=d^{19}=[c,d]=1, b^{a}=b^{2}c^{9}d^{11}, c^{a}= \!\cdots\! \rangle}$ 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, -3, -3, -19, 19, 12, 43, 283395, 217305, 37671, 176584, 4870, 1906, 652, 184685, 110819, 43109, 24215]); a,b,c,d := Explode([G.1, G.4, G.5, G.6]); AssignNames(~G, ["a", "a2", "a6", "b", "c", "d"]);
 
Copy content gap:G := PcGroupCode(31686707130062717648555099371584944858577313409520693506635509237791,19494); a := G.1; b := G.4; c := G.5; d := G.6;
 
Copy content sage:# This uses Sage's interface to GAP, as Sage (currently) has no native support for PC groups G = gap.new('PcGroupCode(31686707130062717648555099371584944858577313409520693506635509237791,19494)'); a = G.1; b = G.4; c = G.5; d = G.6;
 
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(31686707130062717648555099371584944858577313409520693506635509237791,19494)'); a = G.1; b = G.4; c = G.5; d = G.6;
 
Permutation group:Degree $38$ $\langle(1,26,18,34,4,33)(2,22,6,25,15,27)(3,37,13,35,7,21)(5,29,8,36,10,28)(9,32,17,38,16,23) \!\cdots\! \rangle$ Copy content Toggle raw display
Copy content comment:Define the group as a permutation group
 
Copy content magma:G := PermutationGroup< 38 | (1,26,18,34,4,33)(2,22,6,25,15,27)(3,37,13,35,7,21)(5,29,8,36,10,28)(9,32,17,38,16,23)(11,24,12,20,19,30)(14,31), (1,2,6,3,10,19,17,9,15)(4,14,16,5,18,13,12,8,11)(20,27,31,36,28,37,34,35,22)(21,33,29,24,32,23,26,25,38) >;
 
Copy content gap:G := Group( (1,26,18,34,4,33)(2,22,6,25,15,27)(3,37,13,35,7,21)(5,29,8,36,10,28)(9,32,17,38,16,23)(11,24,12,20,19,30)(14,31), (1,2,6,3,10,19,17,9,15)(4,14,16,5,18,13,12,8,11)(20,27,31,36,28,37,34,35,22)(21,33,29,24,32,23,26,25,38) );
 
Copy content sage:G = PermutationGroup(['(1,26,18,34,4,33)(2,22,6,25,15,27)(3,37,13,35,7,21)(5,29,8,36,10,28)(9,32,17,38,16,23)(11,24,12,20,19,30)(14,31)', '(1,2,6,3,10,19,17,9,15)(4,14,16,5,18,13,12,8,11)(20,27,31,36,28,37,34,35,22)(21,33,29,24,32,23,26,25,38)'])
 
Transitive group: 38T32 more information
Direct product: not isomorphic to a non-trivial direct product
Semidirect product: $(C_{19}^2:C_9)$ $\,\rtimes\,$ $S_3$ $(C_{19}^2:S_3)$ $\,\rtimes\,$ $C_9$ $C_{19}^2$ $\,\rtimes\,$ $(S_3\times C_9)$ $(C_{19}^2:C_3)$ $\,\rtimes\,$ $C_{18}$ more information
Trans. wreath product: not isomorphic to a non-trivial transitive wreath product
Non-split product: $((C_{19}:C_3)^2)$ . $C_6$ $(C_{19}^2:(C_3\times S_3))$ . $C_3$ $(C_{19}^2:C_3)$ . $(C_3\times S_3)$ more information

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

Homology

Abelianization: $C_{18} \simeq C_{2} \times C_{9}$
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 7192 subgroups in 56 conjugacy classes, 10 normal, and all normal subgroups are characteristic.

Characteristic subgroups are shown in this color.

Special subgroups

Center: $Z \simeq$ $C_1$ $G/Z \simeq$ $C_{19}^2:(S_3\times C_9)$
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: $G' \simeq$ $C_{19}^2:C_3$ $G/G' \simeq$ $C_{18}$
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: $\Phi \simeq$ $C_1$ $G/\Phi \simeq$ $C_{19}^2:(S_3\times C_9)$
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: $\operatorname{Fit} \simeq$ $C_{19}^2$ $G/\operatorname{Fit} \simeq$ $S_3\times C_9$
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: $R \simeq$ $C_{19}^2:(S_3\times C_9)$ $G/R \simeq$ $C_1$
Copy content comment:Radical of the group G
 
Copy content magma:Radical(G);
 
Copy content gap:SolvableRadical(G);
 
Copy content sage_gap:G.SolvableRadical()
 
Socle: $\operatorname{soc} \simeq$ $C_{19}^2$ $G/\operatorname{soc} \simeq$ $S_3\times C_9$
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: $P_{ 2 } \simeq$ $C_2$
3-Sylow subgroup: $P_{ 3 } \simeq$ $C_3\times C_9$
19-Sylow subgroup: $P_{ 19 } \simeq$ $C_{19}^2$

Subgroup diagram and profile

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Subgroup information

Click on a subgroup in the diagram to see information about it.

Series

Derived series $C_{19}^2:(S_3\times C_9)$ $\rhd$ $C_{19}^2:C_3$ $\rhd$ $C_{19}^2$ $\rhd$ $C_1$
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 $C_{19}^2:(S_3\times C_9)$ $\rhd$ $C_{19}^2:(C_3\times C_9)$ $\rhd$ $(C_{19}:C_3)^2$ $\rhd$ $C_{19}^2:C_3$ $\rhd$ $C_{19}^2$ $\rhd$ $C_1$
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 $C_{19}^2:(S_3\times C_9)$ $\rhd$ $C_{19}^2:C_3$
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 $C_1$
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 2 larger groups in the database.

This group is a maximal quotient of 0 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 $42 \times 42$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

1A 2A 3A 3B 3C 6A 9A 9B 18A 19A 19B 19C 19D 19E 38A 57A
Size 1 57 76 722 722 2166 2166 4332 6498 36 54 54 108 108 1026 1368
2 P 1A 1A 3A 3B 3C 3B 9A 9B 9A 19A 19B 19C 19D 19E 19B 57A
3 P 1A 2A 1A 1A 1A 2A 3B 3B 6A 19A 19B 19C 19D 19E 38A 19A
19 P 1A 2A 3A 3B 3C 6A 9A 9B 18A 1A 1A 1A 1A 1A 2A 3A
19494.i.1a 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
19494.i.1b 1 −1 1 1 1 −1 1 1 −1 1 1 1 1 1 −1 1
19494.i.1c 2 2 2 2 2 2 −1 −1 −1 2 2 2 2 2 2 2
19494.i.1d 2 −2 2 2 2 −2 −1 −1 1 2 2 2 2 2 −2 2
19494.i.1e 6 6 −3 −3 6 −3 0 0 0 6 6 6 6 6 6 −3
19494.i.1f 6 −6 −3 −3 6 3 0 0 0 6 6 6 6 6 −6 −3
19494.i.2a 2 0 −1 2 −1 0 2 −1 0 2 2 2 2 2 0 −1
19494.i.2b 4 0 −2 4 −2 0 −2 1 0 4 4 4 4 4 0 −2
19494.i.2c 12 0 3 −6 −6 0 0 0 0 12 12 12 12 12 0 3
19494.i.18a 36 0 18 0 0 0 0 0 0 17 −2 −2 −2 −2 0 −1
19494.i.18b 72 0 −18 0 0 0 0 0 0 34 −4 −4 −4 −4 0 1
19494.i.27a 54 18 0 0 0 0 0 0 0 −3 −3 −3 −3 −3 −1 0
19494.i.27b 54 −18 0 0 0 0 0 0 0 −3 −3 −3 −3 −3 1 0
19494.i.54a 54 0 0 0 0 0 0 0 0 −3 16 −3 −3 −3 0 0
19494.i.54b 108 0 0 0 0 0 0 0 0 −6 −6 −6 13 −6 0 0
19494.i.54c 108 0 0 0 0 0 0 0 0 −6 −6 13 −6 13 0 0