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Properties

Label 10584.a
Order \( 2^{3} \cdot 3^{3} \cdot 7^{2} \)
Exponent \( 2 \cdot 3^{2} \cdot 7 \)
Nilpotent no
Solvable no
$\card{G^{\mathrm{ab}}}$ \( 3 \cdot 7 \)
$\card{Z(G)}$ \( 3 \cdot 7 \)
$\card{\Aut(G)}$ \( 2^{5} \cdot 3^{4} \cdot 7 \)
$\card{\mathrm{Out}(G)}$ \( 2^{2} \cdot 3^{2} \)
Perm deg. $19$
Trans deg. $189$
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< 19 | (1,2,3,4,5,7,6)(10,11,14,15,16,13,12), (2,3,4,6,8,9,7)(10,12,13,16,15,14,11), (10,13,15,11,12,16,14)(17,18,19), (10,14,16,12,11,15,13) >;
 
Copy content gap:G := Group( (1,2,3,4,5,7,6)(10,11,14,15,16,13,12), (2,3,4,6,8,9,7)(10,12,13,16,15,14,11), (10,13,15,11,12,16,14)(17,18,19), (10,14,16,12,11,15,13) );
 
Copy content sage:G = PermutationGroup(['(1,2,3,4,5,7,6)(10,11,14,15,16,13,12)', '(2,3,4,6,8,9,7)(10,12,13,16,15,14,11)', '(10,13,15,11,12,16,14)(17,18,19)', '(10,14,16,12,11,15,13)'])
 

Group information

Description:$C_3\times \GL(2,8)$
Order: \(10584\)\(\medspace = 2^{3} \cdot 3^{3} \cdot 7^{2} \)
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: \(126\)\(\medspace = 2 \cdot 3^{2} \cdot 7 \)
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:$C_2\times C_6\times \SL(2,8).C_3$, of order \(18144\)\(\medspace = 2^{5} \cdot 3^{4} \cdot 7 \)
Copy content comment:Automorphism group
 
Copy content gap:AutomorphismGroup(G);
 
Copy content magma:AutomorphismGroup(G);
 
Copy content sage_gap:G.AutomorphismGroup()
 
Composition factors:$C_3$, $C_7$, $\SL(2,8)$
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:$1$
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, an A-group, and nonsolvable.

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 7 9 14 21 42 63
Elements 1 63 170 126 1518 504 378 4044 756 3024 10584
Conjugacy classes   1 1 5 2 27 9 6 72 12 54 189
Divisions 1 1 3 1 5 2 1 7 1 2 24
Autjugacy classes 1 1 3 1 3 2 1 5 1 2 20

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 6 7 8 9 12 14 16 21 27 42 48 54 84 96 108 126 252
Irr. complex chars.   21 0 0 84 21 63 0 0 0 0 0 0 0 0 0 0 0 0 0 189
Irr. rational chars. 1 1 1 1 1 0 1 1 1 1 1 2 1 4 1 1 3 1 1 24

Minimal presentations

Permutation degree:$19$
Transitive degree:$189$
Rank: $2$
Inequivalent generating pairs: $4544$

Minimal degrees of faithful linear representations

Over $\mathbb{C}$ Over $\mathbb{R}$ Over $\mathbb{Q}$
Irreducible 7 14 84
Arbitrary 7 9 15

Constructions

Show commands: Gap / Magma / SageMath


Permutation group:Degree $19$ $\langle(1,2,3,4,5,7,6)(10,11,14,15,16,13,12), (2,3,4,6,8,9,7)(10,12,13,16,15,14,11), (10,13,15,11,12,16,14)(17,18,19), (10,14,16,12,11,15,13)\rangle$ Copy content Toggle raw display
Copy content comment:Define the group as a permutation group
 
Copy content magma:G := PermutationGroup< 19 | (1,2,3,4,5,7,6)(10,11,14,15,16,13,12), (2,3,4,6,8,9,7)(10,12,13,16,15,14,11), (10,13,15,11,12,16,14)(17,18,19), (10,14,16,12,11,15,13) >;
 
Copy content gap:G := Group( (1,2,3,4,5,7,6)(10,11,14,15,16,13,12), (2,3,4,6,8,9,7)(10,12,13,16,15,14,11), (10,13,15,11,12,16,14)(17,18,19), (10,14,16,12,11,15,13) );
 
Copy content sage:G = PermutationGroup(['(1,2,3,4,5,7,6)(10,11,14,15,16,13,12)', '(2,3,4,6,8,9,7)(10,12,13,16,15,14,11)', '(10,13,15,11,12,16,14)(17,18,19)', '(10,14,16,12,11,15,13)'])
 
Matrix group:$\left\langle \left(\begin{array}{ll}\alpha^{52} & \alpha \\ \alpha^{33} & \alpha^{46} \\ \end{array}\right), \left(\begin{array}{ll}\alpha^{38} & \alpha^{37} \\ \alpha^{3} & \alpha^{33} \\ \end{array}\right), \left(\begin{array}{ll}\alpha^{57} & 0 \\ 0 & \alpha^{57} \\ \end{array}\right), \left(\begin{array}{ll}\alpha^{27} & 0 \\ 0 & \alpha^{27} \\ \end{array}\right) \right\rangle \subseteq \GL_{2}(\F_{64}) = \GL_{2}(\F_{2}[\alpha]/(\alpha^{6} + \alpha^{4} + \alpha^{3} + \alpha + 1))$
Copy content comment:Define the group as a matrix group with coefficients in GLFq
 
Copy content magma:F:=GF(64); al:=F.1; G := MatrixGroup< 2, F | [[al^52, al^1], [al^33, al^46]],[[al^38, al^37], [al^3, al^33]],[[al^57, 0], [0, al^57]],[[al^27, 0], [0, al^27]] >;
 
Copy content gap:G := Group([[[Z(64)^52, Z(64)^1], [Z(64)^33, Z(64)^46]],[[Z(64)^38, Z(64)^37], [Z(64)^3, Z(64)^33]],[[Z(64)^57, 0*Z(64)], [0*Z(64), Z(64)^57]],[[Z(64)^27, 0*Z(64)], [0*Z(64), Z(64)^27]]]);
 
Copy content sage:F = GF(64); al = F.0; MS = MatrixSpace(F, 2, 2) G = MatrixGroup([MS([[al^52, al^1], [al^33, al^46]]), MS([[al^38, al^37], [al^3, al^33]]), MS([[al^57, 0], [0, al^57]]), MS([[al^27, 0], [0, al^27]])])
 
Direct product: $C_3$ $\, \times\, $ $C_7$ $\, \times\, $ $\SL(2,8)$
Semidirect product: not computed
Trans. wreath product: not isomorphic to a non-trivial transitive wreath product

Elements of the group are displayed as matrices in $\GL_{2}(\F_{64})$.

Homology

Abelianization: $C_{21} \simeq C_{3} \times C_{7}$
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 2308 subgroups in 70 conjugacy classes, 8 normal, and all normal subgroups are characteristic.

Characteristic subgroups are shown in this color.

Special subgroups

Center: $Z \simeq$ $C_{21}$ $G/Z \simeq$ $\SL(2,8)$
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: $G' \simeq$ $\SL(2,8)$ $G/G' \simeq$ $C_{21}$
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: $\Phi \simeq$ $C_1$ $G/\Phi \simeq$ $C_3\times \GL(2,8)$
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: $\operatorname{Fit} \simeq$ $C_{21}$ $G/\operatorname{Fit} \simeq$ $\SL(2,8)$
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: $R \simeq$ $C_{21}$ $G/R \simeq$ $\SL(2,8)$
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: $\operatorname{soc} \simeq$ $C_3\times \GL(2,8)$ $G/\operatorname{soc} \simeq$ $C_1$
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: $P_{ 2 } \simeq$ $C_2^3$
3-Sylow subgroup: $P_{ 3 } \simeq$ $C_3\times C_9$
7-Sylow subgroup: $P_{ 7 } \simeq$ $C_7^2$

Subgroup diagram and profile

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Subgroup information

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Series

Derived series $C_3\times \GL(2,8)$ $\rhd$ $\SL(2,8)$
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 $C_3\times \GL(2,8)$ $\rhd$ $C_{21}$ $\rhd$ $C_7$ $\rhd$ $C_1$
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 $C_3\times \GL(2,8)$ $\rhd$ $\SL(2,8)$
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 $C_1$ $\lhd$ $C_{21}$
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 1 larger groups in the database.

This group is a maximal quotient of 1 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 $189 \times 189$ 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 $24 \times 24$ rational character table.