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Properties

Label 1024.dhg
Order \( 2^{10} \)
Exponent \( 2^{4} \)
Nilpotent yes
Solvable yes
$\card{G^{\mathrm{ab}}}$ \( 2^{3} \)
$\card{Z(G)}$ \( 2 \)
$\card{\Aut(G)}$ \( 2^{13} \)
$\card{\mathrm{Out}(G)}$ \( 2^{4} \)
Perm deg. $16$
Trans deg. $16$
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< 16 | (1,4,2,3)(5,7)(6,8)(9,14,10,13)(15,16), (1,10,4,12,5,14,8,16,2,9,3,11,6,13,7,15) >;
 
Copy content gap:G := Group( (1,4,2,3)(5,7)(6,8)(9,14,10,13)(15,16), (1,10,4,12,5,14,8,16,2,9,3,11,6,13,7,15) );
 
Copy content sage:G = PermutationGroup(['(1,4,2,3)(5,7)(6,8)(9,14,10,13)(15,16)', '(1,10,4,12,5,14,8,16,2,9,3,11,6,13,7,15)'])
 
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(30383902621016833058827572899707802992256306489651377678546313046874520519777039793,1024)'); a = G.1; b = G.3; c = G.4; d = G.5; e = G.7; f = G.8;
 

Group information

Description:$C_2^6.\SD_{16}$
Order: \(1024\)\(\medspace = 2^{10} \)
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: \(16\)\(\medspace = 2^{4} \)
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:$C_2^5:C_4.D_4^2$, of order \(8192\)\(\medspace = 2^{13} \)
Copy content comment:Automorphism group
 
Copy content gap:AutomorphismGroup(G);
 
Copy content magma:AutomorphismGroup(G);
 
Copy content sage_gap:G.AutomorphismGroup()
 
Composition factors:$C_2$ x 10
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()
 
Nilpotency class:$7$
Copy content comment:Nilpotency class of the group
 
Copy content magma:NilpotencyClass(G);
 
Copy content gap:NilpotencyClassOfGroup(G);
 
Copy content sage_gap:G.NilpotencyClassOfGroup()
 
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 and a $p$-group (hence nilpotent, solvable, supersolvable, monomial, 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 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 4 8 16
Elements 1 79 432 256 256 1024
Conjugacy classes   1 11 13 5 4 34
Divisions 1 11 8 4 1 25
Autjugacy classes 1 8 5 3 1 18

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 8 16
Irr. complex chars.   8 10 5 10 1 34
Irr. rational chars. 4 4 7 7 3 25

Minimal presentations

Permutation degree:$16$
Transitive degree:$16$
Rank: $2$
Inequivalent generating pairs: $48$

Minimal degrees of faithful linear representations

Over $\mathbb{C}$ Over $\mathbb{R}$ Over $\mathbb{Q}$
Irreducible 8 8 8
Arbitrary 8 8 8

Constructions

Show commands: Gap / Magma / SageMath


Presentation: ${\langle a, b, c, d, e, f \mid a^{4}=b^{2}=c^{2}=e^{2}=f^{8}=[a,c]=[a,e]= \!\cdots\! \rangle}$ Copy content Toggle raw display
Copy content comment:Define the group with the given generators and relations
 
Copy content magma:G := PCGroup([10, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 20, 2282, 14204, 7814, 4624, 3734, 144, 15365, 2895, 8185, 1015, 81607, 25777, 3867, 4197, 1967, 867, 237, 8648, 14418, 17308, 4368, 268]); a,b,c,d,e,f := Explode([G.1, G.3, G.4, G.5, G.7, G.8]); AssignNames(~G, ["a", "a2", "b", "c", "d", "d2", "e", "f", "f2", "f4"]);
 
Copy content gap:G := PcGroupCode(30383902621016833058827572899707802992256306489651377678546313046874520519777039793,1024); a := G.1; b := G.3; c := G.4; d := G.5; e := G.7; f := G.8;
 
Copy content sage:# This uses Sage's interface to GAP, as Sage (currently) has no native support for PC groups G = gap.new('PcGroupCode(30383902621016833058827572899707802992256306489651377678546313046874520519777039793,1024)'); a = G.1; b = G.3; c = G.4; d = G.5; e = G.7; f = G.8;
 
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(30383902621016833058827572899707802992256306489651377678546313046874520519777039793,1024)'); a = G.1; b = G.3; c = G.4; d = G.5; e = G.7; f = G.8;
 
Permutation group:Degree $16$ $\langle(1,4,2,3)(5,7)(6,8)(9,14,10,13)(15,16), (1,10,4,12,5,14,8,16,2,9,3,11,6,13,7,15)\rangle$ Copy content Toggle raw display
Copy content comment:Define the group as a permutation group
 
Copy content magma:G := PermutationGroup< 16 | (1,4,2,3)(5,7)(6,8)(9,14,10,13)(15,16), (1,10,4,12,5,14,8,16,2,9,3,11,6,13,7,15) >;
 
Copy content gap:G := Group( (1,4,2,3)(5,7)(6,8)(9,14,10,13)(15,16), (1,10,4,12,5,14,8,16,2,9,3,11,6,13,7,15) );
 
Copy content sage:G = PermutationGroup(['(1,4,2,3)(5,7)(6,8)(9,14,10,13)(15,16)', '(1,10,4,12,5,14,8,16,2,9,3,11,6,13,7,15)'])
 
Transitive group: 16T1250 16T1264 more information
Direct product: not isomorphic to a non-trivial direct product
Semidirect product: $(C_2^5.Q_8)$ $\,\rtimes\,$ $C_4$ $(C_2^5.D_4)$ $\,\rtimes\,$ $C_4$ more information
Trans. wreath product: not isomorphic to a non-trivial transitive wreath product
Non-split product: $C_2^6$ . $\SD_{16}$ $(C_2^6.Q_8)$ . $C_2$ $C_2^5$ . $(Q_8:C_4)$ $(C_2^4:D_4)$ . $D_4$ all 19

Elements of the group are displayed as permutations of degree 16.

Homology

Abelianization: $C_{2} \times C_{4} $
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: $C_{2}^{2}$
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()
 

There are 6613 subgroups in 622 conjugacy classes, 23 normal, and all normal subgroups are characteristic.

Characteristic subgroups are shown in this color.

Special subgroups

Center: $Z \simeq$ $C_2$ $G/Z \simeq$ $C_2^5.\SD_{16}$
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: $G' \simeq$ $C_2^5.C_4$ $G/G' \simeq$ $C_2\times C_4$
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: $\Phi \simeq$ $C_2^6.C_4$ $G/\Phi \simeq$ $C_2^2$
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: $\operatorname{Fit} \simeq$ $C_2^6.\SD_{16}$ $G/\operatorname{Fit} \simeq$ $C_1$
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()
 
Radical: $R \simeq$ $C_2^6.\SD_{16}$ $G/R \simeq$ $C_1$
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: $\operatorname{soc} \simeq$ $C_2$ $G/\operatorname{soc} \simeq$ $C_2^5.\SD_{16}$
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: $P_{ 2 } \simeq$ $C_2^6.\SD_{16}$

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 $C_2^6.\SD_{16}$ $\rhd$ $C_2^5.C_4$ $\rhd$ $C_2^3$ $\rhd$ $C_1$
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 $C_2^6.\SD_{16}$ $\rhd$ $C_2^5.C_4^2$ $\rhd$ $C_2^6.C_4$ $\rhd$ $C_2^5.C_4$ $\rhd$ $D_4\times C_2^3$ $\rhd$ $C_2^3\times C_4$ $\rhd$ $C_2^2\times C_4$ $\rhd$ $C_2^3$ $\rhd$ $C_2^2$ $\rhd$ $C_2$ $\rhd$ $C_1$
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 $C_2^6.\SD_{16}$ $\rhd$ $C_2^5.C_4$ $\rhd$ $D_4\times C_2^3$ $\rhd$ $C_2^4$ $\rhd$ $C_2^3$ $\rhd$ $C_2^2$ $\rhd$ $C_2$ $\rhd$ $C_1$
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()
 
Upper central series $C_1$ $\lhd$ $C_2$ $\lhd$ $C_2^2$ $\lhd$ $C_2^3$ $\lhd$ $C_2^3\times C_4$ $\lhd$ $D_4\times C_2^3$ $\lhd$ $C_2^6.C_4$ $\lhd$ $C_2^6.\SD_{16}$
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 4 larger groups in the database.

This group is a maximal quotient of 5 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 $34 \times 34$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $25 \times 25$ rational character table.