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Properties

Label 1024.djz
Order \( 2^{10} \)
Exponent \( 2^{2} \)
Nilpotent yes
Solvable yes
$\card{G^{\mathrm{ab}}}$ \( 2^{4} \)
$\card{Z(G)}$ 4
$\card{\Aut(G)}$ \( 2^{18} \)
$\card{\mathrm{Out}(G)}$ \( 2^{10} \)
Perm deg. $16$
Trans deg. not computed
Rank $4$

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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 | (1,2)(3,5)(4,6)(7,8)(9,10)(11,13)(12,14)(15,16), (4,6), (1,2)(3,5)(9,10)(12,14), (1,3)(2,5)(4,8,6,7)(11,15)(13,16), (11,13)(15,16), (1,4,2,6)(3,7)(5,8)(9,11)(10,13)(12,16)(14,15), (3,5)(7,8)(9,10)(11,13)(12,14)(15,16), (9,12,10,14)(11,15,13,16), (12,14)(15,16) >;
 
Copy content gap:G := Group( (1,2)(3,5)(4,6)(7,8)(9,10)(11,13)(12,14)(15,16), (4,6), (1,2)(3,5)(9,10)(12,14), (1,3)(2,5)(4,8,6,7)(11,15)(13,16), (11,13)(15,16), (1,4,2,6)(3,7)(5,8)(9,11)(10,13)(12,16)(14,15), (3,5)(7,8)(9,10)(11,13)(12,14)(15,16), (9,12,10,14)(11,15,13,16), (12,14)(15,16) );
 
Copy content sage:G = PermutationGroup(['(1,2)(3,5)(4,6)(7,8)(9,10)(11,13)(12,14)(15,16)', '(4,6)', '(1,2)(3,5)(9,10)(12,14)', '(1,3)(2,5)(4,8,6,7)(11,15)(13,16)', '(11,13)(15,16)', '(1,4,2,6)(3,7)(5,8)(9,11)(10,13)(12,16)(14,15)', '(3,5)(7,8)(9,10)(11,13)(12,14)(15,16)', '(9,12,10,14)(11,15,13,16)', '(12,14)(15,16)'])
 
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(54184049342784094549401235244527422733195996786110878448802984581643,1024)'); a = G.1; b = G.2; c = G.4; d = G.6; e = G.7; f = G.8; g = G.9; h = G.10;
 

Group information

Description:$C_2^6.C_2^4$
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: \(4\)\(\medspace = 2^{2} \)
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()
 
Automorphism group:$C_2^9.C_2^6.C_2^3$, of order \(262144\)\(\medspace = 2^{18} \)
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:$3$
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:$2$
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, a $p$-group (hence nilpotent, solvable, supersolvable, monomial, elementary, and hyperelementary), and metabelian. Whether it is almost simple has not been computed.

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
Elements 1 255 768 1024
Conjugacy classes   1 51 42 94
Divisions 1 51 40 92
Autjugacy classes 1 22 13 36

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
Irr. complex chars.   16 44 28 6 94
Irr. rational chars. 16 40 30 6 92

Minimal presentations

Permutation degree:$16$
Transitive degree:not computed
Rank: $4$
Inequivalent generating quadruples: $1290240$

Minimal degrees of faithful linear representations

Over $\mathbb{C}$ Over $\mathbb{R}$ Over $\mathbb{Q}$
Irreducible none none none
Arbitrary not computed not computed not computed

Constructions

Show commands: Gap / Magma / SageMath


Presentation: ${\langle a, b, c, d, e, f, g, h \mid b^{4}=c^{4}=d^{2}=e^{2}=f^{2}=g^{2}= \!\cdots\! \rangle}$ 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, 1280, 4281, 51, 20163, 8493, 2823, 113, 16804, 8414, 3375, 17926, 8976, 4517, 5768, 2898, 64009, 36019, 9639]); a,b,c,d,e,f,g,h := Explode([G.1, G.2, G.4, G.6, G.7, G.8, G.9, G.10]); AssignNames(~G, ["a", "b", "b2", "c", "c2", "d", "e", "f", "g", "h"]);
 
Copy content gap:G := PcGroupCode(54184049342784094549401235244527422733195996786110878448802984581643,1024); a := G.1; b := G.2; c := G.4; d := G.6; e := G.7; f := G.8; g := G.9; h := G.10;
 
Copy content sage:# This uses Sage's interface to GAP, as Sage (currently) has no native support for PC groups G = gap.new('PcGroupCode(54184049342784094549401235244527422733195996786110878448802984581643,1024)'); a = G.1; b = G.2; c = G.4; d = G.6; e = G.7; f = G.8; g = G.9; h = G.10;
 
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(54184049342784094549401235244527422733195996786110878448802984581643,1024)'); a = G.1; b = G.2; c = G.4; d = G.6; e = G.7; f = G.8; g = G.9; h = G.10;
 
Permutation group:Degree $16$ $\langle(1,2)(3,5)(4,6)(7,8)(9,10)(11,13)(12,14)(15,16), (4,6), (1,2)(3,5)(9,10) \!\cdots\! \rangle$ Copy content Toggle raw display
Copy content comment:Define the group as a permutation group
 
Copy content magma:G := PermutationGroup< 16 | (1,2)(3,5)(4,6)(7,8)(9,10)(11,13)(12,14)(15,16), (4,6), (1,2)(3,5)(9,10)(12,14), (1,3)(2,5)(4,8,6,7)(11,15)(13,16), (11,13)(15,16), (1,4,2,6)(3,7)(5,8)(9,11)(10,13)(12,16)(14,15), (3,5)(7,8)(9,10)(11,13)(12,14)(15,16), (9,12,10,14)(11,15,13,16), (12,14)(15,16) >;
 
Copy content gap:G := Group( (1,2)(3,5)(4,6)(7,8)(9,10)(11,13)(12,14)(15,16), (4,6), (1,2)(3,5)(9,10)(12,14), (1,3)(2,5)(4,8,6,7)(11,15)(13,16), (11,13)(15,16), (1,4,2,6)(3,7)(5,8)(9,11)(10,13)(12,16)(14,15), (3,5)(7,8)(9,10)(11,13)(12,14)(15,16), (9,12,10,14)(11,15,13,16), (12,14)(15,16) );
 
Copy content sage:G = PermutationGroup(['(1,2)(3,5)(4,6)(7,8)(9,10)(11,13)(12,14)(15,16)', '(4,6)', '(1,2)(3,5)(9,10)(12,14)', '(1,3)(2,5)(4,8,6,7)(11,15)(13,16)', '(11,13)(15,16)', '(1,4,2,6)(3,7)(5,8)(9,11)(10,13)(12,16)(14,15)', '(3,5)(7,8)(9,10)(11,13)(12,14)(15,16)', '(9,12,10,14)(11,15,13,16)', '(12,14)(15,16)'])
 
Direct product: not computed
Semidirect product: not computed
Trans. wreath product: not isomorphic to a non-trivial transitive wreath product
Aut. group: $\Aut(C_8:D_4)$ $\Aut(C_8.D_4)$

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

Homology

Abelianization: $C_{2}^{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: $C_{2}^{10}$
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: $2$
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()
 

Subgroup data has not been computed.

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 $94 \times 94$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $92 \times 92$ rational character table.