↑

Properties

Label 117936000000.a
Order \( 2^{10} \cdot 3^{4} \cdot 5^{6} \cdot 7 \cdot 13 \)
Exponent \( 2^{4} \cdot 3^{2} \cdot 5 \cdot 7 \cdot 13 \)
Nilpotent no
Solvable no
$\card{G^{\mathrm{ab}}}$ \( 2^{2} \)
$\card{Z(G)}$ 4
$\card{\Aut(G)}$ \( 2^{11} \cdot 3^{4} \cdot 5^{6} \cdot 7 \cdot 13 \)
$\card{\mathrm{Out}(G)}$ \( 2^{3} \)
Perm deg. not computed
Trans deg. not computed
Rank $2$

Related objects

Downloads

Learn more

Show commands: Gap / Magma / Oscar / SageMath

Copy content comment:Construction of abstract group
 
Copy content magma:G := CSU(4,5);
 
Copy content gap:G := Group([[[ Z(5), 0*Z(5), Z(5), Z(5) ], [ Z(5), Z(5)^3, Z(5)^2, Z(5)^0 ], [ Z(5)^0, Z(5), Z(5)^0, Z(5) ], [ Z(5)^0, Z(5)^2, Z(5), Z(5)^3 ]], [[ Z(5)^0, 0*Z(5), 0*Z(5), 0*Z(5) ], [ Z(5)^3, Z(5)^2, 0*Z(5), 0*Z(5) ], [ Z(5)^0, 0*Z(5), 0*Z(5), 0*Z(5) ], [ 0*Z(5), 0*Z(5), 0*Z(5), 0*Z(5) ]], [[ 0*Z(5), Z(5)^0, 0*Z(5), 0*Z(5) ], [ 0*Z(5), 0*Z(5), 0*Z(5), 0*Z(5) ], [ 0*Z(5), 0*Z(5), Z(5)^3, Z(5) ], [ 0*Z(5), 0*Z(5), 0*Z(5), 0*Z(5) ]]]);
 
Copy content sage:MS = MatrixSpace(GF(5), 4, 4) G = MatrixGroup([MS([[2, 0, 2, 2], [2, 3, 4, 1], [1, 2, 1, 2], [1, 4, 2, 3]]), MS([[1, 0, 0, 0], [3, 4, 0, 0], [1, 0, 0, 0], [0, 0, 0, 0]]), MS([[0, 1, 0, 0], [0, 0, 0, 0], [0, 0, 3, 2], [0, 0, 0, 0]])])
 
Copy content oscar:G = matrix_group([matrix(GF(5), [[2, 0, 2, 2], [2, 3, 4, 1], [1, 2, 1, 2], [1, 4, 2, 3]]), matrix(GF(5), [[1, 0, 0, 0], [3, 4, 0, 0], [1, 0, 0, 0], [0, 0, 0, 0]]), matrix(GF(5), [[0, 1, 0, 0], [0, 0, 0, 0], [0, 0, 3, 2], [0, 0, 0, 0]])])
 

Group information

Description:$\GSU(4,5)$
Order: \(117936000000\)\(\medspace = 2^{10} \cdot 3^{4} \cdot 5^{6} \cdot 7 \cdot 13 \)
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: \(65520\)\(\medspace = 2^{4} \cdot 3^{2} \cdot 5 \cdot 7 \cdot 13 \)
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:Group of order \(235872000000\)\(\medspace = 2^{11} \cdot 3^{4} \cdot 5^{6} \cdot 7 \cdot 13 \)
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, $\PSU(4,5)$
Copy content comment:Composition factors of the group
 
Copy content magma:CompositionFactors(G);
 
Copy content gap:CompositionSeries(G);
 
Copy content sage_gap:G.CompositionSeries()
 
Copy content oscar:composition_series(G)
 
Derived length:$1$
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 and nonsolvable. 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 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_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 4 5 6 7 8 9 10 12 13 14 15 16 18 20 21 24 26 28 30 36 40 42 52 56 60 63 72 84 104 120 126 168 208 252 504
Elements 1 813751 7192250 41763752 244140624 765449750 468000000 3155092000 468000000 751920624 3311542000 850500000 468000000 782964000 1134000000 468000000 1978861248 936000000 20529704000 2551500000 936000000 2191644000 936000000 3957408000 936000000 3402000000 1872000000 4940208000 2808000000 1872000000 1872000000 6804000000 9487296000 2808000000 3744000000 13608000000 5616000000 11232000000 117936000000
Conjugacy classes   1 3 4 8 4 16 2 28 2 8 40 3 2 8 4 2 16 4 140 9 4 20 4 20 4 12 8 36 12 8 8 24 56 12 16 48 24 48 668
Divisions 1 3 3 6 4 11 1 12 1 8 18 1 1 5 1 1 10 1 27 2 1 12 1 6 1 2 1 12 1 1 1 2 7 1 1 1 1 1 170
Autjugacy classes 1 3 3 6 4 11 1 10 1 8 18 3 1 5 1 1 10 2 32 6 1 12 1 5 2 6 1 12 6 1 2 9 7 6 2 6 6 6 218

Minimal presentations

Permutation degree:not computed
Transitive degree:not computed
Rank: $2$
Inequivalent generating pairs: not computed

Minimal degrees of faithful linear representations

Over $\mathbb{C}$ Over $\mathbb{R}$ Over $\mathbb{Q}$
Irreducible 104 not computed not computed
Arbitrary not computed not computed not computed

Constructions

Show commands: Gap / Magma / Oscar / SageMath


Groups of Lie type:$\GSU(4,5)$
Copy content magma:G := CSU(4,5);
 
Copy content gap:G := Group([[[ Z(5), 0*Z(5), Z(5), Z(5) ], [ Z(5), Z(5)^3, Z(5)^2, Z(5)^0 ], [ Z(5)^0, Z(5), Z(5)^0, Z(5) ], [ Z(5)^0, Z(5)^2, Z(5), Z(5)^3 ]], [[ Z(5)^0, 0*Z(5), 0*Z(5), 0*Z(5) ], [ Z(5)^3, Z(5)^2, 0*Z(5), 0*Z(5) ], [ Z(5)^0, 0*Z(5), 0*Z(5), 0*Z(5) ], [ 0*Z(5), 0*Z(5), 0*Z(5), 0*Z(5) ]], [[ 0*Z(5), Z(5)^0, 0*Z(5), 0*Z(5) ], [ 0*Z(5), 0*Z(5), 0*Z(5), 0*Z(5) ], [ 0*Z(5), 0*Z(5), Z(5)^3, Z(5) ], [ 0*Z(5), 0*Z(5), 0*Z(5), 0*Z(5) ]]]);
 
Copy content sage:MS = MatrixSpace(GF(5), 4, 4) G = MatrixGroup([MS([[2, 0, 2, 2], [2, 3, 4, 1], [1, 2, 1, 2], [1, 4, 2, 3]]), MS([[1, 0, 0, 0], [3, 4, 0, 0], [1, 0, 0, 0], [0, 0, 0, 0]]), MS([[0, 1, 0, 0], [0, 0, 0, 0], [0, 0, 3, 2], [0, 0, 0, 0]])])
 
Copy content oscar:G = matrix_group([matrix(GF(5), [[2, 0, 2, 2], [2, 3, 4, 1], [1, 2, 1, 2], [1, 4, 2, 3]]), matrix(GF(5), [[1, 0, 0, 0], [3, 4, 0, 0], [1, 0, 0, 0], [0, 0, 0, 0]]), matrix(GF(5), [[0, 1, 0, 0], [0, 0, 0, 0], [0, 0, 3, 2], [0, 0, 0, 0]])])
 
Direct product: not computed
Semidirect product: not computed
Trans. wreath product: not computed

Elements of the group are displayed as matrices in $\GSU(4,5)$.

Homology

Abelianization: $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_gap:G.FactorGroup(G.DerivedSubgroup())
 
Copy content oscar:quo(G, derived_subgroup(G)[1])
 
Schur multiplier: not computed
Copy content comment:The Schur multiplier of the group
 
Copy content gap:AbelianInvariantsMultiplier(G);
 
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)
 

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
 
Copy content oscar:character_table(G) # Output not guaranteed to exactly match the LMFDB table
 

Complex character table

The $668 \times 668$ character table is not available for this group.

Rational character table

The $170 \times 170$ rational character table is not available for this group.