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Properties

Label 174182400.a
Order \( 2^{12} \cdot 3^{5} \cdot 5^{2} \cdot 7 \)
Exponent \( 2^{3} \cdot 3^{2} \cdot 5 \cdot 7 \)
Simple yes
$\card{G^{\mathrm{ab}}}$ \( 1 \)
$\card{Z(G)}$ 1
$\card{\Aut(G)}$ \( 2^{13} \cdot 3^{6} \cdot 5^{2} \cdot 7 \)
$\card{\mathrm{Out}(G)}$ \( 2 \cdot 3 \)
Perm deg. $120$
Trans deg. not computed
Rank $2$

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Show commands: Gap / Magma / SageMath

Copy content magma:G := ChevalleyGroup("D", 4, 2);
 
Copy content comment:Define the group as a permutation group
 
Copy content gap:G := Group( (2,3,5,9)(4,7)(6,11)(8,14,18,30)(10,17,28,45)(12,20,27,38)(13,22,37,57)(15,24,40,61)(16,26,43,66)(19,32,29,47)(21,35)(23,39,60,78)(25,42)(31,50,73,97)(33,53)(34,51)(36,55)(41,63,44,48)(46,70,93,109)(49,69,92,76)(52,75)(54,56,65,74)(58,80,101,115)(59,82,62,85)(67,91,98,84)(68,83)(71,95,110,104)(72,87,103,105)(79,100,89,107)(81,102,94,90)(88,99,114,108)(96,111,119,116)(106,118,120,117)(112,113), (1,2,4,8,15,25)(3,6,12,21,36,39)(5,10,18,31,51,74)(7,13,23,32,52,76)(9,16,27,44,68,80)(11,19,33,54,78,99)(14,20,34,55,79,100)(17,29,48,72,96,50)(22,38,59,83,104,107)(24,41,64,88,35,56)(26,37,58,81,57,47)(28,46,71,91,43,67)(30,49,42,65,89,101)(40,62,86,97,112,102)(45,69)(53,77,61)(60,84,75,98,113,93)(63,87,106,118,119,114)(66,90,108)(70,94)(73,85,105,117,111,115)(82,103,116,95,110,92) );
 
Copy content sage:G = PermutationGroup(['(2,3,5,9)(4,7)(6,11)(8,14,18,30)(10,17,28,45)(12,20,27,38)(13,22,37,57)(15,24,40,61)(16,26,43,66)(19,32,29,47)(21,35)(23,39,60,78)(25,42)(31,50,73,97)(33,53)(34,51)(36,55)(41,63,44,48)(46,70,93,109)(49,69,92,76)(52,75)(54,56,65,74)(58,80,101,115)(59,82,62,85)(67,91,98,84)(68,83)(71,95,110,104)(72,87,103,105)(79,100,89,107)(81,102,94,90)(88,99,114,108)(96,111,119,116)(106,118,120,117)(112,113)', '(1,2,4,8,15,25)(3,6,12,21,36,39)(5,10,18,31,51,74)(7,13,23,32,52,76)(9,16,27,44,68,80)(11,19,33,54,78,99)(14,20,34,55,79,100)(17,29,48,72,96,50)(22,38,59,83,104,107)(24,41,64,88,35,56)(26,37,58,81,57,47)(28,46,71,91,43,67)(30,49,42,65,89,101)(40,62,86,97,112,102)(45,69)(53,77,61)(60,84,75,98,113,93)(63,87,106,118,119,114)(66,90,108)(70,94)(73,85,105,117,111,115)(82,103,116,95,110,92)'])
 

Group information

Description:$\OmegaPlus(8,2)$
Order: \(174182400\)\(\medspace = 2^{12} \cdot 3^{5} \cdot 5^{2} \cdot 7 \)
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: \(2520\)\(\medspace = 2^{3} \cdot 3^{2} \cdot 5 \cdot 7 \)
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:$\OmegaPlus(8,2).S_3$, of order \(1045094400\)\(\medspace = 2^{13} \cdot 3^{6} \cdot 5^{2} \cdot 7 \)
Copy content comment:Automorphism group
 
Copy content gap:AutomorphismGroup(G);
 
Copy content magma:AutomorphismGroup(G);
 
Copy content sage_gap:G.AutomorphismGroup()
 
Composition factors:$\OmegaPlus(8,2)$
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:$0$
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, simple (hence nonsolvable, perfect, quasisimple, and almost simple), and rational.

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 4 5 6 7 8 9 10 12 15
Elements 1 69615 365120 5821200 1741824 19857600 24883200 10886400 19353600 26127360 30240000 34836480 174182400
Conjugacy classes   1 5 5 6 3 14 1 2 3 3 7 3 53
Divisions 1 5 5 6 3 14 1 2 3 3 7 3 53
Autjugacy classes 1 3 3 4 1 6 1 2 1 1 3 1 27

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 28 35 50 84 175 210 300 350 525 567 700 840 972 1050 1344 1400 1575 2100 2240 2268 2835 3200 4096 4200 6075
Irr. complex chars.   1 1 3 1 3 1 3 1 1 1 3 4 3 1 3 3 1 3 3 3 3 3 1 1 1 1 53
Irr. rational chars. 1 1 3 1 3 1 3 1 1 1 3 4 3 1 3 3 1 3 3 3 3 3 1 1 1 1 53

Minimal presentations

Permutation degree:$120$
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 28 28 28
Arbitrary not computed not computed not computed

Constructions

Show commands: Gap / Magma / SageMath


Groups of Lie type:$\OmegaPlus(8,2)$, $\POmegaPlus(8,2)$, $\SpinPlus(8,2)$
Permutation group:Degree $120$ $\langle(2,3,5,9)(4,7)(6,11)(8,14,18,30)(10,17,28,45)(12,20,27,38)(13,22,37,57) \!\cdots\! \rangle$ Copy content Toggle raw display
Copy content comment:Define the group as a permutation group
 
Copy content magma:G := PermutationGroup< 120 | (2,3,5,9)(4,7)(6,11)(8,14,18,30)(10,17,28,45)(12,20,27,38)(13,22,37,57)(15,24,40,61)(16,26,43,66)(19,32,29,47)(21,35)(23,39,60,78)(25,42)(31,50,73,97)(33,53)(34,51)(36,55)(41,63,44,48)(46,70,93,109)(49,69,92,76)(52,75)(54,56,65,74)(58,80,101,115)(59,82,62,85)(67,91,98,84)(68,83)(71,95,110,104)(72,87,103,105)(79,100,89,107)(81,102,94,90)(88,99,114,108)(96,111,119,116)(106,118,120,117)(112,113), (1,2,4,8,15,25)(3,6,12,21,36,39)(5,10,18,31,51,74)(7,13,23,32,52,76)(9,16,27,44,68,80)(11,19,33,54,78,99)(14,20,34,55,79,100)(17,29,48,72,96,50)(22,38,59,83,104,107)(24,41,64,88,35,56)(26,37,58,81,57,47)(28,46,71,91,43,67)(30,49,42,65,89,101)(40,62,86,97,112,102)(45,69)(53,77,61)(60,84,75,98,113,93)(63,87,106,118,119,114)(66,90,108)(70,94)(73,85,105,117,111,115)(82,103,116,95,110,92) >;
 
Copy content gap:G := Group( (2,3,5,9)(4,7)(6,11)(8,14,18,30)(10,17,28,45)(12,20,27,38)(13,22,37,57)(15,24,40,61)(16,26,43,66)(19,32,29,47)(21,35)(23,39,60,78)(25,42)(31,50,73,97)(33,53)(34,51)(36,55)(41,63,44,48)(46,70,93,109)(49,69,92,76)(52,75)(54,56,65,74)(58,80,101,115)(59,82,62,85)(67,91,98,84)(68,83)(71,95,110,104)(72,87,103,105)(79,100,89,107)(81,102,94,90)(88,99,114,108)(96,111,119,116)(106,118,120,117)(112,113), (1,2,4,8,15,25)(3,6,12,21,36,39)(5,10,18,31,51,74)(7,13,23,32,52,76)(9,16,27,44,68,80)(11,19,33,54,78,99)(14,20,34,55,79,100)(17,29,48,72,96,50)(22,38,59,83,104,107)(24,41,64,88,35,56)(26,37,58,81,57,47)(28,46,71,91,43,67)(30,49,42,65,89,101)(40,62,86,97,112,102)(45,69)(53,77,61)(60,84,75,98,113,93)(63,87,106,118,119,114)(66,90,108)(70,94)(73,85,105,117,111,115)(82,103,116,95,110,92) );
 
Copy content sage:G = PermutationGroup(['(2,3,5,9)(4,7)(6,11)(8,14,18,30)(10,17,28,45)(12,20,27,38)(13,22,37,57)(15,24,40,61)(16,26,43,66)(19,32,29,47)(21,35)(23,39,60,78)(25,42)(31,50,73,97)(33,53)(34,51)(36,55)(41,63,44,48)(46,70,93,109)(49,69,92,76)(52,75)(54,56,65,74)(58,80,101,115)(59,82,62,85)(67,91,98,84)(68,83)(71,95,110,104)(72,87,103,105)(79,100,89,107)(81,102,94,90)(88,99,114,108)(96,111,119,116)(106,118,120,117)(112,113)', '(1,2,4,8,15,25)(3,6,12,21,36,39)(5,10,18,31,51,74)(7,13,23,32,52,76)(9,16,27,44,68,80)(11,19,33,54,78,99)(14,20,34,55,79,100)(17,29,48,72,96,50)(22,38,59,83,104,107)(24,41,64,88,35,56)(26,37,58,81,57,47)(28,46,71,91,43,67)(30,49,42,65,89,101)(40,62,86,97,112,102)(45,69)(53,77,61)(60,84,75,98,113,93)(63,87,106,118,119,114)(66,90,108)(70,94)(73,85,105,117,111,115)(82,103,116,95,110,92)'])
 
Direct product: not isomorphic to a non-trivial direct product
Semidirect product: not isomorphic to a non-trivial semidirect product
Trans. wreath product: not isomorphic to a non-trivial transitive wreath product

Elements of the group are displayed as matrices in $\OmegaPlus(8,2)$.

Homology

Abelianization: $C_1 $
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: not computed
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()
 

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

Every character has rational values, so the complex character table is the same as the rational character table below.

Rational character table

See the $53 \times 53$ rational character table. Alternatively, you may search for characters of this group with desired properties.