Properties

Label 256.540
Order \( 2^{8} \)
Exponent \( 2^{7} \)
Nilpotent yes
Solvable yes
$\card{G^{\mathrm{ab}}}$ \( 2^{2} \)
$\card{Z(G)}$ \( 2 \)
$\card{\Aut(G)}$ \( 2^{12} \)
$\card{\mathrm{Out}(G)}$ \( 2^{5} \)
Perm deg. $128$
Trans deg. $128$
Rank $2$

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Show commands: Gap / Magma / Oscar / SageMath

Copy content comment:Construction of abstract group
 
Copy content magma:G := SmallGroup(256, 540);
 
Copy content gap:G := SmallGroup(256, 540);
 
Copy content sage:G = PermutationGroup(['(1,66,2,65)(3,68,4,67)(5,70,6,69)(7,72,8,71)(9,74,10,73)(11,76,12,75)(13,78,14,77)(15,80,16,79)(17,82,18,81)(19,84,20,83)(21,86,22,85)(23,88,24,87)(25,90,26,89)(27,92,28,91)(29,94,30,93)(31,96,32,95)(33,98,34,97)(35,100,36,99)(37,102,38,101)(39,104,40,103)(41,106,42,105)(43,108,44,107)(45,110,46,109)(47,112,48,111)(49,114,50,113)(51,116,52,115)(53,118,54,117)(55,120,56,119)(57,122,58,121)(59,124,60,123)(61,126,62,125)(63,128,64,127)', '(3,4)(5,7)(6,8)(9,13)(10,14)(11,16)(12,15)(17,25)(18,26)(19,28)(20,27)(21,31)(22,32)(23,29)(24,30)(33,49)(34,50)(35,52)(36,51)(37,55)(38,56)(39,53)(40,54)(41,61)(42,62)(43,64)(44,63)(45,57)(46,58)(47,60)(48,59)(65,113)(66,114)(67,116)(68,115)(69,119)(70,120)(71,117)(72,118)(73,125)(74,126)(75,128)(76,127)(77,121)(78,122)(79,124)(80,123)(81,97)(82,98)(83,100)(84,99)(85,103)(86,104)(87,101)(88,102)(89,109)(90,110)(91,112)(92,111)(93,105)(94,106)(95,108)(96,107)', '(1,49,17,45,13,61,29,37,5,53,21,44,12,60,28,36,4,52,20,48,16,64,32,40,8,56,24,42,10,58,26,34,2,50,18,46,14,62,30,38,6,54,22,43,11,59,27,35,3,51,19,47,15,63,31,39,7,55,23,41,9,57,25,33)(65,97,89,121,73,105,87,119,71,103,95,127,79,111,83,115,67,99,91,123,75,107,86,118,70,102,94,126,78,110,82,114,66,98,90,122,74,106,88,120,72,104,96,128,80,112,84,116,68,100,92,124,76,108,85,117,69,101,93,125,77,109,81,113)', '(1,25,9,23,7,31,15,19,3,27,11,22,6,30,14,18,2,26,10,24,8,32,16,20,4,28,12,21,5,29,13,17)(33,57,41,55,39,63,47,51,35,59,43,54,38,62,46,50,34,58,42,56,40,64,48,52,36,60,44,53,37,61,45,49)(65,81,77,93,69,85,76,92,68,84,80,96,72,88,74,90,66,82,78,94,70,86,75,91,67,83,79,95,71,87,73,89)(97,113,109,125,101,117,108,124,100,116,112,128,104,120,106,122,98,114,110,126,102,118,107,123,99,115,111,127,103,119,105,121)', '(1,13,5,12,4,16,8,10,2,14,6,11,3,15,7,9)(17,29,21,28,20,32,24,26,18,30,22,27,19,31,23,25)(33,45,37,44,36,48,40,42,34,46,38,43,35,47,39,41)(49,61,53,60,52,64,56,58,50,62,54,59,51,63,55,57)(65,73,71,79,67,75,70,78,66,74,72,80,68,76,69,77)(81,89,87,95,83,91,86,94,82,90,88,96,84,92,85,93)(97,105,103,111,99,107,102,110,98,106,104,112,100,108,101,109)(113,121,119,127,115,123,118,126,114,122,120,128,116,124,117,125)', '(1,7,3,6,2,8,4,5)(9,15,11,14,10,16,12,13)(17,23,19,22,18,24,20,21)(25,31,27,30,26,32,28,29)(33,39,35,38,34,40,36,37)(41,47,43,46,42,48,44,45)(49,55,51,54,50,56,52,53)(57,63,59,62,58,64,60,61)(65,69,68,72,66,70,67,71)(73,77,76,80,74,78,75,79)(81,85,84,88,82,86,83,87)(89,93,92,96,90,94,91,95)(97,101,100,104,98,102,99,103)(105,109,108,112,106,110,107,111)(113,117,116,120,114,118,115,119)(121,125,124,128,122,126,123,127)', '(1,4,2,3)(5,8,6,7)(9,12,10,11)(13,16,14,15)(17,20,18,19)(21,24,22,23)(25,28,26,27)(29,32,30,31)(33,36,34,35)(37,40,38,39)(41,44,42,43)(45,48,46,47)(49,52,50,51)(53,56,54,55)(57,60,58,59)(61,64,62,63)(65,67,66,68)(69,71,70,72)(73,75,74,76)(77,79,78,80)(81,83,82,84)(85,87,86,88)(89,91,90,92)(93,95,94,96)(97,99,98,100)(101,103,102,104)(105,107,106,108)(109,111,110,112)(113,115,114,116)(117,119,118,120)(121,123,122,124)(125,127,126,128)', '(1,2)(3,4)(5,6)(7,8)(9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22)(23,24)(25,26)(27,28)(29,30)(31,32)(33,34)(35,36)(37,38)(39,40)(41,42)(43,44)(45,46)(47,48)(49,50)(51,52)(53,54)(55,56)(57,58)(59,60)(61,62)(63,64)(65,66)(67,68)(69,70)(71,72)(73,74)(75,76)(77,78)(79,80)(81,82)(83,84)(85,86)(87,88)(89,90)(91,92)(93,94)(95,96)(97,98)(99,100)(101,102)(103,104)(105,106)(107,108)(109,110)(111,112)(113,114)(115,116)(117,118)(119,120)(121,122)(123,124)(125,126)(127,128)'])
 
Copy content sage_gap:G = libgap.SmallGroup(256, 540)
 
Copy content oscar:G = small_group(256, 540)
 

Group information

Description:$\SD_{256}$
Order: \(256\)\(\medspace = 2^{8} \)
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: \(128\)\(\medspace = 2^{7} \)
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()
 
Copy content oscar:exponent(G)
 
Automorphism group:$C_{32}.C_{16}.C_2^3$, of order \(4096\)\(\medspace = 2^{12} \)
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 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()
 
Copy content oscar:composition_series(G)
 
Nilpotency class:$7$
Copy content comment:Nilpotency class of the group
 
Copy content magma:NilpotencyClass(G);
 
Copy content gap:if IsNilpotentGroup(G) then NilpotencyClassOfGroup(G); fi;
 
Copy content sage:libgap(G).NilpotencyClassOfGroup() if G.is_nilpotent() else -1
 
Copy content sage_gap:G.NilpotencyClassOfGroup() if G.IsNilpotentGroup() else -1
 
Copy content oscar:if is_nilpotent(G) nilpotency_class(G) end
 
Derived length:$2$
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, a $p$-group (hence nilpotent, solvable, supersolvable, monomial, elementary, and hyperelementary), and metacyclic (hence metabelian).

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:G.is_cyclic()
 
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 4 8 16 32 64 128
Elements 1 65 66 4 8 16 32 64 256
Conjugacy classes   1 2 2 2 4 8 16 32 67
Divisions 1 2 2 1 1 1 1 1 10
Autjugacy classes 1 2 2 1 1 1 1 1 10

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:# 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 G.CharacterDegrees()
 
Copy content oscar:# Outputs an MSet containing the absolutely irreducible degrees of G and their multiplicities. character_degrees(G)
 

Dimension 1 2 4 8 16 32 64
Irr. complex chars.   4 63 0 0 0 0 0 67
Irr. rational chars. 4 1 1 1 1 1 1 10

Minimal presentations

Permutation degree:$128$
Transitive degree:$128$
Rank: $2$
Inequivalent generating pairs: $6$

Minimal degrees of faithful linear representations

Over $\mathbb{C}$ Over $\mathbb{R}$ Over $\mathbb{Q}$
Irreducible 2 4 64
Arbitrary 2 4 64

Constructions

Show commands: Gap / Magma / Oscar / SageMath


Presentation: $\langle a, b \mid a^{2}=b^{128}=1, b^{a}=b^{63} \rangle$ Copy content Toggle raw display
Copy content comment:Define the group with the given generators and relations
 
Copy content magma:G := PCGroup([8, -2, 2, -2, -2, -2, -2, -2, -2, 2017, 41, 6050, 66, 7939, 91, 9604, 116, 10757, 141, 10758, 166]); a,b := Explode([G.1, G.2]); AssignNames(~G, ["a", "b", "b2", "b4", "b8", "b16", "b32", "b64"]);
 
Copy content gap:G := PcGroupCode(10759584490553416770205641161127763966,256); a := G.1; b := G.2;
 
Copy content sage:# This uses Sage's interface to GAP, as Sage (currently) has no native support for PC groups G = gap.new('PcGroupCode(10759584490553416770205641161127763966,256)'); a = G.1; b = G.2;
 
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(10759584490553416770205641161127763966,256)'); a = G.1; b = G.2;
 
Copy content oscar:# This uses Oscar's interface to GAP, as Oscar (currently) has no native support for PC groups G = PcGroup(GAP.evalstr("PcGroupCode(10759584490553416770205641161127763966,256)")); a = gen(G, 1); b = gen(G, 2);
 
Permutation group:Degree $128$ $\langle(1,66,2,65)(3,68,4,67)(5,70,6,69)(7,72,8,71)(9,74,10,73)(11,76,12,75)(13,78,14,77) \!\cdots\! \rangle$ Copy content Toggle raw display
Copy content comment:Define the group as a permutation group
 
Copy content magma:G := PermutationGroup< 128 | (1,66,2,65)(3,68,4,67)(5,70,6,69)(7,72,8,71)(9,74,10,73)(11,76,12,75)(13,78,14,77)(15,80,16,79)(17,82,18,81)(19,84,20,83)(21,86,22,85)(23,88,24,87)(25,90,26,89)(27,92,28,91)(29,94,30,93)(31,96,32,95)(33,98,34,97)(35,100,36,99)(37,102,38,101)(39,104,40,103)(41,106,42,105)(43,108,44,107)(45,110,46,109)(47,112,48,111)(49,114,50,113)(51,116,52,115)(53,118,54,117)(55,120,56,119)(57,122,58,121)(59,124,60,123)(61,126,62,125)(63,128,64,127), (3,4)(5,7)(6,8)(9,13)(10,14)(11,16)(12,15)(17,25)(18,26)(19,28)(20,27)(21,31)(22,32)(23,29)(24,30)(33,49)(34,50)(35,52)(36,51)(37,55)(38,56)(39,53)(40,54)(41,61)(42,62)(43,64)(44,63)(45,57)(46,58)(47,60)(48,59)(65,113)(66,114)(67,116)(68,115)(69,119)(70,120)(71,117)(72,118)(73,125)(74,126)(75,128)(76,127)(77,121)(78,122)(79,124)(80,123)(81,97)(82,98)(83,100)(84,99)(85,103)(86,104)(87,101)(88,102)(89,109)(90,110)(91,112)(92,111)(93,105)(94,106)(95,108)(96,107), (1,49,17,45,13,61,29,37,5,53,21,44,12,60,28,36,4,52,20,48,16,64,32,40,8,56,24,42,10,58,26,34,2,50,18,46,14,62,30,38,6,54,22,43,11,59,27,35,3,51,19,47,15,63,31,39,7,55,23,41,9,57,25,33)(65,97,89,121,73,105,87,119,71,103,95,127,79,111,83,115,67,99,91,123,75,107,86,118,70,102,94,126,78,110,82,114,66,98,90,122,74,106,88,120,72,104,96,128,80,112,84,116,68,100,92,124,76,108,85,117,69,101,93,125,77,109,81,113), (1,25,9,23,7,31,15,19,3,27,11,22,6,30,14,18,2,26,10,24,8,32,16,20,4,28,12,21,5,29,13,17)(33,57,41,55,39,63,47,51,35,59,43,54,38,62,46,50,34,58,42,56,40,64,48,52,36,60,44,53,37,61,45,49)(65,81,77,93,69,85,76,92,68,84,80,96,72,88,74,90,66,82,78,94,70,86,75,91,67,83,79,95,71,87,73,89)(97,113,109,125,101,117,108,124,100,116,112,128,104,120,106,122,98,114,110,126,102,118,107,123,99,115,111,127,103,119,105,121), (1,13,5,12,4,16,8,10,2,14,6,11,3,15,7,9)(17,29,21,28,20,32,24,26,18,30,22,27,19,31,23,25)(33,45,37,44,36,48,40,42,34,46,38,43,35,47,39,41)(49,61,53,60,52,64,56,58,50,62,54,59,51,63,55,57)(65,73,71,79,67,75,70,78,66,74,72,80,68,76,69,77)(81,89,87,95,83,91,86,94,82,90,88,96,84,92,85,93)(97,105,103,111,99,107,102,110,98,106,104,112,100,108,101,109)(113,121,119,127,115,123,118,126,114,122,120,128,116,124,117,125), (1,7,3,6,2,8,4,5)(9,15,11,14,10,16,12,13)(17,23,19,22,18,24,20,21)(25,31,27,30,26,32,28,29)(33,39,35,38,34,40,36,37)(41,47,43,46,42,48,44,45)(49,55,51,54,50,56,52,53)(57,63,59,62,58,64,60,61)(65,69,68,72,66,70,67,71)(73,77,76,80,74,78,75,79)(81,85,84,88,82,86,83,87)(89,93,92,96,90,94,91,95)(97,101,100,104,98,102,99,103)(105,109,108,112,106,110,107,111)(113,117,116,120,114,118,115,119)(121,125,124,128,122,126,123,127), (1,4,2,3)(5,8,6,7)(9,12,10,11)(13,16,14,15)(17,20,18,19)(21,24,22,23)(25,28,26,27)(29,32,30,31)(33,36,34,35)(37,40,38,39)(41,44,42,43)(45,48,46,47)(49,52,50,51)(53,56,54,55)(57,60,58,59)(61,64,62,63)(65,67,66,68)(69,71,70,72)(73,75,74,76)(77,79,78,80)(81,83,82,84)(85,87,86,88)(89,91,90,92)(93,95,94,96)(97,99,98,100)(101,103,102,104)(105,107,106,108)(109,111,110,112)(113,115,114,116)(117,119,118,120)(121,123,122,124)(125,127,126,128), (1,2)(3,4)(5,6)(7,8)(9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22)(23,24)(25,26)(27,28)(29,30)(31,32)(33,34)(35,36)(37,38)(39,40)(41,42)(43,44)(45,46)(47,48)(49,50)(51,52)(53,54)(55,56)(57,58)(59,60)(61,62)(63,64)(65,66)(67,68)(69,70)(71,72)(73,74)(75,76)(77,78)(79,80)(81,82)(83,84)(85,86)(87,88)(89,90)(91,92)(93,94)(95,96)(97,98)(99,100)(101,102)(103,104)(105,106)(107,108)(109,110)(111,112)(113,114)(115,116)(117,118)(119,120)(121,122)(123,124)(125,126)(127,128) >;
 
Copy content gap:G := Group( (1,66,2,65)(3,68,4,67)(5,70,6,69)(7,72,8,71)(9,74,10,73)(11,76,12,75)(13,78,14,77)(15,80,16,79)(17,82,18,81)(19,84,20,83)(21,86,22,85)(23,88,24,87)(25,90,26,89)(27,92,28,91)(29,94,30,93)(31,96,32,95)(33,98,34,97)(35,100,36,99)(37,102,38,101)(39,104,40,103)(41,106,42,105)(43,108,44,107)(45,110,46,109)(47,112,48,111)(49,114,50,113)(51,116,52,115)(53,118,54,117)(55,120,56,119)(57,122,58,121)(59,124,60,123)(61,126,62,125)(63,128,64,127), (3,4)(5,7)(6,8)(9,13)(10,14)(11,16)(12,15)(17,25)(18,26)(19,28)(20,27)(21,31)(22,32)(23,29)(24,30)(33,49)(34,50)(35,52)(36,51)(37,55)(38,56)(39,53)(40,54)(41,61)(42,62)(43,64)(44,63)(45,57)(46,58)(47,60)(48,59)(65,113)(66,114)(67,116)(68,115)(69,119)(70,120)(71,117)(72,118)(73,125)(74,126)(75,128)(76,127)(77,121)(78,122)(79,124)(80,123)(81,97)(82,98)(83,100)(84,99)(85,103)(86,104)(87,101)(88,102)(89,109)(90,110)(91,112)(92,111)(93,105)(94,106)(95,108)(96,107), (1,49,17,45,13,61,29,37,5,53,21,44,12,60,28,36,4,52,20,48,16,64,32,40,8,56,24,42,10,58,26,34,2,50,18,46,14,62,30,38,6,54,22,43,11,59,27,35,3,51,19,47,15,63,31,39,7,55,23,41,9,57,25,33)(65,97,89,121,73,105,87,119,71,103,95,127,79,111,83,115,67,99,91,123,75,107,86,118,70,102,94,126,78,110,82,114,66,98,90,122,74,106,88,120,72,104,96,128,80,112,84,116,68,100,92,124,76,108,85,117,69,101,93,125,77,109,81,113), (1,25,9,23,7,31,15,19,3,27,11,22,6,30,14,18,2,26,10,24,8,32,16,20,4,28,12,21,5,29,13,17)(33,57,41,55,39,63,47,51,35,59,43,54,38,62,46,50,34,58,42,56,40,64,48,52,36,60,44,53,37,61,45,49)(65,81,77,93,69,85,76,92,68,84,80,96,72,88,74,90,66,82,78,94,70,86,75,91,67,83,79,95,71,87,73,89)(97,113,109,125,101,117,108,124,100,116,112,128,104,120,106,122,98,114,110,126,102,118,107,123,99,115,111,127,103,119,105,121), (1,13,5,12,4,16,8,10,2,14,6,11,3,15,7,9)(17,29,21,28,20,32,24,26,18,30,22,27,19,31,23,25)(33,45,37,44,36,48,40,42,34,46,38,43,35,47,39,41)(49,61,53,60,52,64,56,58,50,62,54,59,51,63,55,57)(65,73,71,79,67,75,70,78,66,74,72,80,68,76,69,77)(81,89,87,95,83,91,86,94,82,90,88,96,84,92,85,93)(97,105,103,111,99,107,102,110,98,106,104,112,100,108,101,109)(113,121,119,127,115,123,118,126,114,122,120,128,116,124,117,125), (1,7,3,6,2,8,4,5)(9,15,11,14,10,16,12,13)(17,23,19,22,18,24,20,21)(25,31,27,30,26,32,28,29)(33,39,35,38,34,40,36,37)(41,47,43,46,42,48,44,45)(49,55,51,54,50,56,52,53)(57,63,59,62,58,64,60,61)(65,69,68,72,66,70,67,71)(73,77,76,80,74,78,75,79)(81,85,84,88,82,86,83,87)(89,93,92,96,90,94,91,95)(97,101,100,104,98,102,99,103)(105,109,108,112,106,110,107,111)(113,117,116,120,114,118,115,119)(121,125,124,128,122,126,123,127), (1,4,2,3)(5,8,6,7)(9,12,10,11)(13,16,14,15)(17,20,18,19)(21,24,22,23)(25,28,26,27)(29,32,30,31)(33,36,34,35)(37,40,38,39)(41,44,42,43)(45,48,46,47)(49,52,50,51)(53,56,54,55)(57,60,58,59)(61,64,62,63)(65,67,66,68)(69,71,70,72)(73,75,74,76)(77,79,78,80)(81,83,82,84)(85,87,86,88)(89,91,90,92)(93,95,94,96)(97,99,98,100)(101,103,102,104)(105,107,106,108)(109,111,110,112)(113,115,114,116)(117,119,118,120)(121,123,122,124)(125,127,126,128), (1,2)(3,4)(5,6)(7,8)(9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22)(23,24)(25,26)(27,28)(29,30)(31,32)(33,34)(35,36)(37,38)(39,40)(41,42)(43,44)(45,46)(47,48)(49,50)(51,52)(53,54)(55,56)(57,58)(59,60)(61,62)(63,64)(65,66)(67,68)(69,70)(71,72)(73,74)(75,76)(77,78)(79,80)(81,82)(83,84)(85,86)(87,88)(89,90)(91,92)(93,94)(95,96)(97,98)(99,100)(101,102)(103,104)(105,106)(107,108)(109,110)(111,112)(113,114)(115,116)(117,118)(119,120)(121,122)(123,124)(125,126)(127,128) );
 
Copy content sage:G = PermutationGroup(['(1,66,2,65)(3,68,4,67)(5,70,6,69)(7,72,8,71)(9,74,10,73)(11,76,12,75)(13,78,14,77)(15,80,16,79)(17,82,18,81)(19,84,20,83)(21,86,22,85)(23,88,24,87)(25,90,26,89)(27,92,28,91)(29,94,30,93)(31,96,32,95)(33,98,34,97)(35,100,36,99)(37,102,38,101)(39,104,40,103)(41,106,42,105)(43,108,44,107)(45,110,46,109)(47,112,48,111)(49,114,50,113)(51,116,52,115)(53,118,54,117)(55,120,56,119)(57,122,58,121)(59,124,60,123)(61,126,62,125)(63,128,64,127)', '(3,4)(5,7)(6,8)(9,13)(10,14)(11,16)(12,15)(17,25)(18,26)(19,28)(20,27)(21,31)(22,32)(23,29)(24,30)(33,49)(34,50)(35,52)(36,51)(37,55)(38,56)(39,53)(40,54)(41,61)(42,62)(43,64)(44,63)(45,57)(46,58)(47,60)(48,59)(65,113)(66,114)(67,116)(68,115)(69,119)(70,120)(71,117)(72,118)(73,125)(74,126)(75,128)(76,127)(77,121)(78,122)(79,124)(80,123)(81,97)(82,98)(83,100)(84,99)(85,103)(86,104)(87,101)(88,102)(89,109)(90,110)(91,112)(92,111)(93,105)(94,106)(95,108)(96,107)', '(1,49,17,45,13,61,29,37,5,53,21,44,12,60,28,36,4,52,20,48,16,64,32,40,8,56,24,42,10,58,26,34,2,50,18,46,14,62,30,38,6,54,22,43,11,59,27,35,3,51,19,47,15,63,31,39,7,55,23,41,9,57,25,33)(65,97,89,121,73,105,87,119,71,103,95,127,79,111,83,115,67,99,91,123,75,107,86,118,70,102,94,126,78,110,82,114,66,98,90,122,74,106,88,120,72,104,96,128,80,112,84,116,68,100,92,124,76,108,85,117,69,101,93,125,77,109,81,113)', '(1,25,9,23,7,31,15,19,3,27,11,22,6,30,14,18,2,26,10,24,8,32,16,20,4,28,12,21,5,29,13,17)(33,57,41,55,39,63,47,51,35,59,43,54,38,62,46,50,34,58,42,56,40,64,48,52,36,60,44,53,37,61,45,49)(65,81,77,93,69,85,76,92,68,84,80,96,72,88,74,90,66,82,78,94,70,86,75,91,67,83,79,95,71,87,73,89)(97,113,109,125,101,117,108,124,100,116,112,128,104,120,106,122,98,114,110,126,102,118,107,123,99,115,111,127,103,119,105,121)', '(1,13,5,12,4,16,8,10,2,14,6,11,3,15,7,9)(17,29,21,28,20,32,24,26,18,30,22,27,19,31,23,25)(33,45,37,44,36,48,40,42,34,46,38,43,35,47,39,41)(49,61,53,60,52,64,56,58,50,62,54,59,51,63,55,57)(65,73,71,79,67,75,70,78,66,74,72,80,68,76,69,77)(81,89,87,95,83,91,86,94,82,90,88,96,84,92,85,93)(97,105,103,111,99,107,102,110,98,106,104,112,100,108,101,109)(113,121,119,127,115,123,118,126,114,122,120,128,116,124,117,125)', '(1,7,3,6,2,8,4,5)(9,15,11,14,10,16,12,13)(17,23,19,22,18,24,20,21)(25,31,27,30,26,32,28,29)(33,39,35,38,34,40,36,37)(41,47,43,46,42,48,44,45)(49,55,51,54,50,56,52,53)(57,63,59,62,58,64,60,61)(65,69,68,72,66,70,67,71)(73,77,76,80,74,78,75,79)(81,85,84,88,82,86,83,87)(89,93,92,96,90,94,91,95)(97,101,100,104,98,102,99,103)(105,109,108,112,106,110,107,111)(113,117,116,120,114,118,115,119)(121,125,124,128,122,126,123,127)', '(1,4,2,3)(5,8,6,7)(9,12,10,11)(13,16,14,15)(17,20,18,19)(21,24,22,23)(25,28,26,27)(29,32,30,31)(33,36,34,35)(37,40,38,39)(41,44,42,43)(45,48,46,47)(49,52,50,51)(53,56,54,55)(57,60,58,59)(61,64,62,63)(65,67,66,68)(69,71,70,72)(73,75,74,76)(77,79,78,80)(81,83,82,84)(85,87,86,88)(89,91,90,92)(93,95,94,96)(97,99,98,100)(101,103,102,104)(105,107,106,108)(109,111,110,112)(113,115,114,116)(117,119,118,120)(121,123,122,124)(125,127,126,128)', '(1,2)(3,4)(5,6)(7,8)(9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22)(23,24)(25,26)(27,28)(29,30)(31,32)(33,34)(35,36)(37,38)(39,40)(41,42)(43,44)(45,46)(47,48)(49,50)(51,52)(53,54)(55,56)(57,58)(59,60)(61,62)(63,64)(65,66)(67,68)(69,70)(71,72)(73,74)(75,76)(77,78)(79,80)(81,82)(83,84)(85,86)(87,88)(89,90)(91,92)(93,94)(95,96)(97,98)(99,100)(101,102)(103,104)(105,106)(107,108)(109,110)(111,112)(113,114)(115,116)(117,118)(119,120)(121,122)(123,124)(125,126)(127,128)'])
 
Copy content sage_gap:G = gap.new('Group( (1,66,2,65)(3,68,4,67)(5,70,6,69)(7,72,8,71)(9,74,10,73)(11,76,12,75)(13,78,14,77)(15,80,16,79)(17,82,18,81)(19,84,20,83)(21,86,22,85)(23,88,24,87)(25,90,26,89)(27,92,28,91)(29,94,30,93)(31,96,32,95)(33,98,34,97)(35,100,36,99)(37,102,38,101)(39,104,40,103)(41,106,42,105)(43,108,44,107)(45,110,46,109)(47,112,48,111)(49,114,50,113)(51,116,52,115)(53,118,54,117)(55,120,56,119)(57,122,58,121)(59,124,60,123)(61,126,62,125)(63,128,64,127), (3,4)(5,7)(6,8)(9,13)(10,14)(11,16)(12,15)(17,25)(18,26)(19,28)(20,27)(21,31)(22,32)(23,29)(24,30)(33,49)(34,50)(35,52)(36,51)(37,55)(38,56)(39,53)(40,54)(41,61)(42,62)(43,64)(44,63)(45,57)(46,58)(47,60)(48,59)(65,113)(66,114)(67,116)(68,115)(69,119)(70,120)(71,117)(72,118)(73,125)(74,126)(75,128)(76,127)(77,121)(78,122)(79,124)(80,123)(81,97)(82,98)(83,100)(84,99)(85,103)(86,104)(87,101)(88,102)(89,109)(90,110)(91,112)(92,111)(93,105)(94,106)(95,108)(96,107), (1,49,17,45,13,61,29,37,5,53,21,44,12,60,28,36,4,52,20,48,16,64,32,40,8,56,24,42,10,58,26,34,2,50,18,46,14,62,30,38,6,54,22,43,11,59,27,35,3,51,19,47,15,63,31,39,7,55,23,41,9,57,25,33)(65,97,89,121,73,105,87,119,71,103,95,127,79,111,83,115,67,99,91,123,75,107,86,118,70,102,94,126,78,110,82,114,66,98,90,122,74,106,88,120,72,104,96,128,80,112,84,116,68,100,92,124,76,108,85,117,69,101,93,125,77,109,81,113), (1,25,9,23,7,31,15,19,3,27,11,22,6,30,14,18,2,26,10,24,8,32,16,20,4,28,12,21,5,29,13,17)(33,57,41,55,39,63,47,51,35,59,43,54,38,62,46,50,34,58,42,56,40,64,48,52,36,60,44,53,37,61,45,49)(65,81,77,93,69,85,76,92,68,84,80,96,72,88,74,90,66,82,78,94,70,86,75,91,67,83,79,95,71,87,73,89)(97,113,109,125,101,117,108,124,100,116,112,128,104,120,106,122,98,114,110,126,102,118,107,123,99,115,111,127,103,119,105,121), (1,13,5,12,4,16,8,10,2,14,6,11,3,15,7,9)(17,29,21,28,20,32,24,26,18,30,22,27,19,31,23,25)(33,45,37,44,36,48,40,42,34,46,38,43,35,47,39,41)(49,61,53,60,52,64,56,58,50,62,54,59,51,63,55,57)(65,73,71,79,67,75,70,78,66,74,72,80,68,76,69,77)(81,89,87,95,83,91,86,94,82,90,88,96,84,92,85,93)(97,105,103,111,99,107,102,110,98,106,104,112,100,108,101,109)(113,121,119,127,115,123,118,126,114,122,120,128,116,124,117,125), (1,7,3,6,2,8,4,5)(9,15,11,14,10,16,12,13)(17,23,19,22,18,24,20,21)(25,31,27,30,26,32,28,29)(33,39,35,38,34,40,36,37)(41,47,43,46,42,48,44,45)(49,55,51,54,50,56,52,53)(57,63,59,62,58,64,60,61)(65,69,68,72,66,70,67,71)(73,77,76,80,74,78,75,79)(81,85,84,88,82,86,83,87)(89,93,92,96,90,94,91,95)(97,101,100,104,98,102,99,103)(105,109,108,112,106,110,107,111)(113,117,116,120,114,118,115,119)(121,125,124,128,122,126,123,127), (1,4,2,3)(5,8,6,7)(9,12,10,11)(13,16,14,15)(17,20,18,19)(21,24,22,23)(25,28,26,27)(29,32,30,31)(33,36,34,35)(37,40,38,39)(41,44,42,43)(45,48,46,47)(49,52,50,51)(53,56,54,55)(57,60,58,59)(61,64,62,63)(65,67,66,68)(69,71,70,72)(73,75,74,76)(77,79,78,80)(81,83,82,84)(85,87,86,88)(89,91,90,92)(93,95,94,96)(97,99,98,100)(101,103,102,104)(105,107,106,108)(109,111,110,112)(113,115,114,116)(117,119,118,120)(121,123,122,124)(125,127,126,128), (1,2)(3,4)(5,6)(7,8)(9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22)(23,24)(25,26)(27,28)(29,30)(31,32)(33,34)(35,36)(37,38)(39,40)(41,42)(43,44)(45,46)(47,48)(49,50)(51,52)(53,54)(55,56)(57,58)(59,60)(61,62)(63,64)(65,66)(67,68)(69,70)(71,72)(73,74)(75,76)(77,78)(79,80)(81,82)(83,84)(85,86)(87,88)(89,90)(91,92)(93,94)(95,96)(97,98)(99,100)(101,102)(103,104)(105,106)(107,108)(109,110)(111,112)(113,114)(115,116)(117,118)(119,120)(121,122)(123,124)(125,126)(127,128) )')
 
Copy content oscar:G = @permutation_group(128, (1,66,2,65)(3,68,4,67)(5,70,6,69)(7,72,8,71)(9,74,10,73)(11,76,12,75)(13,78,14,77)(15,80,16,79)(17,82,18,81)(19,84,20,83)(21,86,22,85)(23,88,24,87)(25,90,26,89)(27,92,28,91)(29,94,30,93)(31,96,32,95)(33,98,34,97)(35,100,36,99)(37,102,38,101)(39,104,40,103)(41,106,42,105)(43,108,44,107)(45,110,46,109)(47,112,48,111)(49,114,50,113)(51,116,52,115)(53,118,54,117)(55,120,56,119)(57,122,58,121)(59,124,60,123)(61,126,62,125)(63,128,64,127), (3,4)(5,7)(6,8)(9,13)(10,14)(11,16)(12,15)(17,25)(18,26)(19,28)(20,27)(21,31)(22,32)(23,29)(24,30)(33,49)(34,50)(35,52)(36,51)(37,55)(38,56)(39,53)(40,54)(41,61)(42,62)(43,64)(44,63)(45,57)(46,58)(47,60)(48,59)(65,113)(66,114)(67,116)(68,115)(69,119)(70,120)(71,117)(72,118)(73,125)(74,126)(75,128)(76,127)(77,121)(78,122)(79,124)(80,123)(81,97)(82,98)(83,100)(84,99)(85,103)(86,104)(87,101)(88,102)(89,109)(90,110)(91,112)(92,111)(93,105)(94,106)(95,108)(96,107), (1,49,17,45,13,61,29,37,5,53,21,44,12,60,28,36,4,52,20,48,16,64,32,40,8,56,24,42,10,58,26,34,2,50,18,46,14,62,30,38,6,54,22,43,11,59,27,35,3,51,19,47,15,63,31,39,7,55,23,41,9,57,25,33)(65,97,89,121,73,105,87,119,71,103,95,127,79,111,83,115,67,99,91,123,75,107,86,118,70,102,94,126,78,110,82,114,66,98,90,122,74,106,88,120,72,104,96,128,80,112,84,116,68,100,92,124,76,108,85,117,69,101,93,125,77,109,81,113), (1,25,9,23,7,31,15,19,3,27,11,22,6,30,14,18,2,26,10,24,8,32,16,20,4,28,12,21,5,29,13,17)(33,57,41,55,39,63,47,51,35,59,43,54,38,62,46,50,34,58,42,56,40,64,48,52,36,60,44,53,37,61,45,49)(65,81,77,93,69,85,76,92,68,84,80,96,72,88,74,90,66,82,78,94,70,86,75,91,67,83,79,95,71,87,73,89)(97,113,109,125,101,117,108,124,100,116,112,128,104,120,106,122,98,114,110,126,102,118,107,123,99,115,111,127,103,119,105,121), (1,13,5,12,4,16,8,10,2,14,6,11,3,15,7,9)(17,29,21,28,20,32,24,26,18,30,22,27,19,31,23,25)(33,45,37,44,36,48,40,42,34,46,38,43,35,47,39,41)(49,61,53,60,52,64,56,58,50,62,54,59,51,63,55,57)(65,73,71,79,67,75,70,78,66,74,72,80,68,76,69,77)(81,89,87,95,83,91,86,94,82,90,88,96,84,92,85,93)(97,105,103,111,99,107,102,110,98,106,104,112,100,108,101,109)(113,121,119,127,115,123,118,126,114,122,120,128,116,124,117,125), (1,7,3,6,2,8,4,5)(9,15,11,14,10,16,12,13)(17,23,19,22,18,24,20,21)(25,31,27,30,26,32,28,29)(33,39,35,38,34,40,36,37)(41,47,43,46,42,48,44,45)(49,55,51,54,50,56,52,53)(57,63,59,62,58,64,60,61)(65,69,68,72,66,70,67,71)(73,77,76,80,74,78,75,79)(81,85,84,88,82,86,83,87)(89,93,92,96,90,94,91,95)(97,101,100,104,98,102,99,103)(105,109,108,112,106,110,107,111)(113,117,116,120,114,118,115,119)(121,125,124,128,122,126,123,127), (1,4,2,3)(5,8,6,7)(9,12,10,11)(13,16,14,15)(17,20,18,19)(21,24,22,23)(25,28,26,27)(29,32,30,31)(33,36,34,35)(37,40,38,39)(41,44,42,43)(45,48,46,47)(49,52,50,51)(53,56,54,55)(57,60,58,59)(61,64,62,63)(65,67,66,68)(69,71,70,72)(73,75,74,76)(77,79,78,80)(81,83,82,84)(85,87,86,88)(89,91,90,92)(93,95,94,96)(97,99,98,100)(101,103,102,104)(105,107,106,108)(109,111,110,112)(113,115,114,116)(117,119,118,120)(121,123,122,124)(125,127,126,128), (1,2)(3,4)(5,6)(7,8)(9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22)(23,24)(25,26)(27,28)(29,30)(31,32)(33,34)(35,36)(37,38)(39,40)(41,42)(43,44)(45,46)(47,48)(49,50)(51,52)(53,54)(55,56)(57,58)(59,60)(61,62)(63,64)(65,66)(67,68)(69,70)(71,72)(73,74)(75,76)(77,78)(79,80)(81,82)(83,84)(85,86)(87,88)(89,90)(91,92)(93,94)(95,96)(97,98)(99,100)(101,102)(103,104)(105,106)(107,108)(109,110)(111,112)(113,114)(115,116)(117,118)(119,120)(121,122)(123,124)(125,126)(127,128))
 
Matrix group:$\left\langle \left(\begin{array}{rr} 1 & 0 \\ 0 & 190 \end{array}\right), \left(\begin{array}{rr} 72 & 36 \\ 22 & 72 \end{array}\right) \right\rangle \subseteq \GL_{2}(\F_{191})$
Copy content comment:Define the group as a matrix group with coefficients in GLFp
 
Copy content magma:G := MatrixGroup< 2, GF(191) | [[1, 0, 0, 190], [72, 36, 22, 72]] >;
 
Copy content gap:G := Group([[[ Z(191)^0, 0*Z(191) ], [ 0*Z(191), Z(191)^95 ]], [[ Z(191)^174, Z(191)^130 ], [ Z(191)^129, Z(191)^174 ]]]);
 
Copy content sage:MS = MatrixSpace(GF(191), 2, 2) G = MatrixGroup([MS([[1, 0], [0, 190]]), MS([[72, 36], [22, 72]])])
 
Copy content sage_gap:G = gap.new('Group([[[ Z(191)^0, 0*Z(191) ], [ 0*Z(191), Z(191)^95 ]], [[ Z(191)^174, Z(191)^130 ], [ Z(191)^129, Z(191)^174 ]]])')
 
Copy content oscar:G = matrix_group([matrix(GF(191), [[1, 0], [0, 190]]), matrix(GF(191), [[72, 36], [22, 72]])])
 
Direct product: not isomorphic to a non-trivial direct product
Semidirect product: $Q_{128}$ $\,\rtimes\,$ $C_2$ $C_{128}$ $\,\rtimes\,$ $C_2$ more information
Trans. wreath product: not isomorphic to a non-trivial transitive wreath product
Non-split product: $D_{64}$ . $C_2$ $C_{32}$ . $D_4$ $C_{16}$ . $D_8$ $C_8$ . $D_{16}$ all 7

Elements of the group are displayed as words in the presentation generators from the presentation above.

Homology

Abelianization: $C_{2}^{2} $
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())
 
Copy content sage_gap:G.FactorGroup(G.DerivedSubgroup())
 
Copy content oscar:quo(G, derived_subgroup(G)[1])
 
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()
 
Copy content oscar:subgroups(G)
 

There are 199 subgroups in 22 conjugacy classes, 11 normal, and all normal subgroups are characteristic.

Characteristic subgroups are shown in this color.

Special subgroups

Center: $Z \simeq$ $C_2$ $G/Z \simeq$ $D_{64}$
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()
 
Copy content oscar:center(G)
 
Commutator: $G' \simeq$ $C_{64}$ $G/G' \simeq$ $C_2^2$
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()
 
Copy content oscar:derived_subgroup(G)
 
Frattini: $\Phi \simeq$ $C_{64}$ $G/\Phi \simeq$ $C_2^2$
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()
 
Copy content oscar:frattini_subgroup(G)
 
Fitting: $\operatorname{Fit} \simeq$ $\SD_{256}$ $G/\operatorname{Fit} \simeq$ $C_1$
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()
 
Copy content oscar:fitting_subgroup(G)
 
Radical: $R \simeq$ $\SD_{256}$ $G/R \simeq$ $C_1$
Copy content comment:Radical of the group G
 
Copy content magma:Radical(G);
 
Copy content gap:SolvableRadical(G);
 
Copy content sage_gap:G.SolvableRadical()
 
Copy content oscar:solvable_radical(G)
 
Socle: $\operatorname{soc} \simeq$ $C_2$ $G/\operatorname{soc} \simeq$ $D_{64}$
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()
 
Copy content oscar:socle(G)
 
2-Sylow subgroup: $P_{ 2 } \simeq$ $\SD_{256}$

Subgroup diagram and profile

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Subgroup information

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Series

Derived series $\SD_{256}$ $\rhd$ $C_{64}$ $\rhd$ $C_1$
Copy content comment:Derived series of the group G
 
Copy content magma:DerivedSeries(G);
 
Copy content gap:DerivedSeriesOfGroup(G);
 
Copy content sage:G.derived_series()
 
Copy content sage_gap:G.DerivedSeriesOfGroup()
 
Copy content oscar:derived_series(G)
 
Chief series $\SD_{256}$ $\rhd$ $C_{128}$ $\rhd$ $C_{64}$ $\rhd$ $C_{32}$ $\rhd$ $C_{16}$ $\rhd$ $C_8$ $\rhd$ $C_4$ $\rhd$ $C_2$ $\rhd$ $C_1$
Copy content comment:Chief series of the group G
 
Copy content magma:ChiefSeries(G);
 
Copy content gap:ChiefSeries(G);
 
Copy content sage:libgap(G).ChiefSeries()
 
Copy content sage_gap:G.ChiefSeries()
 
Copy content oscar:chief_series(G)
 
Lower central series $\SD_{256}$ $\rhd$ $C_{64}$ $\rhd$ $C_{32}$ $\rhd$ $C_{16}$ $\rhd$ $C_8$ $\rhd$ $C_4$ $\rhd$ $C_2$ $\rhd$ $C_1$
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()
 
Copy content oscar:lower_central_series(G)
 
Upper central series $C_1$ $\lhd$ $C_2$ $\lhd$ $C_4$ $\lhd$ $C_8$ $\lhd$ $C_{16}$ $\lhd$ $C_{32}$ $\lhd$ $C_{64}$ $\lhd$ $\SD_{256}$
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()
 
Copy content oscar:upper_central_series(G)
 

Supergroups

This group is a maximal subgroup of 4 larger groups in the database.

This group is a maximal quotient of 4 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
 
Copy content oscar:character_table(G) # Output not guaranteed to exactly match the LMFDB table
 

Complex character table

See the $67 \times 67$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

1A 2A 2B 4A 4B 8A 16A 32A 64A 128A
Size 1 1 64 2 64 4 8 16 32 64
2 P 1A 1A 1A 2A 2A 4A 8A 16A 32A 64A
256.540.1a 1 1 1 1 1 1 1 1 1 1
256.540.1b 1 1 −1 1 −1 1 1 1 1 1
256.540.1c 1 1 −1 1 1 1 1 1 1 −1
256.540.1d 1 1 1 1 −1 1 1 1 1 −1
256.540.2a 2 2 0 2 0 2 2 2 −2 0
256.540.2b 4 4 0 4 0 4 4 −4 0 0
256.540.2c 8 8 0 8 0 8 −8 0 0 0
256.540.2d 16 16 0 16 0 −16 0 0 0 0
256.540.2e 32 32 0 −32 0 0 0 0 0 0
256.540.2f 64 −64 0 0 0 0 0 0 0 0