Show commands: Gap / Magma / Oscar / SageMath

Copy content comment:Define group as a cyclic group
 
Copy content magma:G := CyclicGroup(128);
 
Copy content gap:G := CyclicGroup(128);
 
Copy content sage:G = CyclicPermutationGroup(128)
 
Copy content sage_gap:G = libgap.eval('CyclicGroup(128)')
 
Copy content oscar:G = cyclic_group(128)
 

Group information

Description:$C_{128}$
Order: \(128\)\(\medspace = 2^{7} \)
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()
 
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:$C_2\times C_{32}$, of order \(64\)\(\medspace = 2^{6} \)
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 7
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:$1$
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:$1$
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 cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary, hyperelementary, metacyclic, metabelian, a Z-group, and an A-group) and a $p$-group.

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 1 2 4 8 16 32 64 128
Conjugacy classes   1 1 2 4 8 16 32 64 128
Divisions 1 1 1 1 1 1 1 1 8
Autjugacy classes 1 1 1 1 1 1 1 1 8

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.   128 0 0 0 0 0 0 128
Irr. rational chars. 2 1 1 1 1 1 1 8

Minimal presentations

Permutation degree:$128$
Transitive degree:$128$
Rank: $1$
Inequivalent generators: $1$

Minimal degrees of faithful linear representations

Over $\mathbb{C}$ Over $\mathbb{R}$ Over $\mathbb{Q}$
Irreducible 1 2 64
Arbitrary 1 2 64

Constructions

Show commands: Gap / Magma / Oscar / SageMath


Presentation: $\langle a \mid a^{128}=1 \rangle$ Copy content Toggle raw display
Copy content comment:Define the group with the given generators and relations
 
Copy content magma:G := PCGroup([7, -2, -2, -2, -2, -2, -2, -2, 14, 36, 58, 80, 102, 124]); a := Explode([G.1]); AssignNames(~G, ["a", "a2", "a4", "a8", "a16", "a32", "a64"]);
 
Copy content gap:G := PcGroupCode(4684887383731077183,128); a := G.1;
 
Copy content sage:# This uses Sage's interface to GAP, as Sage (currently) has no native support for PC groups G = gap.new('PcGroupCode(4684887383731077183,128)'); a = G.1;
 
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(4684887383731077183,128)'); a = G.1;
 
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(4684887383731077183,128)")); a = gen(G, 1);
 
Permutation group:Degree $128$ $\langle(1,128,64,96,32,112,48,80,16,120,56,88,24,104,40,72,8,124,60,92,28,108,44,76,12,116,52,84,20,100,36,68,4,126,62,94,30,110,46,78,14,118,54,86,22,102,38,70,6,122,58,90,26,106,42,74,10,114,50,82,18,98,34,66,2,127,63,95,31,111,47,79,15,119,55,87,23,103,39,71,7,123,59,91,27,107,43,75,11,115,51,83,19,99,35,67,3,125,61,93,29,109,45,77,13,117,53,85,21,101,37,69,5,121,57,89,25,105,41,73,9,113,49,81,17,97,33,65) \!\cdots\! \rangle$ Copy content Toggle raw display
Copy content comment:Define the group as a permutation group
 
Copy content magma:G := PermutationGroup< 128 | (1,128,64,96,32,112,48,80,16,120,56,88,24,104,40,72,8,124,60,92,28,108,44,76,12,116,52,84,20,100,36,68,4,126,62,94,30,110,46,78,14,118,54,86,22,102,38,70,6,122,58,90,26,106,42,74,10,114,50,82,18,98,34,66,2,127,63,95,31,111,47,79,15,119,55,87,23,103,39,71,7,123,59,91,27,107,43,75,11,115,51,83,19,99,35,67,3,125,61,93,29,109,45,77,13,117,53,85,21,101,37,69,5,121,57,89,25,105,41,73,9,113,49,81,17,97,33,65), (1,64,32,48,16,56,24,40,8,60,28,44,12,52,20,36,4,62,30,46,14,54,22,38,6,58,26,42,10,50,18,34,2,63,31,47,15,55,23,39,7,59,27,43,11,51,19,35,3,61,29,45,13,53,21,37,5,57,25,41,9,49,17,33)(65,128,96,112,80,120,88,104,72,124,92,108,76,116,84,100,68,126,94,110,78,118,86,102,70,122,90,106,74,114,82,98,66,127,95,111,79,119,87,103,71,123,91,107,75,115,83,99,67,125,93,109,77,117,85,101,69,121,89,105,73,113,81,97), (1,32,16,24,8,28,12,20,4,30,14,22,6,26,10,18,2,31,15,23,7,27,11,19,3,29,13,21,5,25,9,17)(33,64,48,56,40,60,44,52,36,62,46,54,38,58,42,50,34,63,47,55,39,59,43,51,35,61,45,53,37,57,41,49)(65,96,80,88,72,92,76,84,68,94,78,86,70,90,74,82,66,95,79,87,71,91,75,83,67,93,77,85,69,89,73,81)(97,128,112,120,104,124,108,116,100,126,110,118,102,122,106,114,98,127,111,119,103,123,107,115,99,125,109,117,101,121,105,113), (1,16,8,12,4,14,6,10,2,15,7,11,3,13,5,9)(17,32,24,28,20,30,22,26,18,31,23,27,19,29,21,25)(33,48,40,44,36,46,38,42,34,47,39,43,35,45,37,41)(49,64,56,60,52,62,54,58,50,63,55,59,51,61,53,57)(65,80,72,76,68,78,70,74,66,79,71,75,67,77,69,73)(81,96,88,92,84,94,86,90,82,95,87,91,83,93,85,89)(97,112,104,108,100,110,102,106,98,111,103,107,99,109,101,105)(113,128,120,124,116,126,118,122,114,127,119,123,115,125,117,121), (1,8,4,6,2,7,3,5)(9,16,12,14,10,15,11,13)(17,24,20,22,18,23,19,21)(25,32,28,30,26,31,27,29)(33,40,36,38,34,39,35,37)(41,48,44,46,42,47,43,45)(49,56,52,54,50,55,51,53)(57,64,60,62,58,63,59,61)(65,72,68,70,66,71,67,69)(73,80,76,78,74,79,75,77)(81,88,84,86,82,87,83,85)(89,96,92,94,90,95,91,93)(97,104,100,102,98,103,99,101)(105,112,108,110,106,111,107,109)(113,120,116,118,114,119,115,117)(121,128,124,126,122,127,123,125), (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)(65,68,66,67)(69,72,70,71)(73,76,74,75)(77,80,78,79)(81,84,82,83)(85,88,86,87)(89,92,90,91)(93,96,94,95)(97,100,98,99)(101,104,102,103)(105,108,106,107)(109,112,110,111)(113,116,114,115)(117,120,118,119)(121,124,122,123)(125,128,126,127), (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( (1,128,64,96,32,112,48,80,16,120,56,88,24,104,40,72,8,124,60,92,28,108,44,76,12,116,52,84,20,100,36,68,4,126,62,94,30,110,46,78,14,118,54,86,22,102,38,70,6,122,58,90,26,106,42,74,10,114,50,82,18,98,34,66,2,127,63,95,31,111,47,79,15,119,55,87,23,103,39,71,7,123,59,91,27,107,43,75,11,115,51,83,19,99,35,67,3,125,61,93,29,109,45,77,13,117,53,85,21,101,37,69,5,121,57,89,25,105,41,73,9,113,49,81,17,97,33,65), (1,64,32,48,16,56,24,40,8,60,28,44,12,52,20,36,4,62,30,46,14,54,22,38,6,58,26,42,10,50,18,34,2,63,31,47,15,55,23,39,7,59,27,43,11,51,19,35,3,61,29,45,13,53,21,37,5,57,25,41,9,49,17,33)(65,128,96,112,80,120,88,104,72,124,92,108,76,116,84,100,68,126,94,110,78,118,86,102,70,122,90,106,74,114,82,98,66,127,95,111,79,119,87,103,71,123,91,107,75,115,83,99,67,125,93,109,77,117,85,101,69,121,89,105,73,113,81,97), (1,32,16,24,8,28,12,20,4,30,14,22,6,26,10,18,2,31,15,23,7,27,11,19,3,29,13,21,5,25,9,17)(33,64,48,56,40,60,44,52,36,62,46,54,38,58,42,50,34,63,47,55,39,59,43,51,35,61,45,53,37,57,41,49)(65,96,80,88,72,92,76,84,68,94,78,86,70,90,74,82,66,95,79,87,71,91,75,83,67,93,77,85,69,89,73,81)(97,128,112,120,104,124,108,116,100,126,110,118,102,122,106,114,98,127,111,119,103,123,107,115,99,125,109,117,101,121,105,113), (1,16,8,12,4,14,6,10,2,15,7,11,3,13,5,9)(17,32,24,28,20,30,22,26,18,31,23,27,19,29,21,25)(33,48,40,44,36,46,38,42,34,47,39,43,35,45,37,41)(49,64,56,60,52,62,54,58,50,63,55,59,51,61,53,57)(65,80,72,76,68,78,70,74,66,79,71,75,67,77,69,73)(81,96,88,92,84,94,86,90,82,95,87,91,83,93,85,89)(97,112,104,108,100,110,102,106,98,111,103,107,99,109,101,105)(113,128,120,124,116,126,118,122,114,127,119,123,115,125,117,121), (1,8,4,6,2,7,3,5)(9,16,12,14,10,15,11,13)(17,24,20,22,18,23,19,21)(25,32,28,30,26,31,27,29)(33,40,36,38,34,39,35,37)(41,48,44,46,42,47,43,45)(49,56,52,54,50,55,51,53)(57,64,60,62,58,63,59,61)(65,72,68,70,66,71,67,69)(73,80,76,78,74,79,75,77)(81,88,84,86,82,87,83,85)(89,96,92,94,90,95,91,93)(97,104,100,102,98,103,99,101)(105,112,108,110,106,111,107,109)(113,120,116,118,114,119,115,117)(121,128,124,126,122,127,123,125), (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)(65,68,66,67)(69,72,70,71)(73,76,74,75)(77,80,78,79)(81,84,82,83)(85,88,86,87)(89,92,90,91)(93,96,94,95)(97,100,98,99)(101,104,102,103)(105,108,106,107)(109,112,110,111)(113,116,114,115)(117,120,118,119)(121,124,122,123)(125,128,126,127), (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(['(1,128,64,96,32,112,48,80,16,120,56,88,24,104,40,72,8,124,60,92,28,108,44,76,12,116,52,84,20,100,36,68,4,126,62,94,30,110,46,78,14,118,54,86,22,102,38,70,6,122,58,90,26,106,42,74,10,114,50,82,18,98,34,66,2,127,63,95,31,111,47,79,15,119,55,87,23,103,39,71,7,123,59,91,27,107,43,75,11,115,51,83,19,99,35,67,3,125,61,93,29,109,45,77,13,117,53,85,21,101,37,69,5,121,57,89,25,105,41,73,9,113,49,81,17,97,33,65)', '(1,64,32,48,16,56,24,40,8,60,28,44,12,52,20,36,4,62,30,46,14,54,22,38,6,58,26,42,10,50,18,34,2,63,31,47,15,55,23,39,7,59,27,43,11,51,19,35,3,61,29,45,13,53,21,37,5,57,25,41,9,49,17,33)(65,128,96,112,80,120,88,104,72,124,92,108,76,116,84,100,68,126,94,110,78,118,86,102,70,122,90,106,74,114,82,98,66,127,95,111,79,119,87,103,71,123,91,107,75,115,83,99,67,125,93,109,77,117,85,101,69,121,89,105,73,113,81,97)', '(1,32,16,24,8,28,12,20,4,30,14,22,6,26,10,18,2,31,15,23,7,27,11,19,3,29,13,21,5,25,9,17)(33,64,48,56,40,60,44,52,36,62,46,54,38,58,42,50,34,63,47,55,39,59,43,51,35,61,45,53,37,57,41,49)(65,96,80,88,72,92,76,84,68,94,78,86,70,90,74,82,66,95,79,87,71,91,75,83,67,93,77,85,69,89,73,81)(97,128,112,120,104,124,108,116,100,126,110,118,102,122,106,114,98,127,111,119,103,123,107,115,99,125,109,117,101,121,105,113)', '(1,16,8,12,4,14,6,10,2,15,7,11,3,13,5,9)(17,32,24,28,20,30,22,26,18,31,23,27,19,29,21,25)(33,48,40,44,36,46,38,42,34,47,39,43,35,45,37,41)(49,64,56,60,52,62,54,58,50,63,55,59,51,61,53,57)(65,80,72,76,68,78,70,74,66,79,71,75,67,77,69,73)(81,96,88,92,84,94,86,90,82,95,87,91,83,93,85,89)(97,112,104,108,100,110,102,106,98,111,103,107,99,109,101,105)(113,128,120,124,116,126,118,122,114,127,119,123,115,125,117,121)', '(1,8,4,6,2,7,3,5)(9,16,12,14,10,15,11,13)(17,24,20,22,18,23,19,21)(25,32,28,30,26,31,27,29)(33,40,36,38,34,39,35,37)(41,48,44,46,42,47,43,45)(49,56,52,54,50,55,51,53)(57,64,60,62,58,63,59,61)(65,72,68,70,66,71,67,69)(73,80,76,78,74,79,75,77)(81,88,84,86,82,87,83,85)(89,96,92,94,90,95,91,93)(97,104,100,102,98,103,99,101)(105,112,108,110,106,111,107,109)(113,120,116,118,114,119,115,117)(121,128,124,126,122,127,123,125)', '(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)(65,68,66,67)(69,72,70,71)(73,76,74,75)(77,80,78,79)(81,84,82,83)(85,88,86,87)(89,92,90,91)(93,96,94,95)(97,100,98,99)(101,104,102,103)(105,108,106,107)(109,112,110,111)(113,116,114,115)(117,120,118,119)(121,124,122,123)(125,128,126,127)', '(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( (1,128,64,96,32,112,48,80,16,120,56,88,24,104,40,72,8,124,60,92,28,108,44,76,12,116,52,84,20,100,36,68,4,126,62,94,30,110,46,78,14,118,54,86,22,102,38,70,6,122,58,90,26,106,42,74,10,114,50,82,18,98,34,66,2,127,63,95,31,111,47,79,15,119,55,87,23,103,39,71,7,123,59,91,27,107,43,75,11,115,51,83,19,99,35,67,3,125,61,93,29,109,45,77,13,117,53,85,21,101,37,69,5,121,57,89,25,105,41,73,9,113,49,81,17,97,33,65), (1,64,32,48,16,56,24,40,8,60,28,44,12,52,20,36,4,62,30,46,14,54,22,38,6,58,26,42,10,50,18,34,2,63,31,47,15,55,23,39,7,59,27,43,11,51,19,35,3,61,29,45,13,53,21,37,5,57,25,41,9,49,17,33)(65,128,96,112,80,120,88,104,72,124,92,108,76,116,84,100,68,126,94,110,78,118,86,102,70,122,90,106,74,114,82,98,66,127,95,111,79,119,87,103,71,123,91,107,75,115,83,99,67,125,93,109,77,117,85,101,69,121,89,105,73,113,81,97), (1,32,16,24,8,28,12,20,4,30,14,22,6,26,10,18,2,31,15,23,7,27,11,19,3,29,13,21,5,25,9,17)(33,64,48,56,40,60,44,52,36,62,46,54,38,58,42,50,34,63,47,55,39,59,43,51,35,61,45,53,37,57,41,49)(65,96,80,88,72,92,76,84,68,94,78,86,70,90,74,82,66,95,79,87,71,91,75,83,67,93,77,85,69,89,73,81)(97,128,112,120,104,124,108,116,100,126,110,118,102,122,106,114,98,127,111,119,103,123,107,115,99,125,109,117,101,121,105,113), (1,16,8,12,4,14,6,10,2,15,7,11,3,13,5,9)(17,32,24,28,20,30,22,26,18,31,23,27,19,29,21,25)(33,48,40,44,36,46,38,42,34,47,39,43,35,45,37,41)(49,64,56,60,52,62,54,58,50,63,55,59,51,61,53,57)(65,80,72,76,68,78,70,74,66,79,71,75,67,77,69,73)(81,96,88,92,84,94,86,90,82,95,87,91,83,93,85,89)(97,112,104,108,100,110,102,106,98,111,103,107,99,109,101,105)(113,128,120,124,116,126,118,122,114,127,119,123,115,125,117,121), (1,8,4,6,2,7,3,5)(9,16,12,14,10,15,11,13)(17,24,20,22,18,23,19,21)(25,32,28,30,26,31,27,29)(33,40,36,38,34,39,35,37)(41,48,44,46,42,47,43,45)(49,56,52,54,50,55,51,53)(57,64,60,62,58,63,59,61)(65,72,68,70,66,71,67,69)(73,80,76,78,74,79,75,77)(81,88,84,86,82,87,83,85)(89,96,92,94,90,95,91,93)(97,104,100,102,98,103,99,101)(105,112,108,110,106,111,107,109)(113,120,116,118,114,119,115,117)(121,128,124,126,122,127,123,125), (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)(65,68,66,67)(69,72,70,71)(73,76,74,75)(77,80,78,79)(81,84,82,83)(85,88,86,87)(89,92,90,91)(93,96,94,95)(97,100,98,99)(101,104,102,103)(105,108,106,107)(109,112,110,111)(113,116,114,115)(117,120,118,119)(121,124,122,123)(125,128,126,127), (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, (1,128,64,96,32,112,48,80,16,120,56,88,24,104,40,72,8,124,60,92,28,108,44,76,12,116,52,84,20,100,36,68,4,126,62,94,30,110,46,78,14,118,54,86,22,102,38,70,6,122,58,90,26,106,42,74,10,114,50,82,18,98,34,66,2,127,63,95,31,111,47,79,15,119,55,87,23,103,39,71,7,123,59,91,27,107,43,75,11,115,51,83,19,99,35,67,3,125,61,93,29,109,45,77,13,117,53,85,21,101,37,69,5,121,57,89,25,105,41,73,9,113,49,81,17,97,33,65), (1,64,32,48,16,56,24,40,8,60,28,44,12,52,20,36,4,62,30,46,14,54,22,38,6,58,26,42,10,50,18,34,2,63,31,47,15,55,23,39,7,59,27,43,11,51,19,35,3,61,29,45,13,53,21,37,5,57,25,41,9,49,17,33)(65,128,96,112,80,120,88,104,72,124,92,108,76,116,84,100,68,126,94,110,78,118,86,102,70,122,90,106,74,114,82,98,66,127,95,111,79,119,87,103,71,123,91,107,75,115,83,99,67,125,93,109,77,117,85,101,69,121,89,105,73,113,81,97), (1,32,16,24,8,28,12,20,4,30,14,22,6,26,10,18,2,31,15,23,7,27,11,19,3,29,13,21,5,25,9,17)(33,64,48,56,40,60,44,52,36,62,46,54,38,58,42,50,34,63,47,55,39,59,43,51,35,61,45,53,37,57,41,49)(65,96,80,88,72,92,76,84,68,94,78,86,70,90,74,82,66,95,79,87,71,91,75,83,67,93,77,85,69,89,73,81)(97,128,112,120,104,124,108,116,100,126,110,118,102,122,106,114,98,127,111,119,103,123,107,115,99,125,109,117,101,121,105,113), (1,16,8,12,4,14,6,10,2,15,7,11,3,13,5,9)(17,32,24,28,20,30,22,26,18,31,23,27,19,29,21,25)(33,48,40,44,36,46,38,42,34,47,39,43,35,45,37,41)(49,64,56,60,52,62,54,58,50,63,55,59,51,61,53,57)(65,80,72,76,68,78,70,74,66,79,71,75,67,77,69,73)(81,96,88,92,84,94,86,90,82,95,87,91,83,93,85,89)(97,112,104,108,100,110,102,106,98,111,103,107,99,109,101,105)(113,128,120,124,116,126,118,122,114,127,119,123,115,125,117,121), (1,8,4,6,2,7,3,5)(9,16,12,14,10,15,11,13)(17,24,20,22,18,23,19,21)(25,32,28,30,26,31,27,29)(33,40,36,38,34,39,35,37)(41,48,44,46,42,47,43,45)(49,56,52,54,50,55,51,53)(57,64,60,62,58,63,59,61)(65,72,68,70,66,71,67,69)(73,80,76,78,74,79,75,77)(81,88,84,86,82,87,83,85)(89,96,92,94,90,95,91,93)(97,104,100,102,98,103,99,101)(105,112,108,110,106,111,107,109)(113,120,116,118,114,119,115,117)(121,128,124,126,122,127,123,125), (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)(65,68,66,67)(69,72,70,71)(73,76,74,75)(77,80,78,79)(81,84,82,83)(85,88,86,87)(89,92,90,91)(93,96,94,95)(97,100,98,99)(101,104,102,103)(105,108,106,107)(109,112,110,111)(113,116,114,115)(117,120,118,119)(121,124,122,123)(125,128,126,127), (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:$\left\langle \left(\begin{array}{rr} 82 & 22 \\ 92 & 82 \end{array}\right) \right\rangle \subseteq \GL_{2}(\F_{127})$
Copy content comment:Define the group as a matrix group with coefficients in GLFp
 
Copy content magma:G := MatrixGroup< 2, GF(127) | [[82, 22, 92, 82]] >;
 
Copy content gap:G := Group([[[ Z(127)^26, Z(127)^14 ], [ Z(127)^13, Z(127)^26 ]]]);
 
Copy content sage:MS = MatrixSpace(GF(127), 2, 2) G = MatrixGroup([MS([[82, 22], [92, 82]])])
 
Copy content sage_gap:G = gap.new('Group([[[ Z(127)^26, Z(127)^14 ], [ Z(127)^13, Z(127)^26 ]]])')
 
Copy content oscar:G = matrix_group([matrix(GF(127), [[82, 22], [92, 82]])])
 
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
Non-split product: $C_{64}$ . $C_2$ $C_{32}$ . $C_4$ $C_{16}$ . $C_8$ $C_8$ . $C_{16}$ all 6

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

Homology

Primary decomposition: $C_{128}$
Copy content comment:The primary decomposition of the group
 
Copy content magma:PrimaryInvariants(G);
 
Copy content gap:AbelianInvariants(G);
 
Copy content sage_gap:G.AbelianInvariants()
 
Copy content oscar:abelian_invariants(G)
 
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: $0$
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 8 subgroups, all normal, and all normal subgroups are characteristic.

Characteristic subgroups are shown in this color.

Special subgroups

Center: $Z \simeq$ $C_{128}$ $G/Z \simeq$ $C_1$
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: $G' \simeq$ $C_1$ $G/G' \simeq$ $C_{128}$
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: $\Phi \simeq$ $C_{64}$ $G/\Phi \simeq$ $C_2$
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: $\operatorname{Fit} \simeq$ $C_{128}$ $G/\operatorname{Fit} \simeq$ $C_1$
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: $R \simeq$ $C_{128}$ $G/R \simeq$ $C_1$
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: $\operatorname{soc} \simeq$ $C_2$ $G/\operatorname{soc} \simeq$ $C_{64}$
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: $P_{ 2 } \simeq$ $C_{128}$

Subgroup diagram and profile

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Subgroup information

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Series

Derived series $C_{128}$ $\rhd$ $C_1$
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 $C_{128}$ $\rhd$ $C_{64}$ $\rhd$ $C_{32}$ $\rhd$ $C_{16}$ $\rhd$ $C_8$ $\rhd$ $C_4$ $\rhd$ $C_2$ $\rhd$ $C_1$
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 $C_{128}$ $\rhd$ $C_1$
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 $C_1$ $\lhd$ $C_{128}$
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 26 larger groups in the database.

This group is a maximal quotient of 22 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 $128 \times 128$ character table (warning: may be slow to load). Alternatively, you may search for characters of this group with desired properties.

Rational character table

1A 2A 4A 8A 16A 32A 64A 128A
Size 1 1 2 4 8 16 32 64
2 P 1A 1A 2A 4A 8A 16A 32A 64A
128.1.1a 1 1 1 1 1 1 1 1
128.1.1b 1 1 1 1 1 1 1 −1
128.1.1c 2 2 2 2 2 2 −2 0
128.1.1d 4 4 4 4 4 −4 0 0
128.1.1e 8 8 8 8 −8 0 0 0
128.1.1f 16 16 16 −16 0 0 0 0
128.1.1g 32 32 −32 0 0 0 0 0
128.1.1h 64 −64 0 0 0 0 0 0