# Gap code for working with abstract group 256.541. # Some of these functions may take a long time to execute (this depends on the group). # Define group as a dicyclic group: G := DicyclicGroup(256); # Order of the group: Order(G); # Exponent of the group: Exponent(G); # Automorphism group: AutomorphismGroup(G); # The outer automorphism group of G: FactorGroup(AutomorphismGroup(G), InnerAutomorphismGroup(G)); # Composition factors of the group: CompositionSeries(G); # Nilpotency class of the group: if IsNilpotentGroup(G) then NilpotencyClassOfGroup(G); fi; # Derived length of the group: DerivedLength(G); # Determine if the group G is abelian: IsAbelian(G); # Determine if the group G is cyclic: IsCyclic(G); # Determine if the group G is elementary abelian: IsElementaryAbelian(G); # Determine if the group G is a monomial group: IsMonomialGroup(G); # Determine if the group G is nilpotent: IsNilpotentGroup(G); # Determine if the group G is perfect: IsPerfectGroup(G); # Determine if the group G is a p-group: IsPGroup(G); # Determine if the group G is polycyclic: IsPolycyclicGroup(G); # Determine if the group G is simple: IsSimpleGroup(G); # Determine if the group G is solvable: IsSolvableGroup(G); # Determine if the group G is supersolvable: IsSupersolvableGroup(G); # Compute statistics for the group G: # 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"); # List of conjugacy classes of the group: ConjugacyClasses(G); # Output not guaranteed to exactly match the LMFDB table # Compute statistics about the characters of G: # 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); # Define group as Lie representations: Group([[[ Z(127)^8, Z(127)^62 ], [ Z(127)^0, Z(127)^63 ]], [[ Z(127)^63, 0*Z(127) ], [ 0*Z(127), Z(127)^63 ]], [[ Z(127)^63, Z(127)^6 ], [ Z(127)^3, Z(127)^0 ]], [[ Z(127)^20, Z(127)^19 ], [ Z(127)^83, Z(127)^37 ]]]); # Define the group with the given generators and relations: GPC := PcGroupCode(2754453629584150573251214897832863473663,256); a := GPC.1; b := GPC.2; # Define the group as a permutation group: Group( (1,130,2,129)(3,132,4,131)(5,134,6,133)(7,136,8,135)(9,138,10,137)(11,140,12,139)(13,142,14,141)(15,144,16,143)(17,146,18,145)(19,148,20,147)(21,150,22,149)(23,152,24,151)(25,154,26,153)(27,156,28,155)(29,158,30,157)(31,160,32,159)(33,162,34,161)(35,164,36,163)(37,166,38,165)(39,168,40,167)(41,170,42,169)(43,172,44,171)(45,174,46,173)(47,176,48,175)(49,178,50,177)(51,180,52,179)(53,182,54,181)(55,184,56,183)(57,186,58,185)(59,188,60,187)(61,190,62,189)(63,192,64,191)(65,194,66,193)(67,196,68,195)(69,198,70,197)(71,200,72,199)(73,202,74,201)(75,204,76,203)(77,206,78,205)(79,208,80,207)(81,210,82,209)(83,212,84,211)(85,214,86,213)(87,216,88,215)(89,218,90,217)(91,220,92,219)(93,222,94,221)(95,224,96,223)(97,226,98,225)(99,228,100,227)(101,230,102,229)(103,232,104,231)(105,234,106,233)(107,236,108,235)(109,238,110,237)(111,240,112,239)(113,242,114,241)(115,244,116,243)(117,246,118,245)(119,248,120,247)(121,250,122,249)(123,252,124,251)(125,254,126,253)(127,256,128,255), (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)(129,226,130,225)(131,228,132,227)(133,230,134,229)(135,232,136,231)(137,234,138,233)(139,236,140,235)(141,238,142,237)(143,240,144,239)(145,242,146,241)(147,244,148,243)(149,246,150,245)(151,248,152,247)(153,250,154,249)(155,252,156,251)(157,254,158,253)(159,256,160,255)(161,218,162,217)(163,220,164,219)(165,222,166,221)(167,224,168,223)(169,216,170,215)(171,213,172,214)(173,210,174,209)(175,212,176,211)(177,194,178,193)(179,196,180,195)(181,198,182,197)(183,200,184,199)(185,202,186,201)(187,204,188,203)(189,206,190,205)(191,208,192,207), (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)(129,161,153,185,137,169,151,183,135,167,159,191,143,175,147,179,131,163,155,187,139,171,150,182,134,166,158,190,142,174,146,178,130,162,154,186,138,170,152,184,136,168,160,192,144,176,148,180,132,164,156,188,140,172,149,181,133,165,157,189,141,173,145,177)(193,241,209,237,205,253,221,229,197,245,213,236,204,252,220,228,196,244,212,240,208,256,224,232,200,248,216,234,202,250,218,226,194,242,210,238,206,254,222,230,198,246,214,235,203,251,219,227,195,243,211,239,207,255,223,231,199,247,215,233,201,249,217,225), (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)(129,145,141,157,133,149,140,156,132,148,144,160,136,152,138,154,130,146,142,158,134,150,139,155,131,147,143,159,135,151,137,153)(161,177,173,189,165,181,172,188,164,180,176,192,168,184,170,186,162,178,174,190,166,182,171,187,163,179,175,191,167,183,169,185)(193,217,201,215,199,223,207,211,195,219,203,214,198,222,206,210,194,218,202,216,200,224,208,212,196,220,204,213,197,221,205,209)(225,249,233,247,231,255,239,243,227,251,235,246,230,254,238,242,226,250,234,248,232,256,240,244,228,252,236,245,229,253,237,241), (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)(129,137,135,143,131,139,134,142,130,138,136,144,132,140,133,141)(145,153,151,159,147,155,150,158,146,154,152,160,148,156,149,157)(161,169,167,175,163,171,166,174,162,170,168,176,164,172,165,173)(177,185,183,191,179,187,182,190,178,186,184,192,180,188,181,189)(193,205,197,204,196,208,200,202,194,206,198,203,195,207,199,201)(209,221,213,220,212,224,216,218,210,222,214,219,211,223,215,217)(225,237,229,236,228,240,232,234,226,238,230,235,227,239,231,233)(241,253,245,252,244,256,248,250,242,254,246,251,243,255,247,249), (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)(129,133,132,136,130,134,131,135)(137,141,140,144,138,142,139,143)(145,149,148,152,146,150,147,151)(153,157,156,160,154,158,155,159)(161,165,164,168,162,166,163,167)(169,173,172,176,170,174,171,175)(177,181,180,184,178,182,179,183)(185,189,188,192,186,190,187,191)(193,199,195,198,194,200,196,197)(201,207,203,206,202,208,204,205)(209,215,211,214,210,216,212,213)(217,223,219,222,218,224,220,221)(225,231,227,230,226,232,228,229)(233,239,235,238,234,240,236,237)(241,247,243,246,242,248,244,245)(249,255,251,254,250,256,252,253), (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)(129,131,130,132)(133,135,134,136)(137,139,138,140)(141,143,142,144)(145,147,146,148)(149,151,150,152)(153,155,154,156)(157,159,158,160)(161,163,162,164)(165,167,166,168)(169,171,170,172)(173,175,174,176)(177,179,178,180)(181,183,182,184)(185,187,186,188)(189,191,190,192)(193,196,194,195)(197,200,198,199)(201,204,202,203)(205,208,206,207)(209,212,210,211)(213,216,214,215)(217,220,218,219)(221,224,222,223)(225,228,226,227)(229,232,230,231)(233,236,234,235)(237,240,238,239)(241,244,242,243)(245,248,246,247)(249,252,250,251)(253,256,254,255), (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)(129,130)(131,132)(133,134)(135,136)(137,138)(139,140)(141,142)(143,144)(145,146)(147,148)(149,150)(151,152)(153,154)(155,156)(157,158)(159,160)(161,162)(163,164)(165,166)(167,168)(169,170)(171,172)(173,174)(175,176)(177,178)(179,180)(181,182)(183,184)(185,186)(187,188)(189,190)(191,192)(193,194)(195,196)(197,198)(199,200)(201,202)(203,204)(205,206)(207,208)(209,210)(211,212)(213,214)(215,216)(217,218)(219,220)(221,222)(223,224)(225,226)(227,228)(229,230)(231,232)(233,234)(235,236)(237,238)(239,240)(241,242)(243,244)(245,246)(247,248)(249,250)(251,252)(253,254)(255,256) ); # Define the group as a matrix group with coefficients in GLFp: Group([[[ Z(127)^50, Z(127)^125 ], [ Z(127)^61, Z(127)^113 ]], [[ Z(127)^26, Z(127)^14 ], [ Z(127)^13, Z(127)^26 ]]]); # The primary decomposition of the group: AbelianInvariants(G); # The abelianization of the group: FactorGroup(G, DerivedSubgroup(G)); # The Schur multiplier of the group: AbelianInvariantsMultiplier(G); # The commutator length of the group: CommutatorLength(G); # List of subgroups of the group: AllSubgroups(G); # Center of the group: Center(G); # Commutator subgroup of the group G: DerivedSubgroup(G); # Frattini subgroup of the group G: FrattiniSubgroup(G); # Fitting subgroup of the group G: FittingSubgroup(G); # Radical of the group G: SolvableRadical(G); # Socle of the group G: Socle(G); # Derived series of the group G: DerivedSeriesOfGroup(G); # Chief series of the group G: ChiefSeries(G); # The lower central series of the group G: LowerCentralSeriesOfGroup(G); # The upper central series of the group G: UpperCentralSeriesOfGroup(G); # Character table: CharacterTable(G); # Output not guaranteed to exactly match the LMFDB table