# Oscar code for working with abstract group 23328.jz. # If you have not already loaded the Oscar package, you should type "using Oscar;" before running the code below. # Some of these functions may take a long time to execute (this depends on the group). # Construction of abstract group: G = @permutation_group(81, (2,9)(4,17)(5,8)(6,21)(7,28)(10,33)(13,40)(15,48)(16,26)(18,32)(19,50)(20,55)(23,52)(24,39)(25,27)(29,63)(31,34)(35,54)(37,45)(38,43)(41,71)(44,76)(46,79)(47,65)(49,80)(51,56)(53,81)(57,70)(58,62)(59,69)(60,64)(61,68)(66,74)(67,75)(72,77)(73,78), (2,6)(3,12)(4,17)(5,8)(7,24)(9,21)(10,32)(11,22)(13,25)(14,36)(15,45)(16,43)(18,33)(19,50)(20,55)(23,34)(26,38)(27,40)(28,39)(29,61)(30,42)(31,52)(35,56)(37,48)(41,69)(44,74)(46,77)(47,62)(49,75)(51,54)(53,78)(57,64)(58,65)(59,71)(60,70)(63,68)(66,76)(67,80)(72,79)(73,81), (2,9,6,21)(3,12)(4,17)(5,22,8,11)(7,27,13,39)(10,33,23,52)(14,43,30,38)(15,45)(16,42,26,36)(18,34,31,32)(19,54,20,56)(24,40,25,28)(29,62,41,70)(35,50,51,55)(37,48)(44,75,46,78)(47,69,60,61)(49,77,53,74)(57,71,58,63)(59,65,68,64)(66,81,72,80)(67,76,73,79), (3,14,30)(4,19,20)(7,26,40)(10,31,35)(12,36,42)(13,28,16)(15,47,60)(17,50,55)(18,23,51)(24,38,27)(25,39,43)(29,49,44)(32,52,56)(33,34,54)(37,58,57)(41,46,53)(45,62,70)(48,65,64)(59,72,73)(61,75,74)(63,80,76)(66,68,67)(69,77,78)(71,79,81), (2,6)(5,8)(7,13)(9,21)(10,23)(11,22)(14,30)(16,26)(18,31)(19,20)(24,25)(27,39)(28,40)(29,41)(32,34)(33,52)(35,51)(36,42)(38,43)(44,46)(47,60)(49,53)(50,55)(54,56)(57,58)(59,68)(61,69)(62,70)(63,71)(64,65)(66,72)(67,73)(74,77)(75,78)(76,79)(80,81), (1,4,17)(2,10,34)(3,15,48)(5,20,50)(6,23,32)(7,29,64)(8,19,55)(9,31,33)(11,35,54)(12,37,45)(13,41,65)(14,44,76)(16,49,81)(18,52,21)(22,51,56)(24,57,61)(25,58,69)(26,53,80)(27,59,62)(28,60,63)(30,46,79)(36,66,74)(38,67,78)(39,68,70)(40,47,71)(42,72,77)(43,73,75), (2,8,9,11,6,5,21,22)(7,26,28,30,13,16,40,14)(10,19,31,35,23,20,18,51)(24,38,39,42,25,43,27,36)(29,53,60,46,41,49,47,44)(32,50,52,56,34,55,33,54)(57,67,68,72,58,73,59,66)(61,78,70,77,69,75,62,74)(63,79,65,81,71,76,64,80), (3,13,7)(4,18,31)(10,20,51)(12,25,24)(14,28,26)(15,46,44)(16,40,30)(17,33,52)(19,23,35)(27,42,43)(29,47,53)(32,55,54)(34,56,50)(36,39,38)(37,67,73)(41,49,60)(45,77,74)(48,80,81)(57,68,72)(58,66,59)(61,62,78)(63,79,64)(65,76,71)(69,75,70), (1,3,12)(2,7,25)(4,15,37)(5,16,38)(6,13,24)(8,26,43)(9,28,27)(10,29,58)(11,30,36)(14,42,22)(17,48,45)(18,47,68)(19,53,73)(20,49,67)(21,40,39)(23,41,57)(31,60,59)(32,65,61)(33,63,62)(34,64,69)(35,46,66)(44,72,51)(50,81,78)(52,71,70)(54,79,74)(55,80,75)(56,76,77), (1,5,8)(2,11,9)(3,16,26)(4,20,19)(6,21,22)(7,30,28)(10,35,31)(12,38,43)(13,40,14)(15,49,53)(17,50,55)(18,51,23)(24,39,42)(25,36,27)(29,46,60)(32,52,56)(33,34,54)(37,67,73)(41,47,44)(45,78,75)(48,81,80)(57,68,72)(58,66,59)(61,70,77)(62,69,74)(63,64,79)(65,71,76), (1,2,6)(3,7,13)(4,10,23)(5,11,21)(8,9,22)(12,25,24)(14,26,28)(15,29,41)(16,30,40)(17,34,32)(18,20,35)(19,31,51)(27,42,43)(33,56,55)(36,39,38)(37,58,57)(44,53,60)(45,69,61)(46,47,49)(48,64,65)(50,54,52)(59,72,73)(62,77,75)(63,76,80)(66,68,67)(70,78,74)(71,81,79)) # Order of the group: order(G) # Exponent of the group: exponent(G) # Automorphism group: automorphism_group(G) # Composition factors of the group: composition_series(G) # Nilpotency class of the group: if is_nilpotent(G) nilpotency_class(G) end # Derived length of the group: derived_length(G) # Determine if the group G is abelian: is_abelian(G) # Determine if the group G is cyclic: is_cyclic(G) # Determine if the group G is elementary abelian: is_elementary_abelian(G) # Determine if the group G is nilpotent: is_nilpotent(G) # Determine if the group G is perfect: is_perfect(G) # Determine if the group G is a p-group: is_pgroup(G) # Determine if the group G is simple: is_simple(G) # Determine if the group G is solvable: is_solvable(G) # Determine if the group G is supersolvable: is_supersolvable(G) # Compute statistics for the group G: # 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)) # List of conjugacy classes of the group: conjugacy_classes(G) # Output not guaranteed to exactly match the LMFDB table # Compute statistics about the characters of G: # Outputs an MSet containing the absolutely irreducible degrees of G and their multiplicities. character_degrees(G) # Define the group as a permutation group: @permutation_group(81, (2,9)(4,17)(5,8)(6,21)(7,28)(10,33)(13,40)(15,48)(16,26)(18,32)(19,50)(20,55)(23,52)(24,39)(25,27)(29,63)(31,34)(35,54)(37,45)(38,43)(41,71)(44,76)(46,79)(47,65)(49,80)(51,56)(53,81)(57,70)(58,62)(59,69)(60,64)(61,68)(66,74)(67,75)(72,77)(73,78), (2,6)(3,12)(4,17)(5,8)(7,24)(9,21)(10,32)(11,22)(13,25)(14,36)(15,45)(16,43)(18,33)(19,50)(20,55)(23,34)(26,38)(27,40)(28,39)(29,61)(30,42)(31,52)(35,56)(37,48)(41,69)(44,74)(46,77)(47,62)(49,75)(51,54)(53,78)(57,64)(58,65)(59,71)(60,70)(63,68)(66,76)(67,80)(72,79)(73,81), (2,9,6,21)(3,12)(4,17)(5,22,8,11)(7,27,13,39)(10,33,23,52)(14,43,30,38)(15,45)(16,42,26,36)(18,34,31,32)(19,54,20,56)(24,40,25,28)(29,62,41,70)(35,50,51,55)(37,48)(44,75,46,78)(47,69,60,61)(49,77,53,74)(57,71,58,63)(59,65,68,64)(66,81,72,80)(67,76,73,79), (3,14,30)(4,19,20)(7,26,40)(10,31,35)(12,36,42)(13,28,16)(15,47,60)(17,50,55)(18,23,51)(24,38,27)(25,39,43)(29,49,44)(32,52,56)(33,34,54)(37,58,57)(41,46,53)(45,62,70)(48,65,64)(59,72,73)(61,75,74)(63,80,76)(66,68,67)(69,77,78)(71,79,81), (2,6)(5,8)(7,13)(9,21)(10,23)(11,22)(14,30)(16,26)(18,31)(19,20)(24,25)(27,39)(28,40)(29,41)(32,34)(33,52)(35,51)(36,42)(38,43)(44,46)(47,60)(49,53)(50,55)(54,56)(57,58)(59,68)(61,69)(62,70)(63,71)(64,65)(66,72)(67,73)(74,77)(75,78)(76,79)(80,81), (1,4,17)(2,10,34)(3,15,48)(5,20,50)(6,23,32)(7,29,64)(8,19,55)(9,31,33)(11,35,54)(12,37,45)(13,41,65)(14,44,76)(16,49,81)(18,52,21)(22,51,56)(24,57,61)(25,58,69)(26,53,80)(27,59,62)(28,60,63)(30,46,79)(36,66,74)(38,67,78)(39,68,70)(40,47,71)(42,72,77)(43,73,75), (2,8,9,11,6,5,21,22)(7,26,28,30,13,16,40,14)(10,19,31,35,23,20,18,51)(24,38,39,42,25,43,27,36)(29,53,60,46,41,49,47,44)(32,50,52,56,34,55,33,54)(57,67,68,72,58,73,59,66)(61,78,70,77,69,75,62,74)(63,79,65,81,71,76,64,80), (3,13,7)(4,18,31)(10,20,51)(12,25,24)(14,28,26)(15,46,44)(16,40,30)(17,33,52)(19,23,35)(27,42,43)(29,47,53)(32,55,54)(34,56,50)(36,39,38)(37,67,73)(41,49,60)(45,77,74)(48,80,81)(57,68,72)(58,66,59)(61,62,78)(63,79,64)(65,76,71)(69,75,70), (1,3,12)(2,7,25)(4,15,37)(5,16,38)(6,13,24)(8,26,43)(9,28,27)(10,29,58)(11,30,36)(14,42,22)(17,48,45)(18,47,68)(19,53,73)(20,49,67)(21,40,39)(23,41,57)(31,60,59)(32,65,61)(33,63,62)(34,64,69)(35,46,66)(44,72,51)(50,81,78)(52,71,70)(54,79,74)(55,80,75)(56,76,77), (1,5,8)(2,11,9)(3,16,26)(4,20,19)(6,21,22)(7,30,28)(10,35,31)(12,38,43)(13,40,14)(15,49,53)(17,50,55)(18,51,23)(24,39,42)(25,36,27)(29,46,60)(32,52,56)(33,34,54)(37,67,73)(41,47,44)(45,78,75)(48,81,80)(57,68,72)(58,66,59)(61,70,77)(62,69,74)(63,64,79)(65,71,76), (1,2,6)(3,7,13)(4,10,23)(5,11,21)(8,9,22)(12,25,24)(14,26,28)(15,29,41)(16,30,40)(17,34,32)(18,20,35)(19,31,51)(27,42,43)(33,56,55)(36,39,38)(37,58,57)(44,53,60)(45,69,61)(46,47,49)(48,64,65)(50,54,52)(59,72,73)(62,77,75)(63,76,80)(66,68,67)(70,78,74)(71,81,79)) # The primary decomposition of the group: abelian_invariants(G) # The abelianization of the group: quo(G, derived_subgroup(G)[1]) # List of subgroups of the group: subgroups(G) # Center of the group: center(G) # Commutator subgroup of the group G: derived_subgroup(G) # Frattini subgroup of the group G: frattini_subgroup(G) # Fitting subgroup of the group G: fitting_subgroup(G) # Radical of the group G: solvable_radical(G) # Socle of the group G: socle(G) # Derived series of the group G: derived_series(G) # Chief series of the group G: chief_series(G) # The lower central series of the group G: lower_central_series(G) # The upper central series of the group G: upper_central_series(G) # Character table: character_table(G) # Output not guaranteed to exactly match the LMFDB table