# Oscar code for working with abstract group 122304.a. # 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(119, (1,2)(3,13)(4,9)(5,14)(6,15)(7,16)(8,17)(10,18)(11,19)(12,20)(21,55)(22,23)(24,56)(25,57)(26,31)(27,58)(28,36)(29,42)(30,43)(32,50)(33,59)(34,60)(35,61)(37,62)(38,63)(39,51)(40,64)(41,52)(44,65)(45,66)(46,67)(47,68)(48,69)(49,70)(53,71)(54,72)(73,97)(74,82)(75,86)(76,98)(77,81)(78,99)(79,100)(80,85)(83,91)(84,94)(87,101)(88,102)(89,93)(90,96)(92,103)(95,104)(105,111)(106,109)(107,112)(108,110)(113,114,118,115,116,119,117), (1,3,21,8)(2,13,55,17)(4,22,37,30)(5,33,29,35)(6,24,36,41)(7,25,73,45)(9,23,62,43)(10,51,81,32)(11,26,74,48)(12,27,75,49)(14,59,42,61)(15,56,28,52)(16,57,97,66)(18,39,77,50)(19,31,82,69)(20,58,86,70)(34,83,65,85)(38,84,40,78)(44,80,60,91)(46,93,104,72)(47,53,96,92)(54,67,89,95)(63,94,64,99)(68,71,90,103)(76,87,98,101)(79,100)(88,102)(105,109,111,106)(107,112)(108,110)(113,115,117,118,119,114,116), (1,4,2,9)(3,22,13,23)(5,28,14,36)(6,29,15,42)(7,20,16,12)(8,30,17,43)(10,31,18,26)(11,32,19,50)(21,37,55,62)(24,35,56,61)(25,58,57,27)(33,52,59,41)(34,63,60,38)(39,74,51,82)(40,65,64,44)(45,70,66,49)(46,71,67,53)(47,72,68,54)(48,81,69,77)(73,86,97,75)(76,87,98,101)(78,85,99,80)(79,88,100,102)(83,94,91,84)(89,96,93,90)(92,104,103,95)(105,109,111,106)(107,110,112,108)(113,116,114,119,118,117,115), (113,117,119,116,115,118,114), (1,5,31,4,28,18,2,14,26,9,36,10)(3,23,13,22)(6,19,8,29,50,30,15,11,17,42,32,43)(7,34,71,20,63,67,16,60,53,12,38,46)(21,39,52,37,74,59,55,51,41,62,82,33)(24,76,88,35,87,100,56,98,102,61,101,79)(25,27,57,58)(40,68,45,65,54,70,64,47,66,44,72,49)(48,77,69,81)(73,89,94,86,96,91,97,93,84,75,90,83)(78,105,110,85,109,112,99,111,108,80,106,107)(92,95,103,104)(113,116,114,119,118,117,115), (1,6,39,62,14,19,2,15,51,37,5,11)(3,24,77,43,59,31,13,56,81,30,33,26)(4,29,74,21,36,50,9,42,82,55,28,32)(7,40,89,75,60,68,16,64,93,86,34,47)(8,41,18,23,61,69,17,52,10,22,35,48)(12,44,90,97,63,72,20,65,96,73,38,54)(25,78,95,49,91,71,57,99,104,70,83,53)(27,80,103,66,94,46,58,85,92,45,84,67)(76,101,98,87)(79,88,100,102)(105,106,111,109)(107,110,112,108), (1,7,4,20,2,16,9,12)(3,25,22,58,13,57,23,27)(5,34,28,63,14,60,36,38)(6,40,29,65,15,64,42,44)(8,45,30,70,17,66,43,49)(10,46,31,71,18,67,26,53)(11,47,32,72,19,68,50,54)(21,73,37,86,55,97,62,75)(24,78,35,85,56,99,61,80)(33,83,52,94,59,91,41,84)(39,89,74,96,51,93,82,90)(48,92,81,104,69,103,77,95)(76,105,87,109,98,111,101,106)(79,107,88,110,100,112,102,108)(113,118,116,117,114,115,119)) # 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(119, (1,2)(3,13)(4,9)(5,14)(6,15)(7,16)(8,17)(10,18)(11,19)(12,20)(21,55)(22,23)(24,56)(25,57)(26,31)(27,58)(28,36)(29,42)(30,43)(32,50)(33,59)(34,60)(35,61)(37,62)(38,63)(39,51)(40,64)(41,52)(44,65)(45,66)(46,67)(47,68)(48,69)(49,70)(53,71)(54,72)(73,97)(74,82)(75,86)(76,98)(77,81)(78,99)(79,100)(80,85)(83,91)(84,94)(87,101)(88,102)(89,93)(90,96)(92,103)(95,104)(105,111)(106,109)(107,112)(108,110)(113,114,118,115,116,119,117), (1,3,21,8)(2,13,55,17)(4,22,37,30)(5,33,29,35)(6,24,36,41)(7,25,73,45)(9,23,62,43)(10,51,81,32)(11,26,74,48)(12,27,75,49)(14,59,42,61)(15,56,28,52)(16,57,97,66)(18,39,77,50)(19,31,82,69)(20,58,86,70)(34,83,65,85)(38,84,40,78)(44,80,60,91)(46,93,104,72)(47,53,96,92)(54,67,89,95)(63,94,64,99)(68,71,90,103)(76,87,98,101)(79,100)(88,102)(105,109,111,106)(107,112)(108,110)(113,115,117,118,119,114,116), (1,4,2,9)(3,22,13,23)(5,28,14,36)(6,29,15,42)(7,20,16,12)(8,30,17,43)(10,31,18,26)(11,32,19,50)(21,37,55,62)(24,35,56,61)(25,58,57,27)(33,52,59,41)(34,63,60,38)(39,74,51,82)(40,65,64,44)(45,70,66,49)(46,71,67,53)(47,72,68,54)(48,81,69,77)(73,86,97,75)(76,87,98,101)(78,85,99,80)(79,88,100,102)(83,94,91,84)(89,96,93,90)(92,104,103,95)(105,109,111,106)(107,110,112,108)(113,116,114,119,118,117,115), (113,117,119,116,115,118,114), (1,5,31,4,28,18,2,14,26,9,36,10)(3,23,13,22)(6,19,8,29,50,30,15,11,17,42,32,43)(7,34,71,20,63,67,16,60,53,12,38,46)(21,39,52,37,74,59,55,51,41,62,82,33)(24,76,88,35,87,100,56,98,102,61,101,79)(25,27,57,58)(40,68,45,65,54,70,64,47,66,44,72,49)(48,77,69,81)(73,89,94,86,96,91,97,93,84,75,90,83)(78,105,110,85,109,112,99,111,108,80,106,107)(92,95,103,104)(113,116,114,119,118,117,115), (1,6,39,62,14,19,2,15,51,37,5,11)(3,24,77,43,59,31,13,56,81,30,33,26)(4,29,74,21,36,50,9,42,82,55,28,32)(7,40,89,75,60,68,16,64,93,86,34,47)(8,41,18,23,61,69,17,52,10,22,35,48)(12,44,90,97,63,72,20,65,96,73,38,54)(25,78,95,49,91,71,57,99,104,70,83,53)(27,80,103,66,94,46,58,85,92,45,84,67)(76,101,98,87)(79,88,100,102)(105,106,111,109)(107,110,112,108), (1,7,4,20,2,16,9,12)(3,25,22,58,13,57,23,27)(5,34,28,63,14,60,36,38)(6,40,29,65,15,64,42,44)(8,45,30,70,17,66,43,49)(10,46,31,71,18,67,26,53)(11,47,32,72,19,68,50,54)(21,73,37,86,55,97,62,75)(24,78,35,85,56,99,61,80)(33,83,52,94,59,91,41,84)(39,89,74,96,51,93,82,90)(48,92,81,104,69,103,77,95)(76,105,87,109,98,111,101,106)(79,107,88,110,100,112,102,108)(113,118,116,117,114,115,119)) # Define the group as a matrix group with coefficients in GLFq: F = GF(169); al = gen(F); matrix_group([matrix(GF(169), [[al^132, 0], [0, al^132]]), matrix(GF(169), [[al^17, al^6], [al^64, al^6]]), matrix(GF(169), [[al^150, 0], [0, al^150]]), matrix(GF(169), [[al^120, 0], [0, al^120]]), matrix(GF(169), [[al^36, al^158], [al^72, al^12]]), matrix(GF(169), [[al^132, al^2], [al^60, al^89]]), matrix(GF(169), [[al^159, 0], [0, al^159]])]) # 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