Formats: - HTML - YAML - JSON - 2026-07-22T11:56:16.178746
  • gps_subgroup_search • Show schema
    {'Agroup': False, 'Zgroup': False, 'abelian': False, 'ambient': '139968.kc', 'ambient_counter': 263, 'ambient_order': 139968, 'ambient_tex': 'C_3^3.S_3\\wr C_2^2', 'central': False, 'central_factor': False, 'centralizer_order': 1, 'characteristic': True, 'core_order': 69984, 'counter': 5, 'cyclic': False, 'direct': False, 'hall': 0, 'label': '139968.kc.2.D', 'maximal': True, 'maximal_normal': True, 'metabelian': False, 'metacyclic': False, 'minimal': False, 'minimal_normal': False, 'nilpotent': False, 'normal': True, 'old_label': '2.d1', 'outer_equivalence': True, 'perfect': False, 'proper': True, 'quotient': '2.1', 'quotient_Agroup': True, 'quotient_abelian': True, 'quotient_cyclic': True, 'quotient_hash': 1, 'quotient_metabelian': True, 'quotient_nilpotent': True, 'quotient_order': 2, 'quotient_simple': True, 'quotient_solvable': True, 'quotient_supersolvable': True, 'quotient_tex': 'C_2', 'simple': False, 'solvable': True, 'special_labels': [], 'split': True, 'standard_generators': False, 'stem': False, 'subgroup': '69984.ix', 'subgroup_hash': 8107025147652671397, 'subgroup_order': 69984, 'subgroup_tex': '\\He_3^2:D_4:D_6', 'supersolvable': False, 'sylow': 0}
  • gps_subgroup_data • Show schema
    {'ambient': '139968.kc', 'aut_centralizer_order': None, 'aut_label': '2.D', 'aut_quo_index': None, 'aut_stab_index': None, 'aut_weyl_group': None, 'aut_weyl_index': None, 'centralizer': '139968.a1', 'complements': ['69984.D', '69984.E'], 'conjugacy_class_count': 1, 'contained_in': ['1.a1'], 'contains': ['4.B', '4.C', '4.D', '4.J', '4.K', '4.P', '4.Q', '4.R', '4.S', '6.D', '162.D'], 'core': '2.D', 'coset_action_label': None, 'count': 1, 'diagramx': [1558, 1229, 1978, 2441], 'generators': [20961536954955840139656736438147555838909, 376037022144416627773680, 6790235433014835680382490898302599603, 20666296051810669041754456827766345726080, 159854461832359344867486173895269161302357, 526261674673969987453451405246672644, 10873194945918990975816082533, 20666295941918940249388752344471771097744, 265919259904428298369518418529547779063184, 255293883785725491639816944492215251726963, 63824277360366462727540207146212601471956, 138273964646219699874365094639707866061040], 'label': '139968.kc.2.D', 'mobius_quo': 0, 'mobius_sub': None, 'normal_closure': '2.D', 'normal_contained_in': ['1.a1'], 'normal_contains': ['4.B', '4.C', '4.D'], 'normalizer': '1.a1', 'old_label': '2.d1', 'projective_image': '139968.kc', 'quotient_action_image': '2.1', 'quotient_action_kernel': '1.1', 'quotient_action_kernel_order': 1, 'quotient_fusion': None, 'short_label': '2.D', 'subgroup_fusion': None, 'weyl_group': '139968.kc'}
  • gps_groups • Show schema
    {'Agroup': False, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '16.14', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 4, 'aut_exponent': 12, 'aut_gen_orders': [4, 6, 12, 6, 12], 'aut_gens': [[159559492163853914885729822189521812324934, 127645592204040143640350209620394770311305, 329746270577411527948071813100072035917064, 371989804112883273849870024673570922222156], [244347734668580491647094941410870842920505, 20968318967558201908450059395980868226746, 63830541288960273116892042902275776708483, 297835638167171469783089172358534771267170], [106068625826196722919227494366907094946318, 20961791972846267246757500708725309232906, 63824548705498046353780786196060279424148, 42250307508696160263452473941608584236214], [52873255602826220088921208195756644111330, 276251889832293748441283840219447496061919, 191475836675048593769231493506444979196800, 233713394987869128926170566215353523530945], [371986544759618702711339441766091077304916, 276252144850174142276179721608332522475705, 74456188989328541572575620433554332770484, 287203989769454702918985083635094244069718], [351326767708608051099097785207322464894476, 302277942784191537743222942704846869386, 202104217431327692526135418466081817977668, 42244348474977750398615401859983906590690]], 'aut_group': None, 'aut_hash': 6395131201938535785, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 279936, 'aut_permdeg': 432, 'aut_perms': 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'aut_stats': [[1, 1, 1, 1], [2, 18, 1, 1], [2, 54, 2, 1], [2, 81, 1, 1], [2, 162, 1, 2], [2, 162, 2, 1], [2, 486, 2, 1], [3, 2, 1, 1], [3, 2, 2, 1], [3, 4, 1, 1], [3, 4, 2, 1], [3, 8, 1, 1], [3, 24, 1, 2], [3, 24, 2, 1], [3, 48, 1, 3], [3, 48, 2, 2], [3, 72, 1, 2], [3, 72, 2, 2], [3, 144, 1, 1], [3, 144, 2, 1], [3, 288, 1, 1], [3, 288, 2, 1], [4, 486, 2, 2], [4, 1458, 2, 1], [6, 36, 1, 3], [6, 36, 2, 1], [6, 72, 1, 2], [6, 72, 2, 1], [6, 108, 2, 2], [6, 162, 1, 1], [6, 162, 2, 2], [6, 216, 1, 2], [6, 216, 2, 2], [6, 324, 1, 2], [6, 324, 2, 2], [6, 432, 1, 3], [6, 432, 2, 2], [6, 648, 1, 5], [6, 648, 2, 5], [6, 648, 4, 1], [6, 1296, 1, 1], [6, 1296, 2, 1], [6, 1944, 2, 4], [12, 486, 2, 2], [12, 972, 2, 5], [12, 1944, 2, 1], [12, 2916, 2, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_3^2.C_3^5.C_2^3.C_2^4', 'autcent_abelian': True, 'autcent_cyclic': True, 'autcent_exponent': 1, 'autcent_group': '1.1', 'autcent_hash': 1, 'autcent_nilpotent': True, 'autcent_order': 1, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_1', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 12, 'autcentquo_group': None, 'autcentquo_hash': 6395131201938535785, 'autcentquo_nilpotent': False, 'autcentquo_order': 279936, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'C_3^2.C_3^5.C_2^4.C_2^3', 'cc_stats': [[1, 1, 1], [2, 18, 1], [2, 54, 2], [2, 81, 1], [2, 162, 4], [2, 486, 2], [3, 2, 3], [3, 4, 3], [3, 8, 1], [3, 24, 4], [3, 48, 7], [3, 72, 6], [3, 144, 3], [3, 288, 3], [4, 486, 4], [4, 1458, 2], [6, 36, 5], [6, 72, 4], [6, 108, 4], [6, 162, 5], [6, 216, 6], [6, 324, 6], [6, 432, 7], [6, 648, 19], [6, 1296, 3], [6, 1944, 8], [12, 486, 4], [12, 972, 10], [12, 1944, 2], [12, 2916, 2]], 'center_label': '1.1', 'center_order': 1, 'central_product': False, 'central_quotient': '69984.ix', 'commutator_count': 1, 'commutator_label': None, 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '2.1', '3.1', '3.1', '3.1', '3.1', '3.1', '3.1', '3.1'], 'composition_length': 12, 'conjugacy_classes_known': True, 'counter': 232, 'cyclic': False, 'derived_length': 4, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 18, 1, 1], [2, 54, 1, 2], [2, 81, 1, 1], [2, 162, 1, 4], [2, 486, 1, 2], [3, 2, 1, 3], [3, 4, 1, 3], [3, 8, 1, 1], [3, 24, 1, 2], [3, 24, 2, 1], [3, 48, 1, 5], [3, 48, 2, 1], [3, 72, 1, 4], [3, 72, 2, 1], [3, 144, 1, 3], [3, 288, 1, 1], [3, 288, 2, 1], [4, 486, 1, 4], [4, 1458, 1, 2], [6, 36, 1, 5], [6, 72, 1, 4], [6, 108, 1, 4], [6, 162, 1, 3], [6, 162, 2, 1], [6, 216, 1, 4], [6, 216, 2, 1], [6, 324, 1, 6], [6, 432, 1, 5], [6, 432, 2, 1], [6, 648, 1, 9], [6, 648, 2, 5], [6, 1296, 1, 3], [6, 1944, 1, 8], [12, 486, 2, 2], [12, 972, 1, 6], [12, 972, 2, 2], [12, 1944, 1, 2], [12, 2916, 1, 2]], 'element_repr_type': 'Perm', 'elementary': 1, 'eulerian_function': 25383742675200, 'exponent': 12, 'exponents_of_order': [7, 5], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [[24, 1, 4], [48, 0, 4], [48, 1, 4]], 'familial': False, 'frattini_label': '9.2', 'frattini_quotient': '7776.ck', 'hash': 8107025147652671397, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 12, 'inner_gen_orders': [6, 6, 12, 12], 'inner_gens': [[159559492163853914885729822189521812324934, 265623763983402346140622923040554768021695, 212138873180682883074272753335125892217520, 223675479811375937341899449545323801861596], [159555969724918150813307020228796428408770, 127645592204040143640350209620394770311305, 329746015494791384097863121874360614604827, 106074584805269753180431573561524734952243], [159561928556530787557943871596454499171174, 148312142944717273907683240541169412410092, 329746270577411527948071813100072035917064, 233716654241335533966100049232071924658957], [307874079778710719987736680706305382564484, 20968574076401678466383405376508544133241, 63831067752089953404855553212860717139124, 371989804112883273849870024673570922222156]], 'inner_hash': 8107025147652671397, 'inner_nilpotent': False, 'inner_order': 69984, 'inner_split': True, 'inner_tex': '\\He_3^2:D_4:D_6', 'inner_used': [1, 2, 3, 4], 'irrC_degree': 24, 'irrQ_degree': 24, 'irrQ_dim': 24, 'irrR_degree': 24, 'irrep_stats': [[1, 16], [2, 8], [4, 3], [8, 40], [12, 8], [16, 8], [18, 16], [24, 4], [32, 3], [36, 12], [48, 12], [72, 2]], 'label': '69984.ix', 'linC_count': None, 'linC_degree': None, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': None, 'linQ_degree_count': None, 'linQ_dim': None, 'linQ_dim_count': None, 'linR_count': None, 'linR_degree': None, 'maximal_subgroups_known': True, 'metabelian': False, 'metacyclic': False, 'monomial': None, 'name': 'He3^2:D4:D6', 'ngens': 4, 'nilpotency_class': -1, 'nilpotent': False, 'normal_counts': [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 83, 'number_characteristic_subgroups': 29, 'number_conjugacy_classes': 132, 'number_divisions': 116, 'number_normal_subgroups': 93, 'number_subgroup_autclasses': 4035, 'number_subgroup_classes': 7177, 'number_subgroups': 1654386, 'old_label': None, 'order': 69984, 'order_factorization_type': 33, 'order_stats': [[1, 1], [2, 1827], [3, 2186], [4, 4860], [6, 39726], [12, 21384]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': True, 'outer_exponent': 2, 'outer_gen_orders': [2, 2], 'outer_gen_pows': [84792588077604600569672429125677926031237, 0], 'outer_gens': [[233719353829269552291095717724177398661363, 20968047613874225196259411562984070149611, 319114630402522912795470958294619806229760, 297832642035709898903378754852693240462597], [297836164446528054386689437644635662662277, 276252416286268389712366634757462741493949, 191469606388404319621997339831185243737600, 361364937792490163531000338809190931439095]], 'outer_group': '4.2', 'outer_hash': 2, 'outer_nilpotent': True, 'outer_order': 4, 'outer_permdeg': 4, 'outer_perms': [1, 6], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^2', 'pc_rank': 9, 'perfect': False, 'permutation_degree': 36, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 2, 2], 'quasisimple': False, 'rank': 4, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 16], [2, 8], [4, 1], [8, 25], [12, 4], [16, 16], [18, 16], [24, 6], [32, 1], [36, 8], [48, 8], [64, 1], [72, 4], [96, 2]], 'representations': {'PC': {'code': '78100384233339769927388370212529695128170885864582421739304315435526527935368461294901036546110519693081610413472528495857850977462360161547086179364426579905314758221117135272062004102664325678477585504812730261484794047', 'gens': [1, 2, 3, 6, 8, 9, 10, 11, 12], 'pres': [12, 2, 2, 2, 2, 3, 2, 3, 3, 3, 3, 3, 3, 174144, 1493425, 728365, 367346, 389606, 98, 1032579, 196815, 135, 976, 660101, 23345, 797501, 575033, 209, 2024070, 1403154, 1038270, 8106, 13855, 1219, 3359240, 2664596, 847616, 31148, 57092, 27620, 3156489, 2350101, 80673, 777645, 6348682, 190102, 1615714, 651070, 106990, 42850, 2890, 6801419, 850211, 140039]}, 'Perm': {'d': 36, 'gens': [329746270577411527948071813100072035917064, 127645592204040143640350209620394770311305, 371989804112883273849870024673570922222156, 159559492163853914885729822189521812324934]}}, 'schur_multiplier': [2, 2, 2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 2, 2], 'solvability_type': 17, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': '\\He_3^2:D_4:D_6', 'transitive_degree': 36, 'wreath_data': None, 'wreath_product': False}
  • gps_groups • Show schema
    {'Agroup': False, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '8.5', 'all_subgroups_known': False, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 4, 'aut_exponent': 12, 'aut_gen_orders': [12, 4, 6, 12, 6], 'aut_gens': [[116695631750038057511116884221643963965492, 191770534870542345857136736256578962800764, 201804384095843383256611277043493226921996], [170186243334443323617268813228622734584695, 74160684745051814657944626752873306963034, 329446419817020139187192252255250190216894], [170185971871179184993213347672985053804235, 64122769669266808987721392709083779298828, 276256779173841792851964553947498384686904], [297834526583997580324021827493284581376575, 212431398159488177339659375710443349171244, 148608742536480815843098080941981551814974], [170191675904686917519081396210568638727082, 191770806224213288350335240748053732270840, 74162314450084787643944823603030858580109], [170191675969721101698352146844271615462546, 84789065519621245976838185276723453719708, 148611195311988200378246497186054924816078]], 'aut_group': None, 'aut_hash': 6395131201938535785, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 279936, 'aut_permdeg': 468, 'aut_perms': 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'aut_phi_ratio': 6.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 18, 1, 1], [2, 36, 1, 1], [2, 81, 1, 1], [2, 108, 1, 1], [2, 162, 1, 2], [2, 324, 1, 2], [2, 972, 1, 1], [3, 2, 1, 1], [3, 4, 1, 2], [3, 8, 1, 2], [3, 24, 1, 2], [3, 24, 2, 1], [3, 48, 1, 3], [3, 48, 2, 1], [3, 72, 1, 2], [3, 72, 2, 1], [3, 96, 1, 1], [3, 144, 1, 2], [3, 288, 1, 2], [3, 288, 2, 1], [4, 972, 1, 2], [4, 1944, 1, 2], [4, 2916, 1, 1], [4, 5832, 1, 1], [6, 36, 1, 3], [6, 36, 2, 1], [6, 72, 1, 5], [6, 72, 2, 2], [6, 144, 1, 2], [6, 144, 2, 1], [6, 162, 1, 1], [6, 162, 2, 1], [6, 216, 1, 6], [6, 216, 2, 3], [6, 324, 1, 3], [6, 324, 2, 1], [6, 432, 1, 7], [6, 432, 2, 4], [6, 648, 1, 9], [6, 648, 2, 5], [6, 864, 1, 2], [6, 864, 2, 1], [6, 1296, 1, 4], [6, 1296, 2, 2], [6, 2592, 1, 1], [6, 3888, 1, 4], [12, 972, 1, 2], [12, 1944, 1, 7], [12, 1944, 2, 1], [12, 3888, 1, 5], [12, 3888, 2, 1], [12, 5832, 1, 1], [12, 11664, 1, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_3^2.C_3^5.C_2^3.C_2^4', 'autcent_abelian': True, 'autcent_cyclic': True, 'autcent_exponent': 1, 'autcent_group': '1.1', 'autcent_hash': 1, 'autcent_nilpotent': True, 'autcent_order': 1, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_1', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 12, 'autcentquo_group': None, 'autcentquo_hash': 6395131201938535785, 'autcentquo_nilpotent': False, 'autcentquo_order': 279936, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'C_3^2.C_3^5.C_2^3.C_2^4', 'cc_stats': [[1, 1, 1], [2, 18, 1], [2, 36, 1], [2, 81, 1], [2, 108, 1], [2, 162, 2], [2, 324, 2], [2, 972, 1], [3, 2, 1], [3, 4, 2], [3, 8, 2], [3, 24, 4], [3, 48, 5], [3, 72, 4], [3, 96, 1], [3, 144, 2], [3, 288, 4], [4, 972, 2], [4, 1944, 2], [4, 2916, 1], [4, 5832, 1], [6, 36, 5], [6, 72, 9], [6, 144, 4], [6, 162, 3], [6, 216, 12], [6, 324, 5], [6, 432, 15], [6, 648, 19], [6, 864, 4], [6, 1296, 8], [6, 2592, 1], [6, 3888, 4], [12, 972, 2], [12, 1944, 9], [12, 3888, 7], [12, 5832, 1], [12, 11664, 1]], 'center_label': '1.1', 'center_order': 1, 'central_product': False, 'central_quotient': '139968.kc', 'commutator_count': 1, 'commutator_label': None, 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '2.1', '2.1', '3.1', '3.1', '3.1', '3.1', '3.1', '3.1', '3.1'], 'composition_length': 13, 'conjugacy_classes_known': True, 'counter': 263, 'cyclic': False, 'derived_length': 4, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 18, 1, 1], [2, 36, 1, 1], [2, 81, 1, 1], [2, 108, 1, 1], [2, 162, 1, 2], [2, 324, 1, 2], [2, 972, 1, 1], [3, 2, 1, 1], [3, 4, 1, 2], [3, 8, 1, 2], [3, 24, 1, 2], [3, 24, 2, 1], [3, 48, 1, 3], [3, 48, 2, 1], [3, 72, 1, 2], [3, 72, 2, 1], [3, 96, 1, 1], [3, 144, 1, 2], [3, 288, 1, 2], [3, 288, 2, 1], [4, 972, 1, 2], [4, 1944, 1, 2], [4, 2916, 1, 1], [4, 5832, 1, 1], [6, 36, 1, 3], [6, 36, 2, 1], [6, 72, 1, 5], [6, 72, 2, 2], [6, 144, 1, 2], [6, 144, 2, 1], [6, 162, 1, 1], [6, 162, 2, 1], [6, 216, 1, 6], [6, 216, 2, 3], [6, 324, 1, 3], [6, 324, 2, 1], [6, 432, 1, 7], [6, 432, 2, 4], [6, 648, 1, 9], [6, 648, 2, 5], [6, 864, 1, 2], [6, 864, 2, 1], [6, 1296, 1, 4], [6, 1296, 2, 2], [6, 2592, 1, 1], [6, 3888, 1, 4], [12, 972, 2, 1], [12, 1944, 1, 3], [12, 1944, 2, 3], [12, 3888, 1, 3], [12, 3888, 2, 2], [12, 5832, 1, 1], [12, 11664, 1, 1]], 'element_repr_type': 'Perm', 'elementary': 1, 'eulerian_function': None, 'exponent': 12, 'exponents_of_order': [7, 6], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [[24, 0, 4], [24, 1, 4], [48, 0, 8], [72, 1, 4], [96, 1, 2], [144, 1, 1]], 'familial': False, 'frattini_label': '9.2', 'frattini_quotient': '15552.bt', 'hash': 8323456571666149168, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 12, 'inner_gen_orders': [12, 4, 12], 'inner_gens': [[116695631750038057511116884221643963965492, 64119247193705549109762035632842078588748, 148611450155252570138118343695027176287434], [371991162354039041499876262664633802773732, 191770534870542345857136736256578962800764, 64118440550615145320883781350770123934989], [307869182205473451573977511817084635401011, 329451309150031697567931547363966595512587, 201804384095843383256611277043493226921996]], 'inner_hash': 8323456571666149168, 'inner_nilpotent': False, 'inner_order': 139968, 'inner_split': True, 'inner_tex': 'C_3^3.S_3\\wr C_2^2', 'inner_used': [1, 2, 3], 'irrC_degree': 24, 'irrQ_degree': 24, 'irrQ_dim': 24, 'irrR_degree': 24, 'irrep_stats': [[1, 8], [2, 14], [4, 8], [8, 44], [12, 8], [16, 17], [24, 10], [32, 8], [36, 8], [48, 16], [72, 6], [96, 2], [144, 1]], 'label': '139968.kc', 'linC_count': None, 'linC_degree': None, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': None, 'linQ_degree_count': None, 'linQ_dim': None, 'linQ_dim_count': None, 'linR_count': None, 'linR_degree': None, 'maximal_subgroups_known': True, 'metabelian': False, 'metacyclic': False, 'monomial': None, 'name': 'C3^3.S3wrC2^2', 'ngens': 3, 'nilpotency_class': -1, 'nilpotent': False, 'normal_counts': [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 123, 'number_characteristic_subgroups': 43, 'number_conjugacy_classes': 150, 'number_divisions': 119, 'number_normal_subgroups': 43, 'number_subgroup_autclasses': 7459, 'number_subgroup_classes': 7963, 'number_subgroups': 2575292, 'old_label': None, 'order': 139968, 'order_factorization_type': 33, 'order_stats': [[1, 1], [2, 2187], [3, 2186], [4, 14580], [6, 56862], [12, 64152]], 'outer_abelian': True, 'outer_cyclic': True, 'outer_equivalence': True, 'outer_exponent': 2, 'outer_gen_orders': [2], 'outer_gen_pows': [0], 'outer_gens': [[116695631750038057511116884221643963965492, 212437357056452017209923141991442497399624, 191766723827348052817651547303865696691195]], 'outer_group': '2.1', 'outer_hash': 1, 'outer_nilpotent': True, 'outer_order': 2, 'outer_permdeg': 2, 'outer_perms': [1], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2', 'pc_rank': 9, 'perfect': False, 'permutation_degree': 36, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 2], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 8], [2, 10], [4, 6], [8, 22], [12, 4], [16, 23], [24, 8], [32, 7], [36, 8], [48, 8], [64, 2], [72, 4], [96, 7], [144, 2]], 'representations': {'PC': {'code': '238638103437265499469713791232840443710516421857702007892443244806384261443913216132738104175327008313195101108964511437748136641265200137734497447670949636334028420128118769828957204226930824567891344976718149443551881738860528385744843294906965268813035510195904617959656649918699646976462570022103', 'gens': [1, 2, 4, 5, 7, 8, 10, 12, 13], 'pres': [13, 2, 2, 3, 2, 2, 3, 3, 2, 3, 2, 3, 3, 3, 911508, 1095693, 66, 1458602, 5323971, 101104, 4553, 865024, 3760007, 1266360, 517053, 186, 808709, 2426130, 2540, 1415238, 1415251, 1150, 5290279, 4753652, 18753, 827470, 99899, 1320, 306, 134792, 325749, 29518, 471791, 85704, 2881, 16813689, 270682, 2611475, 70261, 41440, 12203, 386, 48058, 3407999, 1297332, 61838, 13829, 6120, 44939, 22488, 11334, 2000, 16086108, 16249, 876160, 8241, 2677]}, 'Perm': {'d': 36, 'gens': [201804384095843383256611277043493226921996, 191770534870542345857136736256578962800764, 116695631750038057511116884221643963965492]}}, 'schur_multiplier': [2, 2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 2], 'solvability_type': 17, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 192, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': 'C_3^3.S_3\\wr C_2^2', 'transitive_degree': 36, 'wreath_data': None, 'wreath_product': False}
  • gps_groups • Show schema
    {'Agroup': True, 'Zgroup': True, 'abelian': True, 'abelian_quotient': '2.1', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': True, 'aut_cyclic': True, 'aut_derived_length': 0, 'aut_exponent': 1, 'aut_gen_orders': [], 'aut_gens': [[1]], 'aut_group': '1.1', 'aut_hash': 1, 'aut_nilpotency_class': 0, 'aut_nilpotent': True, 'aut_order': 1, 'aut_permdeg': 1, 'aut_perms': [], 'aut_phi_ratio': 1.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_1', 'autcent_abelian': True, 'autcent_cyclic': True, 'autcent_exponent': 1, 'autcent_group': '1.1', 'autcent_hash': 1, 'autcent_nilpotent': True, 'autcent_order': 1, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_1', 'autcentquo_abelian': True, 'autcentquo_cyclic': True, 'autcentquo_exponent': 1, 'autcentquo_group': '1.1', 'autcentquo_hash': 1, 'autcentquo_nilpotent': True, 'autcentquo_order': 1, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'C_1', 'cc_stats': [[1, 1, 1], [2, 1, 1]], 'center_label': '2.1', 'center_order': 2, 'central_product': False, 'central_quotient': '1.1', 'commutator_count': 0, 'commutator_label': '1.1', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1'], 'composition_length': 1, 'conjugacy_classes_known': True, 'counter': 1, 'cyclic': True, 'derived_length': 1, 'dihedral': True, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1]], 'element_repr_type': 'PC', 'elementary': 2, 'eulerian_function': 1, 'exponent': 2, 'exponents_of_order': [1], 'factors_of_aut_order': [], 'factors_of_order': [2], 'faithful_reps': [[1, 1, 1]], 'familial': True, 'frattini_label': '1.1', 'frattini_quotient': '2.1', 'hash': 1, 'hyperelementary': 2, 'inner_abelian': True, 'inner_cyclic': True, 'inner_exponent': 1, 'inner_gen_orders': [1], 'inner_gens': [[1]], 'inner_hash': 1, 'inner_nilpotent': True, 'inner_order': 1, 'inner_split': True, 'inner_tex': 'C_1', 'inner_used': [], 'irrC_degree': 1, 'irrQ_degree': 1, 'irrQ_dim': 1, 'irrR_degree': 1, 'irrep_stats': [[1, 2]], 'label': '2.1', 'linC_count': 1, 'linC_degree': 1, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 1, 'linQ_degree_count': 1, 'linQ_dim': 1, 'linQ_dim_count': 1, 'linR_count': 1, 'linR_degree': 1, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C2', 'ngens': 1, 'nilpotency_class': 1, 'nilpotent': True, 'normal_counts': [0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 2, 'number_characteristic_subgroups': 2, 'number_conjugacy_classes': 2, 'number_divisions': 2, 'number_normal_subgroups': 2, 'number_subgroup_autclasses': 2, 'number_subgroup_classes': 2, 'number_subgroups': 2, 'old_label': None, 'order': 2, 'order_factorization_type': 1, 'order_stats': [[1, 1], [2, 1]], 'outer_abelian': True, 'outer_cyclic': True, 'outer_equivalence': False, 'outer_exponent': 1, 'outer_gen_orders': [], 'outer_gen_pows': [], 'outer_gens': [], 'outer_group': '1.1', 'outer_hash': 1, 'outer_nilpotent': True, 'outer_order': 1, 'outer_permdeg': 1, 'outer_perms': [], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_1', 'pc_rank': 1, 'perfect': False, 'permutation_degree': 2, 'pgroup': 2, 'primary_abelian_invariants': [2], 'quasisimple': False, 'rank': 1, 'rational': True, 'rational_characters_known': True, 'ratrep_stats': [[1, 2]], 'representations': {'PC': {'code': 0, 'gens': [1], 'pres': [1, -2]}, 'GLZ': {'b': 3, 'd': 2, 'gens': [12]}, 'Lie': [{'d': 2, 'q': 2, 'gens': [6], 'family': 'CSOPlus'}, {'d': 2, 'q': 2, 'gens': [6], 'family': 'COPlus'}, {'d': 1, 'q': 2, 'gens': [1], 'family': 'AGL'}, {'d': 1, 'q': 2, 'gens': [1], 'family': 'AGammaL'}, {'d': 1, 'q': 2, 'gens': [1], 'family': 'ASigmaL'}], 'GLFp': {'d': 2, 'p': 2, 'gens': [11]}, 'Perm': {'d': 2, 'gens': [1]}}, 'schur_multiplier': [], 'semidirect_product': False, 'simple': True, 'smith_abelian_invariants': [2], 'solvability_type': 0, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_2', 'transitive_degree': 2, 'wreath_data': None, 'wreath_product': False}