Formats: - HTML - YAML - JSON - 2026-07-19T17:50:58.047826
  • gps_subgroup_search • Show schema
    {'Agroup': False, 'Zgroup': False, 'abelian': False, 'ambient': '472392.sf', 'ambient_counter': 474, 'ambient_order': 472392, 'ambient_tex': 'C_3^6.C_3:S_3^3', 'central': False, 'central_factor': False, 'centralizer_order': None, 'characteristic': True, 'core_order': 118098, 'counter': 8, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '472392.sf.4.C', 'maximal': False, 'maximal_normal': False, 'metabelian': False, 'metacyclic': False, 'minimal': False, 'minimal_normal': False, 'nilpotent': False, 'normal': True, 'old_label': '4.c1', 'outer_equivalence': True, 'perfect': False, 'proper': True, 'quotient': '4.2', 'quotient_Agroup': True, 'quotient_abelian': True, 'quotient_cyclic': False, 'quotient_hash': 2, 'quotient_metabelian': True, 'quotient_nilpotent': True, 'quotient_order': 4, 'quotient_simple': False, 'quotient_solvable': True, 'quotient_supersolvable': True, 'quotient_tex': 'C_2^2', 'simple': False, 'solvable': True, 'special_labels': ['C2'], 'split': None, 'standard_generators': False, 'stem': False, 'subgroup': None, 'subgroup_hash': 1829385611185224045, 'subgroup_order': 118098, 'subgroup_tex': 'C_3^6.C_3^4.C_2', 'supersolvable': True, 'sylow': 0}
  • gps_subgroup_data • Show schema
    {'ambient': '472392.sf', 'aut_centralizer_order': None, 'aut_label': '4.C', 'aut_quo_index': None, 'aut_stab_index': None, 'aut_weyl_group': None, 'aut_weyl_index': None, 'centralizer': None, 'complements': None, 'conjugacy_class_count': 1, 'contained_in': None, 'contains': None, 'core': '4.C', 'coset_action_label': None, 'count': 1, 'diagramx': [435, 5315, 523, 2789], 'generators': [276258111896096448822551957946161862781586, 20666295932772708662009338415858721453363, 3267878609843046845879845641849873118, 806582922253211364573312004, 5048669733912579154474904274, 127943795779862844149747235166655023343734, 9147456921273669403019911296244, 10628652136785279783826728729786122960787, 271353676006674129020684387317943043, 376703623296147, 6527104604082525820172220361602581495], 'label': '472392.sf.4.C', 'mobius_quo': None, 'mobius_sub': None, 'normal_closure': '4.C', 'normal_contained_in': [], 'normal_contains': [], 'normalizer': '1.a1', 'old_label': '4.c1', 'projective_image': '472392.sf', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '4.C', 'subgroup_fusion': None, 'weyl_group': '472392.sf'}
  • label None does not appear in gps_groups
  • 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': 36, 'aut_gen_orders': [12, 6, 6, 18], 'aut_gens': [[265629476404889293750750079893211535268082, 223385687824052823967662802977436189585795, 32212137341140166820913317809733406671839], [138276409307844554692055364151157378249996, 74160429919772517170722534191297887306161, 169892631907398873263192580384944802964531], [148314850681919339840870094180040861038691, 371989804104338850664506344387859795336934, 42244874855388883161344143670106713373934], [138280195015591360184585964031794713239705, 361361423338917312552576617875487106194372, 297832370534877483739648175236488567048814], [265623764184545688464968550402550357816964, 191765102300299124331311975698466648473413, 307867289562219288542317821164899841190213]], 'aut_group': None, 'aut_hash': 562857025711197860, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 306110016, 'aut_permdeg': 972, 'aut_perms': 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'aut_phi_ratio': 1944.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 243, 4, 1], [2, 729, 2, 1], [2, 6561, 1, 1], [3, 2, 4, 1], [3, 4, 2, 1], [3, 4, 4, 1], [3, 8, 1, 2], [3, 8, 4, 1], [3, 36, 1, 1], [3, 72, 2, 2], [3, 72, 27, 1], [3, 162, 2, 1], [3, 324, 12, 1], [3, 648, 6, 1], [3, 648, 12, 1], [6, 486, 8, 1], [6, 972, 4, 1], [6, 1458, 4, 1], [6, 2916, 2, 1], [6, 4374, 4, 1], [6, 8748, 12, 1], [6, 13122, 2, 1], [6, 26244, 1, 1], [9, 36, 18, 1], [9, 72, 18, 1], [9, 162, 12, 1], [9, 324, 12, 1], [9, 648, 6, 1], [9, 648, 9, 1], [9, 648, 36, 1], [18, 1458, 12, 1], [18, 2916, 12, 1], [18, 2916, 18, 1], [18, 4374, 24, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_3^6.C_3^4.C_3^3.C_2^3.C_6.C_2^2', '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': 36, 'autcentquo_group': None, 'autcentquo_hash': 562857025711197860, 'autcentquo_nilpotent': False, 'autcentquo_order': 306110016, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'C_3^6.C_3^4.C_3^3.C_2^3.C_6.C_2^2', 'cc_stats': [[1, 1, 1], [2, 243, 4], [2, 729, 2], [2, 6561, 1], [3, 2, 4], [3, 4, 6], [3, 8, 6], [3, 36, 1], [3, 72, 31], [3, 162, 2], [3, 324, 12], [3, 648, 18], [6, 486, 8], [6, 972, 4], [6, 1458, 4], [6, 2916, 2], [6, 4374, 4], [6, 8748, 12], [6, 13122, 2], [6, 26244, 1], [9, 36, 18], [9, 72, 18], [9, 162, 12], [9, 324, 12], [9, 648, 51], [18, 1458, 12], [18, 2916, 30], [18, 4374, 24]], 'center_label': '1.1', 'center_order': 1, 'central_product': False, 'central_quotient': '472392.sf', 'commutator_count': 1, 'commutator_label': None, 'complements_known': False, 'complete': False, 'complex_characters_known': False, 'composition_factors': ['2.1', '2.1', '2.1', '3.1', '3.1', '3.1', '3.1', '3.1', '3.1', '3.1', '3.1', '3.1', '3.1'], 'composition_length': 13, 'conjugacy_classes_known': True, 'counter': 474, 'cyclic': False, 'derived_length': 3, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 243, 1, 4], [2, 729, 1, 2], [2, 6561, 1, 1], [3, 2, 1, 4], [3, 4, 1, 6], [3, 8, 1, 6], [3, 36, 1, 1], [3, 72, 1, 29], [3, 72, 2, 1], [3, 162, 1, 2], [3, 324, 1, 12], [3, 648, 1, 12], [3, 648, 2, 3], [6, 486, 1, 8], [6, 972, 1, 4], [6, 1458, 1, 4], [6, 2916, 1, 2], [6, 4374, 1, 4], [6, 8748, 1, 12], [6, 13122, 1, 2], [6, 26244, 1, 1], [9, 36, 3, 6], [9, 72, 3, 6], [9, 162, 3, 4], [9, 324, 3, 4], [9, 648, 3, 17], [18, 1458, 3, 4], [18, 2916, 3, 10], [18, 4374, 3, 8]], 'element_repr_type': 'Perm', 'elementary': 1, 'eulerian_function': None, 'exponent': 18, 'exponents_of_order': [10, 3], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [[24, 1, 27], [72, 0, 6], [72, 1, 9]], 'familial': False, 'frattini_label': '729.504', 'frattini_quotient': '648.734', 'hash': 5139320985986329917, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 18, 'inner_gen_orders': [6, 6, 18], 'inner_gens': [[265629476404889293750750079893211535268082, 116702447449739774900043559872456137825253, 32209141044834921673710449266762264487755], [20961792100114954634701831451566038416067, 223385687824052823967662802977436189585795, 32206959577846143961789124658770360156638], [10339938403282150767854327583875097567506, 244049530972772032164881850292635715079634, 32212137341140166820913317809733406671839]], 'inner_hash': 5139320985986329917, 'inner_nilpotent': False, 'inner_order': 472392, 'inner_split': True, 'inner_tex': 'C_3^6.C_3:S_3^3', 'inner_used': [1, 2, 3], 'irrC_degree': 24, 'irrQ_degree': 24, 'irrQ_dim': None, 'irrR_degree': None, 'irrep_stats': [[1, 8], [2, 16], [4, 12], [8, 6], [12, 38], [18, 56], [24, 49], [36, 48], [72, 69]], 'label': '472392.sf', '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': True, 'name': 'C3^6.C3:S3^3', '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], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 37, 'number_characteristic_subgroups': 7, 'number_conjugacy_classes': 302, 'number_divisions': 180, 'number_normal_subgroups': 71, 'number_subgroup_autclasses': None, 'number_subgroup_classes': None, 'number_subgroups': None, 'old_label': None, 'order': 472392, 'order_factorization_type': 33, 'order_stats': [[1, 1], [2, 8991], [3, 18224], [6, 194400], [9, 40824], [18, 209952]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': True, 'outer_exponent': 12, 'outer_gen_orders': [6, 6, 6, 3], 'outer_gen_pows': [255586120161191535786048370284879769982645, 0, 180521012791564419921454881260326623698612, 307870812120197594495952748925749406972038], 'outer_gens': [[10333411215965780989315422511232652257275, 244050057234452609809474267980241442605895, 287495963114178478136402576128506327210719], [276258112005982353411655283864245640167533, 95737659373204345730828088556706886053763, 52878959628105724207586262100235865438267], [302023034785912320616840445657660808836, 202100694928599929454254616697252307059440, 180228495825594544773933807326654200018589], [138274227859781454821912362905606444761994, 371989804104343480477466090902374150866612, 32212392350194720101197391306146641831944]], 'outer_group': '648.719', 'outer_hash': 719, 'outer_nilpotent': False, 'outer_order': 648, 'outer_permdeg': 12, 'outer_perms': [373065231, 95016129, 52819708, 127122], 'outer_solvable': True, 'outer_supersolvable': False, 'outer_tex': 'C_3\\wr D_4', 'pc_rank': None, 'perfect': False, 'permutation_degree': 36, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 2], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': False, 'ratrep_stats': [[1, 8], [2, 16], [4, 12], [8, 6], [12, 2], [18, 8], [24, 29], [36, 36], [48, 1], [54, 16], [72, 18], [108, 8], [144, 3], [216, 17]], 'representations': {'PC': {'code': '3231945197257199630207330623762470489827937638329729969029650674242758772892785434567919057766696211843061281933704124225558649107476926108881063991769214010122004218517906457988078696219190317916218879314074405308476157337757440531466651843063988371327703998013810118524371484822311414742465843211786305033956261048575487', 'gens': [1, 2, 4, 6, 8, 10, 11, 12, 13], 'pres': [13, 2, 2, 3, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 384696, 385893, 66, 16224938, 16914771, 3781144, 33881, 146, 23534164, 12448037, 2218740, 568676, 44933, 3788010, 1981075, 660860, 330231, 304, 39318, 196579, 3321, 19184263, 13601972, 10024593, 3452014, 813443, 354816, 410, 454904, 37955, 3088809, 21481222, 1397015, 552288, 55051, 4780, 5003866, 31274123, 14287115, 23368200, 10993357, 2100434, 367911, 644512, 84342, 65451684, 15149185, 6077954, 772719, 1125604, 287546, 100489]}, 'Perm': {'d': 36, 'gens': [265629476404889293750750079893211535268082, 32212137341140166820913317809733406671839, 223385687824052823967662802977436189585795]}}, 'schur_multiplier': [2, 2, 2], 'semidirect_product': None, 'simple': False, 'smith_abelian_invariants': [2, 2, 2], 'solvability_type': 10, 'solvable': True, 'subgroup_inclusions_known': False, 'subgroup_index_bound': 54, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_3^6.C_3:S_3^3', 'transitive_degree': 36, 'wreath_data': None, 'wreath_product': False}
  • gps_groups • Show schema
    {'Agroup': True, 'Zgroup': False, 'abelian': True, 'abelian_quotient': '4.2', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 6, 'aut_gen_orders': [2, 3], 'aut_gens': [[1, 2], [3, 2], [2, 3]], 'aut_group': '6.1', 'aut_hash': 1, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 6, 'aut_permdeg': 3, 'aut_perms': [1, 4], 'aut_phi_ratio': 3.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 3, 1]], 'aut_supersolvable': True, 'aut_tex': 'S_3', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 6, 'autcent_group': '6.1', 'autcent_hash': 1, 'autcent_nilpotent': False, 'autcent_order': 6, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'S_3', '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, 3]], 'center_label': '4.2', 'center_order': 4, 'central_product': True, 'central_quotient': '1.1', 'commutator_count': 0, 'commutator_label': '1.1', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1'], 'composition_length': 2, 'conjugacy_classes_known': True, 'counter': 2, 'cyclic': False, 'derived_length': 1, 'dihedral': True, 'direct_factorization': [['2.1', 2]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3]], 'element_repr_type': 'PC', 'elementary': 2, 'eulerian_function': 1, 'exponent': 2, 'exponents_of_order': [2], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2], 'faithful_reps': [], 'familial': True, 'frattini_label': '1.1', 'frattini_quotient': '4.2', 'hash': 2, 'hyperelementary': 2, 'inner_abelian': True, 'inner_cyclic': True, 'inner_exponent': 1, 'inner_gen_orders': [1, 1], 'inner_gens': [[1, 2], [1, 2]], '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, 4]], 'label': '4.2', 'linC_count': 3, 'linC_degree': 2, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 2, 'linQ_degree_count': 3, 'linQ_dim': 2, 'linQ_dim_count': 3, 'linR_count': 3, 'linR_degree': 2, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C2^2', 'ngens': 2, 'nilpotency_class': 1, 'nilpotent': True, 'normal_counts': [0, 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': 4, 'number_divisions': 4, 'number_normal_subgroups': 5, 'number_subgroup_autclasses': 3, 'number_subgroup_classes': 5, 'number_subgroups': 5, 'old_label': None, 'order': 4, 'order_factorization_type': 2, 'order_stats': [[1, 1], [2, 3]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 6, 'outer_gen_orders': [2, 3], 'outer_gen_pows': [0, 0], 'outer_gens': [[3, 2], [2, 3]], 'outer_group': '6.1', 'outer_hash': 1, 'outer_nilpotent': False, 'outer_order': 6, 'outer_permdeg': 3, 'outer_perms': [1, 4], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'S_3', 'pc_rank': 2, 'perfect': False, 'permutation_degree': 4, 'pgroup': 2, 'primary_abelian_invariants': [2, 2], 'quasisimple': False, 'rank': 2, 'rational': True, 'rational_characters_known': True, 'ratrep_stats': [[1, 4]], 'representations': {'PC': {'code': 0, 'gens': [1, 2], 'pres': [2, -2, 2]}, 'GLZ': {'b': 3, 'd': 2, 'gens': [14, 12]}, 'GLFp': {'d': 2, 'p': 3, 'gens': [55, 56]}, 'Perm': {'d': 4, 'gens': [6, 1]}}, 'schur_multiplier': [2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2], 'solvability_type': 1, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_2^2', 'transitive_degree': 4, 'wreath_data': None, 'wreath_product': False}