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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': False, 'ambient': '1185408.a', 'ambient_counter': 1, 'ambient_order': 1185408, 'ambient_tex': 'D_7^3.C_6^2:D_6', 'central': False, 'central_factor': False, 'centralizer_order': None, 'characteristic': True, 'core_order': 65856, 'counter': 341, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '1185408.a.18._.A', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': False, 'minimal': False, 'minimal_normal': False, 'nilpotent': False, 'normal': True, 'old_label': '18.A', 'outer_equivalence': False, 'perfect': False, 'proper': True, 'quotient': '18.4', 'quotient_Agroup': True, 'quotient_abelian': False, 'quotient_cyclic': False, 'quotient_hash': None, 'quotient_metabelian': True, 'quotient_nilpotent': False, 'quotient_order': 18, 'quotient_simple': False, 'quotient_solvable': True, 'quotient_supersolvable': True, 'quotient_tex': 'C_3:S_3', 'simple': False, 'solvable': True, 'special_labels': [], 'split': None, 'standard_generators': False, 'stem': False, 'subgroup': None, 'subgroup_hash': None, 'subgroup_order': 65856, 'subgroup_tex': 'C_2^3\\times C_7^3:(C_2^2\\times C_6)', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '1185408.a', 'aut_centralizer_order': None, 'aut_label': None, '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': None, 'coset_action_label': None, 'count': 1, 'diagramx': None, 'generators': [858816, 533088, 42336, 592704, 795384, 375024, 6, 12, 492912, 169344], 'label': '1185408.a.18._.A', 'mobius_quo': None, 'mobius_sub': None, 'normal_closure': None, 'normal_contained_in': None, 'normal_contains': None, 'normalizer': None, 'old_label': '18.A', 'projective_image': None, 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '18._.A', 'subgroup_fusion': None, 'weyl_group': None}
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label None does not appear in gps_groups
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gps_groups • Show schema
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{'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': 252, 'aut_gen_orders': [12, 42, 18, 6, 42, 12], 'aut_gens': [[1, 2, 12, 24, 144, 6048, 84672], [656209, 764698, 592716, 723636, 884592, 501984, 145152], [110173, 246146, 12, 174552, 919440, 187488, 931392], [425933, 232814, 592716, 841992, 191640, 554700, 5196], [1117625, 547358, 592716, 834216, 591672, 1063596, 2604], [94245, 1026014, 592704, 587184, 907368, 845856, 145164], [892221, 755566, 12, 36324, 450996, 902028, 596172]], 'aut_group': None, 'aut_hash': 5207978107673879692, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 14224896, 'aut_permdeg': 588, 'aut_perms': 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'aut_phi_ratio': 42.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 3, 1, 2], [2, 21, 4, 1], [2, 42, 2, 1], [2, 147, 2, 2], [2, 252, 2, 1], [2, 294, 1, 2], [2, 343, 2, 1], [2, 1029, 2, 1], [2, 1764, 2, 1], [3, 294, 1, 1], [3, 343, 1, 2], [3, 4704, 3, 1], [4, 252, 2, 1], [4, 1764, 2, 1], [4, 1764, 4, 1], [4, 12348, 4, 1], [6, 294, 1, 7], [6, 294, 2, 4], [6, 294, 4, 4], [6, 343, 1, 2], [6, 343, 2, 2], [6, 1029, 1, 4], [6, 1029, 2, 6], [6, 1029, 4, 2], [6, 2058, 1, 4], [6, 2058, 2, 14], [6, 2058, 4, 2], [6, 4704, 3, 1], [6, 12348, 2, 4], [6, 32928, 6, 1], [7, 18, 1, 1], [7, 72, 3, 1], [7, 108, 1, 1], [12, 12348, 2, 4], [12, 12348, 4, 4], [14, 18, 1, 3], [14, 36, 1, 2], [14, 72, 3, 1], [14, 108, 1, 3], [14, 216, 1, 2], [14, 216, 3, 2], [14, 252, 4, 2], [14, 756, 4, 1], [14, 882, 2, 2], [14, 1512, 2, 3], [14, 1764, 1, 2], [14, 3024, 6, 1], [14, 10584, 2, 1], [21, 1764, 1, 1], [21, 28224, 3, 1], [28, 1512, 2, 2], [28, 3024, 6, 1], [28, 10584, 2, 1], [28, 10584, 4, 1], [42, 1764, 1, 7], [42, 1764, 2, 4], [42, 1764, 4, 4], [42, 28224, 3, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_7^3.C_2^4:\\He_3.C_6.C_2^4', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 2, 'autcent_group': '4.2', 'autcent_hash': 2, 'autcent_nilpotent': True, 'autcent_order': 4, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^2', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 252, 'autcentquo_group': None, 'autcentquo_hash': 3694995943193559889, 'autcentquo_nilpotent': False, 'autcentquo_order': 3556224, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'C_7^3.C_2^4:\\He_3.C_6.C_2^2', 'cc_stats': [[1, 1, 1], [2, 1, 1], [2, 3, 2], [2, 21, 4], [2, 42, 2], [2, 147, 4], [2, 252, 2], [2, 294, 2], [2, 343, 2], [2, 1029, 2], [2, 1764, 2], [3, 294, 1], [3, 343, 2], [3, 4704, 3], [4, 252, 2], [4, 1764, 6], [4, 12348, 4], [6, 294, 31], [6, 343, 6], [6, 1029, 24], [6, 2058, 40], [6, 4704, 3], [6, 12348, 8], [6, 32928, 6], [7, 18, 1], [7, 72, 3], [7, 108, 1], [12, 12348, 24], [14, 18, 3], [14, 36, 2], [14, 72, 3], [14, 108, 3], [14, 216, 8], [14, 252, 8], [14, 756, 4], [14, 882, 4], [14, 1512, 6], [14, 1764, 2], [14, 3024, 6], [14, 10584, 2], [21, 1764, 1], [21, 28224, 3], [28, 1512, 4], [28, 3024, 6], [28, 10584, 6], [42, 1764, 31], [42, 28224, 3]], 'center_label': '2.1', 'center_order': 2, 'central_product': None, 'central_quotient': '592704.e', 'commutator_count': 2, 'commutator_label': '148176.o', 'complements_known': False, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '2.1', '2.1', '2.1', '3.1', '3.1', '3.1', '7.1', '7.1', '7.1'], 'composition_length': 13, 'conjugacy_classes_known': True, 'counter': 1, 'cyclic': False, 'derived_length': 4, 'dihedral': False, 'direct_factorization': None, 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 3, 1, 2], [2, 21, 1, 4], [2, 42, 1, 2], [2, 147, 1, 4], [2, 252, 1, 2], [2, 294, 1, 2], [2, 343, 1, 2], [2, 1029, 1, 2], [2, 1764, 1, 2], [3, 294, 1, 1], [3, 343, 2, 1], [3, 4704, 1, 3], [4, 252, 1, 2], [4, 1764, 1, 6], [4, 12348, 1, 4], [6, 294, 1, 7], [6, 294, 2, 12], [6, 343, 2, 3], [6, 1029, 2, 12], [6, 2058, 1, 8], [6, 2058, 2, 16], [6, 4704, 1, 3], [6, 12348, 2, 4], [6, 32928, 1, 6], [7, 18, 1, 1], [7, 72, 1, 3], [7, 108, 1, 1], [12, 12348, 2, 12], [14, 18, 1, 3], [14, 36, 1, 2], [14, 72, 1, 3], [14, 108, 1, 3], [14, 216, 1, 8], [14, 252, 1, 8], [14, 756, 1, 4], [14, 882, 1, 4], [14, 1512, 1, 6], [14, 1764, 1, 2], [14, 3024, 1, 6], [14, 10584, 1, 2], [21, 1764, 1, 1], [21, 28224, 1, 3], [28, 1512, 1, 4], [28, 3024, 1, 6], [28, 10584, 1, 6], [42, 1764, 1, 7], [42, 1764, 2, 12], [42, 28224, 1, 3]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': None, 'exponent': 84, 'exponents_of_order': [7, 3, 3], 'factors_of_aut_order': [2, 3, 7], 'factors_of_order': [2, 3, 7], 'faithful_reps': [[18, 1, 4], [36, 0, 10], [36, 1, 7], [108, 1, 4], [216, 1, 8]], 'familial': False, 'frattini_label': '1.1', 'frattini_quotient': '1185408.a', 'hash': 5025983948110693805, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 84, 'inner_gen_orders': [12, 6, 2, 6, 42, 14, 14], 'inner_gens': [[1, 130966, 592716, 1012200, 641808, 997920, 266112], [576749, 2, 592716, 976776, 720408, 382764, 876], [592705, 592706, 12, 24, 144, 6048, 84672], [1013905, 976754, 12, 24, 1044720, 1082592, 254016], [332065, 925802, 12, 231576, 144, 356832, 423360], [278209, 899438, 12, 193560, 919440, 6048, 84672], [1088641, 89870, 12, 1016088, 846864, 6048, 84672]], 'inner_hash': 1501378685565645882, 'inner_nilpotent': False, 'inner_order': 592704, 'inner_split': None, 'inner_tex': 'D_7^3.C_3^2:S_4', 'inner_used': [1, 2, 5], 'irrC_degree': 18, 'irrQ_degree': 18, 'irrQ_dim': 18, 'irrR_degree': 18, 'irrep_stats': [[1, 8], [2, 16], [3, 88], [6, 72], [18, 16], [36, 44], [72, 12], [108, 16], [144, 6], [216, 16]], 'label': '1185408.a', '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': 'D7^3.C6^2:D6', 'ngens': 13, 'nilpotency_class': -1, 'nilpotent': False, 'normal_counts': [1, 1, 0, 1, 0, 0, 1, 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, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 3, 0, 0, 0, 9, 0, 1, 0, 0, 0, 0, 4, 0, 0, 4, 0, 3, 0, 0, 0, 9, 0, 0, 0, 3, 0, 4, 0, 4, 0, 0, 0, 1, 0, 9, 0, 3, 0, 0, 0, 7, 0, 1, 0, 0, 12, 0, 1, 0, 4, 7, 0, 7, 1], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 138, 'number_characteristic_subgroups': 37, 'number_conjugacy_classes': 294, 'number_divisions': 222, 'number_normal_subgroups': 110, 'number_subgroup_autclasses': None, 'number_subgroup_classes': None, 'number_subgroups': None, 'old_label': None, 'order': 1185408, 'order_factorization_type': 321, 'order_stats': [[1, 1], [2, 8127], [3, 15092], [4, 60480], [6, 428652], [7, 342], [12, 296352], [14, 62874], [21, 86436], [28, 87696], [42, 139356]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 6, 'outer_gen_orders': [2, 2, 6], 'outer_gen_pows': [85584, 1001860, 64896], 'outer_gens': [[656209, 764698, 592716, 723636, 884592, 501984, 145152], [1117625, 547358, 592716, 834216, 591672, 1063596, 2604], [425933, 232814, 592716, 841992, 191640, 554700, 5196]], 'outer_group': '24.15', 'outer_hash': 15, 'outer_nilpotent': True, 'outer_order': 24, 'outer_permdeg': 24, 'outer_perms': [26066624305540955915536, 111813811581843615722298, 84780562536248466818261], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^2\\times C_6', 'pc_rank': 7, 'perfect': False, 'permutation_degree': 27, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 2], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 8], [2, 16], [3, 24], [6, 48], [12, 28], [18, 16], [36, 20], [72, 24], [108, 16], [144, 6], [216, 16]], 'representations': {'PC': {'code': '1311773825818871817996253679734792305885898547630921570373872177763323583698992079894011212111763616498322501699773146533632478038372992618879281684850309207682842882455744309693912480259026750927154262974203327244919136101192355087243479321626650044492608260463260626582039851043755487396603912738380558808762923338596903352618173448116335463328031207543595808905398944783761561268637636320574727537129', 'gens': [1, 2, 4, 5, 7, 10, 12], 'pres': [13, 2, 2, 3, 2, 2, 3, 2, 3, 7, 2, 7, 2, 7, 12260976, 3405117, 66, 3371318, 30821235, 15410632, 7705181, 65793004, 31745237, 18611610, 186, 39832421, 27363042, 17017447, 58404534, 32778583, 21763319, 3961288, 2005256, 266, 91982599, 38029076, 23326401, 1532603, 2342568, 410, 505448, 24058965, 14859970, 12696, 4285, 129729609, 24879682, 3453875, 5864101, 147494, 322227, 177550, 386, 103783690, 48679511, 7382268, 5333390, 72147, 708796, 348449, 41513483, 68352, 39155257, 1651167, 2476732, 458729, 504606, 466, 89945868, 146041, 31174454, 3577456, 1192541, 993810, 397591]}, 'Perm': {'d': 27, 'gens': [855133419404601712199024252, 436933163868543545356661929, 65328046543234987464924838]}}, 'schur_multiplier': [2, 2, 2, 2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 2], 'solvability_type': 17, 'solvable': True, 'subgroup_inclusions_known': False, 'subgroup_index_bound': 32, 'supersolvable': False, 'sylow_subgroups_known': False, 'tex_name': 'D_7^3.C_6^2:D_6', 'transitive_degree': 42, 'wreath_data': None, 'wreath_product': False}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '2.1', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 5, 'aut_exponent': 24, 'aut_gen_orders': [2, 3, 3], 'aut_gens': [[25, 3, 144], [25, 3, 243], [25, 147, 144], [122, 3, 144]], 'aut_group': '432.734', 'aut_hash': 734, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 432, 'aut_permdeg': 9, 'aut_perms': [42648, 223305, 287216], 'aut_phi_ratio': 72.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 9, 1, 1], [3, 2, 4, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_3^2:\\GL(2,3)', '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': 24, 'autcentquo_group': '432.734', 'autcentquo_hash': 734, 'autcentquo_nilpotent': False, 'autcentquo_order': 432, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'C_3^2:\\GL(2,3)', 'cc_stats': [[1, 1, 1], [2, 9, 1], [3, 2, 4]], 'center_label': '1.1', 'center_order': 1, 'central_product': False, 'central_quotient': '18.4', 'commutator_count': 1, 'commutator_label': '9.2', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '3.1', '3.1'], 'composition_length': 3, 'conjugacy_classes_known': True, 'counter': 4, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 9, 1, 1], [3, 2, 1, 4]], 'element_repr_type': 'Perm', 'elementary': 1, 'eulerian_function': 7, 'exponent': 6, 'exponents_of_order': [2, 1], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [], 'familial': False, 'frattini_label': '1.1', 'frattini_quotient': '18.4', 'hash': 4, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 6, 'inner_gen_orders': [2, 3, 3], 'inner_gens': [[25, 4, 240], [29, 3, 144], [265, 3, 144]], 'inner_hash': 4, 'inner_nilpotent': False, 'inner_order': 18, 'inner_split': False, 'inner_tex': 'C_3:S_3', 'inner_used': [1, 2, 3], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 2], [2, 4]], 'label': '18.4', 'linC_count': 6, 'linC_degree': 4, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 4, 'linQ_degree_count': 6, 'linQ_dim': 4, 'linQ_dim_count': 6, 'linR_count': 6, 'linR_degree': 4, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'C3:S3', 'ngens': 3, 'nilpotency_class': -1, 'nilpotent': False, 'normal_counts': [0, 0, 0, 0, 0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 3, 'number_characteristic_subgroups': 3, 'number_conjugacy_classes': 6, 'number_divisions': 6, 'number_normal_subgroups': 7, 'number_subgroup_autclasses': 6, 'number_subgroup_classes': 12, 'number_subgroups': 28, 'old_label': None, 'order': 18, 'order_factorization_type': 22, 'order_stats': [[1, 1], [2, 9], [3, 8]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 12, 'outer_gen_orders': [2, 3, 2, 2], 'outer_gen_pows': [0, 0, 25, 25], 'outer_gens': [[25, 3, 243], [25, 240, 243], [25, 243, 244], [25, 240, 3]], 'outer_group': '24.12', 'outer_hash': 12, 'outer_nilpotent': False, 'outer_order': 24, 'outer_permdeg': 4, 'outer_perms': [2, 4, 16, 7], 'outer_solvable': True, 'outer_supersolvable': False, 'outer_tex': 'S_4', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 6, 'pgroup': 0, 'primary_abelian_invariants': [2], 'quasisimple': False, 'rank': 3, 'rational': True, 'rational_characters_known': True, 'ratrep_stats': [[1, 2], [2, 4]], 'representations': {'PC': {'code': 5475, 'gens': [1, 2, 3], 'pres': [3, -2, -3, -3, 25, 110]}, 'GLZ': {'b': 3, 'd': 4, 'gens': [26483560, 35931151, 16858733]}, 'GLFp': {'d': 3, 'p': 3, 'gens': [11017, 8101, 13933]}, 'Perm': {'d': 6, 'gens': [25, 3, 144]}}, 'schur_multiplier': [3], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2], 'solvability_type': 7, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_3:S_3', 'transitive_degree': 9, 'wreath_data': None, 'wreath_product': False}
