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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': True, 'ambient': '320.105', 'ambient_counter': 105, 'ambient_order': 320, 'ambient_tex': 'C_{20}.D_8', 'central': False, 'central_factor': False, 'centralizer_order': 80, 'characteristic': True, 'core_order': 80, 'counter': 5, 'cyclic': False, 'direct': False, 'hall': 0, 'label': '320.105.4.a1.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': True, 'old_label': '4.a1.a1', 'outer_equivalence': False, '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': False, 'standard_generators': False, 'stem': False, 'subgroup': '80.20', 'subgroup_hash': 20, 'subgroup_order': 80, 'subgroup_tex': 'C_4\\times C_{20}', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '320.105', 'aut_centralizer_order': 320, 'aut_label': '4.a1', 'aut_quo_index': 6, 'aut_stab_index': 1, 'aut_weyl_group': '16.10', 'aut_weyl_index': 320, 'centralizer': '4.a1.a1', 'complements': [], 'conjugacy_class_count': 1, 'contained_in': ['2.a1.a1', '2.b1.a1', '2.c1.a1'], 'contains': ['8.a1.a1', '8.b1.a1', '8.c1.a1', '20.a1.a1'], 'core': '4.a1.a1', 'coset_action_label': None, 'count': 1, 'diagramx': [5086, 4767, 2492, 5651, 6491, 4978, 7704, 4666], 'generators': [92, 64, 168, 80, 160], 'label': '320.105.4.a1.a1', 'mobius_quo': 0, 'mobius_sub': 2, 'normal_closure': '4.a1.a1', 'normal_contained_in': ['2.a1.a1', '2.c1.a1', '2.b1.a1'], 'normal_contains': ['8.b1.a1', '8.a1.a1', '8.c1.a1', '20.a1.a1'], 'normalizer': '1.a1.a1', 'old_label': '4.a1.a1', 'projective_image': '80.19', 'quotient_action_image': '4.2', 'quotient_action_kernel': '1.1', 'quotient_action_kernel_order': 1, 'quotient_fusion': None, 'short_label': '4.a1.a1', 'subgroup_fusion': None, 'weyl_group': '4.2'}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': True, 'abelian_quotient': '80.20', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 3, 'aut_exponent': 12, 'aut_gen_orders': [2, 4, 12, 2, 2], 'aut_gens': [[1, 4], [43, 6], [63, 6], [61, 11], [3, 46], [43, 44]], 'aut_group': '384.18060', 'aut_hash': 18060, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 384, 'aut_permdeg': 12, 'aut_perms': [40279680, 43908501, 87097100, 7, 23], 'aut_phi_ratio': 12.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 3, 1], [4, 1, 12, 1], [5, 1, 4, 1], [10, 1, 12, 1], [20, 1, 48, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_4\\times \\GL(2,\\mathbb{Z}/4)', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 12, 'autcent_group': '384.18060', 'autcent_hash': 18060, 'autcent_nilpotent': False, 'autcent_order': 384, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': False, 'autcent_tex': 'C_4\\times \\GL(2,\\mathbb{Z}/4)', '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], [4, 1, 12], [5, 1, 4], [10, 1, 12], [20, 1, 48]], 'center_label': '80.20', 'center_order': 80, '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', '2.1', '2.1', '5.1'], 'composition_length': 5, 'conjugacy_classes_known': True, 'counter': 20, 'cyclic': False, 'derived_length': 1, 'dihedral': False, 'direct_factorization': [['4.1', 2], ['5.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [4, 1, 2, 6], [5, 1, 4, 1], [10, 1, 4, 3], [20, 1, 8, 6]], 'element_repr_type': 'PC', 'elementary': 2, 'eulerian_function': 6, 'exponent': 20, 'exponents_of_order': [4, 1], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 5], 'faithful_reps': [], 'familial': False, 'frattini_label': '4.2', 'frattini_quotient': '20.5', 'hash': 20, 'hyperelementary': 2, 'inner_abelian': True, 'inner_cyclic': True, 'inner_exponent': 1, 'inner_gen_orders': [1, 1], 'inner_gens': [[1, 4], [1, 4]], '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, 80]], 'label': '80.20', 'linC_count': 1152, 'linC_degree': 2, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 8, 'linQ_degree_count': 48, 'linQ_dim': 8, 'linQ_dim_count': 48, 'linR_count': 288, 'linR_degree': 4, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C4*C20', 'ngens': 5, 'nilpotency_class': 1, 'nilpotent': True, 'normal_counts': [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': 6, 'number_characteristic_subgroups': 6, 'number_conjugacy_classes': 80, 'number_divisions': 20, 'number_normal_subgroups': 30, 'number_subgroup_autclasses': 12, 'number_subgroup_classes': 30, 'number_subgroups': 30, 'old_label': None, 'order': 80, 'order_factorization_type': 31, 'order_stats': [[1, 1], [2, 3], [4, 12], [5, 4], [10, 12], [20, 48]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 12, 'outer_gen_orders': [2, 4, 12, 2, 2], 'outer_gen_pows': [0, 0, 0, 0, 0], 'outer_gens': [[43, 6], [63, 6], [61, 11], [3, 46], [43, 44]], 'outer_group': '384.18060', 'outer_hash': 18060, 'outer_nilpotent': False, 'outer_order': 384, 'outer_permdeg': 12, 'outer_perms': [40279680, 43908501, 87097100, 7, 23], 'outer_solvable': True, 'outer_supersolvable': False, 'outer_tex': 'C_4\\times \\GL(2,\\mathbb{Z}/4)', 'pc_rank': 2, 'perfect': False, 'permutation_degree': 13, 'pgroup': 0, 'primary_abelian_invariants': [4, 4, 5], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 6], [4, 4], [8, 6]], 'representations': {'PC': {'code': 114420626435, 'gens': [1, 3], 'pres': [5, -2, -2, -2, -2, -5, 10, 42, 58]}, 'GLFp': {'d': 2, 'p': 41, 'gens': [1585222, 620321]}, 'Perm': {'d': 13, 'gens': [1516838400, 131040, 96, 482630400, 41040]}}, 'schur_multiplier': [4], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [4, 20], 'solvability_type': 1, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_4\\times C_{20}', 'transitive_degree': 80, 'wreath_data': None, 'wreath_product': False}
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gps_groups • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '8.2', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 20, 'aut_gen_orders': [4, 20, 4, 4, 2, 4, 4, 2], 'aut_gens': [[1, 2, 16], [161, 10, 272], [81, 50, 16], [253, 210, 144], [85, 314, 144], [89, 106, 304], [249, 258, 48], [13, 50, 304], [13, 190, 304]], 'aut_group': None, 'aut_hash': 5919243039148427669, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 5120, 'aut_permdeg': 96, 'aut_perms': [375315022060181409552564828706875017875839248007786020427899977320319995812818828527659623779144473075512936037512597694250667887232222776646454684025, 304800887597729080994202113583221095892485863541985083147151511707687070733182041131743652338032392883299033088138704077594507834771159587351976084632, 861955497483123264038702426172977405008344606380384948695502444215299290459577328992219432958192972330493978135624346502859031469119763906053616856313, 27632090472713345794445708987398749149753759897383157777248871669162043396459218888214140568046484309048392667863953428295986700191887908853037754364, 651806683666346692304103902955178510037753679211880032608877579232494490536180402782751573004697427934646674092038729501763729647371822123858010698865, 48060191131529940784816011737267923523048183902315002517195550802422592974854852604589617999235539192911486155408410779831527181991980185014141166880, 367889952892351897295686242962695223096761600846957280789150128596962408656151023551342453826030421658037667286201359253998167872605662140133826109400, 108334273590702305370885015245451464681859952881106891381997839895198914015519249739916732220062764769242406407564691448057711593651123548849484818848], 'aut_phi_ratio': 40.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [4, 2, 1, 4], [4, 4, 1, 1], [4, 8, 2, 1], [5, 2, 2, 1], [8, 20, 4, 2], [10, 2, 2, 3], [20, 4, 2, 4], [20, 4, 4, 1], [20, 8, 8, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_5:(C_2\\times C_4\\times C_2^2.C_2^5)', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 2, 'autcent_group': '16.14', 'autcent_hash': 14, 'autcent_nilpotent': True, 'autcent_order': 16, 'autcent_solvable': True, 'autcent_split': False, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^4', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 20, 'autcentquo_group': '320.1638', 'autcentquo_hash': 1638, 'autcentquo_nilpotent': False, 'autcentquo_order': 320, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'C_2^4\\times F_5', 'cc_stats': [[1, 1, 1], [2, 1, 3], [4, 2, 4], [4, 4, 1], [4, 8, 2], [5, 2, 2], [8, 20, 8], [10, 2, 6], [20, 4, 12], [20, 8, 8]], 'center_label': '4.2', 'center_order': 4, 'central_product': False, 'central_quotient': '80.19', 'commutator_count': 2, 'commutator_label': '40.9', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '2.1', '2.1', '5.1'], 'composition_length': 7, 'conjugacy_classes_known': True, 'counter': 105, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [4, 2, 1, 4], [4, 4, 1, 1], [4, 8, 1, 2], [5, 2, 2, 1], [8, 20, 4, 2], [10, 2, 2, 3], [20, 4, 2, 4], [20, 4, 4, 1], [20, 8, 4, 2]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 6, 'exponent': 40, 'exponents_of_order': [6, 1], 'factors_of_aut_order': [2, 5], 'factors_of_order': [2, 5], 'faithful_reps': [], 'familial': False, 'frattini_label': '16.2', 'frattini_quotient': '20.4', 'hash': 105, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 20, 'inner_gen_orders': [2, 4, 10], 'inner_gens': [[1, 94, 176], [245, 2, 304], [161, 34, 16]], 'inner_hash': 19, 'inner_nilpotent': False, 'inner_order': 80, 'inner_split': True, 'inner_tex': 'C_{10}.D_4', 'inner_used': [1, 2, 3], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 8], [2, 26], [4, 13]], 'label': '320.105', 'linC_count': 320, 'linC_degree': 6, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 12, 'linQ_degree_count': 44, 'linQ_dim': 12, 'linQ_dim_count': 4, 'linR_count': 4, 'linR_degree': 6, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'C20.D8', 'ngens': 7, 'nilpotency_class': -1, 'nilpotent': False, 'normal_counts': [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': 22, 'number_characteristic_subgroups': 31, 'number_conjugacy_classes': 47, 'number_divisions': 24, 'number_normal_subgroups': 35, 'number_subgroup_autclasses': 58, 'number_subgroup_classes': 64, 'number_subgroups': 174, 'old_label': None, 'order': 320, 'order_factorization_type': 51, 'order_stats': [[1, 1], [2, 3], [4, 28], [5, 4], [8, 160], [10, 12], [20, 112]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 4, 'outer_gen_orders': [2, 2, 2, 2, 4], 'outer_gen_pows': [0, 0, 0, 0, 0], 'outer_gens': [[169, 2, 16], [1, 170, 16], [81, 2, 176], [1, 82, 176], [1, 2, 272]], 'outer_group': '64.260', 'outer_hash': 260, 'outer_nilpotent': True, 'outer_order': 64, 'outer_permdeg': 12, 'outer_perms': [127, 39916800, 362880, 5040, 17], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^4\\times C_4', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 21, 'pgroup': 0, 'primary_abelian_invariants': [2, 4], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 4], [4, 9], [8, 6], [16, 1]], 'representations': {'PC': {'code': 74588068770217504492828973201849434115, 'gens': [1, 2, 5], 'pres': [7, -2, -2, -2, -2, -2, -2, -5, 1120, 1317, 36, 254, 58, 6164, 5331, 102, 6060, 124, 6285]}, 'GLZN': {'d': 2, 'p': 105, 'gens': [33960739, 82191446, 5136169, 87979576, 1159831, 90854147, 85785094]}, 'Perm': {'d': 21, 'gens': [2690128182443022727, 4886141055427548720, 7940368848461374800, 10493555889062096640, 12414935000230963200, 12414935001350206800, 37]}}, 'schur_multiplier': [2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 4], 'solvability_type': 7, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_{20}.D_8', 'transitive_degree': 320, 'wreath_data': None, 'wreath_product': False}
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gps_groups • Show schema
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{'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}
