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gps_subgroup_search • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'ambient': '945.8', 'ambient_counter': 8, 'ambient_order': 945, 'ambient_tex': 'C_{35}:\\He_3', 'central': False, 'central_factor': False, 'centralizer_order': 15, 'characteristic': True, 'core_order': 945, 'counter': 1, 'cyclic': False, 'direct': True, 'hall': 105, 'label': '945.8.1.a1.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': None, 'minimal': False, 'minimal_normal': False, 'nilpotent': False, 'normal': True, 'old_label': '1.a1.a1', 'outer_equivalence': False, 'perfect': False, 'proper': False, 'quotient': '1.1', 'quotient_Agroup': True, 'quotient_abelian': True, 'quotient_cyclic': True, 'quotient_hash': 1, 'quotient_metabelian': True, 'quotient_nilpotent': True, 'quotient_order': 1, 'quotient_simple': False, 'quotient_solvable': True, 'quotient_supersolvable': True, 'quotient_tex': 'C_1', 'simple': False, 'solvable': True, 'special_labels': ['R', 'L0', 'D0', 'C0'], 'split': True, 'standard_generators': False, 'stem': False, 'subgroup': '945.8', 'subgroup_hash': 8, 'subgroup_order': 945, 'subgroup_tex': 'C_{35}:\\He_3', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '945.8', 'aut_centralizer_order': None, 'aut_label': '1.a1', 'aut_quo_index': None, 'aut_stab_index': None, 'aut_weyl_group': None, 'aut_weyl_index': None, 'centralizer': '63.a1.a1', 'complements': ['945.a1.a1'], 'conjugacy_class_count': 1, 'contained_in': [], 'contains': ['3.a1.a1', '3.b1.a1', '3.b1.b1', '3.b1.c1', '5.a1.a1', '7.a1.a1'], 'core': '1.a1.a1', 'coset_action_label': None, 'count': 1, 'diagramx': [5430, 3282, 4449, 6063, 5452, 4441, 5158, 5014], 'generators': [1, 567, 3, 630, 135], 'label': '945.8.1.a1.a1', 'mobius_quo': 0, 'mobius_sub': 1, 'normal_closure': '1.a1.a1', 'normal_contained_in': [], 'normal_contains': ['3.a1.a1', '3.b1.b1', '3.b1.a1', '3.b1.c1', '5.a1.a1'], 'normalizer': '1.a1.a1', 'old_label': '1.a1.a1', 'projective_image': '63.3', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '1.a1.a1', 'subgroup_fusion': None, 'weyl_group': '63.3'}
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gps_groups • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '45.2', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 84, 'aut_gen_orders': [6, 42, 12, 6], 'aut_gens': [[1, 3, 9], [184, 3, 171], [496, 6, 642], [496, 3, 117], [361, 3, 309]], 'aut_group': None, 'aut_hash': 713841228924550331, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 9072, 'aut_permdeg': 67, 'aut_perms': [15023923797122025678694604270716026404118316383909522464861929824477559662294298598341262817716, 9734568970999500193482538978263147354428982703179152778693046830218604492836162084981683225777, 23346031498056641269236784628757906536729742555253228548448958191515549063449042983583637252978, 27148929979991968423945721823468593432286022155656032377608755637377834866506953533259270087462], 'aut_phi_ratio': 21.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [3, 1, 2, 1], [3, 3, 2, 1], [3, 21, 3, 2], [5, 1, 4, 1], [7, 3, 2, 1], [15, 1, 8, 1], [15, 3, 8, 1], [15, 21, 12, 2], [21, 3, 4, 1], [21, 3, 12, 1], [35, 3, 8, 1], [105, 3, 16, 1], [105, 3, 48, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_4\\times \\He_3:C_2\\times F_7', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 12, 'autcent_group': '36.8', 'autcent_hash': 8, 'autcent_nilpotent': True, 'autcent_order': 36, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_3\\times C_{12}', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 42, 'autcentquo_group': '252.26', 'autcentquo_hash': 26, 'autcentquo_nilpotent': False, 'autcentquo_order': 252, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'S_3\\times F_7', 'cc_stats': [[1, 1, 1], [3, 1, 2], [3, 3, 2], [3, 21, 6], [5, 1, 4], [7, 3, 2], [15, 1, 8], [15, 3, 8], [15, 21, 24], [21, 3, 16], [35, 3, 8], [105, 3, 64]], 'center_label': '15.1', 'center_order': 15, 'central_product': True, 'central_quotient': '63.3', 'commutator_count': 1, 'commutator_label': '21.2', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['3.1', '3.1', '3.1', '5.1', '7.1'], 'composition_length': 5, 'conjugacy_classes_known': True, 'counter': 8, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['189.8', 1], ['5.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [3, 1, 2, 1], [3, 3, 2, 1], [3, 21, 2, 3], [5, 1, 4, 1], [7, 3, 2, 1], [15, 1, 8, 1], [15, 3, 8, 1], [15, 21, 8, 3], [21, 3, 4, 1], [21, 3, 12, 1], [35, 3, 8, 1], [105, 3, 16, 1], [105, 3, 48, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 48, 'exponent': 105, 'exponents_of_order': [3, 1, 1], 'factors_of_aut_order': [2, 3, 7], 'factors_of_order': [3, 5, 7], 'faithful_reps': [[3, 0, 48]], 'familial': False, 'frattini_label': '3.1', 'frattini_quotient': '315.3', 'hash': 8, 'hyperelementary': 3, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 21, 'inner_gen_orders': [3, 1, 21], 'inner_gens': [[1, 3, 150], [1, 3, 9], [814, 3, 9]], 'inner_hash': 3, 'inner_nilpotent': False, 'inner_order': 63, 'inner_split': True, 'inner_tex': 'C_{21}:C_3', 'inner_used': [1, 3], 'irrC_degree': 3, 'irrQ_degree': 144, 'irrQ_dim': 144, 'irrR_degree': 6, 'irrep_stats': [[1, 45], [3, 100]], 'label': '945.8', 'linC_count': 48, 'linC_degree': 3, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 16, 'linQ_degree_count': 1, 'linQ_dim': 16, 'linQ_dim_count': 1, 'linR_count': 24, 'linR_degree': 6, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'C35:He3', 'ngens': 5, 'nilpotency_class': -1, 'nilpotent': False, 'normal_counts': [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': 16, 'number_characteristic_subgroups': 14, 'number_conjugacy_classes': 145, 'number_divisions': 18, 'number_normal_subgroups': 20, 'number_subgroup_autclasses': 28, 'number_subgroup_classes': 44, 'number_subgroups': 232, 'old_label': None, 'order': 945, 'order_factorization_type': 311, 'order_stats': [[1, 1], [3, 134], [5, 4], [7, 6], [15, 536], [21, 48], [35, 24], [105, 192]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 12, 'outer_gen_orders': [4, 6, 6], 'outer_gen_pows': [0, 0, 0], 'outer_gens': [[1, 3, 387], [634, 6, 642], [634, 3, 282]], 'outer_group': '144.159', 'outer_hash': 159, 'outer_nilpotent': False, 'outer_order': 144, 'outer_permdeg': 12, 'outer_perms': [43908480, 10087, 87091231], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_{12}\\times D_6', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 21, 'pgroup': 0, 'primary_abelian_invariants': [3, 3, 5], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 1], [2, 4], [4, 1], [6, 2], [8, 4], [12, 1], [24, 2], [36, 1], [48, 1], [144, 1]], 'representations': {'PC': {'code': 101948826473812199075, 'gens': [1, 2, 3], 'pres': [5, -3, -3, 3, -5, -7, 2252, 57, 8643, 118, 6754]}, 'Perm': {'d': 21, 'gens': [128089319694388624, 57724, 6749568000, 104227, 2568085461317376000]}}, 'schur_multiplier': [3, 3], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [3, 15], 'solvability_type': 7, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_{35}:\\He_3', 'transitive_degree': 315, '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': '45.2', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 84, 'aut_gen_orders': [6, 42, 12, 6], 'aut_gens': [[1, 3, 9], [184, 3, 171], [496, 6, 642], [496, 3, 117], [361, 3, 309]], 'aut_group': None, 'aut_hash': 713841228924550331, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 9072, 'aut_permdeg': 67, 'aut_perms': [15023923797122025678694604270716026404118316383909522464861929824477559662294298598341262817716, 9734568970999500193482538978263147354428982703179152778693046830218604492836162084981683225777, 23346031498056641269236784628757906536729742555253228548448958191515549063449042983583637252978, 27148929979991968423945721823468593432286022155656032377608755637377834866506953533259270087462], 'aut_phi_ratio': 21.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [3, 1, 2, 1], [3, 3, 2, 1], [3, 21, 3, 2], [5, 1, 4, 1], [7, 3, 2, 1], [15, 1, 8, 1], [15, 3, 8, 1], [15, 21, 12, 2], [21, 3, 4, 1], [21, 3, 12, 1], [35, 3, 8, 1], [105, 3, 16, 1], [105, 3, 48, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_4\\times \\He_3:C_2\\times F_7', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 12, 'autcent_group': '36.8', 'autcent_hash': 8, 'autcent_nilpotent': True, 'autcent_order': 36, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_3\\times C_{12}', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 42, 'autcentquo_group': '252.26', 'autcentquo_hash': 26, 'autcentquo_nilpotent': False, 'autcentquo_order': 252, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'S_3\\times F_7', 'cc_stats': [[1, 1, 1], [3, 1, 2], [3, 3, 2], [3, 21, 6], [5, 1, 4], [7, 3, 2], [15, 1, 8], [15, 3, 8], [15, 21, 24], [21, 3, 16], [35, 3, 8], [105, 3, 64]], 'center_label': '15.1', 'center_order': 15, 'central_product': True, 'central_quotient': '63.3', 'commutator_count': 1, 'commutator_label': '21.2', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['3.1', '3.1', '3.1', '5.1', '7.1'], 'composition_length': 5, 'conjugacy_classes_known': True, 'counter': 8, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['189.8', 1], ['5.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [3, 1, 2, 1], [3, 3, 2, 1], [3, 21, 2, 3], [5, 1, 4, 1], [7, 3, 2, 1], [15, 1, 8, 1], [15, 3, 8, 1], [15, 21, 8, 3], [21, 3, 4, 1], [21, 3, 12, 1], [35, 3, 8, 1], [105, 3, 16, 1], [105, 3, 48, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 48, 'exponent': 105, 'exponents_of_order': [3, 1, 1], 'factors_of_aut_order': [2, 3, 7], 'factors_of_order': [3, 5, 7], 'faithful_reps': [[3, 0, 48]], 'familial': False, 'frattini_label': '3.1', 'frattini_quotient': '315.3', 'hash': 8, 'hyperelementary': 3, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 21, 'inner_gen_orders': [3, 1, 21], 'inner_gens': [[1, 3, 150], [1, 3, 9], [814, 3, 9]], 'inner_hash': 3, 'inner_nilpotent': False, 'inner_order': 63, 'inner_split': True, 'inner_tex': 'C_{21}:C_3', 'inner_used': [1, 3], 'irrC_degree': 3, 'irrQ_degree': 144, 'irrQ_dim': 144, 'irrR_degree': 6, 'irrep_stats': [[1, 45], [3, 100]], 'label': '945.8', 'linC_count': 48, 'linC_degree': 3, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 16, 'linQ_degree_count': 1, 'linQ_dim': 16, 'linQ_dim_count': 1, 'linR_count': 24, 'linR_degree': 6, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'C35:He3', 'ngens': 5, 'nilpotency_class': -1, 'nilpotent': False, 'normal_counts': [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': 16, 'number_characteristic_subgroups': 14, 'number_conjugacy_classes': 145, 'number_divisions': 18, 'number_normal_subgroups': 20, 'number_subgroup_autclasses': 28, 'number_subgroup_classes': 44, 'number_subgroups': 232, 'old_label': None, 'order': 945, 'order_factorization_type': 311, 'order_stats': [[1, 1], [3, 134], [5, 4], [7, 6], [15, 536], [21, 48], [35, 24], [105, 192]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 12, 'outer_gen_orders': [4, 6, 6], 'outer_gen_pows': [0, 0, 0], 'outer_gens': [[1, 3, 387], [634, 6, 642], [634, 3, 282]], 'outer_group': '144.159', 'outer_hash': 159, 'outer_nilpotent': False, 'outer_order': 144, 'outer_permdeg': 12, 'outer_perms': [43908480, 10087, 87091231], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_{12}\\times D_6', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 21, 'pgroup': 0, 'primary_abelian_invariants': [3, 3, 5], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 1], [2, 4], [4, 1], [6, 2], [8, 4], [12, 1], [24, 2], [36, 1], [48, 1], [144, 1]], 'representations': {'PC': {'code': 101948826473812199075, 'gens': [1, 2, 3], 'pres': [5, -3, -3, 3, -5, -7, 2252, 57, 8643, 118, 6754]}, 'Perm': {'d': 21, 'gens': [128089319694388624, 57724, 6749568000, 104227, 2568085461317376000]}}, 'schur_multiplier': [3, 3], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [3, 15], 'solvability_type': 7, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_{35}:\\He_3', 'transitive_degree': 315, 'wreath_data': None, 'wreath_product': False}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': True, 'abelian_quotient': '1.1', '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': [], '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]], '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]], 'center_label': '1.1', 'center_order': 1, 'central_product': False, 'central_quotient': '1.1', 'commutator_count': 0, 'commutator_label': '1.1', 'complements_known': True, 'complete': True, 'complex_characters_known': True, 'composition_factors': [], 'composition_length': 0, 'conjugacy_classes_known': True, 'counter': 1, 'cyclic': True, 'derived_length': 0, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 1, 'exponent': 1, 'exponents_of_order': [], 'factors_of_aut_order': [], 'factors_of_order': [], 'faithful_reps': [[1, 1, 1]], 'familial': True, 'frattini_label': '1.1', 'frattini_quotient': '1.1', 'hash': 1, 'hyperelementary': 1, 'inner_abelian': True, 'inner_cyclic': True, 'inner_exponent': 1, 'inner_gen_orders': [], 'inner_gens': [], '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, 1]], 'label': '1.1', 'linC_count': 1, 'linC_degree': 0, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 0, 'linQ_degree_count': 1, 'linQ_dim': 0, 'linQ_dim_count': 1, 'linR_count': 1, 'linR_degree': 0, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C1', 'ngens': 0, 'nilpotency_class': 0, 'nilpotent': True, 'normal_counts': [1], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 1, 'number_characteristic_subgroups': 1, 'number_conjugacy_classes': 1, 'number_divisions': 1, 'number_normal_subgroups': 1, 'number_subgroup_autclasses': 1, 'number_subgroup_classes': 1, 'number_subgroups': 1, 'old_label': None, 'order': 1, 'order_factorization_type': 0, 'order_stats': [[1, 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': 0, 'perfect': True, 'permutation_degree': 1, 'pgroup': 1, 'primary_abelian_invariants': [], 'quasisimple': False, 'rank': 0, 'rational': True, 'rational_characters_known': True, 'ratrep_stats': [[1, 1]], 'representations': {'PC': {'code': 0, 'gens': [], 'pres': []}}, 'schur_multiplier': [], 'semidirect_product': False, 'simple': False, 'smith_abelian_invariants': [], 'solvability_type': 0, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_1', 'transitive_degree': 1, 'wreath_data': None, 'wreath_product': False}
