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gps_subgroup_search • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'ambient': '1404.117', 'ambient_counter': 117, 'ambient_order': 1404, 'ambient_tex': '\\He_3:D_{26}', 'central': False, 'central_factor': False, 'centralizer_order': 3, 'characteristic': True, 'core_order': 702, 'counter': 3, 'cyclic': False, 'direct': False, 'hall': 0, 'label': '1404.117.2.b1.a1', 'maximal': True, 'maximal_normal': True, 'metabelian': True, 'metacyclic': None, 'minimal': False, 'minimal_normal': False, 'nilpotent': False, 'normal': True, 'old_label': '2.b1.a1', 'outer_equivalence': False, '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': '702.30', 'subgroup_hash': 30, 'subgroup_order': 702, 'subgroup_tex': 'D_{13}\\times \\He_3', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '1404.117', 'aut_centralizer_order': 1, 'aut_label': '2.b1', 'aut_quo_index': 1, 'aut_stab_index': 1, 'aut_weyl_group': '67392.c', 'aut_weyl_index': 1, 'centralizer': '468.a1.a1', 'complements': ['702.a1.a1', '702.c1.a1'], 'conjugacy_class_count': 1, 'contained_in': ['1.a1.a1'], 'contains': ['4.a1.a1', '6.a1.a1', '6.a1.b1', '6.a1.c1', '6.a1.d1', '26.b1.a1'], 'core': '2.b1.a1', 'coset_action_label': None, 'count': 1, 'diagramx': [865, 1689, 5086, 5034, 3582, 1699, 4718, 4984], 'generators': [6, 468, 12, 4, 108], 'label': '1404.117.2.b1.a1', 'mobius_quo': 0, 'mobius_sub': -1, 'normal_closure': '2.b1.a1', 'normal_contained_in': ['1.a1.a1'], 'normal_contains': ['4.a1.a1', '6.a1.b1', '6.a1.c1', '6.a1.a1', '6.a1.d1'], 'normalizer': '1.a1.a1', 'old_label': '2.b1.a1', 'projective_image': '468.43', 'quotient_action_image': '2.1', 'quotient_action_kernel': '1.1', 'quotient_action_kernel_order': 1, 'quotient_fusion': None, 'short_label': '2.b1.a1', 'subgroup_fusion': None, 'weyl_group': '468.43'}
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gps_groups • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '18.5', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 5, 'aut_exponent': 312, 'aut_gen_orders': [2, 12, 12, 52, 3, 3], 'aut_gens': [[1, 6, 18], [1, 12, 684], [251, 6, 360], [237, 6, 56], [521, 6, 28], [1, 6, 30], [7, 6, 18]], 'aut_group': '67392.c', 'aut_hash': 1849751742543539073, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 67392, 'aut_permdeg': 37, 'aut_perms': [6173583165643696739390647515597184726102034, 8640564340035618487899288344683712732945220, 8057286566301676931243315118264286071189241, 10600618323702157879064991568640532425546831, 62306241804872729805252576379249198976465, 310342413881166561128005974369801153288197], 'aut_phi_ratio': 312.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 13, 1, 1], [3, 1, 2, 1], [3, 3, 8, 1], [6, 13, 2, 1], [6, 39, 8, 1], [13, 2, 6, 1], [39, 2, 12, 1], [39, 6, 48, 1]], 'aut_supersolvable': False, 'aut_tex': 'F_{13}\\times C_3^2:\\GL(2,3)', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 3, 'autcent_group': '9.2', 'autcent_hash': 2, 'autcent_nilpotent': True, 'autcent_order': 9, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_3^2', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 312, 'autcentquo_group': '7488.g', 'autcentquo_hash': 4198507505901823907, 'autcentquo_nilpotent': False, 'autcentquo_order': 7488, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'F_{13}\\times \\GL(2,3)', 'cc_stats': [[1, 1, 1], [2, 13, 1], [3, 1, 2], [3, 3, 8], [6, 13, 2], [6, 39, 8], [13, 2, 6], [39, 2, 12], [39, 6, 48]], 'center_label': '3.1', 'center_order': 3, 'central_product': True, 'central_quotient': '234.11', 'commutator_count': 1, 'commutator_label': '39.2', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '3.1', '3.1', '3.1', '13.1'], 'composition_length': 5, 'conjugacy_classes_known': True, 'counter': 30, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['26.1', 1], ['27.3', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 13, 1, 1], [3, 1, 2, 1], [3, 3, 2, 4], [6, 13, 2, 1], [6, 39, 2, 4], [13, 2, 6, 1], [39, 2, 12, 1], [39, 6, 12, 4]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 3, 'exponent': 78, 'exponents_of_order': [3, 1, 1], 'factors_of_aut_order': [2, 3, 13], 'factors_of_order': [2, 3, 13], 'faithful_reps': [[6, 0, 12]], 'familial': False, 'frattini_label': '3.1', 'frattini_quotient': '234.11', 'hash': 30, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 78, 'inner_gen_orders': [6, 1, 39], 'inner_gens': [[1, 6, 462], [1, 6, 18], [277, 6, 18]], 'inner_hash': 11, 'inner_nilpotent': False, 'inner_order': 234, 'inner_split': False, 'inner_tex': 'C_3^2\\times D_{13}', 'inner_used': [1, 3], 'irrC_degree': 6, 'irrQ_degree': 72, 'irrQ_dim': 72, 'irrR_degree': 12, 'irrep_stats': [[1, 18], [2, 54], [3, 4], [6, 12]], 'label': '702.30', 'linC_count': 216, 'linC_degree': 5, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 18, 'linQ_degree_count': 2, 'linQ_dim': 18, 'linQ_dim_count': 2, 'linR_count': 12, 'linR_degree': 8, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'D13*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': 9, 'number_characteristic_subgroups': 9, 'number_conjugacy_classes': 88, 'number_divisions': 18, 'number_normal_subgroups': 21, 'number_subgroup_autclasses': 20, 'number_subgroup_classes': 44, 'number_subgroups': 304, 'old_label': None, 'order': 702, 'order_factorization_type': 311, 'order_stats': [[1, 1], [2, 13], [3, 26], [6, 338], [13, 12], [39, 312]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 24, 'outer_gen_orders': [2, 6, 4, 12], 'outer_gen_pows': [236, 3, 0, 0], 'outer_gens': [[11, 12, 34], [237, 6, 104], [471, 6, 232], [235, 6, 310]], 'outer_group': '288.900', 'outer_hash': 900, 'outer_nilpotent': False, 'outer_order': 288, 'outer_permdeg': 13, 'outer_perms': [215273520, 79914241, 727251840, 2994854448], 'outer_solvable': True, 'outer_supersolvable': False, 'outer_tex': 'C_6\\times \\GL(2,3)', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 22, 'pgroup': 0, 'primary_abelian_invariants': [2, 3, 3], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 2], [2, 8], [6, 2], [12, 1], [24, 4], [72, 1]], 'representations': {'PC': {'code': 38629192029321108337511, 'gens': [1, 3, 4], 'pres': [5, -2, -3, -3, 3, -13, 10, 9243, 248, 78, 16204]}, 'GLZN': {'d': 2, 'p': 117, 'gens': [126531991, 1601677, 26699728, 1602667, 1606177]}, 'Perm': {'d': 22, 'gens': [40284847, 56323168315045632000, 7446673389470515200, 112426079892769843200, 559571293]}}, 'schur_multiplier': [3, 3], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [3, 6], 'solvability_type': 7, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'D_{13}\\times \\He_3', 'transitive_degree': 117, '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': '4.2', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 5, 'aut_exponent': 312, 'aut_gen_orders': [2, 12, 12, 52, 3, 3], 'aut_gens': [[1, 2, 12, 36], [1, 2, 24, 1384], [1, 494, 12, 1216], [1, 478, 12, 596], [1, 1142, 12, 512], [5, 2, 12, 60], [957, 14, 12, 48]], 'aut_group': '67392.c', 'aut_hash': 1849751742543539073, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 67392, 'aut_permdeg': 22, 'aut_perms': [873411360538625953817, 873020686947836176685, 668854062152839551711, 616842658632896360522, 649841719940822079630, 868674683204935303926], 'aut_phi_ratio': 156.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 9, 1, 1], [2, 13, 1, 1], [2, 117, 1, 1], [3, 1, 2, 1], [3, 6, 4, 1], [6, 9, 2, 1], [6, 13, 2, 1], [6, 78, 4, 1], [6, 117, 2, 1], [13, 2, 6, 1], [26, 18, 6, 1], [39, 2, 12, 1], [39, 12, 24, 1], [78, 18, 12, 1]], 'aut_supersolvable': False, 'aut_tex': 'F_{13}\\times 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': 312, 'autcentquo_group': '67392.c', 'autcentquo_hash': 1849751742543539073, 'autcentquo_nilpotent': False, 'autcentquo_order': 67392, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'F_{13}\\times C_3^2:\\GL(2,3)', 'cc_stats': [[1, 1, 1], [2, 9, 1], [2, 13, 1], [2, 117, 1], [3, 1, 2], [3, 6, 4], [6, 9, 2], [6, 13, 2], [6, 78, 4], [6, 117, 2], [13, 2, 6], [26, 18, 6], [39, 2, 12], [39, 12, 24], [78, 18, 12]], 'center_label': '3.1', 'center_order': 3, 'central_product': True, 'central_quotient': '468.43', 'commutator_count': 1, 'commutator_label': '351.10', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '3.1', '3.1', '3.1', '13.1'], 'composition_length': 6, 'conjugacy_classes_known': True, 'counter': 117, 'cyclic': False, 'derived_length': 3, 'dihedral': False, 'direct_factorization': [['26.1', 1], ['54.8', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 9, 1, 1], [2, 13, 1, 1], [2, 117, 1, 1], [3, 1, 2, 1], [3, 6, 1, 4], [6, 9, 2, 1], [6, 13, 2, 1], [6, 78, 1, 4], [6, 117, 2, 1], [13, 2, 6, 1], [26, 18, 6, 1], [39, 2, 12, 1], [39, 12, 6, 4], [78, 18, 12, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 15876, 'exponent': 78, 'exponents_of_order': [3, 2, 1], 'factors_of_aut_order': [2, 3, 13], 'factors_of_order': [2, 3, 13], 'faithful_reps': [[6, 0, 24]], 'familial': False, 'frattini_label': '3.1', 'frattini_quotient': '468.43', 'hash': 117, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 78, 'inner_gen_orders': [2, 6, 1, 39], 'inner_gens': [[1, 10, 12, 516], [5, 2, 12, 924], [1, 2, 12, 36], [961, 554, 12, 36]], 'inner_hash': 43, 'inner_nilpotent': False, 'inner_order': 468, 'inner_split': False, 'inner_tex': 'C_{39}:D_6', 'inner_used': [1, 2, 4], 'irrC_degree': 6, 'irrQ_degree': 72, 'irrQ_dim': 72, 'irrR_degree': 12, 'irrep_stats': [[1, 4], [2, 20], [3, 8], [4, 24], [6, 24]], 'label': '1404.117', 'linC_count': 96, 'linC_degree': 5, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 18, 'linQ_degree_count': 8, 'linQ_dim': 18, 'linQ_dim_count': 8, 'linR_count': 48, 'linR_degree': 8, 'maximal_subgroups_known': True, 'metabelian': False, 'metacyclic': False, 'monomial': True, 'name': 'He3:D26', 'ngens': 6, '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], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 15, 'number_characteristic_subgroups': 13, 'number_conjugacy_classes': 80, 'number_divisions': 24, 'number_normal_subgroups': 25, 'number_subgroup_autclasses': 50, 'number_subgroup_classes': 110, 'number_subgroups': 1594, 'old_label': None, 'order': 1404, 'order_factorization_type': 321, 'order_stats': [[1, 1], [2, 139], [3, 26], [6, 590], [13, 12], [26, 108], [39, 312], [78, 216]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 12, 'outer_gen_orders': [6, 6, 6, 6, 3, 3], 'outer_gen_pows': [1, 6, 6, 7, 7, 6], 'outer_gens': [[1, 486, 12, 352], [1, 486, 24, 884], [1, 2, 12, 684], [1, 478, 12, 1352], [1, 946, 12, 840], [1, 2, 12, 360]], 'outer_group': '144.188', 'outer_hash': 188, 'outer_nilpotent': False, 'outer_order': 144, 'outer_permdeg': 9, 'outer_perms': [41070, 768, 31, 131809, 121728, 48], 'outer_solvable': True, 'outer_supersolvable': False, 'outer_tex': 'C_6\\times S_4', 'pc_rank': 4, 'perfect': False, 'permutation_degree': 22, 'pgroup': 0, 'primary_abelian_invariants': [2, 2], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 8], [6, 4], [12, 2], [24, 4], [72, 2]], 'representations': {'PC': {'code': 1313437689348573504318799098148431719, 'gens': [1, 2, 4, 5], 'pres': [6, -2, -2, -3, -3, 3, -13, 121, 31, 146, 15484, 13870, 376, 118, 23339]}, 'Perm': {'d': 22, 'gens': [40284847, 4988539717185024000, 7441025543875123200, 56201167527208704000, 559571293, 109872559308563251200]}}, 'schur_multiplier': [2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2], 'solvability_type': 10, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': '\\He_3:D_{26}', 'transitive_degree': 117, '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': '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}
