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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': True, 'ambient': '1800.704', 'ambient_counter': 704, 'ambient_order': 1800, 'ambient_tex': 'C_6\\times C_{15}:F_5', 'central': False, 'central_factor': False, 'centralizer_order': 450, 'characteristic': False, 'core_order': 90, 'counter': 21, 'cyclic': False, 'direct': False, 'hall': 0, 'label': '1800.704.20.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': True, 'old_label': '20.a1', 'outer_equivalence': True, 'perfect': False, 'proper': True, 'quotient': '20.3', 'quotient_Agroup': True, 'quotient_abelian': False, 'quotient_cyclic': False, 'quotient_hash': 3, 'quotient_metabelian': True, 'quotient_nilpotent': False, 'quotient_order': 20, 'quotient_simple': False, 'quotient_solvable': True, 'quotient_supersolvable': True, 'quotient_tex': 'F_5', 'simple': False, 'solvable': True, 'special_labels': [], 'split': True, 'standard_generators': False, 'stem': False, 'subgroup': '90.10', 'subgroup_hash': 10, 'subgroup_order': 90, 'subgroup_tex': 'C_3\\times C_{30}', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '1800.704', 'aut_centralizer_order': 1000, 'aut_label': '20.a1', 'aut_quo_index': 1, 'aut_stab_index': 6, 'aut_weyl_group': '192.951', 'aut_weyl_index': 6000, 'centralizer': '4.a1', 'complements': ['90.b1'], 'conjugacy_class_count': 6, 'contained_in': ['4.a1', '10.a1'], 'contains': ['40.a1', '60.a1', '100.a1'], 'core': '20.a1', 'coset_action_label': None, 'count': 6, 'diagramx': [1749, 965, 3100, 787], 'generators': [900, 1200, 12, 40], 'label': '1800.704.20.a1', 'mobius_quo': 3, 'mobius_sub': 0, 'normal_closure': '20.a1', 'normal_contained_in': ['4.a1'], 'normal_contains': ['40.a1', '60.a1', '100.a1'], 'normalizer': '1.a1', 'old_label': '20.a1', 'projective_image': '100.11', 'quotient_action_image': '4.1', 'quotient_action_kernel': '5.1', 'quotient_action_kernel_order': 5, 'quotient_fusion': None, 'short_label': '20.a1', 'subgroup_fusion': None, 'weyl_group': '4.1'}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': True, 'abelian_quotient': '90.10', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 4, 'aut_exponent': 24, 'aut_gen_orders': [2, 12, 4, 4], 'aut_gens': [[1, 3], [2, 3], [1, 23], [61, 35], [30, 65]], 'aut_group': '192.951', 'aut_hash': 951, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 192, 'aut_permdeg': 12, 'aut_perms': [39928327, 262085177, 176163120, 127024560], 'aut_phi_ratio': 8.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [3, 1, 8, 1], [5, 1, 4, 1], [6, 1, 8, 1], [10, 1, 4, 1], [15, 1, 32, 1], [30, 1, 32, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_4\\times \\GL(2,3)', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 24, 'autcent_group': '192.951', 'autcent_hash': 951, 'autcent_nilpotent': False, 'autcent_order': 192, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': False, 'autcent_tex': 'C_4\\times \\GL(2,3)', '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, 1], [3, 1, 8], [5, 1, 4], [6, 1, 8], [10, 1, 4], [15, 1, 32], [30, 1, 32]], 'center_label': '90.10', 'center_order': 90, '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', '3.1', '3.1', '5.1'], 'composition_length': 4, 'conjugacy_classes_known': True, 'counter': 10, 'cyclic': False, 'derived_length': 1, 'dihedral': False, 'direct_factorization': [['2.1', 1], ['3.1', 2], ['5.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [3, 1, 2, 4], [5, 1, 4, 1], [6, 1, 2, 4], [10, 1, 4, 1], [15, 1, 8, 4], [30, 1, 8, 4]], 'element_repr_type': 'PC', 'elementary': 3, 'eulerian_function': 18, 'exponent': 30, 'exponents_of_order': [2, 1, 1], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3, 5], 'faithful_reps': [], 'familial': False, 'frattini_label': '1.1', 'frattini_quotient': '90.10', 'hash': 10, 'hyperelementary': 3, 'inner_abelian': True, 'inner_cyclic': True, 'inner_exponent': 1, 'inner_gen_orders': [1, 1], 'inner_gens': [[1, 3], [1, 3]], '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, 90]], 'label': '90.10', 'linC_count': 1728, 'linC_degree': 2, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 8, 'linQ_degree_count': 42, 'linQ_dim': 8, 'linQ_dim_count': 42, 'linR_count': 432, 'linR_degree': 4, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C3*C30', 'ngens': 4, 'nilpotency_class': 1, 'nilpotent': True, 'normal_counts': [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': 8, 'number_characteristic_subgroups': 8, 'number_conjugacy_classes': 90, 'number_divisions': 20, 'number_normal_subgroups': 24, 'number_subgroup_autclasses': 12, 'number_subgroup_classes': 24, 'number_subgroups': 24, 'old_label': None, 'order': 90, 'order_factorization_type': 222, 'order_stats': [[1, 1], [2, 1], [3, 8], [5, 4], [6, 8], [10, 4], [15, 32], [30, 32]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 24, 'outer_gen_orders': [2, 12, 4, 4], 'outer_gen_pows': [0, 0, 0, 0], 'outer_gens': [[2, 3], [1, 23], [61, 35], [30, 65]], 'outer_group': '192.951', 'outer_hash': 951, 'outer_nilpotent': False, 'outer_order': 192, 'outer_permdeg': 12, 'outer_perms': [39928327, 262085177, 176163120, 127024560], 'outer_solvable': True, 'outer_supersolvable': False, 'outer_tex': 'C_4\\times \\GL(2,3)', 'pc_rank': 2, 'perfect': False, 'permutation_degree': 13, 'pgroup': 0, 'primary_abelian_invariants': [2, 3, 3, 5], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 2], [2, 8], [4, 2], [8, 8]], 'representations': {'PC': {'code': 12453447, 'gens': [1, 2], 'pres': [4, -3, -2, -3, -5, 21, 46]}, 'GLFp': {'d': 2, 'p': 31, 'gens': [148980, 208564]}, 'Perm': {'d': 13, 'gens': [479001600, 7257600, 10080, 96]}}, 'schur_multiplier': [3], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [3, 30], 'solvability_type': 1, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_3\\times C_{30}', 'transitive_degree': 90, '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': '72.36', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 4, 'aut_exponent': 120, 'aut_gen_orders': [40, 20, 24, 24], 'aut_gens': [[1, 4, 60], [1489, 516, 1052], [409, 128, 60], [913, 528, 784], [1285, 1000, 592]], 'aut_group': None, 'aut_hash': 3401745568645604803, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 1152000, 'aut_permdeg': 58, 'aut_perms': [899488600826653558709416665506240553115537661040596530198019815455811705657777, 1622339450439635486222078306590183953646177615966380803316761693567189114541873, 234928708564699181620862789498442034335706154646404735219968698929466361807410, 1084100406615023476507404855624553598154537084075918693917789840942101433184431], 'aut_phi_ratio': 2400.0, 'aut_solvable': False, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 25, 1, 2], [3, 1, 8, 1], [4, 25, 2, 2], [5, 4, 6, 1], [6, 1, 8, 1], [6, 25, 8, 2], [10, 4, 6, 1], [12, 25, 16, 2], [15, 4, 48, 1], [30, 4, 48, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_2\\times \\GL(2,3)\\times C_5^2:C_4.S_5', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 24, 'autcent_group': '96.189', 'autcent_hash': 189, 'autcent_nilpotent': False, 'autcent_order': 96, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': False, 'autcent_tex': 'C_2\\times \\GL(2,3)', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 120, 'autcentquo_group': '12000.a', 'autcentquo_hash': 5912173750379521722, 'autcentquo_nilpotent': False, 'autcentquo_order': 12000, 'autcentquo_solvable': False, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'C_5^2.\\GL(2,5)', 'cc_stats': [[1, 1, 1], [2, 1, 1], [2, 25, 2], [3, 1, 8], [4, 25, 4], [5, 4, 6], [6, 1, 8], [6, 25, 16], [10, 4, 6], [12, 25, 32], [15, 4, 48], [30, 4, 48]], 'center_label': '18.5', 'center_order': 18, 'central_product': True, 'central_quotient': '100.11', 'commutator_count': 1, 'commutator_label': '25.2', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '3.1', '3.1', '5.1', '5.1'], 'composition_length': 7, 'conjugacy_classes_known': True, 'counter': 704, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['100.11', 1], ['2.1', 1], ['3.1', 2]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 25, 1, 2], [3, 1, 2, 4], [4, 25, 2, 2], [5, 4, 1, 6], [6, 1, 2, 4], [6, 25, 2, 8], [10, 4, 1, 6], [12, 25, 4, 8], [15, 4, 2, 24], [30, 4, 2, 24]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 2184, 'exponent': 60, 'exponents_of_order': [3, 2, 2], 'factors_of_aut_order': [2, 3, 5], 'factors_of_order': [2, 3, 5], 'faithful_reps': [], 'familial': False, 'frattini_label': '1.1', 'frattini_quotient': '1800.704', 'hash': 704, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 20, 'inner_gen_orders': [4, 5, 5], 'inner_gens': [[1, 52, 780], [13, 4, 60], [1081, 4, 60]], 'inner_hash': 11, 'inner_nilpotent': False, 'inner_order': 100, 'inner_split': False, 'inner_tex': 'C_5:F_5', 'inner_used': [1, 2, 3], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 72], [4, 108]], 'label': '1800.704', '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': True, 'metacyclic': False, 'monomial': True, 'name': 'C6*C15:F5', 'ngens': 7, '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], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 16, 'number_characteristic_subgroups': 16, 'number_conjugacy_classes': 180, 'number_divisions': 90, 'number_normal_subgroups': 132, 'number_subgroup_autclasses': 63, 'number_subgroup_classes': 384, 'number_subgroups': 2112, 'old_label': None, 'order': 1800, 'order_factorization_type': 321, 'order_stats': [[1, 1], [2, 51], [3, 8], [4, 100], [5, 24], [6, 408], [10, 24], [12, 800], [15, 192], [30, 192]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': True, 'outer_exponent': 120, 'outer_gen_orders': [2, 2, 6, 4, 4, 2, 2, 24], 'outer_gen_pows': [1, 0, 0, 0, 0, 0, 0, 0], 'outer_gens': [[901, 736, 84], [1, 44, 60], [901, 604, 60], [901, 1224, 1300], [1, 644, 100], [901, 4, 60], [1, 44, 660], [901, 1368, 1504]], 'outer_group': '11520.gn', 'outer_hash': 3840190186149365950, 'outer_nilpotent': False, 'outer_order': 11520, 'outer_permdeg': 15, 'outer_perms': [355459104000, 90727, 2310751, 1574911, 1191774, 1, 2039280, 186852241831], 'outer_solvable': False, 'outer_supersolvable': False, 'outer_tex': 'C_2\\times S_5\\times \\GL(2,3)', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 18, 'pgroup': 0, 'primary_abelian_invariants': [2, 4, 3, 3], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 18], [4, 20], [8, 48]], 'representations': {'PC': {'code': 13739603406365247787550394990058656185715278561617461703, 'gens': [1, 3, 5], 'pres': [7, -2, -2, -3, -5, -2, -3, -5, 14, 1094, 177, 79, 1011, 682, 27304, 19961, 102, 65525, 10092, 166, 52926, 35293]}, 'Perm': {'d': 18, 'gens': [44548150711680, 24, 5760, 3, 21010055270400, 397620712339200, 713990731104000]}}, 'schur_multiplier': [6], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [6, 12], 'solvability_type': 7, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_6\\times C_{15}:F_5', 'transitive_degree': 450, 'wreath_data': None, 'wreath_product': False}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': False, 'abelian_quotient': '4.1', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 20, 'aut_gen_orders': [4, 5], 'aut_gens': [[1, 4], [1, 12], [5, 4]], 'aut_group': '20.3', 'aut_hash': 3, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 20, 'aut_permdeg': 5, 'aut_perms': [18, 34], 'aut_phi_ratio': 2.5, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 5, 1, 1], [4, 5, 1, 2], [5, 4, 1, 1]], 'aut_supersolvable': True, 'aut_tex': 'F_5', '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': 20, 'autcentquo_group': '20.3', 'autcentquo_hash': 3, 'autcentquo_nilpotent': False, 'autcentquo_order': 20, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'F_5', 'cc_stats': [[1, 1, 1], [2, 5, 1], [4, 5, 2], [5, 4, 1]], 'center_label': '1.1', 'center_order': 1, 'central_product': False, 'central_quotient': '20.3', 'commutator_count': 1, 'commutator_label': '5.1', 'complements_known': True, 'complete': True, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '5.1'], 'composition_length': 3, 'conjugacy_classes_known': True, 'counter': 3, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 5, 1, 1], [4, 5, 2, 1], [5, 4, 1, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 12, 'exponent': 20, 'exponents_of_order': [2, 1], 'factors_of_aut_order': [2, 5], 'factors_of_order': [2, 5], 'faithful_reps': [[4, 1, 1]], 'familial': True, 'frattini_label': '1.1', 'frattini_quotient': '20.3', 'hash': 3, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 20, 'inner_gen_orders': [4, 5], 'inner_gens': [[1, 12], [13, 4]], 'inner_hash': 3, 'inner_nilpotent': False, 'inner_order': 20, 'inner_split': True, 'inner_tex': 'F_5', 'inner_used': [1, 2], 'irrC_degree': 4, 'irrQ_degree': 4, 'irrQ_dim': 4, 'irrR_degree': 4, 'irrep_stats': [[1, 4], [4, 1]], 'label': '20.3', 'linC_count': 1, 'linC_degree': 4, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 4, 'linQ_degree_count': 1, 'linQ_dim': 4, 'linQ_dim_count': 1, 'linR_count': 1, 'linR_degree': 4, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'F5', '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': 5, 'number_characteristic_subgroups': 4, 'number_conjugacy_classes': 5, 'number_divisions': 4, 'number_normal_subgroups': 4, 'number_subgroup_autclasses': 6, 'number_subgroup_classes': 6, 'number_subgroups': 14, 'old_label': None, 'order': 20, 'order_factorization_type': 22, 'order_stats': [[1, 1], [2, 5], [4, 10], [5, 4]], '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': 2, 'perfect': False, 'permutation_degree': 5, 'pgroup': 0, 'primary_abelian_invariants': [4], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 2], [2, 1], [4, 1]], 'representations': {'PC': {'code': 2551363, 'gens': [1, 3], 'pres': [3, -2, -2, -5, 6, 110, 77]}, 'GLZ': {'b': 3, 'd': 4, 'gens': [23561683, 28172809]}, 'Lie': [{'d': 1, 'q': 5, 'gens': [33, 10], 'family': 'AGL'}, {'d': 1, 'q': 5, 'gens': [33, 10], 'family': 'AGammaL'}], 'GLFp': {'d': 2, 'p': 5, 'gens': [131, 127]}, 'Perm': {'d': 5, 'gens': [10, 23, 96]}}, 'schur_multiplier': [], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [4], 'solvability_type': 6, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'F_5', 'transitive_degree': 5, 'wreath_data': None, 'wreath_product': False}
