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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': False, 'ambient': '6000.co', 'ambient_counter': 67, 'ambient_order': 6000, 'ambient_tex': 'A_5\\times C_5:F_5', 'central': False, 'central_factor': False, 'centralizer_order': 4, 'characteristic': False, 'core_order': 10, 'counter': 57, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '6000.co.75.b1.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': None, 'minimal': False, 'minimal_normal': False, 'nilpotent': False, 'normal': False, 'old_label': '75.b1.a1', 'outer_equivalence': False, 'perfect': False, 'proper': True, 'quotient': None, 'quotient_Agroup': None, 'quotient_abelian': None, 'quotient_cyclic': None, 'quotient_hash': None, 'quotient_metabelian': None, 'quotient_nilpotent': None, 'quotient_order': 75, 'quotient_simple': None, 'quotient_solvable': None, 'quotient_supersolvable': None, 'quotient_tex': None, 'simple': False, 'solvable': True, 'special_labels': [], 'split': None, 'standard_generators': False, 'stem': False, 'subgroup': '80.50', 'subgroup_hash': 50, 'subgroup_order': 80, 'subgroup_tex': 'C_2^2\\times F_5', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '6000.co', 'aut_centralizer_order': None, 'aut_label': '75.b1', 'aut_quo_index': None, 'aut_stab_index': None, 'aut_weyl_group': None, 'aut_weyl_index': None, 'centralizer': '1500.e1.a1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['15.a1.a1', '25.c1.a1'], 'contains': ['150.e1.a1', '150.f1.a1', '150.g1.a1', '375.a1.a1'], 'core': '600.a1.a1', 'coset_action_label': None, 'count': 25, 'diagramx': [9442, -1, 1660, -1, 8887, -1, 2022, -1], 'generators': [361686124829, 274028832000, 7185114720, 268719897711, 87218208000], 'label': '6000.co.75.b1.a1', 'mobius_quo': None, 'mobius_sub': 0, 'normal_closure': '1.a1.a1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '25.c1.a1', 'old_label': '75.b1.a1', 'projective_image': '6000.co', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '75.b1.a1', 'subgroup_fusion': None, 'weyl_group': '60.6'}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '16.10', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 3, 'aut_exponent': 60, 'aut_gen_orders': [4, 2, 2, 3, 5, 2, 2], 'aut_gens': [[1, 2, 8], [1, 2, 56], [40, 43, 49], [1, 2, 72], [41, 42, 49], [1, 66, 8], [1, 42, 8], [1, 3, 8]], 'aut_group': '480.1189', 'aut_hash': 1189, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 480, 'aut_permdeg': 9, 'aut_perms': [13, 10800, 23, 10080, 100, 41040, 131760], 'aut_phi_ratio': 15.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 3, 1], [2, 5, 1, 1], [2, 5, 3, 1], [4, 5, 4, 2], [5, 4, 1, 1], [10, 4, 3, 1]], 'aut_supersolvable': False, 'aut_tex': 'F_5\\times S_4', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 12, 'autcent_group': '24.12', 'autcent_hash': 12, 'autcent_nilpotent': False, 'autcent_order': 24, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': False, 'autcent_tex': 'S_4', '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, 1, 3], [2, 5, 4], [4, 5, 8], [5, 4, 1], [10, 4, 3]], 'center_label': '4.2', 'center_order': 4, 'central_product': True, 'central_quotient': '20.3', 'commutator_count': 1, 'commutator_label': '5.1', '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': 50, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['2.1', 2], ['20.3', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [2, 5, 1, 4], [4, 5, 2, 4], [5, 4, 1, 1], [10, 4, 1, 3]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 336, 'exponent': 20, 'exponents_of_order': [4, 1], 'factors_of_aut_order': [2, 3, 5], 'factors_of_order': [2, 5], 'faithful_reps': [], 'familial': False, 'frattini_label': '1.1', 'frattini_quotient': '80.50', 'hash': 50, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 20, 'inner_gen_orders': [1, 4, 5], 'inner_gens': [[1, 2, 8], [1, 2, 24], [1, 66, 8]], 'inner_hash': 3, 'inner_nilpotent': False, 'inner_order': 20, 'inner_split': True, 'inner_tex': 'F_5', 'inner_used': [2, 3], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 16], [4, 4]], 'label': '80.50', 'linC_count': 24, 'linC_degree': 5, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 5, 'linQ_degree_count': 12, 'linQ_dim': 5, 'linQ_dim_count': 12, 'linR_count': 12, 'linR_degree': 5, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'C2^2*F5', 'ngens': 5, 'nilpotency_class': -1, 'nilpotent': False, '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': 8, 'number_characteristic_subgroups': 7, 'number_conjugacy_classes': 20, 'number_divisions': 16, 'number_normal_subgroups': 32, 'number_subgroup_autclasses': 22, 'number_subgroup_classes': 54, 'number_subgroups': 142, 'old_label': None, 'order': 80, 'order_factorization_type': 31, 'order_stats': [[1, 1], [2, 23], [4, 40], [5, 4], [10, 12]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 12, 'outer_gen_orders': [2, 3, 2, 2], 'outer_gen_pows': [0, 0, 0, 0], 'outer_gens': [[1, 2, 9], [41, 42, 49], [1, 43, 8], [1, 42, 8]], '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': 9, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 4], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 8], [2, 4], [4, 4]], 'representations': {'PC': {'code': 167692811997030403, 'gens': [1, 2, 4], 'pres': [5, -2, -2, -2, -2, -5, 26, 248, 373, 58, 609, 414]}, 'GLZ': {'b': 3, 'd': 5, 'gens': [141602720900, 706086597970, 706303244379]}, 'GLFp': {'d': 3, 'p': 5, 'gens': [1152128, 1114095, 497805, 272416, 1565004]}, 'Perm': {'d': 9, 'gens': [5887, 16, 23, 11520, 46800]}}, 'schur_multiplier': [2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 4], 'solvability_type': 7, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_2^2\\times F_5', 'transitive_degree': 20, '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': '4.1', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 60, 'aut_gen_orders': [5, 4, 20, 10, 60, 4, 4, 2], 'aut_gens': [[100630252800, 368352594011, 7185114720, 1099440, 87218208000, 7185387626], [100630252800, 368353404731, 7185871440, 1099440, 87218208000, 7186198346], [193516646400, 105899634011, 368352321120, 1099440, 361686124800, 368352594026], [100630252800, 105900117851, 261534964320, 1532160, 361686124800, 261535721066], [349671168000, 261536376251, 105899634000, 1099440, 186810624000, 105900773066], [100630252800, 181102612376, 368353733040, 1532160, 268719897600, 368352321136], [274028832000, 181102612419, 7186526640, 1532160, 268719897600, 7185114806], [100630252800, 368352321131, 7186526640, 1532160, 87218208000, 7185114746], [100630252800, 368353733051, 7185871440, 493920, 87218208000, 7186526666]], 'aut_group': '48000.bb', 'aut_hash': 1488085773162693681, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 48000, 'aut_permdeg': 40, 'aut_perms': [580583402454223323737047660825209145711666609117, 629747531569703575840122853122048632264695807861, 405370826505978636154371236831247782575465573447, 383403871726815544318374966988300885063004733847, 235994121569839702997150449370545216756515195583, 378973923792177405625904826257560084100558006059, 800775920308861752699194615299182213961668546249, 662174926645479202247864093815743436061465149209], 'aut_phi_ratio': 30.0, 'aut_solvable': False, 'aut_stats': [[1, 1, 1, 1], [2, 5, 1, 1], [2, 15, 1, 1], [2, 75, 1, 1], [3, 20, 1, 1], [4, 25, 1, 2], [4, 375, 1, 2], [5, 2, 2, 1], [5, 4, 1, 1], [5, 4, 4, 1], [5, 12, 2, 1], [5, 24, 4, 1], [5, 48, 2, 1], [5, 48, 8, 1], [6, 100, 1, 1], [10, 10, 2, 1], [10, 30, 2, 1], [10, 60, 1, 1], [10, 60, 2, 1], [10, 60, 4, 1], [10, 120, 4, 1], [10, 150, 2, 1], [12, 500, 1, 2], [15, 40, 2, 1], [15, 80, 1, 1], [15, 80, 4, 1], [20, 300, 2, 2], [30, 200, 2, 1]], 'aut_supersolvable': False, 'aut_tex': 'S_5\\times F_5^2', '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': 60, 'autcentquo_group': '48000.bb', 'autcentquo_hash': 1488085773162693681, 'autcentquo_nilpotent': False, 'autcentquo_order': 48000, 'autcentquo_solvable': False, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'S_5\\times F_5^2', 'cc_stats': [[1, 1, 1], [2, 5, 1], [2, 15, 1], [2, 75, 1], [3, 20, 1], [4, 25, 2], [4, 375, 2], [5, 2, 2], [5, 4, 5], [5, 12, 2], [5, 24, 4], [5, 48, 10], [6, 100, 1], [10, 10, 2], [10, 30, 2], [10, 60, 7], [10, 120, 4], [10, 150, 2], [12, 500, 2], [15, 40, 2], [15, 80, 5], [20, 300, 4], [30, 200, 2]], 'center_label': '1.1', 'center_order': 1, 'central_product': True, 'central_quotient': '6000.co', 'commutator_count': 1, 'commutator_label': '1500.112', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '5.1', '5.1', '60.5'], 'composition_length': 5, 'conjugacy_classes_known': True, 'counter': 67, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['100.10', 1], ['60.5', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 5, 1, 1], [2, 15, 1, 1], [2, 75, 1, 1], [3, 20, 1, 1], [4, 25, 2, 1], [4, 375, 2, 1], [5, 2, 2, 1], [5, 4, 1, 1], [5, 4, 4, 1], [5, 12, 2, 1], [5, 24, 2, 2], [5, 48, 2, 1], [5, 48, 4, 2], [6, 100, 1, 1], [10, 10, 2, 1], [10, 30, 2, 1], [10, 60, 1, 1], [10, 60, 2, 1], [10, 60, 4, 1], [10, 120, 2, 2], [10, 150, 2, 1], [12, 500, 2, 1], [15, 40, 2, 1], [15, 80, 1, 1], [15, 80, 4, 1], [20, 300, 4, 1], [30, 200, 2, 1]], 'element_repr_type': 'Perm', 'elementary': 1, 'eulerian_function': 228, 'exponent': 60, 'exponents_of_order': [4, 3, 1], 'factors_of_aut_order': [2, 3, 5], 'factors_of_order': [2, 3, 5], 'faithful_reps': [[12, 0, 8], [16, 0, 4], [20, 0, 4]], 'familial': False, 'frattini_label': '1.1', 'frattini_quotient': '6000.co', 'hash': 5120809764824495460, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 60, 'inner_gen_orders': [5, 12, 4, 5, 2, 4], 'inner_gens': [[100630252800, 261535237211, 181102612320, 1099440, 186810624000, 181102885226], [193516646400, 368352594011, 261536376240, 493920, 87218208000, 261535237255], [193516646400, 261536047931, 7185114720, 493920, 186810624000, 7186198346], [100630252800, 368353404731, 7185871440, 1099440, 87218208000, 7186198346], [349671168000, 368352594011, 181102612320, 1099440, 87218208000, 181102885226], [193516646400, 261535237281, 7186526640, 493920, 186810624000, 7185387626]], 'inner_hash': 5120809764824495460, 'inner_nilpotent': False, 'inner_order': 6000, 'inner_split': True, 'inner_tex': 'A_5\\times C_5:F_5', 'inner_used': [1, 2, 3, 6], 'irrC_degree': 12, 'irrQ_degree': 48, 'irrQ_dim': 48, 'irrR_degree': 24, 'irrep_stats': [[1, 4], [2, 4], [3, 8], [4, 9], [5, 4], [6, 8], [8, 4], [10, 4], [12, 10], [16, 5], [20, 5]], 'label': '6000.co', 'linC_count': 32, 'linC_degree': 7, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 12, 'linQ_degree_count': 4, 'linQ_dim': 12, 'linQ_dim_count': 2, 'linR_count': 8, 'linR_degree': 9, 'maximal_subgroups_known': True, 'metabelian': False, 'metacyclic': False, 'monomial': False, 'name': 'A5*C5:F5', '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, 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': 32, 'number_characteristic_subgroups': 14, 'number_conjugacy_classes': 65, 'number_divisions': 31, 'number_normal_subgroups': 14, 'number_subgroup_autclasses': 161, 'number_subgroup_classes': 169, 'number_subgroups': 6904, 'old_label': None, 'order': 6000, 'order_factorization_type': 321, 'order_stats': [[1, 1], [2, 95], [3, 20], [4, 800], [5, 624], [6, 100], [10, 1280], [12, 1000], [15, 480], [20, 1200], [30, 400]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 4, 'outer_gen_orders': [2, 4], 'outer_gen_pows': [119, 0], 'outer_gens': [[349671168000, 105899634003, 7185114720, 1099440, 361686124800, 7185387667], [100630252800, 368353733051, 7185114720, 812160, 87218208000, 7186526666]], 'outer_group': '8.2', 'outer_hash': 2, 'outer_nilpotent': True, 'outer_order': 8, 'outer_permdeg': 6, 'outer_perms': [120, 17], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2\\times C_4', 'pc_rank': None, 'perfect': False, 'permutation_degree': 15, 'pgroup': 0, 'primary_abelian_invariants': [4], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 2], [2, 1], [4, 5], [5, 2], [6, 2], [8, 1], [10, 1], [12, 5], [16, 4], [20, 3], [24, 1], [48, 2], [64, 1], [80, 1]], 'representations': {'Perm': {'d': 15, 'gens': [100630252800, 368352594011, 7185114720, 1099440, 87218208000, 7185387626]}}, 'schur_multiplier': [2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [4], 'solvability_type': 13, 'solvable': False, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': 'A_5\\times C_5:F_5', 'transitive_degree': 100, 'wreath_data': None, 'wreath_product': False}
