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gps_subgroup_search • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'ambient': '320.976', 'ambient_counter': 976, 'ambient_order': 320, 'ambient_tex': 'C_{20}.\\SD_{16}', 'central': False, 'central_factor': False, 'centralizer_order': 20, 'characteristic': False, 'core_order': 40, 'counter': 17, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '320.976.4.g1.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': False, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': False, 'old_label': '4.g1.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': 4, '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': '80.47', 'subgroup_hash': 47, 'subgroup_order': 80, 'subgroup_tex': 'Q_8\\times C_{10}', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '320.976', 'aut_centralizer_order': 8, 'aut_label': '4.g1', 'aut_quo_index': None, 'aut_stab_index': 2, 'aut_weyl_group': '128.1031', 'aut_weyl_index': 16, 'centralizer': '16.a1.a1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['2.b1.a1'], 'contains': ['8.a1.a1', '8.g1.a1', '8.g1.b1', '8.h1.a1', '8.h1.b1', '20.g1.a1'], 'core': '8.a1.a1', 'coset_action_label': None, 'count': 2, 'diagramx': [1489, -1, 1275, -1, 1282, -1, 1530, -1], 'generators': [3, 80, 64, 8, 160], 'label': '320.976.4.g1.a1', 'mobius_quo': None, 'mobius_sub': 0, 'normal_closure': '2.b1.a1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '2.b1.a1', 'old_label': '4.g1.a1', 'projective_image': '16.11', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '4.g1.a1', 'subgroup_fusion': None, 'weyl_group': '8.5'}
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gps_groups • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '40.14', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 3, 'aut_exponent': 12, 'aut_gen_orders': [2, 2, 4, 2, 3, 2, 2, 2, 2], 'aut_gens': [[1, 2, 4], [1, 63, 76], [41, 42, 76], [1, 2, 68], [1, 2, 36], [1, 23, 66], [1, 43, 45], [1, 43, 4], [1, 42, 44], [1, 42, 4]], 'aut_group': '768.1087219', 'aut_hash': 1087219, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 768, 'aut_permdeg': 12, 'aut_perms': [298746840, 299185320, 17, 7, 18235440, 143354280, 167368440, 179788320, 223347720], 'aut_phi_ratio': 24.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 1, 2, 1], [4, 2, 6, 1], [5, 1, 4, 1], [10, 1, 4, 1], [10, 1, 8, 1], [20, 2, 24, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_4\\times C_2^3:S_4', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 4, 'autcent_group': '128.1031', 'autcent_hash': 1031, 'autcent_nilpotent': True, 'autcent_order': 128, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^3.C_2^4', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 6, 'autcentquo_group': '6.1', 'autcentquo_hash': 1, 'autcentquo_nilpotent': False, 'autcentquo_order': 6, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'S_3', 'cc_stats': [[1, 1, 1], [2, 1, 3], [4, 2, 6], [5, 1, 4], [10, 1, 12], [20, 2, 24]], 'center_label': '20.5', 'center_order': 20, 'central_product': True, 'central_quotient': '4.2', 'commutator_count': 1, 'commutator_label': '2.1', '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': 47, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['2.1', 1], ['5.1', 1], ['8.4', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [4, 2, 1, 6], [5, 1, 4, 1], [10, 1, 4, 3], [20, 2, 4, 6]], 'element_repr_type': 'PC', 'elementary': 2, 'eulerian_function': 217, 'exponent': 20, 'exponents_of_order': [4, 1], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 5], 'faithful_reps': [], 'familial': False, 'frattini_label': '2.1', 'frattini_quotient': '40.14', 'hash': 47, 'hyperelementary': 2, 'inner_abelian': True, 'inner_cyclic': False, 'inner_exponent': 2, 'inner_gen_orders': [1, 2, 2], 'inner_gens': [[1, 2, 4], [1, 2, 44], [1, 42, 4]], 'inner_hash': 2, 'inner_nilpotent': True, 'inner_order': 4, 'inner_split': False, 'inner_tex': 'C_2^2', 'inner_used': [2, 3], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 40], [2, 10]], 'label': '80.47', 'linC_count': 192, 'linC_degree': 3, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 6, 'linQ_degree_count': 8, 'linQ_dim': 8, 'linQ_dim_count': 8, 'linR_count': 16, 'linR_degree': 5, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'Q8*C10', 'ngens': 5, 'nilpotency_class': 2, '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': 8, 'number_characteristic_subgroups': 8, 'number_conjugacy_classes': 50, 'number_divisions': 20, 'number_normal_subgroups': 38, 'number_subgroup_autclasses': 16, 'number_subgroup_classes': 38, 'number_subgroups': 38, '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, 2, 12, 2, 2], 'outer_gen_pows': [4, 0, 0, 0, 0], 'outer_gens': [[1, 22, 4], [41, 2, 4], [1, 20, 70], [1, 3, 5], [1, 3, 4]], 'outer_group': '192.1469', 'outer_hash': 1469, 'outer_nilpotent': False, 'outer_order': 192, 'outer_permdeg': 10, 'outer_perms': [5040, 1, 80791, 1174320, 367920], 'outer_solvable': True, 'outer_supersolvable': False, 'outer_tex': 'C_2^4.D_6', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 15, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 2, 5], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 8], [2, 2], [4, 8], [8, 2]], 'representations': {'PC': {'code': 60243744929795, 'gens': [1, 2, 3], 'pres': [5, -2, -2, -2, -2, -5, 206, 337, 42, 58]}, 'GLFp': {'d': 3, 'p': 11, 'gens': [1219466810, 326043207, 2184411956, 1234597583, 2143735230]}, 'Perm': {'d': 15, 'gens': [101671758720, 194163338880, 720, 96, 288848891520]}}, 'schur_multiplier': [2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 10], 'solvability_type': 4, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'Q_8\\times C_{10}', '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': '40.14', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 4, 'aut_gen_orders': [4, 4, 2, 4, 4, 2, 4], 'aut_gens': [[1, 2, 16], [5, 246, 152], [13, 86, 208], [1, 6, 144], [165, 170, 16], [13, 6, 112], [161, 254, 176], [5, 170, 272]], 'aut_group': None, 'aut_hash': 2079752091238468872, 'aut_nilpotency_class': 2, 'aut_nilpotent': True, 'aut_order': 2048, 'aut_permdeg': 28, 'aut_perms': [78625550041875772795801990772, 78625446378196794604549552652, 55205715052930840516367038645, 42363414081390272349108899822, 33488933257605600044399347046, 55206155408755127655823643070, 76595390098268414356644823015], 'aut_phi_ratio': 16.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [4, 2, 1, 2], [4, 2, 2, 1], [4, 4, 1, 1], [4, 4, 2, 2], [4, 8, 2, 1], [5, 1, 4, 1], [8, 4, 4, 1], [10, 1, 4, 3], [20, 2, 4, 2], [20, 2, 8, 1], [20, 4, 4, 1], [20, 4, 8, 2], [20, 8, 8, 1], [40, 4, 16, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_2^6.C_2^5', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 4, 'autcent_group': '256.56082', 'autcent_hash': 56082, 'autcent_nilpotent': True, 'autcent_order': 256, 'autcent_solvable': True, 'autcent_split': False, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^6\\times C_4', 'autcentquo_abelian': True, 'autcentquo_cyclic': False, 'autcentquo_exponent': 2, 'autcentquo_group': '8.5', 'autcentquo_hash': 5, 'autcentquo_nilpotent': True, 'autcentquo_order': 8, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'C_2^3', 'cc_stats': [[1, 1, 1], [2, 1, 3], [4, 2, 4], [4, 4, 5], [4, 8, 2], [5, 1, 4], [8, 4, 4], [10, 1, 12], [20, 2, 16], [20, 4, 20], [20, 8, 8], [40, 4, 16]], 'center_label': '20.5', 'center_order': 20, 'central_product': True, 'central_quotient': '16.11', 'commutator_count': 1, 'commutator_label': '8.2', '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': 976, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['5.1', 1], ['64.156', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [4, 2, 1, 4], [4, 4, 1, 3], [4, 4, 2, 1], [4, 8, 1, 2], [5, 1, 4, 1], [8, 4, 2, 2], [10, 1, 4, 3], [20, 2, 4, 4], [20, 4, 4, 3], [20, 4, 8, 1], [20, 8, 4, 2], [40, 4, 8, 2]], 'element_repr_type': 'PC', 'elementary': 2, 'eulerian_function': 5208, 'exponent': 40, 'exponents_of_order': [6, 1], 'factors_of_aut_order': [2], 'factors_of_order': [2, 5], 'faithful_reps': [], 'familial': False, 'frattini_label': '8.2', 'frattini_quotient': '40.14', 'hash': 976, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 4, 'inner_gen_orders': [2, 4, 2], 'inner_gens': [[1, 166, 16], [173, 2, 176], [1, 162, 16]], 'inner_hash': 11, 'inner_nilpotent': True, 'inner_order': 16, 'inner_split': True, 'inner_tex': 'C_2\\times D_4', 'inner_used': [1, 2, 3], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 40], [2, 50], [4, 5]], 'label': '320.976', 'linC_count': 384, 'linC_degree': 4, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 10, 'linQ_degree_count': 48, 'linQ_dim': 12, 'linQ_dim_count': 52, 'linR_count': 104, 'linR_degree': 8, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'C20.SD16', 'ngens': 7, 'nilpotency_class': 3, 'nilpotent': True, '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': 24, 'number_characteristic_subgroups': 34, 'number_conjugacy_classes': 95, 'number_divisions': 32, 'number_normal_subgroups': 58, 'number_subgroup_autclasses': 74, 'number_subgroup_classes': 96, 'number_subgroups': 154, 'old_label': None, 'order': 320, 'order_factorization_type': 51, 'order_stats': [[1, 1], [2, 3], [4, 44], [5, 4], [8, 16], [10, 12], [20, 176], [40, 64]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 4, 'outer_gen_orders': [2, 2, 2, 4, 4], 'outer_gen_pows': [0, 0, 0, 0, 0], 'outer_gens': [[1, 2, 304], [161, 10, 184], [161, 2, 176], [161, 10, 272], [161, 82, 152]], 'outer_group': '128.2154', 'outer_hash': 2154, 'outer_nilpotent': True, 'outer_order': 128, 'outer_permdeg': 12, 'outer_perms': [87096241, 83462401, 87096240, 408, 119755440], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_4^2:C_2^3', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 21, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 2, 5], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 8], [2, 4], [4, 12], [8, 4], [16, 4]], 'representations': {'PC': {'code': 5336620947854534684479219154947, 'gens': [1, 2, 5], 'pres': [7, -2, -2, -2, -2, -2, -2, -5, 56, 2325, 36, 254, 58, 3091, 102, 124]}, 'GLZN': {'d': 2, 'p': 33, 'gens': [898450, 826574, 359380, 1065266, 48302, 794497, 28083]}, 'Perm': {'d': 21, 'gens': [26740633725712080, 2446418218709133360, 5238567056346624000, 96, 1614937680, 7811314463964211200, 2446418217622579200]}}, 'schur_multiplier': [2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 10], 'solvability_type': 4, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_{20}.\\SD_{16}', 'transitive_degree': 320, 'wreath_data': None, 'wreath_product': False}
