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gps_subgroup_search • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'ambient': '480.945', 'ambient_counter': 945, 'ambient_order': 480, 'ambient_tex': 'A_5:Q_8', 'central': False, 'central_factor': False, 'centralizer_order': 2, 'characteristic': False, 'core_order': 4, 'counter': 7, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '480.945.6.a1.a1', 'maximal': True, 'maximal_normal': False, 'metabelian': True, 'metacyclic': None, 'minimal': False, 'minimal_normal': False, 'nilpotent': False, 'normal': False, 'old_label': '6.a1.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': 6, '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': '80.31', 'subgroup_hash': 31, 'subgroup_order': 80, 'subgroup_tex': 'C_{20}:C_4', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '480.945', 'aut_centralizer_order': 1, 'aut_label': '6.a1', 'aut_quo_index': None, 'aut_stab_index': 6, 'aut_weyl_group': '160.207', 'aut_weyl_index': 6, 'centralizer': '240.a1.a1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['1.a1.a1'], 'contains': ['12.a1.a1', '12.b1.a1', '12.b1.b1', '30.b1.a1'], 'core': '120.a1.a1', 'coset_action_label': None, 'count': 6, 'diagramx': [2488, -1, 1548, -1, 1391, -1, 1459, -1], 'generators': [1955946942, 16582, 2003114182, 1484190857, 5167], 'label': '480.945.6.a1.a1', 'mobius_quo': None, 'mobius_sub': -1, 'normal_closure': '1.a1.a1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '6.a1.a1', 'old_label': '6.a1.a1', 'projective_image': '240.189', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '6.a1.a1', 'subgroup_fusion': None, 'weyl_group': '40.12'}
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gps_groups • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '8.2', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 20, 'aut_gen_orders': [2, 4, 20], 'aut_gens': [[1, 4], [61, 44], [1, 52], [45, 4]], 'aut_group': '160.207', 'aut_hash': 207, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 160, 'aut_permdeg': 9, 'aut_perms': [1, 6487, 162742], 'aut_phi_ratio': 5.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 5, 1, 2], [4, 2, 1, 1], [4, 10, 1, 1], [4, 10, 2, 2], [5, 4, 1, 1], [10, 4, 1, 1], [20, 4, 2, 1]], 'aut_supersolvable': True, 'aut_tex': 'D_4\\times F_5', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 2, 'autcent_group': '4.2', 'autcent_hash': 2, 'autcent_nilpotent': True, 'autcent_order': 4, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^2', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 20, 'autcentquo_group': '40.12', 'autcentquo_hash': 12, 'autcentquo_nilpotent': False, 'autcentquo_order': 40, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'C_2\\times F_5', 'cc_stats': [[1, 1, 1], [2, 1, 1], [2, 5, 2], [4, 2, 1], [4, 10, 5], [5, 4, 1], [10, 4, 1], [20, 4, 2]], 'center_label': '2.1', 'center_order': 2, 'central_product': False, 'central_quotient': '40.12', 'commutator_count': 1, 'commutator_label': '10.2', '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': 31, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 5, 1, 2], [4, 2, 1, 1], [4, 10, 1, 1], [4, 10, 2, 2], [5, 4, 1, 1], [10, 4, 1, 1], [20, 4, 2, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 12, 'exponent': 20, 'exponents_of_order': [4, 1], 'factors_of_aut_order': [2, 5], 'factors_of_order': [2, 5], 'faithful_reps': [[4, 0, 2]], 'familial': False, 'frattini_label': '2.1', 'frattini_quotient': '40.12', 'hash': 31, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 20, 'inner_gen_orders': [4, 10], 'inner_gens': [[1, 12], [73, 4]], 'inner_hash': 12, 'inner_nilpotent': False, 'inner_order': 40, 'inner_split': False, 'inner_tex': 'C_2\\times F_5', 'inner_used': [1, 2], 'irrC_degree': 4, 'irrQ_degree': 8, 'irrQ_dim': 8, 'irrR_degree': 8, 'irrep_stats': [[1, 8], [2, 2], [4, 4]], 'label': '80.31', 'linC_count': 2, 'linC_degree': 4, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 6, 'linQ_degree_count': 4, 'linQ_dim': 6, 'linQ_dim_count': 2, 'linR_count': 2, 'linR_degree': 6, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C20:C4', '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': 11, 'number_characteristic_subgroups': 12, 'number_conjugacy_classes': 14, 'number_divisions': 11, 'number_normal_subgroups': 14, 'number_subgroup_autclasses': 22, 'number_subgroup_classes': 26, 'number_subgroups': 78, 'old_label': None, 'order': 80, 'order_factorization_type': 31, 'order_stats': [[1, 1], [2, 11], [4, 52], [5, 4], [10, 4], [20, 8]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 2, 'outer_gen_orders': [2, 2], 'outer_gen_pows': [20, 0], 'outer_gens': [[21, 4], [1, 44]], 'outer_group': '4.2', 'outer_hash': 2, 'outer_nilpotent': True, 'outer_order': 4, 'outer_permdeg': 4, 'outer_perms': [1, 6], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^2', 'pc_rank': 2, 'perfect': False, 'permutation_degree': 9, 'pgroup': 0, 'primary_abelian_invariants': [2, 4], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 4], [4, 2], [8, 1]], 'representations': {'PC': {'code': 85858774671193928610819, 'gens': [1, 3], 'pres': [5, -2, -2, -2, -2, -5, 10, 182, 277, 42, 483, 728, 58, 1204, 809]}, 'GLZ': {'b': 3, 'd': 6, 'gens': [25038649734177915, 91655869359786907]}, 'GLFp': {'d': 4, 'p': 3, 'gens': [33433269, 32577956, 27187973, 28816400, 22871412]}, 'Perm': {'d': 9, 'gens': [5887, 13, 11520, 23, 46800]}}, 'schur_multiplier': [2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 4], 'solvability_type': 6, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_{20}:C_4', 'transitive_degree': 20, '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': 2, 'aut_exponent': 60, 'aut_gen_orders': [20, 4, 2], 'aut_gens': [[1524136312, 386622], [1444286142, 123777239], [2039783639, 479030257], [1085039857, 398039]], 'aut_group': '960.10871', 'aut_hash': 10871, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 960, 'aut_permdeg': 40, 'aut_perms': [617973076486915896897498719060402402688584714653, 158062453747847015690907041072486312109538786906, 220501486763789882328164946655462754172747324296], 'aut_phi_ratio': 7.5, 'aut_solvable': False, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 15, 1, 2], [3, 20, 1, 1], [4, 2, 1, 1], [4, 20, 2, 1], [4, 30, 1, 1], [4, 60, 2, 1], [5, 24, 1, 1], [6, 20, 1, 1], [10, 24, 1, 1], [12, 40, 1, 1], [12, 40, 2, 1], [20, 24, 2, 1]], 'aut_supersolvable': False, 'aut_tex': 'D_4\\times S_5', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 2, 'autcent_group': '4.2', 'autcent_hash': 2, 'autcent_nilpotent': True, 'autcent_order': 4, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^2', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 60, 'autcentquo_group': '240.189', 'autcentquo_hash': 189, 'autcentquo_nilpotent': False, 'autcentquo_order': 240, 'autcentquo_solvable': False, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'C_2\\times S_5', 'cc_stats': [[1, 1, 1], [2, 1, 1], [2, 15, 2], [3, 20, 1], [4, 2, 1], [4, 20, 2], [4, 30, 1], [4, 60, 2], [5, 24, 1], [6, 20, 1], [10, 24, 1], [12, 40, 3], [20, 24, 2]], 'center_label': '2.1', 'center_order': 2, 'central_product': False, 'central_quotient': '240.189', 'commutator_count': 1, 'commutator_label': '120.35', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '60.5'], 'composition_length': 4, 'conjugacy_classes_known': True, 'counter': 945, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 15, 1, 2], [3, 20, 1, 1], [4, 2, 1, 1], [4, 20, 1, 2], [4, 30, 1, 1], [4, 60, 1, 2], [5, 24, 1, 1], [6, 20, 1, 1], [10, 24, 1, 1], [12, 40, 1, 3], [20, 24, 2, 1]], 'element_repr_type': 'Perm', 'elementary': 1, 'eulerian_function': 57, 'exponent': 60, 'exponents_of_order': [5, 1, 1], 'factors_of_aut_order': [2, 3, 5], 'factors_of_order': [2, 3, 5], 'faithful_reps': [[6, 0, 2], [8, -1, 1], [10, -1, 1]], 'familial': False, 'frattini_label': '2.1', 'frattini_quotient': '240.189', 'hash': 945, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 60, 'inner_gen_orders': [6, 2], 'inner_gens': [[1524136312, 479030257], [2003495639, 386622]], 'inner_hash': 189, 'inner_nilpotent': False, 'inner_order': 240, 'inner_split': True, 'inner_tex': 'C_2\\times S_5', 'inner_used': [1, 2], 'irrC_degree': 6, 'irrQ_degree': 8, 'irrQ_dim': 12, 'irrR_degree': 12, 'irrep_stats': [[1, 4], [2, 1], [4, 4], [5, 4], [6, 4], [8, 1], [10, 1]], 'label': '480.945', 'linC_count': 6, 'linC_degree': 6, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 6, 'linQ_degree_count': 4, 'linQ_dim': 8, 'linQ_dim_count': 4, 'linR_count': 4, 'linR_degree': 8, 'maximal_subgroups_known': True, 'metabelian': False, 'metacyclic': False, 'monomial': False, 'name': 'A5:Q8', 'ngens': 2, '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': 7, 'number_conjugacy_classes': 19, 'number_divisions': 18, 'number_normal_subgroups': 9, 'number_subgroup_autclasses': 60, 'number_subgroup_classes': 74, 'number_subgroups': 724, 'old_label': None, 'order': 480, 'order_factorization_type': 311, 'order_stats': [[1, 1], [2, 31], [3, 20], [4, 192], [5, 24], [6, 20], [10, 24], [12, 120], [20, 48]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 2, 'outer_gen_orders': [2, 2], 'outer_gen_pows': [11657, 0], 'outer_gens': [[1524124657, 403192], [1524136312, 391537]], 'outer_group': '4.2', 'outer_hash': 2, 'outer_nilpotent': True, 'outer_order': 4, 'outer_permdeg': 4, 'outer_perms': [1, 6], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^2', 'pc_rank': None, 'perfect': False, 'permutation_degree': 13, 'pgroup': 0, 'primary_abelian_invariants': [2, 2], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 1], [4, 4], [5, 4], [6, 2], [8, 1], [10, 1], [12, 1]], 'representations': {'Perm': {'d': 13, 'gens': [1524136312, 386622]}}, 'schur_multiplier': [2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2], 'solvability_type': 13, 'solvable': False, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': 'A_5:Q_8', 'transitive_degree': 24, 'wreath_data': None, 'wreath_product': False}
