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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': False, 'ambient': '500.45', 'ambient_counter': 45, 'ambient_order': 500, 'ambient_tex': 'C_5^3:C_4', 'central': False, 'central_factor': False, 'centralizer_order': 1, 'characteristic': False, 'core_order': 50, 'counter': 9, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '500.45.5.b1.e1', 'maximal': True, 'maximal_normal': False, 'metabelian': True, 'metacyclic': False, 'minimal': False, 'minimal_normal': False, 'nilpotent': False, 'normal': False, 'old_label': '5.b1.e1', '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': 5, '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': '100.10', 'subgroup_hash': 10, 'subgroup_order': 100, 'subgroup_tex': 'C_5:F_5', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '500.45', 'aut_centralizer_order': 20, 'aut_label': '5.b1', 'aut_quo_index': None, 'aut_stab_index': 30, 'aut_weyl_group': '400.205', 'aut_weyl_index': 600, 'centralizer': '500.a1.a1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['1.a1.a1'], 'contains': ['10.a1.e1', '25.a1.e1', '25.b1.a1'], 'core': '10.a1.e1', 'coset_action_label': None, 'count': 5, 'diagramx': [8364, -1, 2061, -1, 7883, -1, 6207, -1], 'generators': [1, 64, 100, 2], 'label': '500.45.5.b1.e1', 'mobius_quo': None, 'mobius_sub': -1, 'normal_closure': '1.a1.a1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '5.b1.e1', 'old_label': '5.b1.e1', 'projective_image': '500.45', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '5.b1.e1', 'subgroup_fusion': None, 'weyl_group': '100.10'}
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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': 20, 'aut_gen_orders': [4, 20, 5], 'aut_gens': [[1, 4, 20], [1, 12, 20], [13, 4, 60], [21, 4, 20]], 'aut_group': '400.205', 'aut_hash': 205, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 400, 'aut_permdeg': 10, 'aut_perms': [18, 50515, 454320], 'aut_phi_ratio': 10.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 5, 1, 1], [4, 25, 1, 2], [5, 2, 2, 1], [5, 4, 1, 1], [5, 4, 4, 1], [10, 10, 2, 1]], 'aut_supersolvable': True, 'aut_tex': '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': 20, 'autcentquo_group': '400.205', 'autcentquo_hash': 205, 'autcentquo_nilpotent': False, 'autcentquo_order': 400, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'F_5^2', 'cc_stats': [[1, 1, 1], [2, 5, 1], [4, 25, 2], [5, 2, 2], [5, 4, 5], [10, 10, 2]], 'center_label': '1.1', 'center_order': 1, 'central_product': False, 'central_quotient': '100.10', 'commutator_count': 1, 'commutator_label': '25.2', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '5.1', '5.1'], 'composition_length': 4, 'conjugacy_classes_known': True, 'counter': 10, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 5, 1, 1], [4, 25, 2, 1], [5, 2, 2, 1], [5, 4, 1, 1], [5, 4, 4, 1], [10, 10, 2, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 12, 'exponent': 20, 'exponents_of_order': [2, 2], 'factors_of_aut_order': [2, 5], 'factors_of_order': [2, 5], 'faithful_reps': [[4, 0, 4]], 'familial': False, 'frattini_label': '1.1', 'frattini_quotient': '100.10', 'hash': 10, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 20, 'inner_gen_orders': [4, 5, 5], 'inner_gens': [[1, 8, 80], [17, 4, 20], [41, 4, 20]], 'inner_hash': 10, 'inner_nilpotent': False, 'inner_order': 100, 'inner_split': True, 'inner_tex': 'C_5:F_5', 'inner_used': [1, 2, 3], 'irrC_degree': 4, 'irrQ_degree': 16, 'irrQ_dim': 16, 'irrR_degree': 8, 'irrep_stats': [[1, 4], [2, 4], [4, 5]], 'label': '100.10', 'linC_count': 4, 'linC_degree': 4, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 8, 'linQ_degree_count': 2, 'linQ_dim': 8, 'linQ_dim_count': 1, 'linR_count': 2, 'linR_degree': 6, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'C5:F5', 'ngens': 4, 'nilpotency_class': -1, 'nilpotent': False, 'normal_counts': [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': 13, 'number_divisions': 7, 'number_normal_subgroups': 7, 'number_subgroup_autclasses': 13, 'number_subgroup_classes': 13, 'number_subgroups': 56, 'old_label': None, 'order': 100, 'order_factorization_type': 22, 'order_stats': [[1, 1], [2, 5], [4, 50], [5, 24], [10, 20]], 'outer_abelian': True, 'outer_cyclic': True, 'outer_equivalence': False, 'outer_exponent': 4, 'outer_gen_orders': [4], 'outer_gen_pows': [0], 'outer_gens': [[1, 4, 60]], 'outer_group': '4.1', 'outer_hash': 1, 'outer_nilpotent': True, 'outer_order': 4, 'outer_permdeg': 4, 'outer_perms': [17], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_4', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 10, 'pgroup': 0, 'primary_abelian_invariants': [4], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 2], [2, 1], [4, 3], [16, 1]], 'representations': {'PC': {'code': 1970081001743, 'gens': [1, 3, 4], 'pres': [4, -2, -2, -5, -5, 8, 98, 102, 1283]}, 'GLFp': {'d': 3, 'p': 5, 'gens': [118869, 517634, 1786579, 851419]}, 'Perm': {'d': 10, 'gens': [46087, 90720, 37, 493920]}}, 'schur_multiplier': [], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [4], 'solvability_type': 7, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_5: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': 120, 'aut_gen_orders': [30, 4, 24], 'aut_gens': [[1, 4, 20, 100], [129, 56, 44, 100], [205, 84, 88, 200], [37, 60, 24, 400]], 'aut_group': None, 'aut_hash': 1096677362456611443, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 240000, 'aut_permdeg': 125, 'aut_perms': [90188329199825122685400119077962974910882078696631576815857206293392423084537284655802297449963781517920105838715253511824579344645707134078829857428029069346649539708844036075373624625851052317889126023281249, 78255893339403560165866216773153843773740864271627662014566781160638842146831576110288425940874256856823459423760781507504453198286203629135290140455268948060608091736024991152475313813275042584723247977834364, 65957257048922583025691342543405271674596427815169897003891510978702407416953919250738927101071536410733668267480431819090959229455876521014424059935013338354817834589622016672507093020483680950807970330945253], 'aut_phi_ratio': 1200.0, 'aut_solvable': False, 'aut_stats': [[1, 1, 1, 1], [2, 5, 1, 1], [4, 125, 1, 2], [5, 2, 12, 1], [5, 4, 1, 1], [5, 4, 24, 1], [10, 10, 12, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_5^2:C_4.S_5\\times 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': 120, 'autcentquo_group': None, 'autcentquo_hash': 1096677362456611443, 'autcentquo_nilpotent': False, 'autcentquo_order': 240000, 'autcentquo_solvable': False, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'C_5^2:C_4.S_5\\times F_5', 'cc_stats': [[1, 1, 1], [2, 5, 1], [4, 125, 2], [5, 2, 12], [5, 4, 25], [10, 10, 12]], 'center_label': '1.1', 'center_order': 1, 'central_product': False, 'central_quotient': '500.45', 'commutator_count': 1, 'commutator_label': '125.5', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '5.1', '5.1', '5.1'], 'composition_length': 5, 'conjugacy_classes_known': True, 'counter': 45, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 5, 1, 1], [4, 125, 2, 1], [5, 2, 2, 6], [5, 4, 1, 1], [5, 4, 4, 6], [10, 10, 2, 6]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 336, 'exponent': 20, 'exponents_of_order': [3, 2], 'factors_of_aut_order': [2, 3, 5], 'factors_of_order': [2, 5], 'faithful_reps': [], 'familial': False, 'frattini_label': '1.1', 'frattini_quotient': '500.45', 'hash': 45, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 20, 'inner_gen_orders': [4, 5, 5, 5], 'inner_gens': [[1, 16, 80, 200], [9, 4, 20, 100], [41, 4, 20, 100], [401, 4, 20, 100]], 'inner_hash': 45, 'inner_nilpotent': False, 'inner_order': 500, 'inner_split': True, 'inner_tex': 'C_5^3:C_4', 'inner_used': [1, 2, 3, 4], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 4], [2, 24], [4, 25]], 'label': '500.45', 'linC_count': 480, 'linC_degree': 6, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 12, 'linQ_degree_count': 60, 'linQ_dim': 12, 'linQ_dim_count': 15, 'linR_count': 60, 'linR_degree': 8, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'C5^3:C4', 'ngens': 5, 'nilpotency_class': -1, 'nilpotent': False, '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': 7, 'number_conjugacy_classes': 53, 'number_divisions': 22, 'number_normal_subgroups': 25, 'number_subgroup_autclasses': 20, 'number_subgroup_classes': 60, 'number_subgroups': 448, 'old_label': None, 'order': 500, 'order_factorization_type': 32, 'order_stats': [[1, 1], [2, 5], [4, 250], [5, 124], [10, 120]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 120, 'outer_gen_orders': [4, 3], 'outer_gen_pows': [0, 3], 'outer_gens': [[1, 4, 56, 200], [1, 52, 76, 200]], 'outer_group': '480.218', 'outer_hash': 218, 'outer_nilpotent': False, 'outer_order': 480, 'outer_permdeg': 24, 'outer_perms': [310661336687256535137984, 203105797384360893997185], 'outer_solvable': False, 'outer_supersolvable': False, 'outer_tex': '\\GL(2,5)', 'pc_rank': 4, 'perfect': False, 'permutation_degree': 15, 'pgroup': 0, 'primary_abelian_invariants': [4], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 2], [2, 1], [4, 13], [16, 6]], 'representations': {'PC': {'code': 3147888697804031039, 'gens': [1, 3, 4, 5], 'pres': [5, -2, -2, -5, -5, -5, 10, 242, 1603, 5004, 5009]}, 'GLFp': {'d': 4, 'p': 5, 'gens': [110605558821, 30568177402, 119191223885, 132496993126, 3929827501]}, 'Perm': {'d': 15, 'gens': [6266978649, 16, 444240, 99672693840, 46]}}, 'schur_multiplier': [5], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [4], 'solvability_type': 7, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_5^3:C_4', 'transitive_degree': 100, 'wreath_data': None, 'wreath_product': False}
