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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': True, 'ambient': '900.81', 'ambient_counter': 81, 'ambient_order': 900, 'ambient_tex': 'C_{15}:C_{60}', 'central': False, 'central_factor': False, 'centralizer_order': 300, 'characteristic': False, 'core_order': 150, 'counter': 4, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '900.81.3.b1.a1', 'maximal': True, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': False, 'old_label': '3.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': 3, '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': '300.21', 'subgroup_hash': 21, 'subgroup_order': 300, 'subgroup_tex': 'C_5\\times C_{60}', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '900.81', 'aut_centralizer_order': 2, 'aut_label': '3.b1', 'aut_quo_index': None, 'aut_stab_index': 3, 'aut_weyl_group': '1920.240757', 'aut_weyl_index': 6, 'centralizer': '3.b1.a1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['1.a1.a1'], 'contains': ['6.a1.a1', '9.a1.a1', '15.b1.a1', '15.b1.b1', '15.b1.c1', '15.b1.d1', '15.b1.e1', '15.b1.f1'], 'core': '6.a1.a1', 'coset_action_label': None, 'count': 3, 'diagramx': [7762, -1, 8436, -1, 6804, -1, 6064, -1], 'generators': [15, 12, 180, 40, 30], 'label': '900.81.3.b1.a1', 'mobius_quo': None, 'mobius_sub': -1, 'normal_closure': '1.a1.a1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '3.b1.a1', 'old_label': '3.b1.a1', 'projective_image': '6.1', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '3.b1.a1', 'subgroup_fusion': None, 'weyl_group': '1.1'}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': True, 'abelian_quotient': '300.21', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 1, 'aut_exponent': 120, 'aut_gen_orders': [20, 2, 24], 'aut_gens': [[1, 5], [242, 298], [1, 205], [62, 126]], 'aut_group': '1920.240757', 'aut_hash': 2558370768748321728, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 1920, 'aut_permdeg': 28, 'aut_perms': [10904887679788755130643966718, 15511210043330985984000000, 86305757055930750015387], 'aut_phi_ratio': 24.0, 'aut_solvable': False, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [3, 1, 2, 1], [4, 1, 2, 1], [5, 1, 24, 1], [6, 1, 2, 1], [10, 1, 24, 1], [12, 1, 4, 1], [15, 1, 48, 1], [20, 1, 48, 1], [30, 1, 48, 1], [60, 1, 96, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_2^2\\times \\GL(2,5)', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 120, 'autcent_group': '1920.240757', 'autcent_hash': 2558370768748321728, 'autcent_nilpotent': False, 'autcent_order': 1920, 'autcent_solvable': False, 'autcent_split': True, 'autcent_supersolvable': False, 'autcent_tex': 'C_2^2\\times \\GL(2,5)', '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, 2], [4, 1, 2], [5, 1, 24], [6, 1, 2], [10, 1, 24], [12, 1, 4], [15, 1, 48], [20, 1, 48], [30, 1, 48], [60, 1, 96]], 'center_label': '300.21', 'center_order': 300, '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', '2.1', '3.1', '5.1', '5.1'], 'composition_length': 5, 'conjugacy_classes_known': True, 'counter': 21, 'cyclic': False, 'derived_length': 1, 'dihedral': False, 'direct_factorization': [['3.1', 1], ['4.1', 1], ['5.1', 2]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [3, 1, 2, 1], [4, 1, 2, 1], [5, 1, 4, 6], [6, 1, 2, 1], [10, 1, 4, 6], [12, 1, 4, 1], [15, 1, 8, 6], [20, 1, 8, 6], [30, 1, 8, 6], [60, 1, 16, 6]], 'element_repr_type': 'PC', 'elementary': 5, 'eulerian_function': 24, 'exponent': 60, 'exponents_of_order': [2, 2, 1], 'factors_of_aut_order': [2, 3, 5], 'factors_of_order': [2, 3, 5], 'faithful_reps': [], 'familial': False, 'frattini_label': '2.1', 'frattini_quotient': '150.13', 'hash': 21, 'hyperelementary': 5, 'inner_abelian': True, 'inner_cyclic': True, 'inner_exponent': 1, 'inner_gen_orders': [1, 1], 'inner_gens': [[1, 5], [1, 5]], '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, 300]], 'label': '300.21', 'linC_count': 23040, 'linC_degree': 2, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 12, 'linQ_degree_count': 180, 'linQ_dim': 12, 'linQ_dim_count': 180, 'linR_count': 5760, 'linR_degree': 4, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C5*C60', 'ngens': 5, 'nilpotency_class': 1, 'nilpotent': True, 'normal_counts': [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': 12, 'number_characteristic_subgroups': 12, 'number_conjugacy_classes': 300, 'number_divisions': 42, 'number_normal_subgroups': 48, 'number_subgroup_autclasses': 18, 'number_subgroup_classes': 48, 'number_subgroups': 48, 'old_label': None, 'order': 300, 'order_factorization_type': 222, 'order_stats': [[1, 1], [2, 1], [3, 2], [4, 2], [5, 24], [6, 2], [10, 24], [12, 4], [15, 48], [20, 48], [30, 48], [60, 96]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 120, 'outer_gen_orders': [20, 2, 24], 'outer_gen_pows': [0, 0, 0], 'outer_gens': [[242, 298], [1, 205], [62, 126]], 'outer_group': '1920.240757', 'outer_hash': 2558370768748321728, 'outer_nilpotent': False, 'outer_order': 1920, 'outer_permdeg': 28, 'outer_perms': [10904887679788755130643966718, 15511210043330985984000000, 86305757055930750015387], 'outer_solvable': False, 'outer_supersolvable': False, 'outer_tex': 'C_2^2\\times \\GL(2,5)', 'pc_rank': 2, 'perfect': False, 'permutation_degree': 17, 'pgroup': 0, 'primary_abelian_invariants': [4, 3, 5, 5], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 2], [2, 3], [4, 13], [8, 18], [16, 6]], 'representations': {'PC': {'code': 1535366937351, 'gens': [1, 2], 'pres': [5, -5, -2, -2, -3, -5, 26, 42, 78]}, 'GLFp': {'d': 2, 'p': 61, 'gens': [10441130, 7263394]}, 'Perm': {'d': 17, 'gens': [65383718400000, 958003200, 1451520, 96, 21009968179200]}}, 'schur_multiplier': [5], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [5, 60], 'solvability_type': 1, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_5\\times C_{60}', 'transitive_degree': 300, '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': '300.21', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 120, 'aut_gen_orders': [4, 12, 20, 6, 24], 'aut_gens': [[1, 60], [745, 96], [443, 264], [115, 876], [863, 792], [113, 612]], 'aut_group': None, 'aut_hash': 4906371794315990026, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 11520, 'aut_permdeg': 32, 'aut_perms': [148289191364446483147814043043104621, 260489460934150560129032075647921045, 219566746225314142624289783916137491, 115738306187999572358500895689861791, 117869218935384229592675972229835260], 'aut_phi_ratio': 48.0, 'aut_solvable': False, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [3, 1, 2, 1], [3, 2, 1, 1], [3, 2, 2, 1], [4, 3, 2, 1], [5, 1, 24, 1], [6, 1, 2, 1], [6, 2, 1, 1], [6, 2, 2, 1], [10, 1, 24, 1], [12, 3, 4, 1], [15, 1, 48, 1], [15, 2, 24, 1], [15, 2, 48, 1], [20, 3, 48, 1], [30, 1, 48, 1], [30, 2, 24, 1], [30, 2, 48, 1], [60, 3, 96, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_2^2\\times \\GL(2,5)\\times S_3', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 120, 'autcent_group': '1920.240757', 'autcent_hash': 2558370768748321728, 'autcent_nilpotent': False, 'autcent_order': 1920, 'autcent_solvable': False, 'autcent_split': True, 'autcent_supersolvable': False, 'autcent_tex': 'C_2^2\\times \\GL(2,5)', '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, 1], [3, 1, 2], [3, 2, 3], [4, 3, 2], [5, 1, 24], [6, 1, 2], [6, 2, 3], [10, 1, 24], [12, 3, 4], [15, 1, 48], [15, 2, 72], [20, 3, 48], [30, 1, 48], [30, 2, 72], [60, 3, 96]], 'center_label': '150.13', 'center_order': 150, 'central_product': True, 'central_quotient': '6.1', 'commutator_count': 1, 'commutator_label': '3.1', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '3.1', '3.1', '5.1', '5.1'], 'composition_length': 6, 'conjugacy_classes_known': True, 'counter': 81, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['12.1', 1], ['3.1', 1], ['5.1', 2]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [3, 1, 2, 1], [3, 2, 1, 1], [3, 2, 2, 1], [4, 3, 2, 1], [5, 1, 4, 6], [6, 1, 2, 1], [6, 2, 1, 1], [6, 2, 2, 1], [10, 1, 4, 6], [12, 3, 4, 1], [15, 1, 8, 6], [15, 2, 4, 6], [15, 2, 8, 6], [20, 3, 8, 6], [30, 1, 8, 6], [30, 2, 4, 6], [30, 2, 8, 6], [60, 3, 16, 6]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 24, 'exponent': 60, 'exponents_of_order': [2, 2, 2], 'factors_of_aut_order': [2, 3, 5], 'factors_of_order': [2, 3, 5], 'faithful_reps': [], 'familial': False, 'frattini_label': '2.1', 'frattini_quotient': '450.28', 'hash': 81, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 6, 'inner_gen_orders': [2, 3], 'inner_gens': [[1, 660], [301, 60]], 'inner_hash': 1, 'inner_nilpotent': False, 'inner_order': 6, 'inner_split': True, 'inner_tex': 'S_3', 'inner_used': [1, 2], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 300], [2, 150]], 'label': '900.81', 'linC_count': 23040, 'linC_degree': 3, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 12, 'linQ_degree_count': 180, 'linQ_dim': 12, 'linQ_dim_count': 60, 'linR_count': 11520, 'linR_degree': 6, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C15:C60', '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], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 20, 'number_characteristic_subgroups': 20, 'number_conjugacy_classes': 450, 'number_divisions': 70, 'number_normal_subgroups': 80, 'number_subgroup_autclasses': 42, 'number_subgroup_classes': 112, 'number_subgroups': 160, 'old_label': None, 'order': 900, 'order_factorization_type': 222, 'order_stats': [[1, 1], [2, 1], [3, 8], [4, 6], [5, 24], [6, 8], [10, 24], [12, 12], [15, 192], [20, 144], [30, 192], [60, 288]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 120, 'outer_gen_orders': [20, 2, 24], 'outer_gen_pows': [0, 0, 0], 'outer_gens': [[23, 828], [31, 60], [725, 84]], 'outer_group': '1920.240757', 'outer_hash': 2558370768748321728, 'outer_nilpotent': False, 'outer_order': 1920, 'outer_permdeg': 28, 'outer_perms': [113069567010406815409947863783, 7, 185309093810724208726001096880], 'outer_solvable': False, 'outer_supersolvable': False, 'outer_tex': 'C_2^2\\times \\GL(2,5)', 'pc_rank': 2, 'perfect': False, 'permutation_degree': 20, 'pgroup': 0, 'primary_abelian_invariants': [4, 3, 5, 5], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 2], [2, 5], [4, 15], [8, 30], [16, 18]], 'representations': {'PC': {'code': 22546607849132891664503, 'gens': [1, 5], 'pres': [6, -2, -2, -3, -5, -3, -5, 12, 31, 68, 19804, 118]}, 'GLZN': {'d': 2, 'p': 50, 'gens': [3875031, 188776, 6125049, 2284481, 202031, 4063782]}, 'Perm': {'d': 20, 'gens': [6424691348275200, 518918400, 408960, 816513, 44460928512000, 128047474114560000]}}, 'schur_multiplier': [5], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [5, 60], 'solvability_type': 6, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_{15}:C_{60}', 'transitive_degree': 300, 'wreath_data': None, 'wreath_product': False}
