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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': True, 'ambient': '360.91', 'ambient_counter': 91, 'ambient_order': 360, 'ambient_tex': 'C_{60}:C_6', 'central': True, 'central_factor': False, 'centralizer_order': 360, 'characteristic': True, 'core_order': 1, 'counter': 96, 'cyclic': True, 'direct': True, 'hall': 1, 'label': '360.91.360.a1.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': True, 'old_label': '360.a1.a1', 'outer_equivalence': False, 'perfect': True, 'proper': False, 'quotient': '360.91', 'quotient_Agroup': True, 'quotient_abelian': False, 'quotient_cyclic': False, 'quotient_hash': 91, 'quotient_metabelian': True, 'quotient_nilpotent': False, 'quotient_order': 360, 'quotient_simple': False, 'quotient_solvable': True, 'quotient_supersolvable': True, 'quotient_tex': 'C_{60}:C_6', 'simple': False, 'solvable': True, 'special_labels': ['PC', 'U1', 'D2', 'C6'], 'split': True, 'standard_generators': False, 'stem': True, 'subgroup': '1.1', 'subgroup_hash': 1, 'subgroup_order': 1, 'subgroup_tex': 'C_1', 'supersolvable': True, 'sylow': 1}
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gps_subgroup_data • Show schema
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{'ambient': '360.91', 'aut_centralizer_order': None, 'aut_label': '360.a1', 'aut_quo_index': None, 'aut_stab_index': None, 'aut_weyl_group': None, 'aut_weyl_index': None, 'centralizer': '1.a1.a1', 'complements': ['1.a1.a1'], 'conjugacy_class_count': 1, 'contained_in': ['72.a1.a1', '120.a1.a1', '120.a1.b1', '120.a1.c1', '120.a1.d1', '180.a1.a1', '180.b1.a1', '180.b1.b1'], 'contains': [], 'core': '360.a1.a1', 'coset_action_label': None, 'count': 1, 'diagramx': [4021, 5311, 2261, 3988, 5091, 4621, 3543, 5377], 'generators': [], 'label': '360.91.360.a1.a1', 'mobius_quo': 1, 'mobius_sub': 0, 'normal_closure': '360.a1.a1', 'normal_contained_in': ['72.a1.a1', '120.a1.c1', '120.a1.d1', '120.a1.b1', '120.a1.a1', '180.a1.a1'], 'normal_contains': [], 'normalizer': '1.a1.a1', 'old_label': '360.a1.a1', 'projective_image': '360.91', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '360.a1.a1', 'subgroup_fusion': None, 'weyl_group': '1.1'}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': True, 'abelian_quotient': '1.1', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': True, 'aut_cyclic': True, 'aut_derived_length': 0, 'aut_exponent': 1, 'aut_gen_orders': [], 'aut_gens': [], 'aut_group': '1.1', 'aut_hash': 1, 'aut_nilpotency_class': 0, 'aut_nilpotent': True, 'aut_order': 1, 'aut_permdeg': 1, 'aut_perms': [], 'aut_phi_ratio': 1.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_1', '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': 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]], 'center_label': '1.1', 'center_order': 1, 'central_product': False, 'central_quotient': '1.1', 'commutator_count': 0, 'commutator_label': '1.1', 'complements_known': True, 'complete': True, 'complex_characters_known': True, 'composition_factors': [], 'composition_length': 0, 'conjugacy_classes_known': True, 'counter': 1, 'cyclic': True, 'derived_length': 0, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 1, 'exponent': 1, 'exponents_of_order': [], 'factors_of_aut_order': [], 'factors_of_order': [], 'faithful_reps': [[1, 1, 1]], 'familial': True, 'frattini_label': '1.1', 'frattini_quotient': '1.1', 'hash': 1, 'hyperelementary': 1, 'inner_abelian': True, 'inner_cyclic': True, 'inner_exponent': 1, 'inner_gen_orders': [], 'inner_gens': [], '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, 1]], 'label': '1.1', 'linC_count': 1, 'linC_degree': 0, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 0, 'linQ_degree_count': 1, 'linQ_dim': 0, 'linQ_dim_count': 1, 'linR_count': 1, 'linR_degree': 0, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C1', 'ngens': 0, 'nilpotency_class': 0, 'nilpotent': True, 'normal_counts': [1], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 1, 'number_characteristic_subgroups': 1, 'number_conjugacy_classes': 1, 'number_divisions': 1, 'number_normal_subgroups': 1, 'number_subgroup_autclasses': 1, 'number_subgroup_classes': 1, 'number_subgroups': 1, 'old_label': None, 'order': 1, 'order_factorization_type': 0, 'order_stats': [[1, 1]], 'outer_abelian': True, 'outer_cyclic': True, 'outer_equivalence': False, 'outer_exponent': 1, 'outer_gen_orders': [], 'outer_gen_pows': [], 'outer_gens': [], 'outer_group': '1.1', 'outer_hash': 1, 'outer_nilpotent': True, 'outer_order': 1, 'outer_permdeg': 1, 'outer_perms': [], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_1', 'pc_rank': 0, 'perfect': True, 'permutation_degree': 1, 'pgroup': 1, 'primary_abelian_invariants': [], 'quasisimple': False, 'rank': 0, 'rational': True, 'rational_characters_known': True, 'ratrep_stats': [[1, 1]], 'representations': {'PC': {'code': 0, 'gens': [], 'pres': []}}, 'schur_multiplier': [], 'semidirect_product': False, 'simple': False, 'smith_abelian_invariants': [], 'solvability_type': 0, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_1', 'transitive_degree': 1, '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': '72.36', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 4, 'aut_exponent': 120, 'aut_gen_orders': [5, 8, 6, 8, 12], 'aut_gens': [[1, 6], [289, 6], [245, 140], [207, 356], [51, 356], [301, 202]], 'aut_group': None, 'aut_hash': 9123710861761989481, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 3840, 'aut_permdeg': 20, 'aut_perms': [615049234505730651, 798625487423288171, 987180568900246013, 438208205573942939, 771950237957877609], 'aut_phi_ratio': 40.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 5, 2, 1], [3, 1, 8, 1], [4, 1, 2, 1], [4, 5, 2, 1], [5, 2, 2, 1], [6, 1, 8, 1], [6, 5, 16, 1], [10, 2, 2, 1], [12, 1, 16, 1], [12, 5, 16, 1], [15, 2, 16, 1], [20, 2, 4, 1], [30, 2, 16, 1], [60, 2, 32, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_2^2\\times \\GL(2,3)\\times F_5', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 24, 'autcent_group': '192.1475', 'autcent_hash': 1475, 'autcent_nilpotent': False, 'autcent_order': 192, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': False, 'autcent_tex': 'C_2^2\\times \\GL(2,3)', '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, 1], [2, 5, 2], [3, 1, 8], [4, 1, 2], [4, 5, 2], [5, 2, 2], [6, 1, 8], [6, 5, 16], [10, 2, 2], [12, 1, 16], [12, 5, 16], [15, 2, 16], [20, 2, 4], [30, 2, 16], [60, 2, 32]], 'center_label': '36.8', 'center_order': 36, 'central_product': True, 'central_quotient': '10.1', 'commutator_count': 1, 'commutator_label': '5.1', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '3.1', '3.1', '5.1'], 'composition_length': 6, 'conjugacy_classes_known': True, 'counter': 91, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['10.1', 1], ['3.1', 2], ['4.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 5, 1, 2], [3, 1, 2, 4], [4, 1, 2, 1], [4, 5, 2, 1], [5, 2, 2, 1], [6, 1, 2, 4], [6, 5, 2, 8], [10, 2, 2, 1], [12, 1, 4, 4], [12, 5, 4, 4], [15, 2, 4, 4], [20, 2, 4, 1], [30, 2, 4, 4], [60, 2, 8, 4]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 6, 'exponent': 60, 'exponents_of_order': [3, 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': '180.32', 'hash': 91, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 10, 'inner_gen_orders': [2, 5], 'inner_gens': [[1, 294], [73, 6]], 'inner_hash': 1, 'inner_nilpotent': False, 'inner_order': 10, 'inner_split': False, 'inner_tex': 'D_5', 'inner_used': [1, 2], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 72], [2, 72]], 'label': '360.91', 'linC_count': 2304, 'linC_degree': 3, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 10, 'linQ_degree_count': 576, 'linQ_dim': 10, 'linQ_dim_count': 576, 'linR_count': 1728, 'linR_degree': 6, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C60:C6', '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], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 16, 'number_characteristic_subgroups': 18, 'number_conjugacy_classes': 144, 'number_divisions': 45, 'number_normal_subgroups': 66, 'number_subgroup_autclasses': 42, 'number_subgroup_classes': 96, 'number_subgroups': 216, 'old_label': None, 'order': 360, 'order_factorization_type': 321, 'order_stats': [[1, 1], [2, 11], [3, 8], [4, 12], [5, 4], [6, 88], [10, 4], [12, 96], [15, 32], [20, 8], [30, 32], [60, 64]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 24, 'outer_gen_orders': [2, 2, 2, 6, 4, 4], 'outer_gen_pows': [0, 0, 0, 1, 0, 0], 'outer_gens': [[181, 6], [1, 186], [5, 6], [1, 44], [125, 8], [245, 10]], 'outer_group': '384.20055', 'outer_hash': 20055, 'outer_nilpotent': False, 'outer_order': 384, 'outer_permdeg': 14, 'outer_perms': [126, 6267744001, 135359406, 167696208, 13452369840, 34184430000], 'outer_solvable': True, 'outer_supersolvable': False, 'outer_tex': 'C_2^3\\times \\GL(2,3)', 'pc_rank': 2, 'perfect': False, 'permutation_degree': 15, 'pgroup': 0, 'primary_abelian_invariants': [2, 4, 3, 3], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 18], [4, 10], [8, 9], [16, 4]], 'representations': {'PC': {'code': 10106573066898107046993711367, 'gens': [1, 3], 'pres': [6, -2, -3, -2, -2, -3, -5, 12, 5294, 50, 5475, 69, 2884, 118, 10373]}, 'GLZN': {'d': 2, 'p': 30, 'gens': [27301, 189007, 27019, 40966, 27181, 513019]}, 'Perm': {'d': 15, 'gens': [6266937600, 864, 3, 403203, 1680, 99672249600]}}, 'schur_multiplier': [6], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [6, 12], 'solvability_type': 6, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_{60}:C_6', 'transitive_degree': 180, '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': '72.36', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 4, 'aut_exponent': 120, 'aut_gen_orders': [5, 8, 6, 8, 12], 'aut_gens': [[1, 6], [289, 6], [245, 140], [207, 356], [51, 356], [301, 202]], 'aut_group': None, 'aut_hash': 9123710861761989481, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 3840, 'aut_permdeg': 20, 'aut_perms': [615049234505730651, 798625487423288171, 987180568900246013, 438208205573942939, 771950237957877609], 'aut_phi_ratio': 40.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 5, 2, 1], [3, 1, 8, 1], [4, 1, 2, 1], [4, 5, 2, 1], [5, 2, 2, 1], [6, 1, 8, 1], [6, 5, 16, 1], [10, 2, 2, 1], [12, 1, 16, 1], [12, 5, 16, 1], [15, 2, 16, 1], [20, 2, 4, 1], [30, 2, 16, 1], [60, 2, 32, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_2^2\\times \\GL(2,3)\\times F_5', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 24, 'autcent_group': '192.1475', 'autcent_hash': 1475, 'autcent_nilpotent': False, 'autcent_order': 192, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': False, 'autcent_tex': 'C_2^2\\times \\GL(2,3)', '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, 1], [2, 5, 2], [3, 1, 8], [4, 1, 2], [4, 5, 2], [5, 2, 2], [6, 1, 8], [6, 5, 16], [10, 2, 2], [12, 1, 16], [12, 5, 16], [15, 2, 16], [20, 2, 4], [30, 2, 16], [60, 2, 32]], 'center_label': '36.8', 'center_order': 36, 'central_product': True, 'central_quotient': '10.1', 'commutator_count': 1, 'commutator_label': '5.1', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '3.1', '3.1', '5.1'], 'composition_length': 6, 'conjugacy_classes_known': True, 'counter': 91, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['10.1', 1], ['3.1', 2], ['4.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 5, 1, 2], [3, 1, 2, 4], [4, 1, 2, 1], [4, 5, 2, 1], [5, 2, 2, 1], [6, 1, 2, 4], [6, 5, 2, 8], [10, 2, 2, 1], [12, 1, 4, 4], [12, 5, 4, 4], [15, 2, 4, 4], [20, 2, 4, 1], [30, 2, 4, 4], [60, 2, 8, 4]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 6, 'exponent': 60, 'exponents_of_order': [3, 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': '180.32', 'hash': 91, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 10, 'inner_gen_orders': [2, 5], 'inner_gens': [[1, 294], [73, 6]], 'inner_hash': 1, 'inner_nilpotent': False, 'inner_order': 10, 'inner_split': False, 'inner_tex': 'D_5', 'inner_used': [1, 2], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 72], [2, 72]], 'label': '360.91', 'linC_count': 2304, 'linC_degree': 3, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 10, 'linQ_degree_count': 576, 'linQ_dim': 10, 'linQ_dim_count': 576, 'linR_count': 1728, 'linR_degree': 6, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C60:C6', '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], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 16, 'number_characteristic_subgroups': 18, 'number_conjugacy_classes': 144, 'number_divisions': 45, 'number_normal_subgroups': 66, 'number_subgroup_autclasses': 42, 'number_subgroup_classes': 96, 'number_subgroups': 216, 'old_label': None, 'order': 360, 'order_factorization_type': 321, 'order_stats': [[1, 1], [2, 11], [3, 8], [4, 12], [5, 4], [6, 88], [10, 4], [12, 96], [15, 32], [20, 8], [30, 32], [60, 64]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 24, 'outer_gen_orders': [2, 2, 2, 6, 4, 4], 'outer_gen_pows': [0, 0, 0, 1, 0, 0], 'outer_gens': [[181, 6], [1, 186], [5, 6], [1, 44], [125, 8], [245, 10]], 'outer_group': '384.20055', 'outer_hash': 20055, 'outer_nilpotent': False, 'outer_order': 384, 'outer_permdeg': 14, 'outer_perms': [126, 6267744001, 135359406, 167696208, 13452369840, 34184430000], 'outer_solvable': True, 'outer_supersolvable': False, 'outer_tex': 'C_2^3\\times \\GL(2,3)', 'pc_rank': 2, 'perfect': False, 'permutation_degree': 15, 'pgroup': 0, 'primary_abelian_invariants': [2, 4, 3, 3], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 18], [4, 10], [8, 9], [16, 4]], 'representations': {'PC': {'code': 10106573066898107046993711367, 'gens': [1, 3], 'pres': [6, -2, -3, -2, -2, -3, -5, 12, 5294, 50, 5475, 69, 2884, 118, 10373]}, 'GLZN': {'d': 2, 'p': 30, 'gens': [27301, 189007, 27019, 40966, 27181, 513019]}, 'Perm': {'d': 15, 'gens': [6266937600, 864, 3, 403203, 1680, 99672249600]}}, 'schur_multiplier': [6], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [6, 12], 'solvability_type': 6, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_{60}:C_6', 'transitive_degree': 180, 'wreath_data': None, 'wreath_product': False}
