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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': True, 'ambient': '360.91', 'ambient_counter': 91, 'ambient_order': 360, 'ambient_tex': 'C_{60}:C_6', 'central': False, 'central_factor': False, 'centralizer_order': 72, 'characteristic': False, 'core_order': 6, 'counter': 62, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '360.91.30.b1.c1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': False, 'old_label': '30.b1.c1', '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': 30, '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': '12.5', 'subgroup_hash': 5, 'subgroup_order': 12, 'subgroup_tex': 'C_2\\times C_6', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '360.91', 'aut_centralizer_order': 48, 'aut_label': '30.b1', 'aut_quo_index': None, 'aut_stab_index': 20, 'aut_weyl_group': '4.2', 'aut_weyl_index': 960, 'centralizer': '5.a1.a1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['6.a1.c1', '10.b1.a1', '15.a1.c1'], 'contains': ['60.a1.c1', '60.b1.c1', '60.b1.g1', '90.b1.a1'], 'core': '60.a1.c1', 'coset_action_label': None, 'count': 5, 'diagramx': [4225, -1, 5003, -1, 4578, -1, 5892, -1], 'generators': [3, 180, 242], 'label': '360.91.30.b1.c1', 'mobius_quo': None, 'mobius_sub': -1, 'normal_closure': '6.a1.c1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '5.a1.a1', 'old_label': '30.b1.c1', 'projective_image': '60.10', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '30.b1.c1', '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': '12.5', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 6, 'aut_gen_orders': [2, 6], 'aut_gens': [[1, 2], [6, 9], [6, 11]], 'aut_group': '12.4', 'aut_hash': 4, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 12, 'aut_permdeg': 5, 'aut_perms': [6, 31], 'aut_phi_ratio': 3.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 3, 1], [3, 1, 2, 1], [6, 1, 6, 1]], 'aut_supersolvable': True, 'aut_tex': 'D_6', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 6, 'autcent_group': '12.4', 'autcent_hash': 4, 'autcent_nilpotent': False, 'autcent_order': 12, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'D_6', '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, 3], [3, 1, 2], [6, 1, 6]], 'center_label': '12.5', 'center_order': 12, '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'], 'composition_length': 3, 'conjugacy_classes_known': True, 'counter': 5, 'cyclic': False, 'derived_length': 1, 'dihedral': False, 'direct_factorization': [['2.1', 2], ['3.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [3, 1, 2, 1], [6, 1, 2, 3]], 'element_repr_type': 'PC', 'elementary': 2, 'eulerian_function': 4, 'exponent': 6, 'exponents_of_order': [2, 1], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [], 'familial': False, 'frattini_label': '1.1', 'frattini_quotient': '12.5', 'hash': 5, 'hyperelementary': 2, 'inner_abelian': True, 'inner_cyclic': True, 'inner_exponent': 1, 'inner_gen_orders': [1, 1], 'inner_gens': [[1, 2], [1, 2]], '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, 12]], 'label': '12.5', 'linC_count': 24, 'linC_degree': 2, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 3, 'linQ_degree_count': 6, 'linQ_dim': 3, 'linQ_dim_count': 6, 'linR_count': 6, 'linR_degree': 3, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C2*C6', 'ngens': 3, 'nilpotency_class': 1, 'nilpotent': True, 'normal_counts': [0, 0, 0, 0, 0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 4, 'number_characteristic_subgroups': 4, 'number_conjugacy_classes': 12, 'number_divisions': 8, 'number_normal_subgroups': 10, 'number_subgroup_autclasses': 6, 'number_subgroup_classes': 10, 'number_subgroups': 10, 'old_label': None, 'order': 12, 'order_factorization_type': 22, 'order_stats': [[1, 1], [2, 3], [3, 2], [6, 6]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 6, 'outer_gen_orders': [2, 6], 'outer_gen_pows': [0, 0], 'outer_gens': [[6, 9], [6, 11]], 'outer_group': '12.4', 'outer_hash': 4, 'outer_nilpotent': False, 'outer_order': 12, 'outer_permdeg': 5, 'outer_perms': [6, 31], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'D_6', 'pc_rank': 2, 'perfect': False, 'permutation_degree': 7, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 3], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 4]], 'representations': {'PC': {'code': 273, 'gens': [1, 2], 'pres': [3, -2, -2, -3, 16]}, 'GLZ': {'b': 3, 'd': 3, 'gens': [3362, 16507]}, 'GLFp': {'d': 2, 'p': 7, 'gens': [1030, 2064]}, 'Perm': {'d': 7, 'gens': [720, 24, 4]}}, 'schur_multiplier': [2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 6], 'solvability_type': 1, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_2\\times C_6', 'transitive_degree': 12, '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}
