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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': False, 'ambient': '1080.538', 'ambient_counter': 538, 'ambient_order': 1080, 'ambient_tex': 'C_5\\times S_3^3', 'central': False, 'central_factor': False, 'centralizer_order': 30, 'characteristic': False, 'core_order': 45, 'counter': 26, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '1080.538.12.f1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': False, 'minimal_normal': False, 'nilpotent': False, 'normal': False, 'old_label': '12.f1', 'outer_equivalence': True, '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': 12, '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': '90.6', 'subgroup_hash': 6, 'subgroup_order': 90, 'subgroup_tex': 'S_3\\times C_{15}', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '1080.538', 'aut_centralizer_order': 6, 'aut_label': '12.f1', 'aut_quo_index': None, 'aut_stab_index': 18, 'aut_weyl_group': '48.35', 'aut_weyl_index': 108, 'centralizer': '36.b1', 'complements': None, 'conjugacy_class_count': 6, 'contained_in': ['4.b1', '6.d1', '6.e1', '6.f1'], 'contains': ['24.a1', '36.d1', '36.e1', '60.f1'], 'core': '24.a1', 'coset_action_label': None, 'count': 18, 'diagramx': [1904, -1, 7020, -1], 'generators': [37, 216, 720, 24], 'label': '1080.538.12.f1', 'mobius_quo': None, 'mobius_sub': -2, 'normal_closure': '4.b1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '3.a1', 'old_label': '12.f1', 'projective_image': '216.162', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '12.f1', 'subgroup_fusion': None, 'weyl_group': '12.4'}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '30.4', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 12, 'aut_gen_orders': [2, 2, 12], 'aut_gens': [[1, 6], [1, 66], [5, 6], [31, 78]], 'aut_group': '48.35', 'aut_hash': 35, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 48, 'aut_permdeg': 9, 'aut_perms': [1, 24, 46108], 'aut_phi_ratio': 2.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 3, 1, 1], [3, 1, 2, 1], [3, 2, 1, 1], [3, 2, 2, 1], [5, 1, 4, 1], [6, 3, 2, 1], [10, 3, 4, 1], [15, 1, 8, 1], [15, 2, 4, 1], [15, 2, 8, 1], [30, 3, 8, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_4\\times D_6', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 4, 'autcent_group': '8.2', 'autcent_hash': 2, 'autcent_nilpotent': True, 'autcent_order': 8, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2\\times C_4', '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, 3, 1], [3, 1, 2], [3, 2, 3], [5, 1, 4], [6, 3, 2], [10, 3, 4], [15, 1, 8], [15, 2, 12], [30, 3, 8]], 'center_label': '15.1', 'center_order': 15, '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', '3.1', '3.1', '5.1'], 'composition_length': 4, 'conjugacy_classes_known': True, 'counter': 6, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['3.1', 1], ['5.1', 1], ['6.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 3, 1, 1], [3, 1, 2, 1], [3, 2, 1, 1], [3, 2, 2, 1], [5, 1, 4, 1], [6, 3, 2, 1], [10, 3, 4, 1], [15, 1, 8, 1], [15, 2, 4, 1], [15, 2, 8, 1], [30, 3, 8, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 72, 'exponent': 30, 'exponents_of_order': [2, 1, 1], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3, 5], 'faithful_reps': [[2, 0, 8]], 'familial': False, 'frattini_label': '1.1', 'frattini_quotient': '90.6', 'hash': 6, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 6, 'inner_gen_orders': [2, 3], 'inner_gens': [[1, 66], [31, 6]], 'inner_hash': 1, 'inner_nilpotent': False, 'inner_order': 6, 'inner_split': True, 'inner_tex': 'S_3', 'inner_used': [1, 2], 'irrC_degree': 2, 'irrQ_degree': 16, 'irrQ_dim': 16, 'irrR_degree': 4, 'irrep_stats': [[1, 30], [2, 15]], 'label': '90.6', 'linC_count': 8, 'linC_degree': 2, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 8, 'linQ_degree_count': 6, 'linQ_dim': 8, 'linQ_dim_count': 6, 'linR_count': 12, 'linR_degree': 4, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'S3*C15', 'ngens': 4, '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': 12, 'number_characteristic_subgroups': 12, 'number_conjugacy_classes': 45, 'number_divisions': 12, 'number_normal_subgroups': 12, 'number_subgroup_autclasses': 18, 'number_subgroup_classes': 18, 'number_subgroups': 28, 'old_label': None, 'order': 90, 'order_factorization_type': 222, 'order_stats': [[1, 1], [2, 3], [3, 8], [5, 4], [6, 6], [10, 12], [15, 32], [30, 24]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 4, 'outer_gen_orders': [2, 4], 'outer_gen_pows': [0, 0], 'outer_gens': [[5, 24], [1, 78]], 'outer_group': '8.2', 'outer_hash': 2, 'outer_nilpotent': True, 'outer_order': 8, 'outer_permdeg': 6, 'outer_perms': [120, 9], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2\\times C_4', 'pc_rank': 2, 'perfect': False, 'permutation_degree': 11, 'pgroup': 0, 'primary_abelian_invariants': [2, 3, 5], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 2], [2, 3], [4, 3], [8, 3], [16, 1]], 'representations': {'PC': {'code': 8507233559, 'gens': [1, 3], 'pres': [4, -2, -3, -3, -5, 8, 794, 46]}, 'GLFp': {'d': 2, 'p': 31, 'gens': [992, 536254, 148980]}, 'Perm': {'d': 11, 'gens': [1, 144, 4037040, 3]}}, 'schur_multiplier': [], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [30], 'solvability_type': 6, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'S_3\\times C_{15}', 'transitive_degree': 30, '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': '40.14', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 4, 'aut_exponent': 36, 'aut_gen_orders': [4, 12, 6], 'aut_gens': [[1, 2, 12, 72], [6, 25, 40, 144], [30, 780, 361, 220], [54, 372, 369, 656]], 'aut_group': '5184.og', 'aut_hash': 2215668191011695968, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 5184, 'aut_permdeg': 13, 'aut_perms': [2684832444, 2275820402, 1665357914], 'aut_phi_ratio': 18.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 3, 3, 1], [2, 9, 3, 1], [2, 27, 1, 1], [3, 2, 3, 1], [3, 4, 3, 1], [3, 8, 1, 1], [5, 1, 4, 1], [6, 6, 6, 1], [6, 12, 3, 1], [6, 18, 3, 1], [10, 3, 12, 1], [10, 9, 12, 1], [10, 27, 4, 1], [15, 2, 12, 1], [15, 4, 12, 1], [15, 8, 4, 1], [30, 6, 24, 1], [30, 12, 12, 1], [30, 18, 12, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_4\\times S_3\\wr S_3', 'autcent_abelian': True, 'autcent_cyclic': True, 'autcent_exponent': 4, 'autcent_group': '4.1', 'autcent_hash': 1, 'autcent_nilpotent': True, 'autcent_order': 4, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_4', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 36, 'autcentquo_group': '1296.3490', 'autcentquo_hash': 3490, 'autcentquo_nilpotent': False, 'autcentquo_order': 1296, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'S_3\\wr S_3', 'cc_stats': [[1, 1, 1], [2, 3, 3], [2, 9, 3], [2, 27, 1], [3, 2, 3], [3, 4, 3], [3, 8, 1], [5, 1, 4], [6, 6, 6], [6, 12, 3], [6, 18, 3], [10, 3, 12], [10, 9, 12], [10, 27, 4], [15, 2, 12], [15, 4, 12], [15, 8, 4], [30, 6, 24], [30, 12, 12], [30, 18, 12]], 'center_label': '5.1', 'center_order': 5, 'central_product': True, 'central_quotient': '216.162', 'commutator_count': 1, 'commutator_label': '27.5', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '3.1', '3.1', '3.1', '5.1'], 'composition_length': 7, 'conjugacy_classes_known': True, 'counter': 538, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['5.1', 1], ['6.1', 3]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 3, 1, 3], [2, 9, 1, 3], [2, 27, 1, 1], [3, 2, 1, 3], [3, 4, 1, 3], [3, 8, 1, 1], [5, 1, 4, 1], [6, 6, 1, 6], [6, 12, 1, 3], [6, 18, 1, 3], [10, 3, 4, 3], [10, 9, 4, 3], [10, 27, 4, 1], [15, 2, 4, 3], [15, 4, 4, 3], [15, 8, 4, 1], [30, 6, 4, 6], [30, 12, 4, 3], [30, 18, 4, 3]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 55552, 'exponent': 30, 'exponents_of_order': [3, 3, 1], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3, 5], 'faithful_reps': [[8, 0, 4]], 'familial': False, 'frattini_label': '1.1', 'frattini_quotient': '1080.538', 'hash': 538, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 6, 'inner_gen_orders': [2, 6, 6, 3], 'inner_gens': [[1, 10, 12, 72], [5, 2, 60, 72], [1, 26, 12, 792], [1, 2, 372, 72]], 'inner_hash': 162, 'inner_nilpotent': False, 'inner_order': 216, 'inner_split': True, 'inner_tex': 'S_3^3', 'inner_used': [1, 2, 3, 4], 'irrC_degree': 8, 'irrQ_degree': 32, 'irrQ_dim': 32, 'irrR_degree': 16, 'irrep_stats': [[1, 40], [2, 60], [4, 30], [8, 5]], 'label': '1080.538', 'linC_count': None, 'linC_degree': None, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': None, 'linQ_degree_count': None, 'linQ_dim': None, 'linQ_dim_count': None, 'linR_count': None, 'linR_degree': None, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'C5*S3^3', 'ngens': 7, '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, 0, 0, 0, 0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 20, 'number_characteristic_subgroups': 10, 'number_conjugacy_classes': 135, 'number_divisions': 54, 'number_normal_subgroups': 76, 'number_subgroup_autclasses': 108, 'number_subgroup_classes': 324, 'number_subgroups': 1808, 'old_label': None, 'order': 1080, 'order_factorization_type': 321, 'order_stats': [[1, 1], [2, 63], [3, 26], [5, 4], [6, 126], [10, 252], [15, 104], [30, 504]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': True, 'outer_exponent': 12, 'outer_gen_orders': [2, 12], 'outer_gen_pows': [0, 0], 'outer_gens': [[1, 40, 366, 456], [36, 361, 2, 912]], 'outer_group': '24.5', 'outer_hash': 5, 'outer_nilpotent': False, 'outer_order': 24, 'outer_permdeg': 7, 'outer_perms': [136, 849], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_4\\times S_3', 'pc_rank': 4, 'perfect': False, 'permutation_degree': 14, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 2, 5], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 8], [2, 12], [4, 14], [8, 13], [16, 6], [32, 1]], 'representations': {'PC': {'code': 30202356189255582628332840999878935, 'gens': [1, 2, 4, 6], 'pres': [7, -2, -2, -3, -2, -3, -3, -5, 141, 36, 170, 850, 80, 851, 2798, 166]}, 'GLZN': {'d': 2, 'p': 66, 'gens': [7187425, 14220025, 17552749, 433456, 6612409, 8063983, 288949]}, 'Perm': {'d': 14, 'gens': [479001600, 24, 1, 4037040, 144, 6706022400, 3]}}, 'schur_multiplier': [2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 10], 'solvability_type': 7, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_5\\times S_3^3', 'transitive_degree': 60, 'wreath_data': None, 'wreath_product': False}
