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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': False, 'ambient': '2700.q', 'ambient_counter': 17, 'ambient_order': 2700, 'ambient_tex': 'C_{15}^2:D_6', 'central': False, 'central_factor': False, 'centralizer_order': 6, 'characteristic': False, 'core_order': 1, 'counter': 114, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '2700.q.225.b1.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': None, 'minimal': False, 'minimal_normal': False, 'nilpotent': False, 'normal': False, 'old_label': '225.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': 225, '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': '12.4', 'subgroup_hash': 4, 'subgroup_order': 12, 'subgroup_tex': 'D_6', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '2700.q', 'aut_centralizer_order': 24, 'aut_label': '225.b1', 'aut_quo_index': None, 'aut_stab_index': 75, 'aut_weyl_group': '12.4', 'aut_weyl_index': 1800, 'centralizer': '450.a1.a1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['45.b1.a1', '45.b1.b1', '75.a1.a1'], 'contains': ['450.b1.a1', '450.e1.a1', '450.e1.b1', '675.a1.a1'], 'core': '2700.a1.a1', 'coset_action_label': None, 'count': 75, 'diagramx': [5009, -1, 779, -1, 6964, -1, 8379, -1], 'generators': [109, 6, 1920], 'label': '2700.q.225.b1.a1', 'mobius_quo': None, 'mobius_sub': 0, 'normal_closure': '1.a1.a1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '75.a1.a1', 'old_label': '225.b1.a1', 'projective_image': '2700.q', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '225.b1.a1', 'subgroup_fusion': None, 'weyl_group': '6.1'}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '4.2', '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], [1, 10], [3, 2]], 'aut_group': '12.4', 'aut_hash': 4, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 12, 'aut_permdeg': 5, 'aut_perms': [6, 49], 'aut_phi_ratio': 3.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 3, 2, 1], [3, 2, 1, 1], [6, 2, 1, 1]], 'aut_supersolvable': True, 'aut_tex': 'D_6', 'autcent_abelian': True, 'autcent_cyclic': True, 'autcent_exponent': 2, 'autcent_group': '2.1', 'autcent_hash': 1, 'autcent_nilpotent': True, 'autcent_order': 2, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2', '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], [2, 3, 2], [3, 2, 1], [6, 2, 1]], 'center_label': '2.1', 'center_order': 2, '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'], 'composition_length': 3, 'conjugacy_classes_known': True, 'counter': 4, 'cyclic': False, 'derived_length': 2, 'dihedral': True, 'direct_factorization': [['2.1', 1], ['6.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 3, 1, 2], [3, 2, 1, 1], [6, 2, 1, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 3, 'exponent': 6, 'exponents_of_order': [2, 1], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [[2, 1, 1]], 'familial': True, 'frattini_label': '1.1', 'frattini_quotient': '12.4', 'hash': 4, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 6, 'inner_gen_orders': [2, 3], 'inner_gens': [[1, 10], [5, 2]], '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': 2, 'irrQ_dim': 2, 'irrR_degree': 2, 'irrep_stats': [[1, 4], [2, 2]], 'label': '12.4', 'linC_count': 1, 'linC_degree': 2, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 2, 'linQ_degree_count': 1, 'linQ_dim': 2, 'linQ_dim_count': 1, 'linR_count': 1, 'linR_degree': 2, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'D6', 'ngens': 3, 'nilpotency_class': -1, 'nilpotent': False, 'normal_counts': [0, 0, 0, 0, 0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 5, 'number_characteristic_subgroups': 5, 'number_conjugacy_classes': 6, 'number_divisions': 6, 'number_normal_subgroups': 7, 'number_subgroup_autclasses': 8, 'number_subgroup_classes': 10, 'number_subgroups': 16, 'old_label': None, 'order': 12, 'order_factorization_type': 22, 'order_stats': [[1, 1], [2, 7], [3, 2], [6, 2]], 'outer_abelian': True, 'outer_cyclic': True, 'outer_equivalence': False, 'outer_exponent': 2, 'outer_gen_orders': [2], 'outer_gen_pows': [0], 'outer_gens': [[7, 2]], 'outer_group': '2.1', 'outer_hash': 1, 'outer_nilpotent': True, 'outer_order': 2, 'outer_permdeg': 2, 'outer_perms': [1], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2', 'pc_rank': 2, 'perfect': False, 'permutation_degree': 5, 'pgroup': 0, 'primary_abelian_invariants': [2, 2], 'quasisimple': False, 'rank': 2, 'rational': True, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 2]], 'representations': {'PC': {'code': 43377, 'gens': [1, 2], 'pres': [3, -2, -2, -3, 61, 16, 74]}, 'GLZ': {'b': 3, 'd': 2, 'gens': [17, 35]}, 'GLFp': {'d': 2, 'p': 3, 'gens': [31, 55, 56]}, 'Perm': {'d': 5, 'gens': [6, 1, 30]}}, 'schur_multiplier': [2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2], 'solvability_type': 6, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'D_6', 'transitive_degree': 6, 'wreath_data': None, 'wreath_product': False}
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gps_groups • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '4.2', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 3, 'aut_exponent': 120, 'aut_gen_orders': [30, 8, 24, 24], 'aut_gens': [[1, 2, 12, 180], [2261, 1402, 168, 264], [431, 694, 2244, 1608], [2431, 302, 2244, 2484], [2119, 1510, 1644, 2364]], 'aut_group': None, 'aut_hash': 7509908978670635385, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 21600, 'aut_permdeg': 90, 'aut_perms': [1223718560084616643378011208422965467506668793203639699838220593818746746451832335686901635315283618356704229363990232635125174632593337037, 1181674562415681112563614535571498175673032522628224692185802545653006757524643646115437306083046741477899046391560581345618446675707747838, 365485412783665425823287248426271448329414864055784104381614255917074554446401272649710462951220353102249428940907350473684225225077400145, 427414949721339587139027811969459071248455897039281133220302817737683668206477290996457252490043882635733454882249136064436272803425588382], 'aut_phi_ratio': 30.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 25, 1, 1], [2, 45, 2, 1], [3, 1, 2, 1], [3, 6, 1, 1], [3, 150, 3, 1], [5, 6, 4, 1], [6, 25, 2, 1], [6, 45, 4, 1], [6, 150, 1, 1], [6, 150, 3, 1], [10, 90, 4, 1], [15, 6, 8, 1], [15, 12, 4, 1], [15, 12, 8, 1], [30, 90, 8, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_{15}^2.C_{12}.C_2^3', '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': 7509908978670635385, 'autcentquo_nilpotent': False, 'autcentquo_order': 21600, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'C_{15}^2.C_{12}.C_2^3', 'cc_stats': [[1, 1, 1], [2, 25, 1], [2, 45, 2], [3, 1, 2], [3, 6, 1], [3, 150, 3], [5, 6, 4], [6, 25, 2], [6, 45, 4], [6, 150, 4], [10, 90, 4], [15, 6, 8], [15, 12, 12], [30, 90, 8]], 'center_label': '3.1', 'center_order': 3, 'central_product': False, 'central_quotient': '900.96', 'commutator_count': 1, 'commutator_label': '675.12', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '3.1', '3.1', '3.1', '5.1', '5.1'], 'composition_length': 7, 'conjugacy_classes_known': True, 'counter': 17, 'cyclic': False, 'derived_length': 3, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 25, 1, 1], [2, 45, 1, 2], [3, 1, 2, 1], [3, 6, 1, 1], [3, 150, 1, 3], [5, 6, 2, 2], [6, 25, 2, 1], [6, 45, 2, 2], [6, 150, 1, 4], [10, 90, 2, 2], [15, 6, 4, 2], [15, 12, 2, 2], [15, 12, 4, 2], [30, 90, 4, 2]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 353808, 'exponent': 30, 'exponents_of_order': [3, 2, 2], 'factors_of_aut_order': [2, 3, 5], 'factors_of_order': [2, 3, 5], 'faithful_reps': [[6, 0, 16], [12, 0, 8]], 'familial': False, 'frattini_label': '3.1', 'frattini_quotient': '900.96', 'hash': 8099013394032210383, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 30, 'inner_gen_orders': [10, 6, 5, 15], 'inner_gens': [[1, 2290, 552, 2640], [137, 2, 2280, 204], [2161, 614, 12, 180], [421, 158, 12, 180]], 'inner_hash': 96, 'inner_nilpotent': False, 'inner_order': 900, 'inner_split': False, 'inner_tex': '(C_5\\times C_{15}):D_6', 'inner_used': [1, 2, 4], 'irrC_degree': 6, 'irrQ_degree': 24, 'irrQ_dim': 24, 'irrR_degree': 12, 'irrep_stats': [[1, 4], [2, 8], [3, 8], [6, 24], [12, 12]], 'label': '2700.q', 'linC_count': 16, 'linC_degree': 6, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 18, 'linQ_degree_count': 16, 'linQ_dim': 18, 'linQ_dim_count': 16, 'linR_count': 40, 'linR_degree': 12, 'maximal_subgroups_known': True, 'metabelian': False, 'metacyclic': False, 'monomial': True, 'name': 'C15^2:D6', '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, 0, 0, 0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 16, 'number_characteristic_subgroups': 12, 'number_conjugacy_classes': 56, 'number_divisions': 28, 'number_normal_subgroups': 20, 'number_subgroup_autclasses': 78, 'number_subgroup_classes': 154, 'number_subgroups': 3982, 'old_label': None, 'order': 2700, 'order_factorization_type': 321, 'order_stats': [[1, 1], [2, 115], [3, 458], [5, 24], [6, 830], [10, 360], [15, 192], [30, 720]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 12, 'outer_gen_orders': [2, 12], 'outer_gen_pows': [0, 2274], 'outer_gens': [[1, 1234, 672, 780], [547, 2686, 588, 2028]], 'outer_group': '24.5', 'outer_hash': 5, 'outer_nilpotent': False, 'outer_order': 24, 'outer_permdeg': 7, 'outer_perms': [120, 1457], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_4\\times S_3', 'pc_rank': 4, 'perfect': False, 'permutation_degree': 24, 'pgroup': 0, 'primary_abelian_invariants': [2, 2], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 8], [6, 4], [12, 4], [24, 6], [48, 2]], 'representations': {'PC': {'code': '1612142572079446564926954310253041976083279094358905608142674983885914487', 'gens': [1, 2, 4, 6], 'pres': [7, 2, 2, 3, 3, 5, 3, 5, 4788, 32061, 36, 48302, 15459, 31930, 15473, 108, 57964, 18911, 10728, 110885, 4296, 19801, 166, 105846, 15007, 902]}, 'Perm': {'d': 24, 'gens': [28304768306459701734966, 4817187026792126393844, 55346845900579008695499]}}, 'schur_multiplier': [2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2], 'solvability_type': 11, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': 'C_{15}^2:D_6', 'transitive_degree': 45, 'wreath_data': None, 'wreath_product': False}
