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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': False, 'ambient': '864.4673', 'ambient_counter': 4673, 'ambient_order': 864, 'ambient_tex': 'S_4\\times S_3^2', 'central': False, 'central_factor': False, 'centralizer_order': 9, 'characteristic': False, 'core_order': 18, 'counter': 128, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '864.4673.16.a1.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': None, 'minimal': False, 'minimal_normal': False, 'nilpotent': False, 'normal': False, 'old_label': '16.a1.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': 16, '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': '54.12', 'subgroup_hash': 12, 'subgroup_order': 54, 'subgroup_tex': 'S_3\\times C_3^2', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '864.4673', 'aut_centralizer_order': 9, 'aut_label': '16.a1', 'aut_quo_index': None, 'aut_stab_index': 8, 'aut_weyl_group': '24.14', 'aut_weyl_index': 72, 'centralizer': '96.b1.a1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['4.a1.a1', '8.b1.a1', '8.c1.a1', '8.d1.a1'], 'contains': ['32.a1.a1', '48.a1.a1', '48.c1.a1', '48.d1.a1', '48.e1.a1'], 'core': '48.a1.a1', 'coset_action_label': None, 'count': 4, 'diagramx': [363, -1, 8230, -1, 774, -1, 1543, -1], 'generators': [374400, 4, 126840, 897120], 'label': '864.4673.16.a1.a1', 'mobius_quo': None, 'mobius_sub': -2, 'normal_closure': '4.a1.a1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '4.f1.a1', 'old_label': '16.a1.a1', 'projective_image': '864.4673', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '16.a1.a1', 'subgroup_fusion': None, 'weyl_group': '24.14'}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '18.5', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 4, 'aut_exponent': 24, 'aut_gen_orders': [2, 6, 4, 12], 'aut_gens': [[1, 3, 18], [2, 4, 18], [12, 17, 36], [14, 4, 18], [6, 46, 18]], 'aut_group': '288.851', 'aut_hash': 851, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 288, 'aut_permdeg': 11, 'aut_perms': [1992360, 727201, 11753520, 5907604], 'aut_phi_ratio': 16.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 3, 1, 1], [3, 1, 8, 1], [3, 2, 1, 1], [3, 2, 8, 1], [6, 3, 8, 1]], 'aut_supersolvable': False, 'aut_tex': 'S_3\\times \\GL(2,3)', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 24, 'autcent_group': '48.29', 'autcent_hash': 29, 'autcent_nilpotent': False, 'autcent_order': 48, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': False, 'autcent_tex': '\\GL(2,3)', '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, 8], [3, 2, 9], [6, 3, 8]], 'center_label': '9.2', 'center_order': 9, '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', '3.1'], 'composition_length': 4, 'conjugacy_classes_known': True, 'counter': 12, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['3.1', 2], ['6.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 3, 1, 1], [3, 1, 2, 4], [3, 2, 1, 1], [3, 2, 2, 4], [6, 3, 2, 4]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 3, 'exponent': 6, 'exponents_of_order': [3, 1], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [], 'familial': False, 'frattini_label': '1.1', 'frattini_quotient': '54.12', 'hash': 12, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 6, 'inner_gen_orders': [1, 2, 3], 'inner_gens': [[1, 3, 18], [1, 3, 36], [1, 39, 18]], 'inner_hash': 1, 'inner_nilpotent': False, 'inner_order': 6, 'inner_split': True, 'inner_tex': 'S_3', 'inner_used': [2, 3], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 18], [2, 9]], 'label': '54.12', 'linC_count': 96, 'linC_degree': 3, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 6, 'linQ_degree_count': 48, 'linQ_dim': 6, 'linQ_dim_count': 48, 'linR_count': 48, 'linR_degree': 6, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'S3*C3^2', 'ngens': 4, 'nilpotency_class': -1, 'nilpotent': False, 'normal_counts': [0, 0, 0, 0, 0, 0, 0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 6, 'number_characteristic_subgroups': 6, 'number_conjugacy_classes': 27, 'number_divisions': 15, 'number_normal_subgroups': 18, 'number_subgroup_autclasses': 14, 'number_subgroup_classes': 32, 'number_subgroups': 52, 'old_label': None, 'order': 54, 'order_factorization_type': 31, 'order_stats': [[1, 1], [2, 3], [3, 26], [6, 24]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 24, 'outer_gen_orders': [2, 3], 'outer_gen_pows': [0, 0], 'outer_gens': [[2, 4, 18], [14, 11, 18]], 'outer_group': '48.29', 'outer_hash': 29, 'outer_nilpotent': False, 'outer_order': 48, 'outer_permdeg': 8, 'outer_perms': [36887, 21251], 'outer_solvable': True, 'outer_supersolvable': False, 'outer_tex': '\\GL(2,3)', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 9, 'pgroup': 0, 'primary_abelian_invariants': [2, 3, 3], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 2], [2, 9], [4, 4]], 'representations': {'PC': {'code': 469035, 'gens': [1, 2, 4], 'pres': [4, -3, -2, -3, -3, 21, 199]}, 'GLZ': {'b': 3, 'd': 6, 'gens': [108470321895114534, 125101736062734304]}, 'GLFp': {'d': 3, 'p': 7, 'gens': [30419024, 36826717, 23068812, 32729290]}, 'Perm': {'d': 9, 'gens': [1, 144, 45600, 3]}}, 'schur_multiplier': [3], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [3, 6], 'solvability_type': 7, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'S_3\\times C_3^2', 'transitive_degree': 18, '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': '8.5', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 3, 'aut_exponent': 12, 'aut_gen_orders': [2, 2, 2, 2, 3, 3, 3, 2, 2], 'aut_gens': [[374400, 379560, 2, 3, 856920, 1210440, 7, 16], [374400, 379560, 2, 3, 856920, 897120, 7, 16], [379560, 374400, 2, 3, 897120, 1250640, 7, 16], [374400, 379560, 2, 4, 856920, 1210440, 16, 7], [374400, 379560, 2, 3, 1250640, 897120, 7, 16], [374400, 379560, 5, 3, 856920, 1210440, 23, 7], [1588320, 379560, 2, 3, 856920, 1210440, 7, 16], [1901640, 1538640, 2, 3, 856920, 1210440, 7, 16], [374400, 379560, 21, 20, 856920, 1210440, 7, 16], [374400, 379560, 2, 12, 856920, 1210440, 7, 16]], 'aut_group': '1728.47847', 'aut_hash': 47847, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 1728, 'aut_permdeg': 12, 'aut_perms': [163699200, 126672057, 29441070, 163861344, 29440352, 163735353, 445295811, 22182248, 22184166], 'aut_phi_ratio': 6.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 3, 1, 1], [2, 3, 2, 1], [2, 6, 1, 1], [2, 9, 1, 1], [2, 9, 2, 1], [2, 18, 2, 1], [2, 27, 1, 1], [2, 54, 1, 1], [3, 2, 2, 1], [3, 4, 1, 1], [3, 8, 1, 1], [3, 16, 2, 1], [3, 32, 1, 1], [4, 6, 1, 1], [4, 18, 2, 1], [4, 54, 1, 1], [6, 6, 2, 2], [6, 12, 1, 1], [6, 12, 2, 1], [6, 18, 2, 1], [6, 24, 1, 1], [6, 24, 2, 1], [6, 36, 2, 1], [6, 48, 2, 1], [6, 72, 1, 1], [12, 12, 2, 1], [12, 24, 1, 1], [12, 36, 2, 1]], 'aut_supersolvable': False, 'aut_tex': 'D_6^2:D_6', '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': 12, 'autcentquo_group': '1728.47847', 'autcentquo_hash': 47847, 'autcentquo_nilpotent': False, 'autcentquo_order': 1728, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'D_6^2:D_6', 'cc_stats': [[1, 1, 1], [2, 3, 3], [2, 6, 1], [2, 9, 3], [2, 18, 2], [2, 27, 1], [2, 54, 1], [3, 2, 2], [3, 4, 1], [3, 8, 1], [3, 16, 2], [3, 32, 1], [4, 6, 1], [4, 18, 2], [4, 54, 1], [6, 6, 4], [6, 12, 3], [6, 18, 2], [6, 24, 3], [6, 36, 2], [6, 48, 2], [6, 72, 1], [12, 12, 2], [12, 24, 1], [12, 36, 2]], 'center_label': '1.1', 'center_order': 1, 'central_product': True, 'central_quotient': '864.4673', 'commutator_count': 1, 'commutator_label': '108.41', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '2.1', '3.1', '3.1', '3.1'], 'composition_length': 8, 'conjugacy_classes_known': True, 'counter': 4673, 'cyclic': False, 'derived_length': 3, 'dihedral': False, 'direct_factorization': [['24.12', 1], ['6.1', 2]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 3, 1, 3], [2, 6, 1, 1], [2, 9, 1, 3], [2, 18, 1, 2], [2, 27, 1, 1], [2, 54, 1, 1], [3, 2, 1, 2], [3, 4, 1, 1], [3, 8, 1, 1], [3, 16, 1, 2], [3, 32, 1, 1], [4, 6, 1, 1], [4, 18, 1, 2], [4, 54, 1, 1], [6, 6, 1, 4], [6, 12, 1, 3], [6, 18, 1, 2], [6, 24, 1, 3], [6, 36, 1, 2], [6, 48, 1, 2], [6, 72, 1, 1], [12, 12, 1, 2], [12, 24, 1, 1], [12, 36, 1, 2]], 'element_repr_type': 'Perm', 'elementary': 1, 'eulerian_function': 80640, 'exponent': 12, 'exponents_of_order': [5, 3], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [[12, 1, 2]], 'familial': False, 'frattini_label': '1.1', 'frattini_quotient': '864.4673', 'hash': 4673, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 12, 'inner_gen_orders': [2, 2, 2, 3, 3, 3, 2, 2], 'inner_gens': [[374400, 379560, 2, 3, 856920, 897120, 7, 16], [374400, 379560, 2, 3, 1250640, 1210440, 7, 16], [374400, 379560, 2, 4, 856920, 1210440, 16, 7], [374400, 379560, 1, 3, 856920, 1210440, 16, 23], [374400, 1951320, 2, 3, 856920, 1210440, 7, 16], [1901640, 379560, 2, 3, 856920, 1210440, 7, 16], [374400, 379560, 21, 20, 856920, 1210440, 7, 16], [374400, 379560, 21, 11, 856920, 1210440, 7, 16]], 'inner_hash': 4673, 'inner_nilpotent': False, 'inner_order': 864, 'inner_split': True, 'inner_tex': 'S_4\\times S_3^2', 'inner_used': [1, 2, 3, 4, 5, 6, 7], 'irrC_degree': 12, 'irrQ_degree': 12, 'irrQ_dim': 12, 'irrR_degree': 12, 'irrep_stats': [[1, 8], [2, 12], [3, 8], [4, 6], [6, 8], [8, 1], [12, 2]], 'label': '864.4673', 'linC_count': 144, 'linC_degree': 7, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 7, 'linQ_degree_count': 144, 'linQ_dim': 7, 'linQ_dim_count': 144, 'linR_count': 144, 'linR_degree': 7, 'maximal_subgroups_known': True, 'metabelian': False, 'metacyclic': False, 'monomial': True, 'name': 'S4*S3^2', 'ngens': 8, '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': 30, 'number_characteristic_subgroups': 18, 'number_conjugacy_classes': 45, 'number_divisions': 45, 'number_normal_subgroups': 48, 'number_subgroup_autclasses': 380, 'number_subgroup_classes': 605, 'number_subgroups': 6740, 'old_label': None, 'order': 864, 'order_factorization_type': 33, 'order_stats': [[1, 1], [2, 159], [3, 80], [4, 96], [6, 408], [12, 120]], 'outer_abelian': True, 'outer_cyclic': True, 'outer_equivalence': False, 'outer_exponent': 2, 'outer_gen_orders': [2], 'outer_gen_pows': [0], 'outer_gens': [[379560, 374400, 2, 4, 897120, 1250640, 16, 7]], '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': 5, 'perfect': False, 'permutation_degree': 10, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 2], 'quasisimple': False, 'rank': 3, 'rational': True, 'rational_characters_known': True, 'ratrep_stats': [[1, 8], [2, 12], [3, 8], [4, 6], [6, 8], [8, 1], [12, 2]], 'representations': {'PC': {'code': 36580756201620780868174227814105063260525675172885, 'gens': [1, 2, 3, 5, 7], 'pres': [8, 2, 2, 2, 3, 2, 3, 2, 3, 250, 66, 267, 5540, 2668, 116, 1173, 40326, 6062, 9094, 1542, 166, 36871]}, 'GLZN': {'d': 2, 'p': 60, 'gens': [10282343, 218731, 346333, 325801, 6804031, 217201, 4269059, 6858916]}, 'Perm': {'d': 10, 'gens': [374400, 379560, 2, 3, 856920, 1210440, 7, 16]}}, 'schur_multiplier': [2, 2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 2], 'solvability_type': 11, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': 'S_4\\times S_3^2', 'transitive_degree': 24, 'wreath_data': None, 'wreath_product': False}
