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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': False, 'ambient': '972.445', 'ambient_counter': 445, 'ambient_order': 972, 'ambient_tex': 'C_3^3.S_3^2', 'central': False, 'central_factor': False, 'centralizer_order': 6, 'characteristic': False, 'core_order': 3, 'counter': 102, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '972.445.54.y1.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': False, 'minimal': False, 'minimal_normal': False, 'nilpotent': False, 'normal': False, 'old_label': '54.y1.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': 54, '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': '18.4', 'subgroup_hash': 4, 'subgroup_order': 18, 'subgroup_tex': 'C_3:S_3', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '972.445', 'aut_centralizer_order': 6, 'aut_label': '54.y1', 'aut_quo_index': None, 'aut_stab_index': 3, 'aut_weyl_group': '54.5', 'aut_weyl_index': 18, 'centralizer': '162.e1.a1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['18.o1.a1', '18.r1.a1', '18.u1.a1', '27.b1.a1'], 'contains': ['108.e1.a1', '162.p1.a1', '162.r1.a1'], 'core': '324.a1.a1', 'coset_action_label': None, 'count': 3, 'diagramx': [1128, -1, 7966, -1, 7285, -1, 3661, -1], 'generators': [16278328, 16041895, 11495158], 'label': '972.445.54.y1.a1', 'mobius_quo': None, 'mobius_sub': 0, 'normal_closure': '6.e1.a1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '3.b1.a1', 'old_label': '54.y1.a1', 'projective_image': '972.445', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '54.y1.a1', 'subgroup_fusion': None, 'weyl_group': '54.5'}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '2.1', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 5, 'aut_exponent': 24, 'aut_gen_orders': [2, 3, 3], 'aut_gens': [[25, 3, 144], [25, 3, 243], [25, 147, 144], [122, 3, 144]], 'aut_group': '432.734', 'aut_hash': 734, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 432, 'aut_permdeg': 9, 'aut_perms': [42648, 223305, 287216], 'aut_phi_ratio': 72.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 9, 1, 1], [3, 2, 4, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_3^2:\\GL(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': 24, 'autcentquo_group': '432.734', 'autcentquo_hash': 734, 'autcentquo_nilpotent': False, 'autcentquo_order': 432, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'C_3^2:\\GL(2,3)', 'cc_stats': [[1, 1, 1], [2, 9, 1], [3, 2, 4]], 'center_label': '1.1', 'center_order': 1, 'central_product': False, 'central_quotient': '18.4', 'commutator_count': 1, 'commutator_label': '9.2', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '3.1', '3.1'], 'composition_length': 3, 'conjugacy_classes_known': True, 'counter': 4, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 9, 1, 1], [3, 2, 1, 4]], 'element_repr_type': 'Perm', 'elementary': 1, 'eulerian_function': 7, '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': '18.4', 'hash': 4, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 6, 'inner_gen_orders': [2, 3, 3], 'inner_gens': [[25, 4, 240], [29, 3, 144], [265, 3, 144]], 'inner_hash': 4, 'inner_nilpotent': False, 'inner_order': 18, 'inner_split': False, 'inner_tex': 'C_3:S_3', 'inner_used': [1, 2, 3], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 2], [2, 4]], 'label': '18.4', 'linC_count': 6, 'linC_degree': 4, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 4, 'linQ_degree_count': 6, 'linQ_dim': 4, 'linQ_dim_count': 6, 'linR_count': 6, 'linR_degree': 4, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'C3:S3', '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': 3, 'number_characteristic_subgroups': 3, 'number_conjugacy_classes': 6, 'number_divisions': 6, 'number_normal_subgroups': 7, 'number_subgroup_autclasses': 6, 'number_subgroup_classes': 12, 'number_subgroups': 28, 'old_label': None, 'order': 18, 'order_factorization_type': 22, 'order_stats': [[1, 1], [2, 9], [3, 8]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 12, 'outer_gen_orders': [2, 3, 2, 2], 'outer_gen_pows': [0, 0, 25, 25], 'outer_gens': [[25, 3, 243], [25, 240, 243], [25, 243, 244], [25, 240, 3]], 'outer_group': '24.12', 'outer_hash': 12, 'outer_nilpotent': False, 'outer_order': 24, 'outer_permdeg': 4, 'outer_perms': [2, 4, 16, 7], 'outer_solvable': True, 'outer_supersolvable': False, 'outer_tex': 'S_4', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 6, 'pgroup': 0, 'primary_abelian_invariants': [2], 'quasisimple': False, 'rank': 3, 'rational': True, 'rational_characters_known': True, 'ratrep_stats': [[1, 2], [2, 4]], 'representations': {'PC': {'code': 5475, 'gens': [1, 2, 3], 'pres': [3, -2, -3, -3, 25, 110]}, 'GLZ': {'b': 3, 'd': 4, 'gens': [26483560, 35931151, 16858733]}, 'GLFp': {'d': 3, 'p': 3, 'gens': [11017, 8101, 13933]}, 'Perm': {'d': 6, 'gens': [25, 3, 144]}}, 'schur_multiplier': [3], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2], 'solvability_type': 7, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_3:S_3', 'transitive_degree': 9, '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': '12.5', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 3, 'aut_exponent': 18, 'aut_gen_orders': [2, 3, 3, 3, 3, 3, 2], 'aut_gens': [[16278328, 24577758, 1305911, 2939606, 36731395, 2794152, 29360962], [16278328, 7138662, 4514243, 2939606, 36731395, 11391087, 29360962], [33540394, 24577758, 1305911, 2939606, 36731395, 42750594, 22071741], [17852974, 24577758, 1305911, 2939606, 11537230, 11172560, 17852999], [16278328, 24577758, 1305911, 2939606, 36731395, 20646627, 16278362], [16278328, 24577758, 41439590, 2939606, 36731395, 2794152, 29360962], [4314670, 32864349, 35469088, 43014227, 36731395, 2794152, 4750707], [16278328, 20201582, 29656017, 32864349, 36731395, 14853248, 29360962]], 'aut_group': '972.445', 'aut_hash': 445, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 972, 'aut_permdeg': 36, 'aut_perms': [331093143824985607273236522129984378314067, 13633213040008548063764286135025081711792, 225232583683157182508650447566751140637897, 53969143714621496993854795574035405836775, 153784469452389964328765297048291929440, 262107881910387795571638951518569941715677, 335860263482059789150588035461216128168276], 'aut_phi_ratio': 3.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 9, 1, 1], [2, 27, 1, 2], [3, 2, 1, 1], [3, 3, 1, 2], [3, 6, 1, 3], [3, 18, 1, 6], [6, 18, 1, 1], [6, 27, 1, 6], [6, 54, 1, 9], [9, 36, 1, 3]], 'aut_supersolvable': True, 'aut_tex': 'C_3^3.S_3^2', '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': 18, 'autcentquo_group': '972.445', 'autcentquo_hash': 445, 'autcentquo_nilpotent': False, 'autcentquo_order': 972, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'C_3^3.S_3^2', 'cc_stats': [[1, 1, 1], [2, 9, 1], [2, 27, 2], [3, 2, 1], [3, 3, 2], [3, 6, 3], [3, 18, 6], [6, 18, 1], [6, 27, 6], [6, 54, 9], [9, 36, 3]], 'center_label': '1.1', 'center_order': 1, 'central_product': False, 'central_quotient': '972.445', 'commutator_count': 1, 'commutator_label': '81.7', 'complements_known': True, 'complete': True, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '3.1', '3.1', '3.1', '3.1', '3.1'], 'composition_length': 7, 'conjugacy_classes_known': True, 'counter': 445, 'cyclic': False, 'derived_length': 3, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 9, 1, 1], [2, 27, 1, 2], [3, 2, 1, 1], [3, 3, 2, 1], [3, 6, 1, 1], [3, 6, 2, 1], [3, 18, 1, 2], [3, 18, 2, 2], [6, 18, 1, 1], [6, 27, 2, 3], [6, 54, 1, 3], [6, 54, 2, 3], [9, 36, 1, 1], [9, 36, 2, 1]], 'element_repr_type': 'GLFp', 'elementary': 1, 'eulerian_function': 144, 'exponent': 18, 'exponents_of_order': [5, 2], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [[18, 1, 2]], 'familial': False, 'frattini_label': '9.2', 'frattini_quotient': '108.38', 'hash': 445, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 18, 'inner_gen_orders': [2, 3, 3, 3, 3, 3, 2], 'inner_gens': [[16278328, 7138662, 4514243, 2939606, 36731395, 11391087, 29360962], [33540394, 24577758, 1305911, 2939606, 36731395, 20646627, 33540338], [17852974, 24577758, 1305911, 2939606, 11537230, 14853248, 1542375], [16278328, 24577758, 1305911, 2939606, 36731395, 42750594, 4750707], [16278328, 24577758, 41439590, 2939606, 36731395, 2794152, 29360962], [26377288, 7138662, 31289712, 20201582, 36731395, 2794152, 16278362], [16278328, 20201582, 29656017, 32864349, 36731395, 14853248, 29360962]], 'inner_hash': 445, 'inner_nilpotent': False, 'inner_order': 972, 'inner_split': True, 'inner_tex': 'C_3^3.S_3^2', 'inner_used': [1, 2, 3, 5, 6, 7], 'irrC_degree': 18, 'irrQ_degree': 18, 'irrQ_dim': 18, 'irrR_degree': 18, 'irrep_stats': [[1, 12], [2, 12], [4, 3], [6, 6], [18, 2]], 'label': '972.445', 'linC_count': 2, 'linC_degree': 18, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 18, 'linQ_degree_count': 2, 'linQ_dim': 18, 'linQ_dim_count': 2, 'linR_count': 2, 'linR_degree': 18, 'maximal_subgroups_known': True, 'metabelian': False, 'metacyclic': False, 'monomial': True, 'name': 'C3^3.S3^2', '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], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 35, 'number_characteristic_subgroups': 23, 'number_conjugacy_classes': 35, 'number_divisions': 24, 'number_normal_subgroups': 23, 'number_subgroup_autclasses': 153, 'number_subgroup_classes': 153, 'number_subgroups': 1840, 'old_label': None, 'order': 972, 'order_factorization_type': 32, 'order_stats': [[1, 1], [2, 63], [3, 134], [6, 666], [9, 108]], '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': 5, 'perfect': False, 'permutation_degree': 27, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 3], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 8], [4, 5], [6, 2], [8, 1], [12, 2], [18, 2]], 'representations': {'PC': {'code': 37775847321378497257394809683840891619295919, 'gens': [1, 3, 5, 6, 7], 'pres': [7, -2, -3, -2, -3, -3, -3, 3, 14, 5168, 2340, 58, 3699, 1690, 5478, 2335, 9077, 31758, 5312, 622]}, 'GLFp': {'d': 4, 'p': 3, 'gens': [16278328, 24577758, 1305911, 2939606, 36731395, 2794152, 29360962]}, 'Perm': {'d': 27, 'gens': [81081325226784534221250, 15512334169126899406221385, 376703666922484, 7550102686338507196661760000, 2516538717432998141660098080, 418802676036117248219142964, 806634631153204606767248884]}}, 'schur_multiplier': [2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 6], 'solvability_type': 10, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_3^3.S_3^2', 'transitive_degree': 27, 'wreath_data': None, 'wreath_product': False}
