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gps_subgroup_search • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'ambient': '5832.iu', 'ambient_counter': 229, 'ambient_order': 5832, 'ambient_tex': 'C_3^3.S_3^3', 'central': False, 'central_factor': False, 'centralizer_order': 9, 'characteristic': False, 'core_order': 54, 'counter': 274, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '5832.iu.36.bu1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': False, 'minimal': False, 'minimal_normal': False, 'nilpotent': False, 'normal': False, 'old_label': '36.bu1', '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': 36, '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': '162.4', 'subgroup_hash': 4, 'subgroup_order': 162, 'subgroup_tex': 'C_3^2:C_{18}', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '5832.iu', 'aut_centralizer_order': None, 'aut_label': '36.bu1', 'aut_quo_index': None, 'aut_stab_index': None, 'aut_weyl_group': None, 'aut_weyl_index': None, 'centralizer': '648.d1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['12.w1', '12.x1', '12.y1', '12.z1'], 'contains': ['72.t1', '108.e1', '108.ck1'], 'core': '108.e1', 'coset_action_label': None, 'count': 4, 'diagramx': [1833, -1, 8657, -1], 'generators': [489, 36, 290, 648, 48], 'label': '5832.iu.36.bu1', 'mobius_quo': None, 'mobius_sub': 0, 'normal_closure': '4.b1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '4.b1', 'old_label': '36.bu1', 'projective_image': '5832.iu', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '36.bu1', 'subgroup_fusion': None, 'weyl_group': '162.34'}
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gps_groups • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '18.2', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 3, 'aut_exponent': 6, 'aut_gen_orders': [6, 6, 3, 3], 'aut_gens': [[1, 18, 54], [13, 36, 108], [89, 18, 108], [1, 126, 54], [55, 18, 54]], 'aut_group': '324.117', 'aut_hash': 117, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 324, 'aut_permdeg': 12, 'aut_perms': [16188507, 56790960, 25966344, 92354424], 'aut_phi_ratio': 6.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 9, 1, 1], [3, 1, 2, 1], [3, 2, 1, 1], [3, 2, 2, 1], [3, 6, 1, 1], [3, 6, 2, 1], [6, 9, 2, 1], [9, 3, 6, 1], [9, 6, 6, 1], [18, 9, 6, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_3^3:D_6', 'autcent_abelian': True, 'autcent_cyclic': True, 'autcent_exponent': 3, 'autcent_group': '3.1', 'autcent_hash': 1, 'autcent_nilpotent': True, 'autcent_order': 3, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_3', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 6, 'autcentquo_group': '108.17', 'autcentquo_hash': 17, 'autcentquo_nilpotent': False, 'autcentquo_order': 108, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'C_3^2:D_6', 'cc_stats': [[1, 1, 1], [2, 9, 1], [3, 1, 2], [3, 2, 3], [3, 6, 3], [6, 9, 2], [9, 3, 6], [9, 6, 6], [18, 9, 6]], 'center_label': '3.1', 'center_order': 3, 'central_product': False, 'central_quotient': '54.5', 'commutator_count': 1, 'commutator_label': '9.2', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '3.1', '3.1', '3.1', '3.1'], 'composition_length': 5, '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, 1, 2, 1], [3, 2, 1, 1], [3, 2, 2, 1], [3, 6, 1, 1], [3, 6, 2, 1], [6, 9, 2, 1], [9, 3, 6, 1], [9, 6, 6, 1], [18, 9, 6, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 36, 'exponent': 18, 'exponents_of_order': [4, 1], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [[6, 0, 2]], 'familial': False, 'frattini_label': '9.2', 'frattini_quotient': '18.3', 'hash': 4, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 6, 'inner_gen_orders': [6, 3, 3], 'inner_gens': [[1, 144, 108], [91, 18, 54], [109, 18, 54]], 'inner_hash': 5, 'inner_nilpotent': False, 'inner_order': 54, 'inner_split': True, 'inner_tex': 'C_3^2:C_6', 'inner_used': [1, 2], 'irrC_degree': 6, 'irrQ_degree': 12, 'irrQ_dim': 12, 'irrR_degree': 12, 'irrep_stats': [[1, 18], [2, 9], [6, 3]], 'label': '162.4', 'linC_count': 2, 'linC_degree': 6, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 12, 'linQ_degree_count': 3, 'linQ_dim': 12, 'linQ_dim_count': 3, 'linR_count': 6, 'linR_degree': 8, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'C3^2:C18', 'ngens': 5, 'nilpotency_class': -1, 'nilpotent': False, 'normal_counts': [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': 11, 'number_characteristic_subgroups': 11, 'number_conjugacy_classes': 30, 'number_divisions': 11, 'number_normal_subgroups': 11, 'number_subgroup_autclasses': 28, 'number_subgroup_classes': 28, 'number_subgroups': 98, 'old_label': None, 'order': 162, 'order_factorization_type': 31, 'order_stats': [[1, 1], [2, 9], [3, 26], [6, 18], [9, 54], [18, 54]], 'outer_abelian': True, 'outer_cyclic': True, 'outer_equivalence': False, 'outer_exponent': 6, 'outer_gen_orders': [6], 'outer_gen_pows': [0], 'outer_gens': [[11, 18, 108]], 'outer_group': '6.2', 'outer_hash': 2, 'outer_nilpotent': True, 'outer_order': 6, 'outer_permdeg': 5, 'outer_perms': [28], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_6', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 18, 'pgroup': 0, 'primary_abelian_invariants': [2, 9], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 2], [2, 3], [4, 1], [6, 3], [12, 2]], 'representations': {'PC': {'code': 724502737082479, 'gens': [1, 4, 5], 'pres': [5, -2, -3, -3, -3, -3, 10, 36, 2883, 1268, 2704]}, 'GLZN': {'d': 2, 'p': 54, 'gens': [3936607, 159515, 746467, 158437, 2991835]}, 'GLZq': {'d': 2, 'q': 27, 'gens': [389992, 196840, 20222, 78754, 19927]}, 'Perm': {'d': 18, 'gens': [3402106385166720, 2644117572417840, 376703666887107, 174400128148, 711549292653507]}}, 'schur_multiplier': [3], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [18], 'solvability_type': 7, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_3^2:C_{18}', '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': '24.15', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 3, 'aut_exponent': 18, 'aut_gen_orders': [6, 6, 3, 18, 3], 'aut_gens': [[1, 6, 36, 108, 1944], [1297, 930, 36, 5112, 1944], [313, 510, 72, 1908, 1944], [1, 78, 36, 108, 1944], [3709, 438, 36, 108, 3888], [1, 6, 36, 3996, 1944]], 'aut_group': '5832.iu', 'aut_hash': 8798309712670076054, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 5832, 'aut_permdeg': 42, 'aut_perms': [466446389790809234360896617784859711400025859580697, 760447119412444256702905003447490200423325456057226, 17971374467734837585225719559823135838594948506934, 667345542177269663861596032032746176749234151818981, 139204709986993234062283280051896320000], 'aut_phi_ratio': 3.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 3, 1, 1], [2, 9, 1, 1], [2, 27, 1, 3], [2, 81, 1, 2], [3, 2, 1, 3], [3, 3, 1, 2], [3, 4, 1, 3], [3, 6, 1, 7], [3, 8, 1, 1], [3, 12, 1, 8], [3, 24, 1, 3], [6, 6, 1, 2], [6, 9, 1, 2], [6, 12, 1, 1], [6, 18, 1, 7], [6, 27, 1, 6], [6, 36, 1, 4], [6, 54, 1, 15], [6, 81, 1, 6], [6, 108, 1, 6], [6, 162, 1, 6], [9, 18, 1, 3], [9, 36, 1, 6], [9, 72, 1, 3], [18, 54, 1, 6], [18, 108, 1, 6], [18, 162, 1, 3]], 'aut_supersolvable': True, 'aut_tex': 'C_3^3.S_3^3', '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': 18, 'autcentquo_group': '5832.iu', 'autcentquo_hash': 8798309712670076054, 'autcentquo_nilpotent': False, 'autcentquo_order': 5832, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'C_3^3.S_3^3', 'cc_stats': [[1, 1, 1], [2, 3, 1], [2, 9, 1], [2, 27, 3], [2, 81, 2], [3, 2, 3], [3, 3, 2], [3, 4, 3], [3, 6, 7], [3, 8, 1], [3, 12, 8], [3, 24, 3], [6, 6, 2], [6, 9, 2], [6, 12, 1], [6, 18, 7], [6, 27, 6], [6, 36, 4], [6, 54, 15], [6, 81, 6], [6, 108, 6], [6, 162, 6], [9, 18, 3], [9, 36, 6], [9, 72, 3], [18, 54, 6], [18, 108, 6], [18, 162, 3]], 'center_label': '1.1', 'center_order': 1, 'central_product': True, 'central_quotient': '5832.iu', 'commutator_count': 1, 'commutator_label': '243.32', 'complements_known': True, 'complete': True, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '3.1', '3.1', '3.1', '3.1', '3.1', '3.1'], 'composition_length': 9, 'conjugacy_classes_known': True, 'counter': 229, 'cyclic': False, 'derived_length': 3, 'dihedral': False, 'direct_factorization': [['6.1', 1], ['972.429', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 3, 1, 1], [2, 9, 1, 1], [2, 27, 1, 3], [2, 81, 1, 2], [3, 2, 1, 3], [3, 3, 2, 1], [3, 4, 1, 3], [3, 6, 1, 1], [3, 6, 2, 3], [3, 8, 1, 1], [3, 12, 1, 2], [3, 12, 2, 3], [3, 24, 1, 1], [3, 24, 2, 1], [6, 6, 1, 2], [6, 9, 2, 1], [6, 12, 1, 1], [6, 18, 1, 3], [6, 18, 2, 2], [6, 27, 2, 3], [6, 36, 1, 2], [6, 36, 2, 1], [6, 54, 1, 5], [6, 54, 2, 5], [6, 81, 2, 3], [6, 108, 1, 2], [6, 108, 2, 2], [6, 162, 1, 2], [6, 162, 2, 2], [9, 18, 1, 1], [9, 18, 2, 1], [9, 36, 1, 2], [9, 36, 2, 2], [9, 72, 1, 1], [9, 72, 2, 1], [18, 54, 1, 2], [18, 54, 2, 2], [18, 108, 1, 2], [18, 108, 2, 2], [18, 162, 1, 1], [18, 162, 2, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 7547904, 'exponent': 18, 'exponents_of_order': [6, 3], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [[24, 0, 2], [24, 1, 1]], 'familial': False, 'frattini_label': '9.2', 'frattini_quotient': '648.731', 'hash': 8798309712670076054, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 18, 'inner_gen_orders': [6, 6, 3, 18, 3], 'inner_gens': [[1, 1578, 36, 5112, 1944], [445, 6, 72, 1872, 1944], [1, 78, 36, 108, 1944], [2881, 294, 36, 108, 3888], [1, 6, 36, 3996, 1944]], 'inner_hash': 8798309712670076054, 'inner_nilpotent': False, 'inner_order': 5832, 'inner_split': True, 'inner_tex': 'C_3^3.S_3^3', 'inner_used': [1, 2, 4], 'irrC_degree': 24, 'irrQ_degree': 24, 'irrQ_dim': 24, 'irrR_degree': 24, 'irrep_stats': [[1, 24], [2, 36], [4, 18], [6, 16], [8, 3], [12, 16], [24, 4]], 'label': '5832.iu', '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': False, 'metacyclic': False, 'monomial': True, 'name': 'C3^3.S3^3', 'ngens': 9, '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], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 117, 'number_characteristic_subgroups': 95, 'number_conjugacy_classes': 117, 'number_divisions': 81, 'number_normal_subgroups': 95, 'number_subgroup_autclasses': 1558, 'number_subgroup_classes': 1558, 'number_subgroups': 32274, 'old_label': None, 'order': 5832, 'order_factorization_type': 33, 'order_stats': [[1, 1], [2, 255], [3, 242], [6, 3390], [9, 486], [18, 1458]], 'outer_abelian': True, 'outer_cyclic': True, 'outer_equivalence': True, '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': 21, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 2, 3], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 8], [2, 20], [4, 18], [6, 8], [8, 7], [12, 12], [16, 1], [24, 6], [48, 1]], 'representations': {'PC': {'code': '558823488896874999332588589268422237420281139056627773685635839064842868018857691831', 'gens': [1, 3, 5, 6, 9], 'pres': [9, 2, 3, 2, 3, 3, 2, 3, 3, 3, 18, 42608, 9812, 74, 2163, 562, 276053, 90410, 16871, 680, 158, 31758, 27231, 18924, 1545, 249, 93319, 15577, 2969]}, 'Perm': {'d': 21, 'gens': [5238230990304753919, 262560264661147073, 2452529073353280126]}}, 'schur_multiplier': [2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 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_3^3', 'transitive_degree': 36, 'wreath_data': None, 'wreath_product': False}
