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gps_subgroup_search • Show schema
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{'Agroup': True, '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': 27, 'counter': 350, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '5832.iu.36.eq1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': False, 'minimal': False, 'minimal_normal': False, 'nilpotent': False, 'normal': False, 'old_label': '36.eq1', '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': '162.53', 'subgroup_hash': 53, 'subgroup_order': 162, 'subgroup_tex': 'C_3^3:C_6', '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.eq1', 'aut_quo_index': None, 'aut_stab_index': None, 'aut_weyl_group': None, 'aut_weyl_index': None, 'centralizer': '648.f1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['12.bn1', '18.bb1', '18.by1', '18.cv1'], 'contains': ['72.bo1', '108.hn1', '108.hp1', '108.hq1', '108.hu1', '108.hv1', '108.ig1', '108.ih1', '108.il1'], 'core': '216.f1', 'coset_action_label': None, 'count': 3, 'diagramx': [4880, -1, 5422, -1], 'generators': [975, 648, 74, 48, 1944], 'label': '5832.iu.36.eq1', 'mobius_quo': None, 'mobius_sub': 0, 'normal_closure': '4.f1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '3.d1', 'old_label': '36.eq1', 'projective_image': '5832.iu', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '36.eq1', 'subgroup_fusion': None, 'weyl_group': '216.162'}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '6.2', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 1, 'aut_exponent': 936, 'aut_gen_orders': [2, 13, 3, 3, 3, 2], 'aut_gens': [[1, 6, 18, 54], [5, 6, 18, 54], [19, 18, 114, 126], [61, 6, 18, 54], [151, 6, 18, 54], [145, 6, 18, 54], [145, 78, 108, 36]], 'aut_group': '606528.a', 'aut_hash': 6817351772196718478, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 606528, 'aut_permdeg': 29, 'aut_perms': [1, 6023421669826496591100443564094, 3852569116134693679973157838248, 2084700862304935458009410705334, 448541197670899512502566745086, 4281374468099257021338616679790], 'aut_phi_ratio': 11232.0, 'aut_solvable': False, 'aut_stats': [[1, 1, 1, 1], [2, 27, 1, 1], [3, 1, 2, 1], [3, 2, 13, 1], [3, 2, 26, 1], [6, 27, 2, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_2\\times C_3^3:\\GL(3,3)', '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': 936, 'autcentquo_group': '303264.a', 'autcentquo_hash': 8091482650397092313, 'autcentquo_nilpotent': False, 'autcentquo_order': 303264, 'autcentquo_solvable': False, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'C_3^3:\\GL(3,3)', 'cc_stats': [[1, 1, 1], [2, 27, 1], [3, 1, 2], [3, 2, 39], [6, 27, 2]], 'center_label': '3.1', 'center_order': 3, 'central_product': True, 'central_quotient': '54.14', 'commutator_count': 1, 'commutator_label': '27.5', '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': 53, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['3.1', 1], ['54.14', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 27, 1, 1], [3, 1, 2, 1], [3, 2, 1, 13], [3, 2, 2, 13], [6, 27, 2, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 600, 'exponent': 6, 'exponents_of_order': [4, 1], 'factors_of_aut_order': [2, 3, 13], 'factors_of_order': [2, 3], 'faithful_reps': [], 'familial': False, 'frattini_label': '1.1', 'frattini_quotient': '162.53', 'hash': 53, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 6, 'inner_gen_orders': [2, 3, 3, 3], 'inner_gens': [[1, 12, 36, 108], [13, 6, 18, 54], [37, 6, 18, 54], [109, 6, 18, 54]], 'inner_hash': 14, 'inner_nilpotent': False, 'inner_order': 54, 'inner_split': True, 'inner_tex': 'C_3^2:S_3', 'inner_used': [1, 2, 3, 4], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 6], [2, 39]], 'label': '162.53', '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': 'C3^3:C6', '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': 6, 'number_characteristic_subgroups': 6, 'number_conjugacy_classes': 45, 'number_divisions': 30, 'number_normal_subgroups': 58, 'number_subgroup_autclasses': 19, 'number_subgroup_classes': 190, 'number_subgroups': 580, 'old_label': None, 'order': 162, 'order_factorization_type': 31, 'order_stats': [[1, 1], [2, 27], [3, 80], [6, 54]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': True, 'outer_exponent': 312, 'outer_gen_orders': [2, 13], 'outer_gen_pows': [0, 0], 'outer_gens': [[5, 102, 156, 30], [1, 126, 132, 84]], 'outer_group': '11232.a', 'outer_hash': 778507202365856770, 'outer_nilpotent': False, 'outer_order': 11232, 'outer_permdeg': 15, 'outer_perms': [549021842335, 892132403040], 'outer_solvable': False, 'outer_supersolvable': False, 'outer_tex': '\\GL(3,3)', 'pc_rank': 4, 'perfect': False, 'permutation_degree': 12, 'pgroup': 0, 'primary_abelian_invariants': [2, 3], 'quasisimple': False, 'rank': 4, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 2], [2, 15], [4, 13]], 'representations': {'PC': {'code': 2271081929775, 'gens': [1, 3, 4, 5], 'pres': [5, -2, -3, -3, -3, -3, 10, 182, 723, 2704]}, 'GLZN': {'d': 2, 'p': 18, 'gens': [24505, 40831, 6083, 5941, 96397]}, 'Perm': {'d': 12, 'gens': [3633976, 325, 45708, 43591395, 43545600]}}, 'schur_multiplier': [3, 3, 3], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [6], 'solvability_type': 7, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_3^3:C_6', 'transitive_degree': 54, '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}
