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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': False, 'ambient': '3888.js', 'ambient_counter': 253, 'ambient_order': 3888, 'ambient_tex': 'C_3:S_3^4', 'central': False, 'central_factor': False, 'centralizer_order': 4, 'characteristic': False, 'core_order': 18, 'counter': 156, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '3888.js.27.b1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': False, 'minimal': False, 'minimal_normal': False, 'nilpotent': False, 'normal': False, 'old_label': '27.b1', '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': 27, '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': '144.192', 'subgroup_hash': 192, 'subgroup_order': 144, 'subgroup_tex': 'D_6^2', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '3888.js', 'aut_centralizer_order': 8, 'aut_label': '27.b1', 'aut_quo_index': None, 'aut_stab_index': 162, 'aut_weyl_group': '72.46', 'aut_weyl_index': 1296, 'centralizer': '972.a1', 'complements': None, 'conjugacy_class_count': 6, 'contained_in': ['9.b1', '9.c1'], 'contains': ['54.a1', '54.b1', '54.e1', '54.k1', '54.o1', '54.p1', '54.q1', '54.r1', '54.u1', '54.x1', '54.z1', '81.a1', '81.b1'], 'core': '216.a1', 'coset_action_label': None, 'count': 162, 'diagramx': None, 'generators': [93405312000, 180667066050, 269, 39916800, 40320, 43545600], 'label': '3888.js.27.b1', 'mobius_quo': None, 'mobius_sub': None, 'normal_closure': '1.a1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '27.b1', 'old_label': '27.b1', 'projective_image': '3888.js', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '27.b1', 'subgroup_fusion': None, 'weyl_group': '36.10'}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '16.14', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 3, 'aut_exponent': 12, 'aut_gen_orders': [6, 6, 4], 'aut_gens': [[41064, 1681, 744, 2424, 3, 403200], [40320, 1681, 1680, 744, 4, 403200], [746, 41064, 2424, 1680, 725760, 4], [1681, 42000, 744, 1680, 403200, 4]], 'aut_group': '6912.bq', 'aut_hash': 8864889281089178955, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 6912, 'aut_permdeg': 14, 'aut_perms': [1444302721, 21284001292, 40436449517], 'aut_phi_ratio': 144.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 3, 1], [2, 3, 8, 1], [2, 9, 4, 1], [3, 2, 2, 1], [3, 4, 1, 1], [6, 2, 6, 1], [6, 4, 3, 1], [6, 6, 8, 1]], 'aut_supersolvable': False, 'aut_tex': 'D_6^2:(C_2\\times S_4)', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 12, 'autcent_group': '96.227', 'autcent_hash': 227, 'autcent_nilpotent': False, 'autcent_order': 96, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': False, 'autcent_tex': 'C_2^2:S_4', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 12, 'autcentquo_group': '72.40', 'autcentquo_hash': 40, 'autcentquo_nilpotent': False, 'autcentquo_order': 72, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': '\\SOPlus(4,2)', 'cc_stats': [[1, 1, 1], [2, 1, 3], [2, 3, 8], [2, 9, 4], [3, 2, 2], [3, 4, 1], [6, 2, 6], [6, 4, 3], [6, 6, 8]], 'center_label': '4.2', 'center_order': 4, 'central_product': True, 'central_quotient': '36.10', 'commutator_count': 1, 'commutator_label': '9.2', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '3.1', '3.1'], 'composition_length': 6, 'conjugacy_classes_known': True, 'counter': 192, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['2.1', 2], ['6.1', 2]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [2, 3, 1, 8], [2, 9, 1, 4], [3, 2, 1, 2], [3, 4, 1, 1], [6, 2, 1, 6], [6, 4, 1, 3], [6, 6, 1, 8]], 'element_repr_type': 'Perm', 'elementary': 1, 'eulerian_function': 17745, 'exponent': 6, 'exponents_of_order': [4, 2], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [], 'familial': False, 'frattini_label': '1.1', 'frattini_quotient': '144.192', 'hash': 192, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 6, 'inner_gen_orders': [2, 2, 1, 1, 3, 3], 'inner_gens': [[41064, 1681, 744, 2424, 3, 725760], [41064, 1681, 744, 2424, 4, 403200], [41064, 1681, 744, 2424, 3, 403200], [41064, 1681, 744, 2424, 3, 403200], [41064, 1685, 744, 2424, 3, 403200], [766824, 1681, 744, 2424, 3, 403200]], 'inner_hash': 10, 'inner_nilpotent': False, 'inner_order': 36, 'inner_split': True, 'inner_tex': 'S_3^2', 'inner_used': [1, 2, 5, 6], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 16], [2, 16], [4, 4]], 'label': '144.192', 'linC_count': None, 'linC_degree': None, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 4, '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': 'D6^2', 'ngens': 6, 'nilpotency_class': -1, 'nilpotent': False, 'normal_counts': [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': 9, 'number_characteristic_subgroups': 6, 'number_conjugacy_classes': 36, 'number_divisions': 36, 'number_normal_subgroups': 104, 'number_subgroup_autclasses': 45, 'number_subgroup_classes': 284, 'number_subgroups': 912, 'old_label': None, 'order': 144, 'order_factorization_type': 32, 'order_stats': [[1, 1], [2, 63], [3, 8], [6, 72]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': True, 'outer_exponent': 12, 'outer_gen_orders': [2, 2, 3, 2, 2, 2, 2], 'outer_gen_pows': [42745, 0, 41064, 0, 0, 0, 0], 'outer_gens': [[1681, 41064, 2424, 744, 403200, 4], [1681, 41064, 744, 2424, 725760, 4], [41064, 1681, 2424, 1680, 3, 725760], [42744, 745, 744, 2424, 4, 725760], [40320, 1, 744, 2424, 4, 725760], [42000, 745, 744, 2424, 4, 403200], [42744, 1, 744, 2424, 3, 403200]], 'outer_group': '192.955', 'outer_hash': 955, 'outer_nilpotent': False, 'outer_order': 192, 'outer_permdeg': 8, 'outer_perms': [1562, 24460, 843, 16680, 5160, 5176, 11543], 'outer_solvable': True, 'outer_supersolvable': False, 'outer_tex': 'C_2^3:S_4', 'pc_rank': 4, 'perfect': False, 'permutation_degree': 10, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 2, 2], 'quasisimple': False, 'rank': 4, 'rational': True, 'rational_characters_known': True, 'ratrep_stats': [[1, 16], [2, 16], [4, 4]], 'representations': {'PC': {'code': 4214134166136227077, 'gens': [1, 2, 3, 5], 'pres': [6, -2, -2, -2, -3, -2, -3, 188, 50, 201, 916, 88, 881]}, 'GLZ': {'b': 3, 'd': 4, 'gens': [12016555, 26483311, 35931072, 31030333]}, 'GLFp': {'d': 4, 'p': 3, 'gens': [24822599, 6594100, 8092916, 16002800, 28816400, 31289712]}, 'Perm': {'d': 10, 'gens': [41064, 1681, 744, 2424, 3, 403200]}}, 'schur_multiplier': [2, 2, 2, 2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 2, 2], 'solvability_type': 7, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'D_6^2', 'transitive_degree': 24, 'wreath_data': None, 'wreath_product': False}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '16.14', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 4, 'aut_exponent': 36, 'aut_gen_orders': [4, 18, 6, 4], 'aut_gens': [[3, 43545600, 45360, 39916800, 93405312000, 180667066290, 269, 40320, 144], [144, 45360, 79833600, 85680, 174356582400, 180583608694, 269, 39916800, 3], [174356582643, 45360, 43545600, 85680, 244, 87178296811, 6227021250, 39916800, 93405312243], [93405312148, 79833600, 45360, 3628800, 147, 87261754315, 180583603798, 5040, 174356582548], [174356582547, 79833600, 80640, 39916800, 148, 87181920694, 6227021275, 85680, 93405312147]], 'aut_group': '93312.is', 'aut_hash': 458068617261292163, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 93312, 'aut_permdeg': 33, 'aut_perms': [7488918491129848130499211209898814287, 7483182473869955279100711230071288207, 3692085122588095920907256051229287602, 715469638848146002050080652783461232], 'aut_phi_ratio': 72.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 3, 2, 1], [2, 9, 1, 1], [2, 9, 3, 1], [2, 27, 6, 1], [2, 81, 3, 1], [3, 2, 2, 1], [3, 2, 3, 1], [3, 4, 1, 1], [3, 4, 2, 1], [3, 4, 3, 1], [3, 4, 6, 1], [3, 8, 3, 1], [3, 8, 4, 1], [3, 8, 6, 1], [3, 16, 2, 1], [3, 16, 3, 1], [6, 6, 2, 1], [6, 6, 6, 1], [6, 12, 4, 1], [6, 12, 6, 2], [6, 18, 3, 2], [6, 18, 6, 1], [6, 24, 4, 1], [6, 24, 6, 1], [6, 36, 2, 1], [6, 36, 3, 2], [6, 36, 6, 1], [6, 54, 6, 2], [6, 72, 3, 1], [6, 108, 6, 1], [6, 162, 3, 1]], 'aut_supersolvable': False, 'aut_tex': 'S_3^5.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': 36, 'autcentquo_group': '93312.is', 'autcentquo_hash': 458068617261292163, 'autcentquo_nilpotent': False, 'autcentquo_order': 93312, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'S_3^5.D_6', 'cc_stats': [[1, 1, 1], [2, 3, 2], [2, 9, 4], [2, 27, 6], [2, 81, 3], [3, 2, 5], [3, 4, 12], [3, 8, 13], [3, 16, 5], [6, 6, 8], [6, 12, 16], [6, 18, 12], [6, 24, 10], [6, 36, 14], [6, 54, 12], [6, 72, 3], [6, 108, 6], [6, 162, 3]], 'center_label': '1.1', 'center_order': 1, 'central_product': True, 'central_quotient': '3888.js', 'commutator_count': 1, 'commutator_label': '243.67', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '3.1', '3.1', '3.1', '3.1', '3.1'], 'composition_length': 9, 'conjugacy_classes_known': True, 'counter': 253, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['108.40', 1], ['6.1', 2]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 3, 1, 2], [2, 9, 1, 4], [2, 27, 1, 6], [2, 81, 1, 3], [3, 2, 1, 5], [3, 4, 1, 10], [3, 4, 2, 1], [3, 8, 1, 9], [3, 8, 2, 2], [3, 16, 1, 3], [3, 16, 2, 1], [6, 6, 1, 8], [6, 12, 1, 12], [6, 12, 2, 2], [6, 18, 1, 12], [6, 24, 1, 6], [6, 24, 2, 2], [6, 36, 1, 12], [6, 36, 2, 1], [6, 54, 1, 12], [6, 72, 1, 3], [6, 108, 1, 6], [6, 162, 1, 3]], 'element_repr_type': 'Perm', 'elementary': 1, 'eulerian_function': 623772240, 'exponent': 6, 'exponents_of_order': [5, 4], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [[16, 0, 2]], 'familial': False, 'frattini_label': '1.1', 'frattini_quotient': '3888.js', 'hash': 528584399898365654, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 6, 'inner_gen_orders': [3, 3, 3, 2, 3, 6, 2, 2, 3], 'inner_gens': [[3, 43545600, 45360, 39916800, 93405312000, 180667066054, 266, 40320, 144], [3, 43545600, 45360, 3628800, 93405312000, 180623520690, 269, 40320, 144], [3, 43545600, 45360, 39916800, 93405312000, 180667066290, 269, 5040, 144], [3, 79833600, 45360, 39916800, 93405312000, 180587232690, 269, 40320, 144], [3, 43545600, 45360, 39916800, 93405312000, 87261754290, 269, 40320, 144], [144, 79833600, 45360, 3628800, 174356582400, 180667066290, 265, 40320, 3], [4, 43545600, 45360, 39916800, 93405312000, 180667066194, 269, 40320, 240], [3, 43545600, 80640, 39916800, 93405312000, 180667066290, 269, 40320, 144], [3, 43545600, 45360, 39916800, 93405312000, 180667066197, 125, 40320, 144]], 'inner_hash': 528584399898365654, 'inner_nilpotent': False, 'inner_order': 3888, 'inner_split': True, 'inner_tex': 'C_3:S_3^4', 'inner_used': [1, 2, 3, 4, 5, 6, 7, 8], 'irrC_degree': 16, 'irrQ_degree': 32, 'irrQ_dim': 32, 'irrR_degree': 32, 'irrep_stats': [[1, 16], [2, 40], [4, 48], [8, 26], [16, 5]], 'label': '3888.js', '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:S3^4', '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, 0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 36, 'number_characteristic_subgroups': 12, 'number_conjugacy_classes': 135, 'number_divisions': 126, 'number_normal_subgroups': 228, 'number_subgroup_autclasses': 766, 'number_subgroup_classes': 4266, 'number_subgroups': 87768, 'old_label': None, 'order': 3888, 'order_factorization_type': 33, 'order_stats': [[1, 1], [2, 447], [3, 242], [6, 3198]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': True, 'outer_exponent': 6, 'outer_gen_orders': [2, 2, 6], 'outer_gen_pows': [0, 269, 0], 'outer_gens': [[3, 43545600, 45360, 39916800, 174356582400, 180667066290, 269, 40320, 144], [240, 79833600, 45360, 39916800, 174356582400, 180587232715, 269, 40320, 3], [174356582548, 45360, 43545600, 40320, 147, 6227106956, 180583603650, 39916800, 93405312148]], 'outer_group': '24.14', 'outer_hash': 14, 'outer_nilpotent': False, 'outer_order': 24, 'outer_permdeg': 7, 'outer_perms': [7, 120, 856], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2\\times D_6', 'pc_rank': 6, 'perfect': False, 'permutation_degree': 15, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 2, 2], 'quasisimple': False, 'rank': 4, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 16], [2, 40], [4, 40], [8, 22], [16, 7], [32, 1]], 'representations': {'PC': {'code': '418182879202542983638411866285743712356080812124173732311110747', 'gens': [1, 2, 4, 6, 7, 9], 'pres': [9, -2, -2, -3, -2, -3, -3, -2, -3, -3, 1477, 46, 4106, 2388, 1425, 102, 1093, 680, 34035, 5703, 186, 31120, 5218, 1034]}, 'Perm': {'d': 15, 'gens': [3, 43545600, 45360, 39916800, 93405312000, 180667066290, 269, 40320, 144]}}, 'schur_multiplier': [2, 2, 2, 2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 2, 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_3^4', 'transitive_degree': 48, 'wreath_data': None, 'wreath_product': False}
