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gps_subgroup_search • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'ambient': '384.4899', 'ambient_counter': 4899, 'ambient_order': 384, 'ambient_tex': 'C_3\\times C_2^4.D_4', 'central': False, 'central_factor': False, 'centralizer_order': 12, 'characteristic': False, 'core_order': 48, 'counter': 22, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '384.4899.4.j1.b1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': False, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': False, 'old_label': '4.j1.b1', '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': 4, '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': '96.169', 'subgroup_hash': 169, 'subgroup_order': 96, 'subgroup_tex': 'C_{12}.D_4', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '384.4899', 'aut_centralizer_order': 4, 'aut_label': '4.j1', 'aut_quo_index': None, 'aut_stab_index': 4, 'aut_weyl_group': '128.2328', 'aut_weyl_index': 16, 'centralizer': '32.a1.a1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['2.a1.a1'], 'contains': ['8.e1.b1', '8.i1.b1', '8.j1.a1', '8.l1.a1', '8.m1.b1', '8.p1.a1', '8.r1.a1', '12.j1.b1'], 'core': '8.e1.b1', 'coset_action_label': None, 'count': 2, 'diagramx': [2064, -1, 4531, -1, 2430, -1, 6981, -1], 'generators': [97, 192, 8, 4, 298, 128], 'label': '384.4899.4.j1.b1', 'mobius_quo': None, 'mobius_sub': 0, 'normal_closure': '2.a1.a1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '2.a1.a1', 'old_label': '4.j1.b1', 'projective_image': '32.27', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '4.j1.b1', 'subgroup_fusion': None, 'weyl_group': '16.14'}
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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': 2, 'aut_exponent': 4, 'aut_gen_orders': [2, 2, 2, 2, 2, 2, 4], 'aut_gens': [[1, 2, 8], [5, 6, 44], [49, 50, 60], [5, 6, 8], [49, 54, 44], [5, 50, 44], [1, 2, 56], [1, 2, 11]], 'aut_group': '256.55643', 'aut_hash': 55643, 'aut_nilpotency_class': 2, 'aut_nilpotent': True, 'aut_order': 256, 'aut_permdeg': 14, 'aut_perms': [14856750847, 126, 14856751008, 12936672414, 2918003760, 26951102640, 6881333760], 'aut_phi_ratio': 8.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [2, 2, 2, 1], [3, 1, 2, 1], [4, 2, 2, 2], [4, 4, 2, 2], [6, 1, 2, 3], [6, 2, 4, 1], [12, 2, 4, 2], [12, 4, 4, 2]], 'aut_supersolvable': True, 'aut_tex': 'C_2^5:D_4', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 2, 'autcent_group': '128.2328', 'autcent_hash': 2328, 'autcent_nilpotent': True, 'autcent_order': 128, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^7', 'autcentquo_abelian': True, 'autcentquo_cyclic': True, 'autcentquo_exponent': 2, 'autcentquo_group': '2.1', 'autcentquo_hash': 1, 'autcentquo_nilpotent': True, 'autcentquo_order': 2, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'C_2', 'cc_stats': [[1, 1, 1], [2, 1, 3], [2, 2, 2], [3, 1, 2], [4, 2, 4], [4, 4, 4], [6, 1, 6], [6, 2, 4], [12, 2, 8], [12, 4, 8]], 'center_label': '12.5', 'center_order': 12, 'central_product': True, 'central_quotient': '8.5', 'commutator_count': 1, 'commutator_label': '4.2', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '2.1', '3.1'], 'composition_length': 6, 'conjugacy_classes_known': True, 'counter': 169, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['3.1', 1], ['32.29', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [2, 2, 1, 2], [3, 1, 2, 1], [4, 2, 1, 2], [4, 2, 2, 1], [4, 4, 1, 4], [6, 1, 2, 3], [6, 2, 2, 2], [12, 2, 2, 2], [12, 2, 4, 1], [12, 4, 2, 4]], 'element_repr_type': 'PC', 'elementary': 2, 'eulerian_function': 1092, 'exponent': 12, 'exponents_of_order': [5, 1], 'factors_of_aut_order': [2], 'factors_of_order': [2, 3], 'faithful_reps': [], 'familial': False, 'frattini_label': '4.2', 'frattini_quotient': '24.15', 'hash': 169, 'hyperelementary': 2, 'inner_abelian': True, 'inner_cyclic': False, 'inner_exponent': 2, 'inner_gen_orders': [2, 2, 2], 'inner_gens': [[1, 2, 60], [1, 2, 56], [53, 50, 8]], 'inner_hash': 5, 'inner_nilpotent': True, 'inner_order': 8, 'inner_split': False, 'inner_tex': 'C_2^3', 'inner_used': [1, 2, 3], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 24], [2, 18]], 'label': '96.169', 'linC_count': 96, 'linC_degree': 4, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 6, 'linQ_degree_count': 40, 'linQ_dim': 6, 'linQ_dim_count': 4, 'linR_count': 8, 'linR_degree': 6, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'C12.D4', 'ngens': 6, 'nilpotency_class': 2, 'nilpotent': True, 'normal_counts': [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': 18, 'number_characteristic_subgroups': 24, 'number_conjugacy_classes': 42, 'number_divisions': 26, 'number_normal_subgroups': 48, 'number_subgroup_autclasses': 52, 'number_subgroup_classes': 74, 'number_subgroups': 100, 'old_label': None, 'order': 96, 'order_factorization_type': 51, 'order_stats': [[1, 1], [2, 7], [3, 2], [4, 24], [6, 14], [12, 48]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 2, 'outer_gen_orders': [2, 2, 2, 2, 2], 'outer_gen_pows': [3, 0, 0, 0, 0], 'outer_gens': [[1, 2, 11], [1, 54, 8], [1, 50, 8], [49, 2, 8], [1, 2, 40]], 'outer_group': '32.51', 'outer_hash': 51, 'outer_nilpotent': True, 'outer_order': 32, 'outer_permdeg': 10, 'outer_perms': [362880, 5040, 120, 6, 1], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^5', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 15, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 2, 3], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 8], [2, 12], [4, 5], [8, 1]], 'representations': {'PC': {'code': 23750089768052353, 'gens': [1, 2, 4], 'pres': [6, -2, -2, -2, 2, -2, -3, 31, 1443, 681, 69, 88]}, 'GLZ': {'b': 3, 'd': 6, 'gens': [25038648970471750, 91793116181224959, 125101754243306125]}, 'GLFq': {'d': 4, 'q': 4, 'gens': [1083311105, 1087522049, 1126927041, 2211481282, 2149582850, 1087570945]}, 'Perm': {'d': 15, 'gens': [87661824504, 186819072744, 12945584904, 4, 12541704, 186810624000]}}, 'schur_multiplier': [2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 6], 'solvability_type': 4, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_{12}.D_4', 'transitive_degree': 48, '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': 2, 'aut_exponent': 4, 'aut_gen_orders': [2, 2, 2, 4, 4, 4, 4, 4], 'aut_gens': [[1, 2, 4, 16], [203, 2, 111, 113], [1, 2, 5, 25], [1, 202, 196, 122], [3, 202, 292, 218], [3, 2, 111, 177], [3, 2, 300, 370], [9, 202, 205, 307], [195, 202, 300, 80]], 'aut_group': None, 'aut_hash': 2644724121517920397, 'aut_nilpotency_class': 3, 'aut_nilpotent': True, 'aut_order': 2048, 'aut_permdeg': 34, 'aut_perms': [89929003039749946034330638050945402793, 166011275397063767230667851766817143323, 73993844772290421900511448289572344753, 262443100180423537838265725212586562567, 116108426839895408365760093782087295705, 99006531604398589033179540205653963785, 44317096241468147958361808818516730965, 23275842176130962650452267427224965394], 'aut_phi_ratio': 16.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [2, 2, 1, 2], [2, 4, 2, 1], [3, 1, 2, 1], [4, 4, 2, 1], [4, 8, 1, 1], [4, 8, 2, 4], [6, 1, 2, 3], [6, 2, 2, 2], [6, 4, 4, 1], [8, 8, 4, 1], [12, 4, 4, 1], [12, 8, 2, 1], [12, 8, 4, 4], [24, 8, 8, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_2^8.C_2^3', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 2, 'autcent_group': '128.2328', 'autcent_hash': 2328, 'autcent_nilpotent': True, 'autcent_order': 128, 'autcent_solvable': True, 'autcent_split': False, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^7', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 4, 'autcentquo_group': '16.11', 'autcentquo_hash': 11, 'autcentquo_nilpotent': True, 'autcentquo_order': 16, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'C_2\\times D_4', 'cc_stats': [[1, 1, 1], [2, 1, 3], [2, 2, 2], [2, 4, 2], [3, 1, 2], [4, 4, 2], [4, 8, 9], [6, 1, 6], [6, 2, 4], [6, 4, 4], [8, 8, 4], [12, 4, 4], [12, 8, 18], [24, 8, 8]], 'center_label': '12.5', 'center_order': 12, 'central_product': True, 'central_quotient': '32.27', 'commutator_count': 1, 'commutator_label': '16.10', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '2.1', '2.1', '2.1', '3.1'], 'composition_length': 8, 'conjugacy_classes_known': True, 'counter': 4899, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['128.340', 1], ['3.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [2, 2, 1, 2], [2, 4, 1, 2], [3, 1, 2, 1], [4, 4, 1, 2], [4, 8, 1, 5], [4, 8, 2, 2], [6, 1, 2, 3], [6, 2, 2, 2], [6, 4, 2, 2], [8, 8, 2, 2], [12, 4, 2, 2], [12, 8, 2, 5], [12, 8, 4, 2], [24, 8, 4, 2]], 'element_repr_type': 'PC', 'elementary': 2, 'eulerian_function': 8736, 'exponent': 24, 'exponents_of_order': [7, 1], 'factors_of_aut_order': [2], 'factors_of_order': [2, 3], 'faithful_reps': [], 'familial': False, 'frattini_label': '16.10', 'frattini_quotient': '24.15', 'hash': 4899, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 4, 'inner_gen_orders': [2, 2, 2, 4], 'inner_gens': [[1, 2, 4, 218], [1, 2, 204, 216], [1, 202, 4, 304], [3, 202, 100, 16]], 'inner_hash': 27, 'inner_nilpotent': True, 'inner_order': 32, 'inner_split': True, 'inner_tex': 'C_2^2\\wr C_2', 'inner_used': [1, 2, 3, 4], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 24], [2, 30], [4, 15]], 'label': '384.4899', 'linC_count': 128, 'linC_degree': 6, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 14, 'linQ_degree_count': 48, 'linQ_dim': 14, 'linQ_dim_count': 32, 'linR_count': 32, 'linR_degree': 12, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'C3*C2^4.D4', 'ngens': 8, 'nilpotency_class': 3, 'nilpotent': True, 'normal_counts': [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': 28, 'number_characteristic_subgroups': 36, 'number_conjugacy_classes': 69, 'number_divisions': 38, 'number_normal_subgroups': 68, 'number_subgroup_autclasses': 168, 'number_subgroup_classes': 228, 'number_subgroups': 568, 'old_label': None, 'order': 384, 'order_factorization_type': 51, 'order_stats': [[1, 1], [2, 15], [3, 2], [4, 80], [6, 30], [8, 32], [12, 160], [24, 64]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 4, 'outer_gen_orders': [2, 2, 2, 2, 4], 'outer_gen_pows': [0, 0, 0, 0, 0], 'outer_gens': [[1, 2, 204, 272], [1, 2, 204, 16], [1, 2, 204, 80], [1, 2, 5, 283], [193, 2, 197, 91]], 'outer_group': '64.261', 'outer_hash': 261, 'outer_nilpotent': True, 'outer_order': 64, 'outer_permdeg': 10, 'outer_perms': [1174320, 136, 368056, 368047, 138], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'D_4\\times C_2^3', 'pc_rank': 4, 'perfect': False, 'permutation_degree': 27, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 2, 3], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 8], [2, 14], [4, 9], [8, 5], [16, 2]], 'representations': {'PC': {'code': 6421070931721932075057295955447280641, 'gens': [1, 2, 3, 5], 'pres': [8, 2, 2, 2, 2, 2, 2, 2, 3, 2458, 66, 8724, 4332, 3060, 116, 11141, 2709, 141, 166]}, 'Perm': {'d': 27, 'gens': [453760580333021203808748240, 870729871040446976577986400, 259710146441275711104000, 4, 1291861510299837637890888480, 4991028205505503569408000, 1711455689043410443547502960, 1711455689043410443535001600]}}, 'schur_multiplier': [2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 6], 'solvability_type': 4, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_3\\times C_2^4.D_4', 'transitive_degree': 96, 'wreath_data': None, 'wreath_product': False}
