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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': False, 'ambient': '11664.kv', 'ambient_counter': 282, 'ambient_order': 11664, 'ambient_tex': '\\He_3^2.(C_2\\times D_4)', 'central': False, 'central_factor': False, 'centralizer_order': 9, 'characteristic': False, 'core_order': 3, 'counter': 252, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '11664.kv.216.BH', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': False, 'minimal': False, 'minimal_normal': False, 'nilpotent': False, 'normal': False, 'old_label': '216.bh1', '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': 216, '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': '54.13', 'subgroup_hash': 13, 'subgroup_order': 54, 'subgroup_tex': 'C_3^2:C_6', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '11664.kv', 'aut_centralizer_order': None, 'aut_label': '216.BH', 'aut_quo_index': None, 'aut_stab_index': None, 'aut_weyl_group': None, 'aut_weyl_index': None, 'centralizer': '1296.A', 'complements': None, 'conjugacy_class_count': 2, 'contained_in': ['72.Q', '108.P', '108.S', '108.BM'], 'contains': ['432.L', '648.Z', '648.BA', '648.BI'], 'core': '3888.B', 'coset_action_label': None, 'count': 36, 'diagramx': [3766, -1, 1580, -1], 'generators': [4, 1296, 7400, 4464], 'label': '11664.kv.216.BH', 'mobius_quo': None, 'mobius_sub': 0, 'normal_closure': '8.B', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '18.A', 'old_label': '216.bh1', 'projective_image': '11664.kv', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '216.BH', 'subgroup_fusion': None, 'weyl_group': '72.46'}
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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': 5, 'aut_exponent': 24, 'aut_gen_orders': [2, 4, 2, 4, 3, 2, 6, 6], 'aut_gens': [[1, 6, 18], [5, 6, 18], [47, 42, 48], [19, 12, 36], [53, 24, 42], [7, 6, 18], [5, 48, 18], [53, 6, 18], [11, 30, 36]], 'aut_group': '864.4661', 'aut_hash': 4661, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 864, 'aut_permdeg': 11, 'aut_perms': [1, 30224881, 7984086, 25453807, 4843488, 2245249, 27748111, 4241545], 'aut_phi_ratio': 48.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 9, 1, 1], [3, 1, 2, 1], [3, 2, 4, 1], [3, 2, 8, 1], [6, 9, 2, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_2\\times C_3^2:\\GL(2,3)', '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': 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, 1, 2], [3, 2, 12], [6, 9, 2]], 'center_label': '3.1', 'center_order': 3, 'central_product': True, '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', '3.1'], 'composition_length': 4, 'conjugacy_classes_known': True, 'counter': 13, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['18.4', 1], ['3.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 9, 1, 1], [3, 1, 2, 1], [3, 2, 1, 4], [3, 2, 2, 4], [6, 9, 2, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 91, 'exponent': 6, 'exponents_of_order': [3, 1], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [], 'familial': False, 'frattini_label': '1.1', 'frattini_quotient': '54.13', 'hash': 13, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 6, 'inner_gen_orders': [2, 3, 3], 'inner_gens': [[1, 12, 36], [13, 6, 18], [37, 6, 18]], '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, 6], [2, 12]], 'label': '54.13', 'linC_count': 48, 'linC_degree': 4, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 6, 'linQ_degree_count': 24, 'linQ_dim': 6, 'linQ_dim_count': 24, 'linR_count': 24, 'linR_degree': 6, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'C3^2:C6', 'ngens': 4, 'nilpotency_class': -1, 'nilpotent': False, 'normal_counts': [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': 18, 'number_divisions': 12, 'number_normal_subgroups': 14, 'number_subgroup_autclasses': 14, 'number_subgroup_classes': 32, 'number_subgroups': 72, 'old_label': None, 'order': 54, 'order_factorization_type': 31, 'order_stats': [[1, 1], [2, 9], [3, 26], [6, 18]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 12, 'outer_gen_orders': [2, 2, 3, 2, 2], 'outer_gen_pows': [0, 0, 0, 1, 1], 'outer_gens': [[5, 36, 12], [5, 6, 18], [1, 36, 42], [1, 48, 30], [1, 30, 24]], 'outer_group': '48.48', 'outer_hash': 48, 'outer_nilpotent': False, 'outer_order': 48, 'outer_permdeg': 6, 'outer_perms': [143, 127, 576, 126, 121], 'outer_solvable': True, 'outer_supersolvable': False, 'outer_tex': 'C_2\\times S_4', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 9, 'pgroup': 0, 'primary_abelian_invariants': [2, 3], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 2], [2, 6], [4, 4]], 'representations': {'PC': {'code': 25289495, 'gens': [1, 3, 4], 'pres': [4, -2, -3, -3, -3, 8, 146, 579]}, 'GLZ': {'b': 3, 'd': 6, 'gens': [125101736062737953, 41624334336939899, 41715841511539063]}, 'GLFp': {'d': 4, 'p': 3, 'gens': [11495158, 22274384, 5504967, 25627796]}, 'Perm': {'d': 9, 'gens': [42001, 97224, 136227, 4]}}, 'schur_multiplier': [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_3^2:C_6', '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': '8.5', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 4, 'aut_exponent': 24, 'aut_gen_orders': [6, 12, 12, 12, 12], 'aut_gens': [[1, 2, 8, 24, 144, 432, 1296, 3888], [10329, 10334, 5728, 9672, 2448, 1728, 1296, 7776], [2827, 298, 7248, 2908, 6576, 5584, 1296, 3888], [9835, 7222, 7592, 10496, 3024, 2880, 1296, 3888], [9805, 7302, 160, 10316, 10520, 2840, 2592, 3888], [11337, 8694, 8896, 3828, 2792, 96, 1296, 3888]], 'aut_group': None, 'aut_hash': 2302517250360555534, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 46656, 'aut_permdeg': 162, 'aut_perms': [4737916610715155664706401302694766945012519924953301976824808089393804293648194579990954855708401038921529071894620255454018529348958821742187275270695945806764351215513489066481636158403019556584192837638862119121133385707112792036951057084806190098427908414588436500510831271111715221668, 7974862195986620764570782997193519899216126092955316737965175675006575162213083837625994692455517747352360410253457681896986818390495762818061786351150192440709605742330913587291023397475729610782719227281892814675137859198102266080399923801657049270871296852140271291839578098418287100290, 5687185783820374802948162372881313873288657029100817877521171592239608525129547531421634988555609410451264108314162733724110875599238783186663663976977501492522999528748597520234911224346052528205006924135794191453459064143148642266678265761894686759015734690537447031245761010229366260312, 6586474084541327125109866304448262014215488733785111117583594827861397168026512246470539610687452602850772565875601767806545280352032774343041733839438622251684704378243091088784497279658999140002073882130055376186822302119601904539951498786240339296298690928769824906441783382117357307430, 2627297330428913050694626691194921378603014867027762817117397868331299490249951100562254549052047074189940586554463690006989256567298722129437919785090383471449797948618513294843022291565809830814791932930737254356274830645592657855370054688147965002285878690089072330711995391990521310230], 'aut_phi_ratio': 12.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 27, 2, 1], [2, 54, 2, 1], [2, 81, 1, 1], [2, 162, 2, 1], [3, 2, 1, 2], [3, 4, 1, 1], [3, 24, 2, 1], [3, 36, 4, 1], [3, 48, 2, 1], [3, 72, 2, 1], [3, 144, 2, 1], [4, 162, 1, 1], [4, 486, 1, 1], [6, 54, 2, 1], [6, 108, 2, 1], [6, 162, 1, 2], [6, 324, 1, 1], [6, 324, 4, 3], [6, 648, 2, 2], [12, 162, 2, 1], [12, 324, 1, 1], [12, 324, 2, 1], [12, 486, 2, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_3^2.C_3^4.D_4.C_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': None, 'autcentquo_hash': 2302517250360555534, 'autcentquo_nilpotent': False, 'autcentquo_order': 46656, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'C_3^2.C_3^4.D_4.C_2^3', 'cc_stats': [[1, 1, 1], [2, 27, 2], [2, 54, 2], [2, 81, 1], [2, 162, 2], [3, 2, 2], [3, 4, 1], [3, 24, 2], [3, 36, 4], [3, 48, 2], [3, 72, 2], [3, 144, 2], [4, 162, 1], [4, 486, 1], [6, 54, 2], [6, 108, 2], [6, 162, 2], [6, 324, 13], [6, 648, 4], [12, 162, 2], [12, 324, 3], [12, 486, 2]], 'center_label': '1.1', 'center_order': 1, 'central_product': False, 'central_quotient': '11664.kv', 'commutator_count': 1, 'commutator_label': '1458.1576', 'complements_known': False, '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', '3.1'], 'composition_length': 10, 'conjugacy_classes_known': True, 'counter': 282, 'cyclic': False, 'derived_length': 4, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 27, 1, 2], [2, 54, 1, 2], [2, 81, 1, 1], [2, 162, 1, 2], [3, 2, 1, 2], [3, 4, 1, 1], [3, 24, 1, 2], [3, 36, 1, 4], [3, 48, 1, 2], [3, 72, 1, 2], [3, 144, 1, 2], [4, 162, 1, 1], [4, 486, 1, 1], [6, 54, 1, 2], [6, 108, 1, 2], [6, 162, 1, 2], [6, 324, 1, 13], [6, 648, 1, 4], [12, 162, 2, 1], [12, 324, 1, 1], [12, 324, 2, 1], [12, 486, 2, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 10886400, 'exponent': 12, 'exponents_of_order': [6, 4], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [[12, 1, 4], [48, 1, 2]], 'familial': False, 'frattini_label': '9.2', 'frattini_quotient': '1296.3531', 'hash': 2955081780199608270, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 12, 'inner_gen_orders': [6, 4, 3, 6, 3, 3, 3, 3], 'inner_gens': [[1, 6110, 5728, 11256, 8928, 10800, 1296, 7776], [4813, 2, 3136, 5504, 11520, 720, 2592, 3888], [10001, 930, 8, 6648, 144, 5616, 1296, 3888], [5617, 1602, 9368, 24, 9360, 6048, 2592, 3888], [8497, 9794, 8, 6792, 144, 8208, 1296, 3888], [5185, 4034, 10376, 11256, 4032, 432, 1296, 3888], [1, 2594, 8, 2616, 144, 432, 1296, 3888], [7777, 2, 8, 24, 144, 432, 1296, 3888]], 'inner_hash': 2955081780199608270, 'inner_nilpotent': False, 'inner_order': 11664, 'inner_split': False, 'inner_tex': '\\He_3^2.(C_2\\times D_4)', 'inner_used': [1, 2, 3, 4], 'irrC_degree': 12, 'irrQ_degree': 12, 'irrQ_dim': 12, 'irrR_degree': 12, 'irrep_stats': [[1, 8], [2, 2], [4, 16], [8, 8], [12, 4], [16, 2], [18, 12], [36, 1], [48, 2]], 'label': '11664.kv', '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': None, 'name': 'He3^2.(C2*D4)', 'ngens': 10, '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, 0, 0, 0, 0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 29, 'number_characteristic_subgroups': 13, 'number_conjugacy_classes': 55, 'number_divisions': 52, 'number_normal_subgroups': 27, 'number_subgroup_autclasses': 526, 'number_subgroup_classes': 1295, 'number_subgroups': 102579, 'old_label': None, 'order': 11664, 'order_factorization_type': 33, 'order_stats': [[1, 1], [2, 567], [3, 728], [4, 648], [6, 7452], [12, 2268]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': True, 'outer_exponent': 2, 'outer_gen_orders': [2, 2], 'outer_gen_pows': [2670, 4], 'outer_gens': [[3051, 2594, 8880, 9488, 2016, 144, 2592, 7776], [2689, 2, 9232, 5692, 2744, 6528, 1296, 7776]], 'outer_group': '4.2', 'outer_hash': 2, 'outer_nilpotent': True, 'outer_order': 4, 'outer_permdeg': 4, 'outer_perms': [1, 6], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^2', 'pc_rank': 8, 'perfect': False, 'permutation_degree': 36, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 2], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 8], [2, 2], [4, 16], [8, 8], [12, 2], [16, 2], [18, 8], [24, 1], [36, 3], [48, 2]], 'representations': {'PC': {'code': '3736206736914572732371176538338573298327729471332021339675670395103035095875381889405422227048755497135150582706638795111865356981490165501023', 'gens': [1, 2, 4, 5, 7, 8, 9, 10], 'pres': [10, 2, 2, 2, 3, 2, 3, 3, 3, 3, 3, 112800, 122201, 51, 260762, 229123, 62733, 103863, 562804, 137614, 1524, 41584, 144, 158405, 49455, 1465, 39275, 624966, 403216, 186506, 27346, 2156, 864007, 28817, 120987, 56197, 20207, 2937, 4627, 116658, 9768, 777609]}, 'Perm': {'d': 36, 'gens': [106363587271560335870164627814774715242610, 329447778369213946313011098570331825082185, 276255404203435341815859230992429384525196]}}, 'schur_multiplier': [2, 2, 2], 'semidirect_product': None, 'simple': False, 'smith_abelian_invariants': [2, 2, 2], 'solvability_type': 17, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': '\\He_3^2.(C_2\\times D_4)', 'transitive_degree': 36, 'wreath_data': None, 'wreath_product': False}
