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gps_subgroup_search • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'ambient': '7776.ga', 'ambient_counter': 157, 'ambient_order': 7776, 'ambient_tex': 'C_6^3:S_3^2', 'central': False, 'central_factor': False, 'centralizer_order': 36, 'characteristic': False, 'core_order': 216, 'counter': 121, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '7776.ga.18.y1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': None, 'minimal': False, 'minimal_normal': False, 'nilpotent': False, 'normal': False, 'old_label': '18.y1', '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': 18, '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': '432.709', 'subgroup_hash': 709, 'subgroup_order': 432, 'subgroup_tex': 'D_6:C_6^2', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '7776.ga', 'aut_centralizer_order': None, 'aut_label': '18.y1', 'aut_quo_index': None, 'aut_stab_index': None, 'aut_weyl_group': None, 'aut_weyl_index': None, 'centralizer': '216.e1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['6.j1', '6.k1', '6.l1', '9.d1'], 'contains': ['36.a1', '36.r1', '36.bc1', '36.cu1', '36.ec1', '54.bh1', '54.bi1', '54.bl1', '54.bm1'], 'core': '36.a1', 'coset_action_label': None, 'count': 9, 'diagramx': None, 'generators': [1110215538432000, 289524, 3628800, 22629, 12454080745, 356995102464000, 397533011508588], 'label': '7776.ga.18.y1', 'mobius_quo': None, 'mobius_sub': -3, 'normal_closure': '2.b1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '9.d1', 'old_label': '18.y1', 'projective_image': '3888.gu', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '18.y1', 'subgroup_fusion': None, 'weyl_group': '24.14'}
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gps_groups • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '72.50', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 4, 'aut_exponent': 24, 'aut_gen_orders': [8, 12, 4, 4, 12, 12], 'aut_gens': [[1, 6, 72], [65, 392, 360], [221, 286, 396], [217, 284, 108], [41, 258, 108], [63, 284, 108], [1, 140, 108]], 'aut_group': None, 'aut_hash': 1822566015781033619, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 9216, 'aut_permdeg': 24, 'aut_perms': [202621586405910432672780, 318733964645526518062825, 311967325460926571468683, 602731053973990640905023, 38429036611139238558750, 311416099544399033083962], 'aut_phi_ratio': 64.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 1, 2, 1], [2, 2, 2, 1], [2, 6, 2, 1], [3, 1, 8, 1], [3, 2, 1, 1], [3, 2, 8, 1], [4, 6, 2, 1], [6, 1, 8, 1], [6, 1, 16, 1], [6, 2, 1, 1], [6, 2, 2, 1], [6, 2, 4, 1], [6, 2, 8, 1], [6, 2, 16, 2], [6, 2, 32, 1], [6, 6, 16, 1], [12, 6, 16, 1]], 'aut_supersolvable': False, 'aut_tex': 'Q_8.D_6^2.C_2^3', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 24, 'autcent_group': '1536.408607494', 'autcent_hash': 998638736221672515, 'autcent_nilpotent': False, 'autcent_order': 1536, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': False, 'autcent_tex': 'C_2^2\\wr C_2\\times \\GL(2,3)', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 6, 'autcentquo_group': '6.1', 'autcentquo_hash': 1, 'autcentquo_nilpotent': False, 'autcentquo_order': 6, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'S_3', 'cc_stats': [[1, 1, 1], [2, 1, 3], [2, 2, 2], [2, 6, 2], [3, 1, 8], [3, 2, 9], [4, 6, 2], [6, 1, 24], [6, 2, 79], [6, 6, 16], [12, 6, 16]], 'center_label': '36.14', 'center_order': 36, 'central_product': True, 'central_quotient': '12.4', 'commutator_count': 1, 'commutator_label': '6.2', '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'], 'composition_length': 7, 'conjugacy_classes_known': True, 'counter': 709, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['2.1', 1], ['24.8', 1], ['3.1', 2]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [2, 2, 1, 2], [2, 6, 1, 2], [3, 1, 2, 4], [3, 2, 1, 1], [3, 2, 2, 4], [4, 6, 1, 2], [6, 1, 2, 12], [6, 2, 1, 3], [6, 2, 2, 38], [6, 6, 2, 8], [12, 6, 2, 8]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 2184, 'exponent': 12, 'exponents_of_order': [4, 3], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [], 'familial': False, 'frattini_label': '2.1', 'frattini_quotient': '216.174', 'hash': 709, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 6, 'inner_gen_orders': [2, 2, 3], 'inner_gens': [[1, 42, 72], [37, 6, 360], [1, 150, 72]], 'inner_hash': 4, 'inner_nilpotent': False, 'inner_order': 12, 'inner_split': False, 'inner_tex': 'D_6', 'inner_used': [1, 2, 3], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 72], [2, 90]], 'label': '432.709', 'linC_count': None, 'linC_degree': None, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 6, '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:C6^2', 'ngens': 7, '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], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 20, 'number_characteristic_subgroups': 22, 'number_conjugacy_classes': 162, 'number_divisions': 88, 'number_normal_subgroups': 162, 'number_subgroup_autclasses': 112, 'number_subgroup_classes': 452, 'number_subgroups': 904, 'old_label': None, 'order': 432, 'order_factorization_type': 33, 'order_stats': [[1, 1], [2, 19], [3, 26], [4, 12], [6, 278], [12, 96]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': True, 'outer_exponent': 24, 'outer_gen_orders': [2, 6, 2, 2, 4, 4], 'outer_gen_pows': [0, 0, 0, 0, 0, 0], 'outer_gens': [[5, 226, 72], [25, 272, 72], [41, 30, 108], [221, 246, 72], [245, 226, 108], [25, 34, 72]], 'outer_group': '768.1089273', 'outer_hash': 1089273, 'outer_nilpotent': False, 'outer_order': 768, 'outer_permdeg': 14, 'outer_perms': [2442625920, 1005908401, 13902013560, 288, 33734096214, 20765162880], 'outer_solvable': True, 'outer_supersolvable': False, 'outer_tex': 'C_2\\times D_4\\times \\GL(2,3)', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 15, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 2, 3, 3], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 8], [2, 38], [4, 42]], 'representations': {'PC': {'code': 112197688920474384586528421, 'gens': [1, 3, 6], 'pres': [7, -2, -3, -2, -2, -3, -2, -3, 14, 884, 58, 80, 2539, 124, 2372]}, 'GLZ': {'b': 3, 'd': 6, 'gens': [24992906222183252, 41624325034614139, 125101743017174879]}, 'GLFp': {'d': 3, 'p': 7, 'gens': [30419024, 36826717, 5579858, 28836015, 10065962, 24600908, 23068812]}, 'Perm': {'d': 15, 'gens': [87657655680, 186810624024, 12933043200, 5760, 10083, 186810624000, 3991680]}}, 'schur_multiplier': [2, 2, 6], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 6, 6], 'solvability_type': 7, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'D_6:C_6^2', 'transitive_degree': 72, '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': 3, 'aut_exponent': 36, 'aut_gen_orders': [6, 6, 6, 6, 12, 6], 'aut_gens': [[712689240229075, 43159484847303, 397533486881388], [377930346465315, 355687907215457, 1067068514991303], [397545462219356, 355693655386442, 712695467535424], [754540568127307, 43153736828103, 1108914091352354], [1067074742162792, 20923269036533, 377924123076313], [712694984609388, 1308157116257, 397539235206322], [712689240348035, 732303873321095, 397533490520275]], 'aut_group': None, 'aut_hash': 4764413270895111565, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 93312, 'aut_permdeg': 114, 'aut_perms': [445998467213430455183371950673501990163526867035329777687108546599505610297755195025631276613995081861702719720921436860047327675464526537470764226061167902733283590300237259780491231273, 2252636996628713525212768301492362197947675963172841745747889094713450384712286140761808055470561574407584924036494687027067479954663836914136868258919610000538285172978661150662308135883, 1859063581631455878972518128096985207862930050331940286768584485968783112605831182878799237150090460995096287547369413817262573763933292402162640954961546407416876774811968056508755106100, 2524192866062423721479573602262645478313627521259806838205910883318904166870960206137445311366380926573835198478803123064589601577786847859046086903388955901395369684222025470210129433963, 2214235725718500773294501335837705359139333232985304803309881249592774743863029083038421772053222411318467557638220630920682685502639893964363674948259538051501079302011162981202208269730, 1244827525649229859888805190936857756283050236461528196346232626574904176395743993702091841143879642202756948347052669588352513016630695482541867782200135892576931995468547033913782714920], 'aut_phi_ratio': 36.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 3, 1, 2], [2, 3, 2, 1], [2, 9, 2, 1], [2, 54, 2, 1], [2, 162, 2, 1], [3, 2, 1, 2], [3, 3, 2, 1], [3, 4, 1, 1], [3, 6, 2, 1], [3, 6, 3, 1], [3, 12, 3, 1], [3, 72, 1, 1], [3, 144, 1, 1], [4, 54, 2, 1], [4, 162, 2, 1], [6, 2, 1, 2], [6, 3, 2, 3], [6, 4, 1, 1], [6, 6, 1, 4], [6, 6, 2, 6], [6, 6, 3, 3], [6, 6, 6, 2], [6, 9, 4, 2], [6, 12, 1, 2], [6, 12, 2, 2], [6, 12, 3, 3], [6, 12, 6, 2], [6, 18, 2, 1], [6, 18, 4, 1], [6, 18, 6, 2], [6, 18, 12, 1], [6, 54, 4, 1], [6, 72, 1, 1], [6, 108, 2, 1], [6, 108, 4, 1], [6, 144, 1, 1], [6, 162, 4, 1], [6, 216, 2, 1], [9, 72, 2, 1], [9, 144, 2, 1], [12, 54, 4, 1], [12, 108, 2, 1], [12, 108, 4, 1], [12, 162, 4, 1], [18, 72, 2, 1], [18, 144, 2, 1], [18, 216, 4, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_2\\times C_6^2.C_3^4.C_2^4', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 2, 'autcent_group': '4.2', 'autcent_hash': 2, 'autcent_nilpotent': True, 'autcent_order': 4, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^2', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 36, 'autcentquo_group': None, 'autcentquo_hash': 2074683354935437783, 'autcentquo_nilpotent': False, 'autcentquo_order': 23328, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'C_6^2.C_3^4.C_2^3', 'cc_stats': [[1, 1, 1], [2, 1, 1], [2, 3, 4], [2, 9, 2], [2, 54, 2], [2, 162, 2], [3, 2, 2], [3, 3, 2], [3, 4, 1], [3, 6, 5], [3, 12, 3], [3, 72, 1], [3, 144, 1], [4, 54, 2], [4, 162, 2], [6, 2, 2], [6, 3, 6], [6, 4, 1], [6, 6, 37], [6, 9, 8], [6, 12, 27], [6, 18, 30], [6, 54, 4], [6, 72, 1], [6, 108, 6], [6, 144, 1], [6, 162, 4], [6, 216, 2], [9, 72, 2], [9, 144, 2], [12, 54, 4], [12, 108, 6], [12, 162, 4], [18, 72, 2], [18, 144, 2], [18, 216, 4]], 'center_label': '2.1', 'center_order': 2, 'central_product': True, 'central_quotient': '3888.gu', 'commutator_count': 2, 'commutator_label': '972.590', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '2.1', '3.1', '3.1', '3.1', '3.1', '3.1'], 'composition_length': 10, 'conjugacy_classes_known': True, 'counter': 157, 'cyclic': False, 'derived_length': 3, 'dihedral': False, 'direct_factorization': [['2.1', 1], ['6.1', 1], ['648.281', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 3, 1, 4], [2, 9, 1, 2], [2, 54, 1, 2], [2, 162, 1, 2], [3, 2, 1, 2], [3, 3, 2, 1], [3, 4, 1, 1], [3, 6, 1, 3], [3, 6, 2, 1], [3, 12, 1, 3], [3, 72, 1, 1], [3, 144, 1, 1], [4, 54, 1, 2], [4, 162, 1, 2], [6, 2, 1, 2], [6, 3, 2, 3], [6, 4, 1, 1], [6, 6, 1, 15], [6, 6, 2, 11], [6, 9, 2, 4], [6, 12, 1, 11], [6, 12, 2, 8], [6, 18, 1, 14], [6, 18, 2, 8], [6, 54, 2, 2], [6, 72, 1, 1], [6, 108, 1, 2], [6, 108, 2, 2], [6, 144, 1, 1], [6, 162, 2, 2], [6, 216, 1, 2], [9, 72, 1, 2], [9, 144, 1, 2], [12, 54, 2, 2], [12, 108, 1, 2], [12, 108, 2, 2], [12, 162, 2, 2], [18, 72, 1, 2], [18, 144, 1, 2], [18, 216, 1, 4]], 'element_repr_type': 'Perm', 'elementary': 1, 'eulerian_function': 816480, 'exponent': 36, 'exponents_of_order': [5, 5], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [[12, 0, 6], [12, 1, 3]], 'familial': False, 'frattini_label': '9.2', 'frattini_quotient': '864.4692', 'hash': 1987921750283067703, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 36, 'inner_gen_orders': [12, 6, 12], 'inner_gens': [[712689240229075, 355687910908032, 397545940930865], [754540568255062, 43159484847303, 1108920797193982], [712694988240368, 1320611144934, 397533486881388]], 'inner_hash': 1441392995906330158, 'inner_nilpotent': False, 'inner_order': 3888, 'inner_split': True, 'inner_tex': 'S_3\\times C_3^3:S_4', 'inner_used': [1, 2, 3], 'irrC_degree': 12, 'irrQ_degree': 12, 'irrQ_dim': 12, 'irrR_degree': 12, 'irrep_stats': [[1, 8], [2, 20], [3, 40], [4, 8], [6, 80], [12, 30]], 'label': '7776.ga', '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': 'C6^3:S3^2', 'ngens': 3, '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, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 72, 'number_characteristic_subgroups': 54, 'number_conjugacy_classes': 186, 'number_divisions': 138, 'number_normal_subgroups': 98, 'number_subgroup_autclasses': 1612, 'number_subgroup_classes': 3972, 'number_subgroups': 82438, 'old_label': None, 'order': 7776, 'order_factorization_type': 33, 'order_stats': [[1, 1], [2, 463], [3, 296], [4, 432], [6, 3344], [9, 432], [12, 1512], [18, 1296]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': True, 'outer_exponent': 6, 'outer_gen_orders': [2, 2, 6], 'outer_gen_pows': [0, 0, 0], 'outer_gens': [[712689240226865, 43159485013151, 397533486899356], [712689240229075, 43159481218503, 397533490510188], [397545462051949, 1320611206411, 712683009569388]], 'outer_group': '24.15', 'outer_hash': 15, 'outer_nilpotent': True, 'outer_order': 24, 'outer_permdeg': 9, 'outer_perms': [24, 40320, 723], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^2\\times C_6', 'pc_rank': None, 'perfect': False, 'permutation_degree': 18, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 2], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 8], [2, 20], [3, 8], [4, 8], [6, 48], [12, 38], [24, 8]], 'representations': {'PC': {'code': '10023141228343465678174942243459781929434705403130831641419086306831607147172956141189781009756844335507721408568485246300543654751418', 'gens': [1, 2, 4, 6, 8, 10], 'pres': [10, 2, 2, 3, 2, 3, 3, 3, 2, 3, 2, 47040, 151401, 51, 84482, 201123, 110653, 32063, 113, 196804, 150614, 3624, 494, 252005, 188655, 9385, 2555, 765, 235, 30246, 552967, 48997, 21647, 7257, 237, 136088, 22718, 194409, 16239, 24349, 8159]}, 'Perm': {'d': 18, 'gens': [712689240229075, 43159484847303, 397533486881388]}}, 'schur_multiplier': [2, 2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 2], 'solvability_type': 11, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': 'C_6^3:S_3^2', 'transitive_degree': 36, 'wreath_data': None, 'wreath_product': False}
