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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': True, 'ambient': '490.9', 'ambient_counter': 9, 'ambient_order': 490, 'ambient_tex': 'C_7:D_{35}', 'central': False, 'central_factor': False, 'centralizer_order': 245, 'characteristic': False, 'core_order': 35, 'counter': 14, 'cyclic': True, 'direct': False, 'hall': 0, 'label': '490.9.14.a1.b1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': True, 'old_label': '14.a1.b1', 'outer_equivalence': False, 'perfect': False, 'proper': True, 'quotient': '14.1', 'quotient_Agroup': True, 'quotient_abelian': False, 'quotient_cyclic': False, 'quotient_hash': 1, 'quotient_metabelian': True, 'quotient_nilpotent': False, 'quotient_order': 14, 'quotient_simple': False, 'quotient_solvable': True, 'quotient_supersolvable': True, 'quotient_tex': 'D_7', 'simple': False, 'solvable': True, 'special_labels': [], 'split': True, 'standard_generators': False, 'stem': False, 'subgroup': '35.1', 'subgroup_hash': 1, 'subgroup_order': 35, 'subgroup_tex': 'C_{35}', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '490.9', 'aut_centralizer_order': 10290, 'aut_label': '14.a1', 'aut_quo_index': 1, 'aut_stab_index': 8, 'aut_weyl_group': '24.9', 'aut_weyl_index': 82320, 'centralizer': '2.a1.a1', 'complements': ['35.a1.a1', '35.a1.g1', '35.a1.h1', '35.a1.f1', '35.a1.d1', '35.a1.c1', '35.a1.e1'], 'conjugacy_class_count': 1, 'contained_in': ['2.a1.a1', '7.a1.b1'], 'contains': ['70.a1.b1', '98.a1.a1'], 'core': '14.a1.b1', 'coset_action_label': None, 'count': 1, 'diagramx': [3249, 6856, 8836, 9647, 4681, 7697, 3976, 8508], 'generators': [294, 212], 'label': '490.9.14.a1.b1', 'mobius_quo': 1, 'mobius_sub': 7, 'normal_closure': '14.a1.b1', 'normal_contained_in': ['2.a1.a1'], 'normal_contains': ['70.a1.b1', '98.a1.a1'], 'normalizer': '1.a1.a1', 'old_label': '14.a1.b1', 'projective_image': '490.9', 'quotient_action_image': '2.1', 'quotient_action_kernel': '7.1', 'quotient_action_kernel_order': 7, 'quotient_fusion': None, 'short_label': '14.a1.b1', 'subgroup_fusion': None, 'weyl_group': '2.1'}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': True, 'abelian_quotient': '35.1', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': True, 'aut_cyclic': False, 'aut_derived_length': 1, 'aut_exponent': 12, 'aut_gen_orders': [2, 12], 'aut_gens': [[1], [6], [3]], 'aut_group': '24.9', 'aut_hash': 9, 'aut_nilpotency_class': 1, 'aut_nilpotent': True, 'aut_order': 24, 'aut_permdeg': 9, 'aut_perms': [42000, 867], 'aut_phi_ratio': 1.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [5, 1, 4, 1], [7, 1, 6, 1], [35, 1, 24, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_2\\times C_{12}', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 12, 'autcent_group': '24.9', 'autcent_hash': 9, 'autcent_nilpotent': True, 'autcent_order': 24, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2\\times C_{12}', 'autcentquo_abelian': True, 'autcentquo_cyclic': True, 'autcentquo_exponent': 1, 'autcentquo_group': '1.1', 'autcentquo_hash': 1, 'autcentquo_nilpotent': True, 'autcentquo_order': 1, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'C_1', 'cc_stats': [[1, 1, 1], [5, 1, 4], [7, 1, 6], [35, 1, 24]], 'center_label': '35.1', 'center_order': 35, 'central_product': True, 'central_quotient': '1.1', 'commutator_count': 0, 'commutator_label': '1.1', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['5.1', '7.1'], 'composition_length': 2, 'conjugacy_classes_known': True, 'counter': 1, 'cyclic': True, 'derived_length': 1, 'dihedral': False, 'direct_factorization': [['5.1', 1], ['7.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [5, 1, 4, 1], [7, 1, 6, 1], [35, 1, 24, 1]], 'element_repr_type': 'PC', 'elementary': 35, 'eulerian_function': 1, 'exponent': 35, 'exponents_of_order': [1, 1], 'factors_of_aut_order': [2, 3], 'factors_of_order': [5, 7], 'faithful_reps': [[1, 0, 24]], 'familial': True, 'frattini_label': '1.1', 'frattini_quotient': '35.1', 'hash': 1, 'hyperelementary': 35, 'inner_abelian': True, 'inner_cyclic': True, 'inner_exponent': 1, 'inner_gen_orders': [1], 'inner_gens': [[1]], 'inner_hash': 1, 'inner_nilpotent': True, 'inner_order': 1, 'inner_split': True, 'inner_tex': 'C_1', 'inner_used': [], 'irrC_degree': 1, 'irrQ_degree': 24, 'irrQ_dim': 24, 'irrR_degree': 2, 'irrep_stats': [[1, 35]], 'label': '35.1', 'linC_count': 24, 'linC_degree': 1, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 10, 'linQ_degree_count': 1, 'linQ_dim': 10, 'linQ_dim_count': 1, 'linR_count': 12, 'linR_degree': 2, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C35', 'ngens': 2, 'nilpotency_class': 1, 'nilpotent': True, 'normal_counts': [0, 0, 0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 4, 'number_characteristic_subgroups': 4, 'number_conjugacy_classes': 35, 'number_divisions': 4, 'number_normal_subgroups': 4, 'number_subgroup_autclasses': 4, 'number_subgroup_classes': 4, 'number_subgroups': 4, 'old_label': None, 'order': 35, 'order_factorization_type': 11, 'order_stats': [[1, 1], [5, 4], [7, 6], [35, 24]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 12, 'outer_gen_orders': [2, 12], 'outer_gen_pows': [0, 0], 'outer_gens': [[6], [3]], 'outer_group': '24.9', 'outer_hash': 9, 'outer_nilpotent': True, 'outer_order': 24, 'outer_permdeg': 9, 'outer_perms': [42000, 867], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2\\times C_{12}', 'pc_rank': 1, 'perfect': False, 'permutation_degree': 12, 'pgroup': 0, 'primary_abelian_invariants': [5, 7], 'quasisimple': False, 'rank': 1, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 1], [4, 1], [6, 1], [24, 1]], 'representations': {'PC': {'code': 203, 'gens': [1], 'pres': [2, -5, -7, 10]}, 'GLFp': {'d': 2, 'p': 29, 'gens': [72212]}, 'Perm': {'d': 12, 'gens': [159667200, 4320]}}, 'schur_multiplier': [], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [35], 'solvability_type': 0, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_{35}', 'transitive_degree': 35, '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': '2.1', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 1680, 'aut_gen_orders': [240, 84, 6], 'aut_gens': [[1, 2, 14], [303, 218, 366], [125, 280, 176], [459, 220, 342]], 'aut_group': None, 'aut_hash': 3941173762477618425, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 1975680, 'aut_permdeg': 245, 'aut_perms': [408164211503684045435972780676423933092253162024245092921677177882208954447926623103815334085962485134527165627664111645042236638552466639560075581139964934869564752521542115638973298867114580985421067394610658827977379850804369254312348339557374669297119130504096319863323597016277506399600351354297747593403001276809273759203096187122127253325306502238299437563927837902980655113563833133008243234458129158042516226563813092438795537504822902009574085798027940138299376421017682, 1548943837081407698390320049273243172629792806012609390539329035495689798043730056168878925918248290342217036350638225052713496894402499104234092152602428422396827996312408975587056776074678254209446623341620183458503406630361629211323096185277321034001168554049574014004361585367047828412858565736947794367225169905709664340010548430467289330590425217508671779044770659605920031727658715311117249188664228801700814663461594714253967136708588535595007799802171309054203366330279595, 412062738620512860711239495817380990096977435538812572011778516887720834566560575391462756962649272122008294144600674614895734192654492565494169394705615452888004493644782446032717729593046847843470558134816438554441410864716944082709622003640025764791708282259247803757530932556909254416268154035954426208351302599157577886200029613811938306617155378365677641768472095701565631364255803121478589073921266859478007520786428260182111779721903652796128735143511683106128552966393515], 'aut_phi_ratio': 11760.0, 'aut_solvable': False, 'aut_stats': [[1, 1, 1, 1], [2, 245, 1, 1], [5, 2, 2, 1], [7, 2, 24, 1], [35, 2, 96, 1]], 'aut_supersolvable': False, 'aut_tex': 'F_5\\times C_7^2:C_3.\\SL(2,7).C_2', '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': 1680, 'autcentquo_group': None, 'autcentquo_hash': 3941173762477618425, 'autcentquo_nilpotent': False, 'autcentquo_order': 1975680, 'autcentquo_solvable': False, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'C_7^2:C_3.\\SL(2,7).C_2\\times F_5', 'cc_stats': [[1, 1, 1], [2, 245, 1], [5, 2, 2], [7, 2, 24], [35, 2, 96]], 'center_label': '1.1', 'center_order': 1, 'central_product': False, 'central_quotient': '490.9', 'commutator_count': 1, 'commutator_label': '245.2', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '5.1', '7.1', '7.1'], 'composition_length': 4, 'conjugacy_classes_known': True, 'counter': 9, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 245, 1, 1], [5, 2, 2, 1], [7, 2, 3, 8], [35, 2, 12, 8]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 42, 'exponent': 70, 'exponents_of_order': [2, 1, 1], 'factors_of_aut_order': [2, 3, 5, 7], 'factors_of_order': [2, 5, 7], 'faithful_reps': [], 'familial': False, 'frattini_label': '1.1', 'frattini_quotient': '490.9', 'hash': 9, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 70, 'inner_gen_orders': [2, 7, 35], 'inner_gens': [[1, 12, 476], [5, 2, 14], [29, 2, 14]], 'inner_hash': 9, 'inner_nilpotent': False, 'inner_order': 490, 'inner_split': False, 'inner_tex': 'C_7:D_{35}', 'inner_used': [1, 2, 3], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 2], [2, 122]], 'label': '490.9', 'linC_count': 6048, 'linC_degree': 4, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 16, 'linQ_degree_count': 28, 'linQ_dim': 16, 'linQ_dim_count': 28, 'linR_count': 6048, 'linR_degree': 4, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'C7:D35', 'ngens': 4, 'nilpotency_class': -1, 'nilpotent': False, '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': 5, 'number_characteristic_subgroups': 5, 'number_conjugacy_classes': 124, 'number_divisions': 19, 'number_normal_subgroups': 21, 'number_subgroup_autclasses': 12, 'number_subgroup_classes': 40, 'number_subgroups': 656, 'old_label': None, 'order': 490, 'order_factorization_type': 222, 'order_stats': [[1, 1], [2, 245], [5, 4], [7, 48], [35, 192]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 336, 'outer_gen_orders': [4, 2, 6, 3, 12], 'outer_gen_pows': [0, 0, 1, 0, 0], 'outer_gens': [[1, 2, 112], [1, 2, 406], [1, 216, 254], [1, 4, 224], [1, 4, 50]], 'outer_group': '4032.l', 'outer_hash': 4821371165894102431, 'outer_nilpotent': False, 'outer_order': 4032, 'outer_permdeg': 35, 'outer_perms': [322098184976819500884068643100631216640, 1252568676375854727932890561450993825080, 9357572277440684802862123707844569799467, 3, 2664759046887252956059888429022696499864], 'outer_solvable': False, 'outer_supersolvable': False, 'outer_tex': '\\GL(2,7):C_2', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 19, 'pgroup': 0, 'primary_abelian_invariants': [2], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 2], [4, 1], [6, 8], [24, 8]], 'representations': {'PC': {'code': 394689655115915, 'gens': [1, 2, 3], 'pres': [4, -2, -7, -5, -7, 97, 5714, 70, 6723]}, 'Perm': {'d': 19, 'gens': [774317586107861, 47214720, 7532210586701957, 14272132408936365]}}, 'schur_multiplier': [7], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2], 'solvability_type': 7, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_7:D_{35}', 'transitive_degree': 245, 'wreath_data': None, 'wreath_product': False}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': False, 'abelian_quotient': '2.1', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 42, 'aut_gen_orders': [6, 7], 'aut_gens': [[1, 2], [13, 10], [7, 2]], 'aut_group': '42.1', 'aut_hash': 1, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 42, 'aut_permdeg': 7, 'aut_perms': [660, 3454], 'aut_phi_ratio': 7.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 7, 1, 1], [7, 2, 3, 1]], 'aut_supersolvable': True, 'aut_tex': 'F_7', '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': 42, 'autcentquo_group': '42.1', 'autcentquo_hash': 1, 'autcentquo_nilpotent': False, 'autcentquo_order': 42, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'F_7', 'cc_stats': [[1, 1, 1], [2, 7, 1], [7, 2, 3]], 'center_label': '1.1', 'center_order': 1, 'central_product': False, 'central_quotient': '14.1', 'commutator_count': 1, 'commutator_label': '7.1', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '7.1'], 'composition_length': 2, 'conjugacy_classes_known': True, 'counter': 1, 'cyclic': False, 'derived_length': 2, 'dihedral': True, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 7, 1, 1], [7, 2, 3, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 3, 'exponent': 14, 'exponents_of_order': [1, 1], 'factors_of_aut_order': [2, 3, 7], 'factors_of_order': [2, 7], 'faithful_reps': [[2, 1, 3]], 'familial': True, 'frattini_label': '1.1', 'frattini_quotient': '14.1', 'hash': 1, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 14, 'inner_gen_orders': [2, 7], 'inner_gens': [[1, 12], [5, 2]], 'inner_hash': 1, 'inner_nilpotent': False, 'inner_order': 14, 'inner_split': True, 'inner_tex': 'D_7', 'inner_used': [1, 2], 'irrC_degree': 2, 'irrQ_degree': 6, 'irrQ_dim': 6, 'irrR_degree': 2, 'irrep_stats': [[1, 2], [2, 3]], 'label': '14.1', 'linC_count': 3, 'linC_degree': 2, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 6, 'linQ_degree_count': 1, 'linQ_dim': 6, 'linQ_dim_count': 1, 'linR_count': 3, 'linR_degree': 2, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'D7', 'ngens': 2, 'nilpotency_class': -1, 'nilpotent': False, 'normal_counts': [0, 0, 0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 3, 'number_characteristic_subgroups': 3, 'number_conjugacy_classes': 5, 'number_divisions': 3, 'number_normal_subgroups': 3, 'number_subgroup_autclasses': 4, 'number_subgroup_classes': 4, 'number_subgroups': 10, 'old_label': None, 'order': 14, 'order_factorization_type': 11, 'order_stats': [[1, 1], [2, 7], [7, 6]], 'outer_abelian': True, 'outer_cyclic': True, 'outer_equivalence': False, 'outer_exponent': 3, 'outer_gen_orders': [3], 'outer_gen_pows': [1], 'outer_gens': [[1, 10]], 'outer_group': '3.1', 'outer_hash': 1, 'outer_nilpotent': True, 'outer_order': 3, 'outer_permdeg': 3, 'outer_perms': [3], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_3', 'pc_rank': 2, 'perfect': False, 'permutation_degree': 7, 'pgroup': 0, 'primary_abelian_invariants': [2], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 2], [6, 1]], 'representations': {'PC': {'code': 797, 'gens': [1, 2], 'pres': [2, -2, -7, 49]}, 'GLZ': {'b': 3, 'd': 6, 'gens': [91793233555822635, 27211943320546358]}, 'Lie': [{'d': 2, 'q': 8, 'gens': [72, 2562], 'family': 'CSOPlus'}], 'GLFp': {'d': 2, 'p': 7, 'gens': [351, 2059]}, 'Perm': {'d': 7, 'gens': [719, 4320]}}, 'schur_multiplier': [], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2], 'solvability_type': 6, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'D_7', 'transitive_degree': 7, 'wreath_data': None, 'wreath_product': False}
