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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': True, 'ambient': '320.1525', 'ambient_counter': 1525, 'ambient_order': 320, 'ambient_tex': 'C_2^2:Q_8\\times C_{10}', 'central': True, 'central_factor': False, 'centralizer_order': 320, 'characteristic': False, 'core_order': 4, 'counter': 91, 'cyclic': False, 'direct': False, 'hall': 0, 'label': '320.1525.80.e1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': True, 'old_label': '80.e1', 'outer_equivalence': True, 'perfect': False, 'proper': True, 'quotient': '80.48', 'quotient_Agroup': False, 'quotient_abelian': False, 'quotient_cyclic': False, 'quotient_hash': 48, 'quotient_metabelian': True, 'quotient_nilpotent': True, 'quotient_order': 80, 'quotient_simple': False, 'quotient_solvable': True, 'quotient_supersolvable': True, 'quotient_tex': 'D_4:C_{10}', 'simple': False, 'solvable': True, 'special_labels': [], 'split': False, 'standard_generators': False, 'stem': False, 'subgroup': '4.2', 'subgroup_hash': 2, 'subgroup_order': 4, 'subgroup_tex': 'C_2^2', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '320.1525', 'aut_centralizer_order': 4096, 'aut_label': '80.e1', 'aut_quo_index': 3, 'aut_stab_index': 2, 'aut_weyl_group': '2.1', 'aut_weyl_index': 8192, 'centralizer': '1.a1', 'complements': [], 'conjugacy_class_count': 2, 'contained_in': ['16.e1', '40.b1', '40.l1', '40.m1'], 'contains': ['160.b1', '160.d1'], 'core': '80.e1', 'coset_action_label': None, 'count': 2, 'diagramx': [8389, 2197, 8030, 3689], 'generators': [2, 160], 'label': '320.1525.80.e1', 'mobius_quo': 2, 'mobius_sub': 0, 'normal_closure': '80.e1', 'normal_contained_in': ['16.e1', '40.b1'], 'normal_contains': ['160.b1', '160.d1'], 'normalizer': '1.a1', 'old_label': '80.e1', 'projective_image': '80.48', 'quotient_action_image': '1.1', 'quotient_action_kernel': '80.48', 'quotient_action_kernel_order': 80, 'quotient_fusion': None, 'short_label': '80.e1', 'subgroup_fusion': None, 'weyl_group': '1.1'}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': True, 'abelian_quotient': '4.2', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 6, 'aut_gen_orders': [2, 3], 'aut_gens': [[1, 2], [3, 2], [2, 3]], 'aut_group': '6.1', 'aut_hash': 1, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 6, 'aut_permdeg': 3, 'aut_perms': [1, 4], 'aut_phi_ratio': 3.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 3, 1]], 'aut_supersolvable': True, 'aut_tex': 'S_3', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 6, 'autcent_group': '6.1', 'autcent_hash': 1, 'autcent_nilpotent': False, 'autcent_order': 6, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'S_3', '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], [2, 1, 3]], 'center_label': '4.2', 'center_order': 4, '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': ['2.1', '2.1'], 'composition_length': 2, 'conjugacy_classes_known': True, 'counter': 2, 'cyclic': False, 'derived_length': 1, 'dihedral': True, 'direct_factorization': [['2.1', 2]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3]], 'element_repr_type': 'PC', 'elementary': 2, 'eulerian_function': 1, 'exponent': 2, 'exponents_of_order': [2], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2], 'faithful_reps': [], 'familial': True, 'frattini_label': '1.1', 'frattini_quotient': '4.2', 'hash': 2, 'hyperelementary': 2, 'inner_abelian': True, 'inner_cyclic': True, 'inner_exponent': 1, 'inner_gen_orders': [1, 1], 'inner_gens': [[1, 2], [1, 2]], 'inner_hash': 1, 'inner_nilpotent': True, 'inner_order': 1, 'inner_split': True, 'inner_tex': 'C_1', 'inner_used': [], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 4]], 'label': '4.2', 'linC_count': 3, 'linC_degree': 2, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 2, 'linQ_degree_count': 3, 'linQ_dim': 2, 'linQ_dim_count': 3, 'linR_count': 3, 'linR_degree': 2, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C2^2', 'ngens': 2, 'nilpotency_class': 1, 'nilpotent': True, 'normal_counts': [0, 0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 2, 'number_characteristic_subgroups': 2, 'number_conjugacy_classes': 4, 'number_divisions': 4, 'number_normal_subgroups': 5, 'number_subgroup_autclasses': 3, 'number_subgroup_classes': 5, 'number_subgroups': 5, 'old_label': None, 'order': 4, 'order_factorization_type': 2, 'order_stats': [[1, 1], [2, 3]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 6, 'outer_gen_orders': [2, 3], 'outer_gen_pows': [0, 0], 'outer_gens': [[3, 2], [2, 3]], 'outer_group': '6.1', 'outer_hash': 1, 'outer_nilpotent': False, 'outer_order': 6, 'outer_permdeg': 3, 'outer_perms': [1, 4], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'S_3', 'pc_rank': 2, 'perfect': False, 'permutation_degree': 4, 'pgroup': 2, 'primary_abelian_invariants': [2, 2], 'quasisimple': False, 'rank': 2, 'rational': True, 'rational_characters_known': True, 'ratrep_stats': [[1, 4]], 'representations': {'PC': {'code': 0, 'gens': [1, 2], 'pres': [2, -2, 2]}, 'GLZ': {'b': 3, 'd': 2, 'gens': [14, 12]}, 'GLFp': {'d': 2, 'p': 3, 'gens': [55, 56]}, 'Perm': {'d': 4, 'gens': [6, 1]}}, 'schur_multiplier': [2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2], 'solvability_type': 1, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_2^2', 'transitive_degree': 4, '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': '80.52', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 4, 'aut_gen_orders': [4, 4, 4, 4, 4, 4, 4], 'aut_gens': [[1, 2, 4, 16], [161, 170, 166, 218], [171, 170, 12, 149], [11, 162, 14, 149], [169, 170, 14, 309], [169, 170, 12, 157], [1, 170, 166, 216], [171, 170, 4, 215]], 'aut_group': None, 'aut_hash': 161445847492853209, 'aut_nilpotency_class': 3, 'aut_nilpotent': True, 'aut_order': 16384, 'aut_permdeg': 28, 'aut_perms': [39910677511831552591329263367, 256970762521151592744522062099, 236620582406845154996824832281, 143886887501751577063455466906, 151611661492202911273474015159, 3367747775500899509728959634, 251402577868001308202197870112], 'aut_phi_ratio': 128.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [2, 1, 4, 1], [2, 2, 4, 1], [4, 2, 4, 2], [4, 4, 4, 2], [5, 1, 4, 1], [10, 1, 4, 3], [10, 1, 16, 1], [10, 2, 16, 1], [20, 2, 16, 2], [20, 4, 16, 2]], 'aut_supersolvable': True, 'aut_tex': 'C_2^8.C_2^6', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 4, 'autcent_group': None, 'autcent_hash': 7903657167058653875, 'autcent_nilpotent': True, 'autcent_order': 8192, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_4\\times C_2^8.C_2^3', '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, 7], [2, 2, 4], [4, 2, 8], [4, 4, 8], [5, 1, 4], [10, 1, 28], [10, 2, 16], [20, 2, 32], [20, 4, 32]], 'center_label': '40.14', 'center_order': 40, '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', '2.1', '5.1'], 'composition_length': 7, 'conjugacy_classes_known': True, 'counter': 1525, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['2.1', 1], ['32.29', 1], ['5.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 7], [2, 2, 1, 4], [4, 2, 1, 4], [4, 2, 2, 2], [4, 4, 1, 8], [5, 1, 4, 1], [10, 1, 4, 7], [10, 2, 4, 4], [20, 2, 4, 4], [20, 2, 8, 2], [20, 4, 4, 8]], 'element_repr_type': 'PC', 'elementary': 2, 'eulerian_function': 196560, 'exponent': 20, 'exponents_of_order': [6, 1], 'factors_of_aut_order': [2], 'factors_of_order': [2, 5], 'faithful_reps': [], 'familial': False, 'frattini_label': '4.2', 'frattini_quotient': '80.52', 'hash': 1525, 'hyperelementary': 2, 'inner_abelian': True, 'inner_cyclic': False, 'inner_exponent': 2, 'inner_gen_orders': [2, 1, 2, 2], 'inner_gens': [[1, 2, 4, 184], [1, 2, 4, 16], [1, 2, 4, 176], [169, 2, 164, 16]], 'inner_hash': 5, 'inner_nilpotent': True, 'inner_order': 8, 'inner_split': False, 'inner_tex': 'C_2^3', 'inner_used': [1, 3, 4], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 80], [2, 60]], 'label': '320.1525', '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': 'C2^2:Q8*C10', 'ngens': 7, 'nilpotency_class': 2, 'nilpotent': True, 'normal_counts': [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': 26, 'number_conjugacy_classes': 140, 'number_divisions': 52, 'number_normal_subgroups': 194, 'number_subgroup_autclasses': 104, 'number_subgroup_classes': 322, 'number_subgroups': 450, 'old_label': None, 'order': 320, 'order_factorization_type': 51, 'order_stats': [[1, 1], [2, 15], [4, 48], [5, 4], [10, 60], [20, 192]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': True, 'outer_exponent': 4, 'outer_gen_orders': [4, 4, 4, 4, 2, 4, 4], 'outer_gen_pows': [0, 0, 0, 0, 1, 0, 0], 'outer_gens': [[161, 170, 166, 218], [171, 170, 12, 149], [11, 162, 14, 149], [169, 170, 14, 309], [169, 170, 12, 157], [1, 170, 166, 216], [171, 170, 4, 215]], 'outer_group': None, 'outer_hash': 938950599931876664, 'outer_nilpotent': True, 'outer_order': 2048, 'outer_permdeg': 256, 'outer_perms': [3443511276905558929928078912555033447117197435204431764279528216928332403569322617953964547990041674049885373333112151832677551589662157686172021927797786531687486341717164313228447436294603685364952090652573496609754078736421600747038116450020232090489845529588414294772708268991944747445294080000390908853845798416854167748571832424330244368212628486923345387921803650567660967676716057039658545206781499332010706637166274549175355892958004595547335078808002616202973840177810112415522274075864432348660, 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20263727355719851011024840726372388051478690605744917208795759774042944920918210937142089317846870328659096860727775777821992641603962873038211303959150807845284374955589012489138739210336725236912689333737473067814840423731097148546817713969484868102787948204445297091039026529893424674398402925519169097432349248848713225117563002649153206107505269034676332847900222165564825577666008550367821130212008425892361840034466435432560428042084289655103902864187746341930730604504789100130313974094243115046149, 23627770571483793111896500619773792310108032582527307385479068406808405226947504261070728796966293885719468775871773952916401415549655086389735347288632540437486450882447065119424608295667244953493633906674822658163738949433449963808396337174458382011772236861466645185746166889152276938421044322296787426412339250138252090498493352283436598394877652791691272002654930868224575054054029789163555581801925934616953616291570566746952451671113166305481890117778199031331700297286928751802787212947266549389746], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^8.C_2^3', 'pc_rank': 4, 'perfect': False, 'permutation_degree': 19, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 2, 2, 5], 'quasisimple': False, 'rank': 4, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 16], [2, 8], [4, 18], [8, 8], [16, 2]], 'representations': {'PC': {'code': 163354794426385977638915, 'gens': [1, 2, 3, 5], 'pres': [7, -2, -2, -2, -2, 2, -2, -5, 58, 6444, 1558, 102, 124]}, 'GLZN': {'d': 2, 'p': 40, 'gens': [1248019, 576409, 2016201, 64801, 1344021, 1888829, 64321]}, 'Perm': {'d': 19, 'gens': [6423384248387280, 13516123354202160, 732384824371200, 13516211060876880, 96, 1614937680, 13516122267648000]}}, 'schur_multiplier': [2, 2, 2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 2, 10], 'solvability_type': 4, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_2^2:Q_8\\times C_{10}', 'transitive_degree': 160, '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': '40.14', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 3, 'aut_exponent': 12, 'aut_gen_orders': [2, 2, 12, 2, 2], 'aut_gens': [[3782, 1815856, 11349061], [43981, 12030046, 2496841], [226920, 12030046, 11349061], [227041, 6355496, 7381], [226920, 1815856, 2496841], [226920, 1815856, 11349061]], 'aut_group': '192.1469', 'aut_hash': 1469, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 192, 'aut_permdeg': 10, 'aut_perms': [5040, 1, 80790, 1174320, 367920], 'aut_phi_ratio': 6.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 2, 3, 1], [4, 1, 2, 1], [4, 2, 3, 1], [5, 1, 4, 1], [10, 1, 4, 1], [10, 2, 12, 1], [20, 1, 8, 1], [20, 2, 12, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_2^4.D_6', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 4, 'autcent_group': '32.45', 'autcent_hash': 45, 'autcent_nilpotent': True, 'autcent_order': 32, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^3\\times C_4', '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, 1], [2, 2, 3], [4, 1, 2], [4, 2, 3], [5, 1, 4], [10, 1, 4], [10, 2, 12], [20, 1, 8], [20, 2, 12]], 'center_label': '20.2', 'center_order': 20, 'central_product': True, 'central_quotient': '4.2', 'commutator_count': 1, 'commutator_label': '2.1', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '5.1'], 'composition_length': 5, 'conjugacy_classes_known': True, 'counter': 48, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['16.13', 1], ['5.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 2, 1, 3], [4, 1, 2, 1], [4, 2, 1, 3], [5, 1, 4, 1], [10, 1, 4, 1], [10, 2, 4, 3], [20, 1, 8, 1], [20, 2, 4, 3]], 'element_repr_type': 'GLFp', 'elementary': 2, 'eulerian_function': 868, 'exponent': 20, 'exponents_of_order': [4, 1], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 5], 'faithful_reps': [[2, 0, 8]], 'familial': False, 'frattini_label': '2.1', 'frattini_quotient': '40.14', 'hash': 48, 'hyperelementary': 2, 'inner_abelian': True, 'inner_cyclic': False, 'inner_exponent': 2, 'inner_gen_orders': [2, 1, 2], 'inner_gens': [[3782, 1815856, 2496841], [3782, 1815856, 11349061], [226920, 1815856, 11349061]], 'inner_hash': 2, 'inner_nilpotent': True, 'inner_order': 4, 'inner_split': True, 'inner_tex': 'C_2^2', 'inner_used': [1, 3], 'irrC_degree': 2, 'irrQ_degree': 16, 'irrQ_dim': 16, 'irrR_degree': 4, 'irrep_stats': [[1, 40], [2, 10]], 'label': '80.48', 'linC_count': 8, 'linC_degree': 2, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 8, 'linQ_degree_count': 8, 'linQ_dim': 8, 'linQ_dim_count': 8, 'linR_count': 4, 'linR_degree': 4, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'D4:C10', 'ngens': 3, 'nilpotency_class': 2, 'nilpotent': True, 'normal_counts': [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': 10, 'number_characteristic_subgroups': 10, 'number_conjugacy_classes': 50, 'number_divisions': 18, 'number_normal_subgroups': 34, 'number_subgroup_autclasses': 20, 'number_subgroup_classes': 40, 'number_subgroups': 46, 'old_label': None, 'order': 80, 'order_factorization_type': 31, 'order_stats': [[1, 1], [2, 7], [4, 8], [5, 4], [10, 28], [20, 32]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 12, 'outer_gen_orders': [2, 2, 12], 'outer_gen_pows': [226982, 226982, 226982], 'outer_gens': [[13618861, 1815856, 41602], [3782, 12030046, 2496841], [13618861, 6355496, 223321]], 'outer_group': '48.35', 'outer_hash': 35, 'outer_nilpotent': False, 'outer_order': 48, 'outer_permdeg': 9, 'outer_perms': [1, 24, 46084], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_4\\times D_6', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 13, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 2, 5], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 8], [4, 9], [16, 1]], 'representations': {'PC': {'code': 753045368835, 'gens': [1, 2, 3], 'pres': [5, -2, -2, -2, -2, -5, 421, 42, 58]}, 'GLFp': {'d': 2, 'p': 61, 'gens': [3782, 1815856, 11349061]}, 'Perm': {'d': 13, 'gens': [3133236960, 2091640320, 1516969440, 96, 482671440]}}, 'schur_multiplier': [2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 10], 'solvability_type': 4, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'D_4:C_{10}', 'transitive_degree': 40, 'wreath_data': None, 'wreath_product': False}
