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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': False, 'ambient': '6600.x', 'ambient_counter': 24, 'ambient_order': 6600, 'ambient_tex': 'C_{66}:C_{10}^2', 'central': False, 'central_factor': False, 'centralizer_order': 20, 'characteristic': False, 'core_order': 220, 'counter': 19, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '6600.x.15.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': False, 'minimal': False, 'minimal_normal': False, 'nilpotent': False, 'normal': False, 'old_label': '15.a1', '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': 15, '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': '440.48', 'subgroup_hash': 48, 'subgroup_order': 440, 'subgroup_tex': 'C_{10}\\times D_{22}', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '6600.x', 'aut_centralizer_order': None, 'aut_label': '15.a1', 'aut_quo_index': None, 'aut_stab_index': None, 'aut_weyl_group': None, 'aut_weyl_index': None, 'centralizer': '330.a1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['3.a1', '5.a1'], 'contains': ['30.a1', '30.c1', '75.a1', '165.a1'], 'core': '30.a1', 'coset_action_label': None, 'count': 3, 'diagramx': [7437, -1, 720, -1], 'generators': [5, 3300, 2, 50, 600], 'label': '6600.x.15.a1', 'mobius_quo': None, 'mobius_sub': 1, 'normal_closure': '5.a1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '3.a1', 'old_label': '15.a1', 'projective_image': '330.3', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '15.a1', 'subgroup_fusion': None, 'weyl_group': '110.1'}
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gps_groups • Show schema
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{'Agroup': True, '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': 660, 'aut_gen_orders': [10, 30, 44, 4], 'aut_gens': [[1, 2, 4], [220, 143, 417], [220, 342, 37], [220, 382, 49], [220, 142, 305]], 'aut_group': None, 'aut_hash': 8962605465823330988, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 10560, 'aut_permdeg': 48, 'aut_perms': [8529339366027413365298767320373867489496435543815842835696405, 1127390792360206690331675168209599590050592303262269184456553, 2451387801402680159649832069879355968741638268728506362922917, 2453961781433264950731273060393460087386901921462144895475696], 'aut_phi_ratio': 66.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 3, 1], [2, 11, 4, 1], [5, 1, 4, 1], [10, 1, 12, 1], [10, 11, 16, 1], [11, 2, 5, 1], [22, 2, 15, 1], [55, 2, 20, 1], [110, 2, 60, 1]], 'aut_supersolvable': False, 'aut_tex': '(C_{11}\\times A_4).C_{20}.C_2^2', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 12, 'autcent_group': '96.186', 'autcent_hash': 186, 'autcent_nilpotent': False, 'autcent_order': 96, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': False, 'autcent_tex': 'C_4\\times S_4', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 110, 'autcentquo_group': '110.1', 'autcentquo_hash': 1, 'autcentquo_nilpotent': False, 'autcentquo_order': 110, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'F_{11}', 'cc_stats': [[1, 1, 1], [2, 1, 3], [2, 11, 4], [5, 1, 4], [10, 1, 12], [10, 11, 16], [11, 2, 5], [22, 2, 15], [55, 2, 20], [110, 2, 60]], 'center_label': '20.5', 'center_order': 20, 'central_product': True, 'central_quotient': '22.1', 'commutator_count': 1, 'commutator_label': '11.1', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '5.1', '11.1'], 'composition_length': 5, 'conjugacy_classes_known': True, 'counter': 48, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['2.1', 2], ['22.1', 1], ['5.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [2, 11, 1, 4], [5, 1, 4, 1], [10, 1, 4, 3], [10, 11, 4, 4], [11, 2, 5, 1], [22, 2, 5, 3], [55, 2, 20, 1], [110, 2, 20, 3]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 2604, 'exponent': 110, 'exponents_of_order': [3, 1, 1], 'factors_of_aut_order': [2, 3, 5, 11], 'factors_of_order': [2, 5, 11], 'faithful_reps': [], 'familial': False, 'frattini_label': '1.1', 'frattini_quotient': '440.48', 'hash': 48, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 22, 'inner_gen_orders': [1, 2, 11], 'inner_gens': [[1, 2, 4], [1, 2, 84], [1, 362, 4]], 'inner_hash': 1, 'inner_nilpotent': False, 'inner_order': 22, 'inner_split': True, 'inner_tex': 'D_{11}', 'inner_used': [2, 3], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 40], [2, 100]], 'label': '440.48', 'linC_count': 1440, 'linC_degree': 3, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 14, 'linQ_degree_count': 12, 'linQ_dim': 14, 'linQ_dim_count': 12, 'linR_count': 120, 'linR_degree': 4, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'C10*D22', 'ngens': 5, 'nilpotency_class': -1, 'nilpotent': False, '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': 10, 'number_characteristic_subgroups': 10, 'number_conjugacy_classes': 140, 'number_divisions': 24, 'number_normal_subgroups': 42, 'number_subgroup_autclasses': 24, 'number_subgroup_classes': 64, 'number_subgroups': 284, 'old_label': None, 'order': 440, 'order_factorization_type': 311, 'order_stats': [[1, 1], [2, 47], [5, 4], [10, 188], [11, 10], [22, 30], [55, 40], [110, 120]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 60, 'outer_gen_orders': [2, 12, 2, 10], 'outer_gen_pows': [0, 0, 0, 0], 'outer_gens': [[221, 222, 84], [221, 222, 129], [1, 3, 4], [1, 223, 164]], 'outer_group': '480.1014', 'outer_hash': 1014, 'outer_nilpotent': False, 'outer_order': 480, 'outer_permdeg': 13, 'outer_perms': [3628800, 79833609, 1520467200, 1037928960], 'outer_solvable': True, 'outer_supersolvable': False, 'outer_tex': 'C_{20}\\times S_4', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 20, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 2, 5], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 8], [4, 8], [10, 4], [40, 4]], 'representations': {'PC': {'code': 553238140765978800039, 'gens': [1, 2, 3], 'pres': [5, -2, -2, -2, -5, -11, 637, 42, 1688, 118, 5009]}, 'GLFp': {'d': 3, 'p': 11, 'gens': [2247819944, 285289266, 911390422, 418103124, 2143735230]}, 'Perm': {'d': 20, 'gens': [6423384156537600, 7, 16, 46200, 134491868718278400]}}, 'schur_multiplier': [2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 10], 'solvability_type': 7, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_{10}\\times D_{22}', 'transitive_degree': 220, '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': '200.52', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 3, 'aut_exponent': 660, 'aut_gen_orders': [60, 30, 10, 20, 20], 'aut_gens': [[1, 10, 100], [3304, 1316, 6405], [1, 5818, 705], [1, 618, 1700], [3307, 614, 2900], [3, 3411, 1700]], 'aut_group': None, 'aut_hash': 5631806002831954820, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 316800, 'aut_permdeg': 187, 'aut_perms': [841129298575923107247515802958921578197730086148346007812258680128336249532978822246340541165541065916361231302014202479585074201562476282580050314601621724698392783130544111123378504476145278928460559129772636990253128041629471533041438318511517521372032445653921907087412880496193841906023161939765598991276029050588958588638885462792443642995, 307064794981905273667842342596986553604211983140451359783852290579472181297110272823019318191762175500884546068228435650067225710802159421250945314322838682496056067104438892154559364361637567336406112371819679670282841392833072611130160023078423431187743924231723673535205621858642889376209935244868483176566180651523449245970445811110311183124, 616886412986211232648130167928100453223413498313998532406992938805739814458205288160184703947339763427185032236195365670997021810544358921010942114029636327384056648175607961428805502734297919091547363015587484425617098793317206505684202453549747774398537706647239804842764336860942802184975523979455356380572606371712775076693782756042339780215, 1108575171424476076809030800034723859335145260764345262426106759185023890336670966685521966473217060931782423382164790732765279595287068651742893652636512314180823741728690128684983658933268804625269109004198219455890967404121387390304710310876660411548309665291803574919025855347668534350125391962047502414182815099184947421690688960600733046314, 294311467074677035541509727783582010743649470952361584260152725346717785121420739760528715255064304238383787032959074053916331162287139174182703881760645514555366838167203920228581313057972333998132007073175546512491475623153388194065940862692312936398584255705522073076939719485263215854740167940840216468267031032122558409942775908862852222590], 'aut_phi_ratio': 198.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 3, 1], [2, 33, 4, 1], [3, 2, 1, 1], [5, 1, 4, 1], [5, 11, 5, 4], [6, 2, 3, 1], [10, 1, 12, 1], [10, 11, 15, 4], [10, 33, 16, 1], [10, 33, 20, 4], [11, 10, 1, 1], [15, 2, 4, 1], [15, 22, 5, 4], [22, 10, 3, 1], [30, 2, 12, 1], [30, 22, 15, 4], [33, 10, 2, 1], [55, 10, 4, 1], [66, 10, 6, 1], [110, 10, 12, 1], [165, 10, 8, 1], [330, 10, 24, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_2^2.C_{165}.C_{60}.C_2^3', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 60, 'autcent_group': '480.1189', 'autcent_hash': 1189, 'autcent_nilpotent': False, 'autcent_order': 480, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': False, 'autcent_tex': 'F_5\\times S_4', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 330, 'autcentquo_group': '660.15', 'autcentquo_hash': 15, 'autcentquo_nilpotent': False, 'autcentquo_order': 660, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'S_3\\times F_{11}', 'cc_stats': [[1, 1, 1], [2, 1, 3], [2, 33, 4], [3, 2, 1], [5, 1, 4], [5, 11, 20], [6, 2, 3], [10, 1, 12], [10, 11, 60], [10, 33, 96], [11, 10, 1], [15, 2, 4], [15, 22, 20], [22, 10, 3], [30, 2, 12], [30, 22, 60], [33, 10, 2], [55, 10, 4], [66, 10, 6], [110, 10, 12], [165, 10, 8], [330, 10, 24]], 'center_label': '20.5', 'center_order': 20, 'central_product': True, 'central_quotient': '330.3', 'commutator_count': 1, 'commutator_label': '33.1', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '3.1', '5.1', '5.1', '11.1'], 'composition_length': 7, 'conjugacy_classes_known': True, 'counter': 24, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['2.1', 2], ['330.3', 1], ['5.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [2, 33, 1, 4], [3, 2, 1, 1], [5, 1, 4, 1], [5, 11, 4, 5], [6, 2, 1, 3], [10, 1, 4, 3], [10, 11, 4, 15], [10, 33, 4, 24], [11, 10, 1, 1], [15, 2, 4, 1], [15, 22, 4, 5], [22, 10, 1, 3], [30, 2, 4, 3], [30, 22, 4, 15], [33, 10, 2, 1], [55, 10, 4, 1], [66, 10, 2, 3], [110, 10, 4, 3], [165, 10, 8, 1], [330, 10, 8, 3]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 249984, 'exponent': 330, 'exponents_of_order': [3, 2, 1, 1], 'factors_of_aut_order': [2, 3, 5, 11], 'factors_of_order': [2, 3, 5, 11], 'faithful_reps': [], 'familial': False, 'frattini_label': '1.1', 'frattini_quotient': '6600.x', 'hash': 6275062025069515071, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 330, 'inner_gen_orders': [1, 10, 33], 'inner_gens': [[1, 10, 100], [1, 10, 1700], [1, 5010, 100]], 'inner_hash': 3, 'inner_nilpotent': False, 'inner_order': 330, 'inner_split': True, 'inner_tex': 'C_{33}:C_{10}', 'inner_used': [2, 3], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 200], [2, 100], [10, 60]], 'label': '6600.x', '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': 'C66:C10^2', '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, 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': 38, 'number_characteristic_subgroups': 23, 'number_conjugacy_classes': 360, 'number_divisions': 100, 'number_normal_subgroups': 188, 'number_subgroup_autclasses': 96, 'number_subgroup_classes': 512, 'number_subgroups': 4984, 'old_label': None, 'order': 6600, 'order_factorization_type': 321, 'order_stats': [[1, 1], [2, 135], [3, 2], [5, 224], [6, 6], [10, 3840], [11, 10], [15, 448], [22, 30], [30, 1344], [33, 20], [55, 40], [66, 60], [110, 120], [165, 80], [330, 240]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': True, 'outer_exponent': 60, 'outer_gen_orders': [2, 2, 12, 2, 10], 'outer_gen_pows': [4400, 0, 0, 0, 2200], 'outer_gens': [[1, 4410, 4300], [3301, 3310, 4300], [3303, 3310, 1005], [1, 15, 100], [1, 1118, 100]], 'outer_group': '960.11361', 'outer_hash': 11361, 'outer_nilpotent': False, 'outer_order': 960, 'outer_permdeg': 11, 'outer_perms': [120, 1, 408244, 23, 15240967], 'outer_solvable': True, 'outer_supersolvable': False, 'outer_tex': 'C_2\\times F_5\\times S_4', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 23, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 2, 5, 5], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 8], [2, 4], [4, 48], [8, 24], [10, 4], [20, 4], [40, 4], [80, 4]], 'representations': {'PC': {'code': '63826120447865781000262618868102686790124050303019', 'gens': [1, 3, 5], 'pres': [7, -2, -5, -2, -5, -2, -3, -11, 14, 58, 5968, 4400, 102, 14299, 10526, 166, 17660, 4437]}, 'GLFp': {'d': 4, 'p': 11, 'gens': [12531822321003912, 38376113094163029, 37668179739389305, 33418192856010432, 42497879883498899, 36687300164533435, 17156201816241749]}, 'Perm': {'d': 23, 'gens': [1175213670737158414990, 1569209241600, 122002357046169600, 2622063340474, 256099058543578927, 51347039741995273374, 5498466907777]}}, 'schur_multiplier': [2, 2, 10], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 10, 10], 'solvability_type': 7, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_{66}:C_{10}^2', 'transitive_degree': 660, 'wreath_data': None, 'wreath_product': False}
