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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': False, 'ambient': '5644682640.a', 'ambient_counter': 1, 'ambient_order': 5644682640, 'ambient_tex': '\\PSL(3,19)', 'central': False, 'central_factor': False, 'centralizer_order': None, 'characteristic': False, 'core_order': 1, 'counter': 166, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '5644682640.a.868680._.J', 'maximal': False, 'maximal_normal': False, 'metabelian': False, 'metacyclic': False, 'minimal': False, 'minimal_normal': False, 'nilpotent': False, 'normal': False, 'old_label': '868680.J', 'outer_equivalence': False, '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': 868680, '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': '6498.m', 'subgroup_hash': None, 'subgroup_order': 6498, 'subgroup_tex': 'C_{19}^2:D_9', 'supersolvable': False, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '5644682640.a', 'aut_centralizer_order': None, 'aut_label': None, 'aut_quo_index': None, 'aut_stab_index': None, 'aut_weyl_group': None, 'aut_weyl_index': None, 'centralizer': None, 'complements': None, 'conjugacy_class_count': 1, 'contained_in': None, 'contains': None, 'core': None, 'coset_action_label': None, 'count': 144780, 'diagramx': None, 'generators': [308989829746, 130190383608, 225115979350, 121353128939, 309948129286], 'label': '5644682640.a.868680._.J', 'mobius_quo': None, 'mobius_sub': None, 'normal_closure': None, 'normal_contained_in': None, 'normal_contains': None, 'normalizer': None, 'old_label': '868680.J', 'projective_image': None, 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '868680._.J', 'subgroup_fusion': None, 'weyl_group': None}
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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': 3, 'aut_exponent': 342, 'aut_gen_orders': [18, 18, 19, 19], 'aut_gens': [[1, 2, 18, 342], [523, 376, 2016, 828], [5249, 6212, 5040, 5130], [2539, 2936, 18, 342], [5293, 4106, 18, 342]], 'aut_group': '116964.i', 'aut_hash': 1212891710756311096, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 116964, 'aut_permdeg': 171, 'aut_perms': [1039386724374649520713876673743552786217543443888196503708332523339632856251124942723488940267635455204426119744790338420316314730909748450046810998632383123310828983040094781337161922693695782596920274634010409514854623383920035184660720650564543759760442853070321422307702939304134239897361234589748646414597, 311854682947226783422979415236241305839754704702446653018585961862471930576132463485978192931388863420442852581597330606397839970834905030880891408809664515595330971260604816781601140885983453230879560913794740915196835644185597759899606718491162296803400202579337185829580861026632200431036835701978580421513, 203281401700178476127294079652138631224777056305401547489499559036422699190307748424060794601115830956682845231524727852715616879974356512100974959892783840349836957918126474680347315950015787740663390483681813264938686433456541629978139155857994639607100280653439089445901705869773860642174638475054976726798, 1145763486295403425321635849704485114387488380519346033038389370337404154257327992432751334816218068376548799356240873273901177955528021846444376570491112646871336376262487432917697883838566828723659734746568160409317441502368504783109627433988212591685118216378047898799254739389798306589215219596387359859251], 'aut_phi_ratio': 57.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 171, 1, 1], [3, 722, 1, 1], [9, 722, 1, 3], [19, 9, 18, 1], [19, 18, 2, 1], [19, 18, 9, 1], [38, 171, 18, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_{19}^2:(C_9\\times D_{18})', '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': 342, 'autcentquo_group': '116964.i', 'autcentquo_hash': 1212891710756311096, 'autcentquo_nilpotent': False, 'autcentquo_order': 116964, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'C_{19}^2:(C_9\\times D_{18})', 'cc_stats': [[1, 1, 1], [2, 171, 1], [3, 722, 1], [9, 722, 3], [19, 9, 18], [19, 18, 11], [38, 171, 18]], 'center_label': '1.1', 'center_order': 1, 'central_product': False, 'central_quotient': '6498.m', 'commutator_count': 1, 'commutator_label': '3249.15', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '3.1', '3.1', '19.1', '19.1'], 'composition_length': 5, 'conjugacy_classes_known': True, 'counter': 13, 'cyclic': False, 'derived_length': 3, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 171, 1, 1], [3, 722, 1, 1], [9, 722, 3, 1], [19, 9, 18, 1], [19, 18, 2, 1], [19, 18, 9, 1], [38, 171, 18, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 180, 'exponent': 342, 'exponents_of_order': [2, 2, 1], 'factors_of_aut_order': [2, 3, 19], 'factors_of_order': [2, 3, 19], 'faithful_reps': [[9, 0, 36], [18, 0, 2], [18, 1, 9]], 'familial': False, 'frattini_label': '1.1', 'frattini_quotient': '6498.m', 'hash': None, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 342, 'inner_gen_orders': [38, 9, 19, 19], 'inner_gens': [[1, 340, 6300, 3870], [4865, 2, 990, 3078], [559, 5870, 18, 342], [3313, 3764, 18, 342]], 'inner_hash': 2188275780713474934, 'inner_nilpotent': False, 'inner_order': 6498, 'inner_split': True, 'inner_tex': 'C_{19}^2:D_9', 'inner_used': [1, 2], 'irrC_degree': 9, 'irrQ_degree': 36, 'irrQ_dim': 36, 'irrR_degree': 18, 'irrep_stats': [[1, 2], [2, 4], [9, 36], [18, 11]], 'label': '6498.m', 'linC_count': 36, 'linC_degree': 9, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 36, 'linQ_degree_count': 1, 'linQ_dim': 36, 'linQ_dim_count': 1, 'linR_count': 27, 'linR_degree': 18, 'maximal_subgroups_known': True, 'metabelian': False, 'metacyclic': False, 'monomial': True, 'name': 'C19^2:D9', 'ngens': 2, '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], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 10, 'number_characteristic_subgroups': 5, 'number_conjugacy_classes': 53, 'number_divisions': 8, 'number_normal_subgroups': 5, 'number_subgroup_autclasses': 19, 'number_subgroup_classes': 19, 'number_subgroups': 2630, 'old_label': None, 'order': 6498, 'order_factorization_type': 222, 'order_stats': [[1, 1], [2, 171], [3, 722], [9, 2166], [19, 360], [38, 3078]], 'outer_abelian': True, 'outer_cyclic': True, 'outer_equivalence': False, 'outer_exponent': 18, 'outer_gen_orders': [18], 'outer_gen_pows': [0], 'outer_gens': [[1075, 3904, 4518, 5184]], 'outer_group': '18.2', 'outer_hash': 2, 'outer_nilpotent': True, 'outer_order': 18, 'outer_permdeg': 11, 'outer_perms': [3989739], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_{18}', 'pc_rank': 4, 'perfect': False, 'permutation_degree': 38, 'pgroup': 0, 'primary_abelian_invariants': [2], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 2], [2, 1], [6, 1], [36, 1], [162, 3]], 'representations': {'PC': {'code': '3855435851509477540077650937891544722449639087', 'gens': [1, 2, 4, 5], 'pres': [5, -2, -3, -3, -19, 19, 10530, 3401, 36, 83342, 126003, 9908, 1813, 96754, 38484, 9989]}, 'Perm': {'d': 38, 'gens': [236744638745043237596536418174214072278461439, 338513300532897801305632202598633190687313440]}}, 'schur_multiplier': [], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2], 'solvability_type': 11, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': 'C_{19}^2:D_9', 'transitive_degree': 38, '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': '1.1', 'all_subgroups_known': False, 'almost_simple': True, 'aut_abelian': None, 'aut_cyclic': None, 'aut_derived_length': None, 'aut_exponent': None, 'aut_gen_orders': None, 'aut_gens': [[16984866253, 16089815143], [176244092267, 69020256000], [163961147004, 219693676963], [194802679949, 67388406293], [305653122232, 98558697144], [93900441739, 33351131835], [67038048300, 96188451285], [221113362884, 77435834938], [69015763169, 31350590020]], 'aut_group': None, 'aut_hash': None, 'aut_nilpotency_class': None, 'aut_nilpotent': None, 'aut_order': 33868095840, 'aut_permdeg': None, 'aut_perms': None, 'aut_phi_ratio': None, 'aut_solvable': None, 'aut_stats': [[1, 1, 1, 1], [2, 137541, 1, 1], [3, 137541, 2, 1], [3, 17421860, 1, 1], [4, 47039022, 1, 1], [5, 47039022, 1, 2], [6, 137541, 2, 1], [6, 52265580, 1, 1], [6, 52265580, 2, 1], [8, 47039022, 2, 1], [9, 52265580, 1, 3], [10, 47039022, 1, 2], [12, 47039022, 2, 1], [15, 47039022, 2, 2], [18, 52265580, 1, 3], [18, 52265580, 2, 3], [19, 137160, 1, 1], [19, 15636240, 3, 1], [20, 47039022, 1, 4], [24, 47039022, 2, 2], [30, 47039022, 2, 2], [38, 49514760, 1, 1], [40, 47039022, 2, 4], [57, 49514760, 2, 1], [60, 47039022, 2, 4], [114, 49514760, 2, 1], [120, 47039022, 2, 8], [127, 44446320, 2, 21]], 'aut_supersolvable': None, 'aut_tex': None, 'autcent_abelian': None, 'autcent_cyclic': None, 'autcent_exponent': None, 'autcent_group': None, 'autcent_hash': None, 'autcent_nilpotent': None, 'autcent_order': None, 'autcent_solvable': None, 'autcent_split': None, 'autcent_supersolvable': None, 'autcent_tex': None, 'autcentquo_abelian': None, 'autcentquo_cyclic': None, 'autcentquo_exponent': None, 'autcentquo_group': None, 'autcentquo_hash': None, 'autcentquo_nilpotent': None, 'autcentquo_order': None, 'autcentquo_solvable': None, 'autcentquo_supersolvable': None, 'autcentquo_tex': None, 'cc_stats': [[1, 1, 1], [2, 137541, 1], [3, 137541, 2], [3, 17421860, 1], [4, 47039022, 1], [5, 47039022, 2], [6, 137541, 2], [6, 52265580, 3], [8, 47039022, 2], [9, 52265580, 3], [10, 47039022, 2], [12, 47039022, 2], [15, 47039022, 4], [18, 52265580, 9], [19, 137160, 1], [19, 15636240, 3], [20, 47039022, 4], [24, 47039022, 4], [30, 47039022, 4], [38, 49514760, 1], [40, 47039022, 8], [57, 49514760, 2], [60, 47039022, 8], [114, 49514760, 2], [120, 47039022, 16], [127, 44446320, 42]], 'center_label': '1.1', 'center_order': None, 'central_product': None, 'central_quotient': '5644682640.a', 'commutator_count': 1, 'commutator_label': '5644682640.a', 'complements_known': False, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['5644682640.a'], 'composition_length': 1, 'conjugacy_classes_known': True, 'counter': 1, 'cyclic': False, 'derived_length': 0, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 137541, 1, 1], [3, 137541, 2, 1], [3, 17421860, 1, 1], [4, 47039022, 1, 1], [5, 47039022, 2, 1], [6, 137541, 2, 1], [6, 52265580, 1, 1], [6, 52265580, 2, 1], [8, 47039022, 2, 1], [9, 52265580, 3, 1], [10, 47039022, 2, 1], [12, 47039022, 2, 1], [15, 47039022, 4, 1], [18, 52265580, 3, 1], [18, 52265580, 6, 1], [19, 137160, 1, 1], [19, 15636240, 1, 3], [20, 47039022, 4, 1], [24, 47039022, 4, 1], [30, 47039022, 4, 1], [38, 49514760, 1, 1], [40, 47039022, 8, 1], [57, 49514760, 2, 1], [60, 47039022, 8, 1], [114, 49514760, 2, 1], [120, 47039022, 16, 1], [127, 44446320, 42, 1]], 'element_repr_type': 'Lie', 'elementary': 1, 'eulerian_function': None, 'exponent': 868680, 'exponents_of_order': [4, 4, 3, 1, 1], 'factors_of_aut_order': [2, 3, 5, 19, 127], 'factors_of_order': [2, 3, 5, 19, 127], 'faithful_reps': [[380, 1, 1], [381, 0, 4], [381, 1, 1], [2540, 1, 3], [6480, 0, 42], [6858, 0, 48], [6858, 1, 9], [6859, 1, 1], [7239, 0, 4], [7239, 1, 1], [7620, 0, 8], [7620, 1, 7]], 'familial': True, 'frattini_label': '1.1', 'frattini_quotient': '5644682640.a', 'hash': 5896861318696862772, 'hyperelementary': 1, 'inner_abelian': None, 'inner_cyclic': None, 'inner_exponent': None, 'inner_gen_orders': None, 'inner_gens': None, 'inner_hash': None, 'inner_nilpotent': None, 'inner_order': None, 'inner_split': None, 'inner_tex': '\\PSL(3,19)', 'inner_used': None, 'irrC_degree': 380, 'irrQ_degree': 380, 'irrQ_dim': 380, 'irrR_degree': 380, 'irrep_stats': [[1, 1], [380, 1], [381, 5], [2540, 3], [6480, 42], [6858, 57], [6859, 1], [7239, 5], [7620, 15]], 'label': '5644682640.a', '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': False, 'name': 'PSL(3,19)', 'ngens': 2, 'nilpotency_class': -1, 'nilpotent': False, 'normal_counts': [1, 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, 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, 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, 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, 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, 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, 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, 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, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 75, 'number_characteristic_subgroups': 2, 'number_conjugacy_classes': 130, 'number_divisions': 30, 'number_normal_subgroups': 2, 'number_subgroup_autclasses': None, 'number_subgroup_classes': 453, 'number_subgroups': 5987090011, 'old_label': None, 'order': 5644682640, 'order_factorization_type': 321, 'order_stats': [[1, 1], [2, 137541], [3, 17696942], [4, 47039022], [5, 94078044], [6, 157071822], [8, 94078044], [9, 156796740], [10, 94078044], [12, 94078044], [15, 188156088], [18, 470390220], [19, 47045880], [20, 188156088], [24, 188156088], [30, 188156088], [38, 49514760], [40, 376312176], [57, 99029520], [60, 376312176], [114, 99029520], [120, 752624352], [127, 1866745440]], 'outer_abelian': None, 'outer_cyclic': None, 'outer_equivalence': False, 'outer_exponent': None, 'outer_gen_orders': None, 'outer_gen_pows': None, 'outer_gens': None, 'outer_group': '6.1', 'outer_hash': None, 'outer_nilpotent': None, 'outer_order': 6, 'outer_permdeg': None, 'outer_perms': None, 'outer_solvable': None, 'outer_supersolvable': None, 'outer_tex': 'S_3', 'pc_rank': None, 'perfect': True, 'permutation_degree': 381, 'pgroup': 0, 'primary_abelian_invariants': [], 'quasisimple': True, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 1], [380, 1], [381, 1], [762, 2], [2540, 3], [6858, 1], [6859, 1], [7239, 1], [7620, 1], [13716, 4], [14478, 2], [15240, 1], [22860, 2], [27432, 4], [45720, 1], [54864, 2], [109728, 1], [272160, 1]], 'representations': {'Lie': [{'d': 3, 'q': 19, 'gens': [16984866253, 16089815143], 'family': 'PSL'}, {'d': 3, 'q': 19, 'family': 'PSigmaL'}], 'Perm': {'d': 381, 'gens': [1762767108574691198711235554880360015470653492055874597649923168084482816130935932123670764138620429640979304664001798569460526762505721275070405645662806060667688208233619455963602505404583798145702462908597995459584984499550418554938286060962283999915285754126842053672168825994973998375078790505052679794258819956765231499357967643463303279176073914469738714532877018739561574995382186650741339614187159609720951399292181096681266204584831622809391642340290672568838256111663231400470016186901983649850806242989430535621133126679709469536917040816000372799214811778485882368910089788048793295655692236902455780026242467823694581221136391703574578927141373322152022019747683387784447433967028831930813634424616445036557322133061820519112125115780846903601602460260530053474774525550109166018169829603868289395, 1991695886552025762465103171241810503663081925731820397153462670990869334577248171436442754484378858136472566663123119789492911771556378718419661615234641380168608536159538590205281623397448779117068637210931684307174201245149497100670485077954447599352400888456914849534288881878500628443027761014486125162012389810781996593747769443870364897583798575223250240386952710388935337909860631273711868380221006741942356095281819295924691190917699297643591711939948021378361341120237743521251825611726011009299926480779988893376789822785561567277857442434359711642180233905578214356930442344992809807577201824184032951327260175835306368775161446205342895305753614913974609117678306807710567960086284503900361333847351862563480259611594208585612225893799971297221810718176387721295716165472491780231655212212944106274648366709]}}, 'schur_multiplier': None, 'semidirect_product': False, 'simple': True, 'smith_abelian_invariants': [], 'solvability_type': 13, 'solvable': False, 'subgroup_inclusions_known': False, 'subgroup_index_bound': 5644682640, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': '\\PSL(3,19)', 'transitive_degree': 381, 'wreath_data': None, 'wreath_product': False}
