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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': True, 'ambient': '122304.a', 'ambient_counter': 1, 'ambient_order': 122304, 'ambient_tex': 'C_{56}.\\PGL(2,13)', 'central': True, 'central_factor': False, 'centralizer_order': 122304, 'characteristic': True, 'core_order': 8, 'counter': 378, 'cyclic': True, 'direct': False, 'hall': 0, 'label': '122304.a.15288.a1.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': True, 'old_label': '15288.a1.a1', 'outer_equivalence': False, 'perfect': False, 'proper': True, 'quotient': None, 'quotient_Agroup': False, 'quotient_abelian': False, 'quotient_cyclic': False, 'quotient_hash': 2970156246303075913, 'quotient_metabelian': True, 'quotient_nilpotent': False, 'quotient_order': 15288, 'quotient_simple': False, 'quotient_solvable': False, 'quotient_supersolvable': False, 'quotient_tex': 'C_7.\\PSL(2,13).C_2', 'simple': False, 'solvable': True, 'special_labels': [], 'split': False, 'standard_generators': False, 'stem': False, 'subgroup': '8.1', 'subgroup_hash': 1, 'subgroup_order': 8, 'subgroup_tex': 'C_8', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '122304.a', 'aut_centralizer_order': None, 'aut_label': '15288.a1', 'aut_quo_index': None, 'aut_stab_index': None, 'aut_weyl_group': None, 'aut_weyl_index': None, 'centralizer': '1.a1.a1', 'complements': [], 'conjugacy_class_count': 1, 'contained_in': ['1176.c1.a1', '2184.a1.a1', '2184.q1.a1', '2184.u1.a1', '2184.u1.b1', '2184.u1.c1', '5096.i1.a1', '7644.a1.a1', '7644.g1.a1'], 'contains': ['30576.a1.a1'], 'core': '15288.a1.a1', 'coset_action_label': None, 'count': 1, 'diagramx': [7957, 3293, 8744, 7775, 7218, 1126, 8102, 1126], 'generators': [545429530, 57921720, 24134050], 'label': '122304.a.15288.a1.a1', 'mobius_quo': 0, 'mobius_sub': -1092, 'normal_closure': '15288.a1.a1', 'normal_contained_in': ['14.c1.a1', '2184.a1.a1'], 'normal_contains': ['30576.a1.a1'], 'normalizer': '1.a1.a1', 'old_label': '15288.a1.a1', 'projective_image': None, 'quotient_action_image': '1.1', 'quotient_action_kernel': None, 'quotient_action_kernel_order': 15288, 'quotient_fusion': None, 'short_label': '15288.a1.a1', 'subgroup_fusion': None, 'weyl_group': '1.1'}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': True, 'abelian_quotient': '8.1', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': True, 'aut_cyclic': False, 'aut_derived_length': 1, 'aut_exponent': 2, 'aut_gen_orders': [2, 2], 'aut_gens': [[1], [5], [3]], 'aut_group': '4.2', 'aut_hash': 2, 'aut_nilpotency_class': 1, 'aut_nilpotent': True, 'aut_order': 4, 'aut_permdeg': 4, 'aut_perms': [1, 6], 'aut_phi_ratio': 1.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [4, 1, 2, 1], [8, 1, 4, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_2^2', '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': 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, 1], [4, 1, 2], [8, 1, 4]], 'center_label': '8.1', 'center_order': 8, 'central_product': False, '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', '2.1'], 'composition_length': 3, 'conjugacy_classes_known': True, 'counter': 1, 'cyclic': True, 'derived_length': 1, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [4, 1, 2, 1], [8, 1, 4, 1]], 'element_repr_type': 'PC', 'elementary': 2, 'eulerian_function': 1, 'exponent': 8, 'exponents_of_order': [3], 'factors_of_aut_order': [2], 'factors_of_order': [2], 'faithful_reps': [[1, 0, 4]], 'familial': True, 'frattini_label': '4.1', 'frattini_quotient': '2.1', 'hash': 1, 'hyperelementary': 2, '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': 4, 'irrQ_dim': 4, 'irrR_degree': 2, 'irrep_stats': [[1, 8]], 'label': '8.1', 'linC_count': 4, 'linC_degree': 1, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 4, 'linQ_degree_count': 1, 'linQ_dim': 4, 'linQ_dim_count': 1, 'linR_count': 2, 'linR_degree': 2, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C8', 'ngens': 1, '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': 8, 'number_divisions': 4, 'number_normal_subgroups': 4, 'number_subgroup_autclasses': 4, 'number_subgroup_classes': 4, 'number_subgroups': 4, 'old_label': None, 'order': 8, 'order_factorization_type': 3, 'order_stats': [[1, 1], [2, 1], [4, 2], [8, 4]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 2, 'outer_gen_orders': [2, 2], 'outer_gen_pows': [0, 0], 'outer_gens': [[5], [3]], 'outer_group': '4.2', 'outer_hash': 2, 'outer_nilpotent': True, 'outer_order': 4, 'outer_permdeg': 4, 'outer_perms': [1, 6], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^2', 'pc_rank': 1, 'perfect': False, 'permutation_degree': 8, 'pgroup': 2, 'primary_abelian_invariants': [8], 'quasisimple': False, 'rank': 1, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 2], [2, 1], [4, 1]], 'representations': {'PC': {'code': 323, 'gens': [1], 'pres': [3, -2, -2, -2, 6, 16]}, 'GLZ': {'b': 3, 'd': 4, 'gens': [26315074]}, 'GLFp': {'d': 2, 'p': 3, 'gens': [77]}, 'Perm': {'d': 8, 'gens': [40176, 16582, 5167]}}, 'schur_multiplier': [], 'semidirect_product': False, 'simple': False, 'smith_abelian_invariants': [8], 'solvability_type': 0, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_8', 'transitive_degree': 8, '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': '56.8', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 1, 'aut_exponent': 1092, 'aut_gen_orders': [42, 6, 12, 6, 42, 12], 'aut_gens': [[801250460, 325178460, 448893330, 772289600, 407719451, 98196940, 390971610], [613004870, 203456389, 608178060, 598524440, 358216170, 412459473, 178591970], [613004870, 223258229, 270301360, 598524440, 685494178, 51903794, 641965730], [106189820, 709988660, 458546950, 77228960, 780160055, 370793039, 593697630], [613004870, 341252326, 608178060, 598524440, 745005220, 230406627, 699887450], [497161430, 232083322, 535775910, 473027380, 362300390, 27655003, 767462790], [279954980, 175307606, 603351250, 265474550, 722759065, 348642382, 188245590]], 'aut_group': None, 'aut_hash': 1123980728386944378, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 104832, 'aut_permdeg': 166, 'aut_perms': [8073678868002122520183138900870225689535912480693521445703446057802539521549641568540472985473516583842309063966461369688803001030103016539156992320634177616006094254790020833137934955061601033197986660680146957597539604978039037257164228833953939897224833568058991649910812031617736578330675180989, 1063026628903214225412754561371083798693271331094175748611552413753337076870057709699130164752910215811879302876381388465807073291713780553597825026112982814468230698502516893587421939663631672888531698353694518530056025200582052092408362500593849766225958930588100017648419948812488419895234862617, 1583207115970351370332018593830814858728265792195576107325981772506491773856159451139798554584086305018437753869668440769126943768818030313720022333155580359466206212001111120145652110789899466111966466331232562335117958483898374531784868474898480258323309184317326409163001895100102960237633623669, 4446801646880926600831779562076679815258783289976894366424098949912857163277345624729230371796938638943133603300414003998991742077832013085849265046234601562286411504882536180847216689351257458799854510799640081496282419483223502509925282826826128357125043598395333045782595870607913751659010225571, 4410323632118713221857089319224309078152213671871801699987353524910221001370984390077004840618106769049685000504939978035596196672111317617609824187622462867983299354732188218594402828352832485309560894098204506850030911827897815402941040138596657464832644304086833669344336496144611246677855707254, 1836296955787605449440172935350906674189417534096693626517512118514324013425960159548422606631469271352089766758503702417215140931653169486092716847679367023936890178615482008014081794506771551762245994205324093761032515291947984065687960905820945336354840508605665661914875886315182751959713798656], 'aut_phi_ratio': 3.25, 'aut_solvable': False, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 156, 1, 1], [2, 182, 1, 1], [3, 182, 1, 1], [4, 1, 2, 1], [4, 156, 1, 1], [4, 182, 1, 1], [4, 182, 4, 1], [6, 182, 1, 1], [6, 182, 2, 1], [7, 1, 6, 1], [7, 156, 1, 3], [7, 156, 6, 3], [8, 1, 4, 1], [8, 156, 2, 1], [8, 182, 2, 3], [12, 182, 1, 2], [12, 182, 2, 1], [12, 182, 4, 2], [13, 168, 1, 1], [14, 1, 6, 1], [14, 156, 1, 3], [14, 156, 2, 3], [14, 156, 6, 4], [14, 156, 12, 3], [14, 182, 6, 1], [21, 182, 6, 1], [24, 182, 2, 4], [24, 182, 4, 2], [26, 168, 1, 1], [28, 1, 12, 1], [28, 156, 2, 6], [28, 156, 6, 1], [28, 156, 12, 6], [28, 182, 6, 1], [28, 182, 24, 1], [42, 182, 6, 1], [42, 182, 12, 1], [52, 168, 2, 1], [56, 1, 24, 1], [56, 156, 4, 6], [56, 156, 12, 1], [56, 156, 24, 6], [56, 182, 12, 3], [84, 182, 6, 2], [84, 182, 12, 1], [84, 182, 24, 2], [91, 168, 6, 1], [104, 168, 4, 1], [168, 182, 12, 4], [168, 182, 24, 2], [182, 168, 6, 1], [364, 168, 12, 1], [728, 168, 24, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_2^3\\times C_6\\times \\PSL(2,13).C_2', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 6, 'autcent_group': '48.52', 'autcent_hash': 52, 'autcent_nilpotent': True, 'autcent_order': 48, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^3\\times C_6', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 1092, 'autcentquo_group': '2184.b', 'autcentquo_hash': 7763277135502868404, 'autcentquo_nilpotent': False, 'autcentquo_order': 2184, 'autcentquo_solvable': False, 'autcentquo_supersolvable': False, 'autcentquo_tex': '\\PGL(2,13)', 'cc_stats': [[1, 1, 1], [2, 1, 1], [2, 156, 1], [2, 182, 1], [3, 182, 1], [4, 1, 2], [4, 156, 1], [4, 182, 5], [6, 182, 3], [7, 1, 6], [7, 156, 21], [8, 1, 4], [8, 156, 2], [8, 182, 6], [12, 182, 12], [13, 168, 1], [14, 1, 6], [14, 156, 69], [14, 182, 6], [21, 182, 6], [24, 182, 16], [26, 168, 1], [28, 1, 12], [28, 156, 90], [28, 182, 30], [42, 182, 18], [52, 168, 2], [56, 1, 24], [56, 156, 180], [56, 182, 36], [84, 182, 72], [91, 168, 6], [104, 168, 4], [168, 182, 96], [182, 168, 6], [364, 168, 12], [728, 168, 24]], 'center_label': '56.2', 'center_order': 56, 'central_product': True, 'central_quotient': '2184.b', 'commutator_count': 1, 'commutator_label': '2184.a', 'complements_known': True, 'complete': False, 'complex_characters_known': False, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '7.1', '1092.25'], 'composition_length': 6, 'conjugacy_classes_known': True, 'counter': 1, 'cyclic': False, 'derived_length': 1, 'dihedral': False, 'direct_factorization': [['17472.a', 1], ['7.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 156, 1, 1], [2, 182, 1, 1], [3, 182, 1, 1], [4, 1, 2, 1], [4, 156, 1, 1], [4, 182, 1, 1], [4, 182, 2, 2], [6, 182, 1, 1], [6, 182, 2, 1], [7, 1, 6, 1], [7, 156, 3, 1], [7, 156, 6, 3], [8, 1, 4, 1], [8, 156, 2, 1], [8, 182, 2, 3], [12, 182, 2, 2], [12, 182, 4, 2], [13, 168, 1, 1], [14, 1, 6, 1], [14, 156, 3, 1], [14, 156, 6, 11], [14, 182, 6, 1], [21, 182, 6, 1], [24, 182, 4, 4], [26, 168, 1, 1], [28, 1, 12, 1], [28, 156, 6, 3], [28, 156, 12, 6], [28, 182, 6, 1], [28, 182, 12, 2], [42, 182, 6, 1], [42, 182, 12, 1], [52, 168, 2, 1], [56, 1, 24, 1], [56, 156, 12, 3], [56, 156, 24, 6], [56, 182, 12, 3], [84, 182, 12, 2], [84, 182, 24, 2], [91, 168, 6, 1], [104, 168, 4, 1], [168, 182, 24, 4], [182, 168, 6, 1], [364, 168, 12, 1], [728, 168, 24, 1]], 'element_repr_type': 'GLFq', 'elementary': 1, 'eulerian_function': 47520, 'exponent': 2184, 'exponents_of_order': [6, 2, 1, 1], 'factors_of_aut_order': [2, 3, 7, 13], 'factors_of_order': [2, 3, 7, 13], 'faithful_reps': [[12, 0, 168], [14, 0, 144]], 'familial': False, 'frattini_label': '4.1', 'frattini_quotient': None, 'hash': 3069938595893683073, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 1092, 'inner_gen_orders': [1, 4, 1, 1, 3, 6, 1], 'inner_gens': [[801250460, 325178460, 448893330, 772289600, 407719451, 98196940, 390971610], [801250460, 325178460, 448893330, 772289600, 322412819, 98196940, 390971610], [801250460, 325178460, 448893330, 772289600, 407719451, 98196940, 390971610], [801250460, 325178460, 448893330, 772289600, 407719451, 98196940, 390971610], [801250460, 265626252, 448893330, 772289600, 407719451, 629557829, 390971610], [801250460, 325178460, 448893330, 772289600, 104816927, 98196940, 390971610], [801250460, 325178460, 448893330, 772289600, 407719451, 98196940, 390971610]], 'inner_hash': 7763277135502868404, 'inner_nilpotent': False, 'inner_order': 2184, 'inner_split': True, 'inner_tex': '\\PGL(2,13)', 'inner_used': [2, 5], 'irrC_degree': 12, 'irrQ_degree': 288, 'irrQ_dim': 288, 'irrR_degree': None, 'irrep_stats': [[1, 56], [12, 364], [13, 56], [14, 308]], 'label': '122304.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': 'C56.PGL(2,13)', '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, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 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': 104, 'number_characteristic_subgroups': 24, 'number_conjugacy_classes': 784, 'number_divisions': 89, 'number_normal_subgroups': 24, 'number_subgroup_autclasses': 364, 'number_subgroup_classes': 411, 'number_subgroups': 43012, 'old_label': None, 'order': 122304, 'order_factorization_type': 321, 'order_stats': [[1, 1], [2, 339], [3, 182], [4, 1068], [6, 546], [7, 3282], [8, 1408], [12, 2184], [13, 168], [14, 11862], [21, 1092], [24, 2912], [26, 168], [28, 19512], [42, 3276], [52, 336], [56, 34656], [84, 13104], [91, 1008], [104, 672], [168, 17472], [182, 1008], [364, 2016], [728, 4032]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 6, 'outer_gen_orders': [2, 2, 2, 6], 'outer_gen_pows': [4826810, 4826810, 4826810, 4826810], 'outer_gens': [[106189820, 608456295, 419932470, 77228960, 802823618, 98196940, 395798420], [801250460, 558529820, 448893330, 772289600, 407719451, 98196940, 390971610], [801250460, 325178460, 448893330, 772289600, 407719451, 98196940, 487507810], [613004870, 282335365, 270301360, 598524440, 6508752, 98196940, 641965730]], 'outer_group': '48.52', 'outer_hash': 52, 'outer_nilpotent': True, 'outer_order': 48, 'outer_permdeg': 11, 'outer_perms': [24, 3628800, 40320, 723], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^3\\times C_6', 'pc_rank': None, 'perfect': False, 'permutation_degree': 119, 'pgroup': 0, 'primary_abelian_invariants': [2, 4, 7], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 2], [6, 4], [12, 2], [13, 4], [14, 6], [26, 2], [28, 5], [36, 4], [48, 1], [56, 3], [72, 14], [78, 4], [84, 6], [112, 2], [144, 6], [156, 2], [168, 5], [288, 8], [336, 3], [672, 2]], 'representations': {'GLFq': {'d': 2, 'q': 169, 'gens': [801250460, 325178460, 448893330, 772289600, 407719451, 98196940, 390971610]}, 'Perm': {'d': 119, 'gens': [468793673994728239708733833840615414648590099282964154730913912320615099091506432343470365305844939011667520488668570180004256790141826713163083354378569344424469194793732392425734601207350941426, 981190575098954688708377818957715765584590674974081906247836368138869438353252641841883297500094001119847816758652223164305362427187514875207383119175389607451247871695112488022696376187354481091, 1433792357229609654239613962052575661983069749600519040633308452047004942761124419546282581321071580585698634961612205661127761023434552290590149885956824173653078748499378354351559810884663608520, 2962, 1922135294288091731581901711177210937309527559174798198175594764056750166983033988189255893671966177576518371203437440996048823537382897448178539967207371202616902039029970871812401336892074519880, 2394591997160533116179675433823000531970077675803853415064760105654344724462344596759241888282519521668734044814274185126772114060486281354191690107658236060384011054232505445211813517579988228480, 2867046161196250376938170532243001060688250037681537540135951363757496713091077334580917085124714817691260606705620484615462865604105762554505757111652530370043188101434170547597283971871065987957]}}, 'schur_multiplier': [2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 28], 'solvability_type': 13, 'solvable': False, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': 'C_{56}.\\PGL(2,13)', 'transitive_degree': 784, 'wreath_data': None, 'wreath_product': False}
