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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': False, 'ambient': '4056.bb', 'ambient_counter': 28, 'ambient_order': 4056, 'ambient_tex': 'C_{13}^2:(C_4\\times S_3)', 'central': False, 'central_factor': False, 'centralizer_order': 2, 'characteristic': False, 'core_order': 1, 'counter': 28, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '4056.bb.78.e1.b1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': False, 'minimal_normal': False, 'nilpotent': False, 'normal': False, 'old_label': '78.e1.b1', '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': 78, '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': '52.3', 'subgroup_hash': 3, 'subgroup_order': 52, 'subgroup_tex': 'C_{13}:C_4', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '4056.bb', 'aut_centralizer_order': 2, 'aut_label': '78.e1', 'aut_quo_index': None, 'aut_stab_index': 78, 'aut_weyl_group': '156.7', 'aut_weyl_index': 156, 'centralizer': '2028.a1.b1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['6.c1.a1', '39.a1.b1'], 'contains': ['156.c1.b1', '1014.c1.a1'], 'core': '4056.a1.a1', 'coset_action_label': None, 'count': 39, 'diagramx': [5011, -1, 3297, -1, 5524, -1, 9759, -1], 'generators': [1, 3456, 3540], 'label': '4056.bb.78.e1.b1', 'mobius_quo': None, 'mobius_sub': 0, 'normal_closure': '2.c1.a1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '39.a1.b1', 'old_label': '78.e1.b1', 'projective_image': '4056.bb', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '78.e1.b1', 'subgroup_fusion': None, 'weyl_group': '52.3'}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': False, 'abelian_quotient': '4.1', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 156, 'aut_gen_orders': [12, 13], 'aut_gens': [[1, 4], [1, 28], [21, 4]], 'aut_group': '156.7', 'aut_hash': 7, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 156, 'aut_permdeg': 13, 'aut_perms': [178063522, 4900431375], 'aut_phi_ratio': 6.5, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 13, 1, 1], [4, 13, 1, 2], [13, 4, 3, 1]], 'aut_supersolvable': True, 'aut_tex': 'F_{13}', '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': 156, 'autcentquo_group': '156.7', 'autcentquo_hash': 7, 'autcentquo_nilpotent': False, 'autcentquo_order': 156, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'F_{13}', 'cc_stats': [[1, 1, 1], [2, 13, 1], [4, 13, 2], [13, 4, 3]], 'center_label': '1.1', 'center_order': 1, 'central_product': False, 'central_quotient': '52.3', 'commutator_count': 1, 'commutator_label': '13.1', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '13.1'], 'composition_length': 3, 'conjugacy_classes_known': True, 'counter': 3, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 13, 1, 1], [4, 13, 2, 1], [13, 4, 3, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 12, 'exponent': 52, 'exponents_of_order': [2, 1], 'factors_of_aut_order': [2, 3, 13], 'factors_of_order': [2, 13], 'faithful_reps': [[4, 1, 3]], 'familial': False, 'frattini_label': '1.1', 'frattini_quotient': '52.3', 'hash': 3, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 52, 'inner_gen_orders': [4, 13], 'inner_gens': [[1, 20], [37, 4]], 'inner_hash': 3, 'inner_nilpotent': False, 'inner_order': 52, 'inner_split': True, 'inner_tex': 'C_{13}:C_4', 'inner_used': [1, 2], 'irrC_degree': 4, 'irrQ_degree': 12, 'irrQ_dim': 12, 'irrR_degree': 4, 'irrep_stats': [[1, 4], [4, 3]], 'label': '52.3', 'linC_count': 3, 'linC_degree': 4, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 12, 'linQ_degree_count': 1, 'linQ_dim': 12, 'linQ_dim_count': 1, 'linR_count': 3, 'linR_degree': 4, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C13:C4', 'ngens': 3, 'nilpotency_class': -1, 'nilpotent': False, 'normal_counts': [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': 4, 'number_conjugacy_classes': 7, 'number_divisions': 4, 'number_normal_subgroups': 4, 'number_subgroup_autclasses': 6, 'number_subgroup_classes': 6, 'number_subgroups': 30, 'old_label': None, 'order': 52, 'order_factorization_type': 22, 'order_stats': [[1, 1], [2, 13], [4, 26], [13, 12]], 'outer_abelian': True, 'outer_cyclic': True, 'outer_equivalence': False, 'outer_exponent': 3, 'outer_gen_orders': [3], 'outer_gen_pows': [3], 'outer_gens': [[1, 24]], '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': 13, 'pgroup': 0, 'primary_abelian_invariants': [4], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 2], [2, 1], [12, 1]], 'representations': {'PC': {'code': 1656987851, 'gens': [1, 3], 'pres': [3, -2, -2, -13, 6, 182, 221]}, 'GLFp': {'d': 2, 'p': 13, 'gens': [2211, 10997]}, 'Perm': {'d': 13, 'gens': [289742629, 479001599, 5748019200]}}, 'schur_multiplier': [], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [4], 'solvability_type': 6, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_{13}:C_4', 'transitive_degree': 13, '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': '8.2', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 3, 'aut_exponent': 156, 'aut_gen_orders': [6, 12, 26, 13], 'aut_gens': [[1, 2, 24, 312], [1249, 1234, 3528, 3048], [2909, 1634, 3600, 624], [2245, 3986, 2208, 3744], [1777, 2186, 24, 312]], 'aut_group': '24336.o', 'aut_hash': 645997265909674319, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 24336, 'aut_permdeg': 247, 'aut_perms': [187080073175515173563079777302435518598188805678355900458249211957657909987884315029565017785886288493958013985350016205060332522946679102154591890096931454463281840220562021058803802333727012558289790800873969749490484673043984490821176337435643206446219309725867811013359261544132786406853531600488612122703398768425923409009775327624172869922488067917791713814125384668778986678309425047188720364022788125007936135081665524022346806992926968588504341212342490353020453868340812231367, 198941177198303712508375713501301810303287767753684285323850954217522552133842763621139242831540075947448503401266513781318932362256990275334032268524408228325880358451367987119448211774520395396095748197259781642823757482913544029268077793490018360532017821749577213526375616818853296957557756199133236171693383984050310668723095605917296313137137796086382247503011431390409732802446660563648938190596858421046154082689386259534368918499368436453992486578034159892678779811948860446937, 178508371199840916843964574561441880914665475415617207821902652389144749066471494547674180436893798467330714782198383099001877589838736267748457328149805793031580575348910250416647060079601877261597656798829313212558878116437081452373161283490393958749653031528443160600647561568722616003494761292374881559651047803843514687934537238183378004316281709812750855215981016908995404330628003284595865372509961413248587747797420825876290781152974269031054247232261788853653240733908897827314, 156175955157497185804324675475084476338308515696235924924971782343182356758648118989579276233665766219571685036027316461435214062421385080870705975681406416299977342599398089738161138662838185603543508739485047924153684933003744416492132674349971438538225965825404499353261428605807714278263980695137933444759908613724811825748493929019029288869811009016872349467081292125612664972643252090474151854736434854077809908324536767291447963964283581561528315232382483232066646101349870081640], 'aut_phi_ratio': 19.5, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 39, 2, 1], [2, 169, 1, 1], [3, 338, 1, 1], [4, 169, 1, 2], [4, 507, 2, 1], [6, 338, 1, 1], [12, 338, 1, 2], [13, 12, 6, 1], [13, 24, 1, 1], [13, 24, 3, 1], [26, 156, 6, 1]], 'aut_supersolvable': False, 'aut_tex': 'D_{13}^2.(C_6\\times S_3)', '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': 156, 'autcentquo_group': '24336.o', 'autcentquo_hash': 645997265909674319, 'autcentquo_nilpotent': False, 'autcentquo_order': 24336, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'D_{13}^2.(C_6\\times S_3)', 'cc_stats': [[1, 1, 1], [2, 39, 2], [2, 169, 1], [3, 338, 1], [4, 169, 2], [4, 507, 2], [6, 338, 1], [12, 338, 2], [13, 12, 6], [13, 24, 4], [26, 156, 6]], 'center_label': '1.1', 'center_order': 1, 'central_product': False, 'central_quotient': '4056.bb', 'commutator_count': 1, 'commutator_label': '507.5', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '3.1', '13.1', '13.1'], 'composition_length': 6, 'conjugacy_classes_known': True, 'counter': 28, 'cyclic': False, 'derived_length': 3, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 39, 1, 2], [2, 169, 1, 1], [3, 338, 1, 1], [4, 169, 2, 1], [4, 507, 2, 1], [6, 338, 1, 1], [12, 338, 2, 1], [13, 12, 3, 2], [13, 24, 1, 1], [13, 24, 3, 1], [26, 156, 3, 2]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 168, 'exponent': 156, 'exponents_of_order': [3, 2, 1], 'factors_of_aut_order': [2, 3, 13], 'factors_of_order': [2, 3, 13], 'faithful_reps': [[12, 1, 12], [24, 1, 4]], 'familial': False, 'frattini_label': '1.1', 'frattini_quotient': '4056.bb', 'hash': 3166737781490808363, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 156, 'inner_gen_orders': [4, 12, 13, 13], 'inner_gens': [[1, 2746, 3792, 3552], [2729, 2, 456, 624], [601, 3938, 24, 312], [1129, 3746, 24, 312]], 'inner_hash': 3166737781490808363, 'inner_nilpotent': False, 'inner_order': 4056, 'inner_split': True, 'inner_tex': 'C_{13}^2:(C_4\\times S_3)', 'inner_used': [1, 2], 'irrC_degree': 12, 'irrQ_degree': 24, 'irrQ_dim': 24, 'irrR_degree': 12, 'irrep_stats': [[1, 8], [2, 4], [12, 12], [24, 4]], 'label': '4056.bb', 'linC_count': 12, 'linC_degree': 12, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 24, 'linQ_degree_count': 1, 'linQ_dim': 24, 'linQ_dim_count': 1, 'linR_count': 12, 'linR_degree': 12, 'maximal_subgroups_known': True, 'metabelian': False, 'metacyclic': False, 'monomial': True, 'name': 'C13^2:(C4*S3)', 'ngens': 6, '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], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 14, 'number_characteristic_subgroups': 10, 'number_conjugacy_classes': 28, 'number_divisions': 15, 'number_normal_subgroups': 12, 'number_subgroup_autclasses': 45, 'number_subgroup_classes': 57, 'number_subgroups': 4090, 'old_label': None, 'order': 4056, 'order_factorization_type': 321, 'order_stats': [[1, 1], [2, 247], [3, 338], [4, 1352], [6, 338], [12, 676], [13, 168], [26, 936]], 'outer_abelian': True, 'outer_cyclic': True, 'outer_equivalence': False, 'outer_exponent': 6, 'outer_gen_orders': [6], 'outer_gen_pows': [0], 'outer_gens': [[677, 1562, 2832, 1248]], 'outer_group': '6.2', 'outer_hash': 2, 'outer_nilpotent': True, 'outer_order': 6, 'outer_permdeg': 5, 'outer_perms': [28], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_6', 'pc_rank': 4, 'perfect': False, 'permutation_degree': 26, 'pgroup': 0, 'primary_abelian_invariants': [2, 4], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 4], [4, 1], [24, 1], [36, 4], [72, 1]], 'representations': {'PC': {'code': '510441394630661718194494619427342127350372583007196380197151', 'gens': [1, 2, 5, 6], 'pres': [6, -2, -2, -2, -3, -13, 13, 21240, 32953, 31, 41402, 50, 49347, 113764, 6850, 20536, 10192, 127877, 11243, 11249, 4235]}, 'Perm': {'d': 26, 'gens': [260943395812615902259770396, 260554401281602754478350309]}}, 'schur_multiplier': [2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 4], 'solvability_type': 11, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': 'C_{13}^2:(C_4\\times S_3)', 'transitive_degree': 26, 'wreath_data': None, 'wreath_product': False}
