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gps_subgroup_search • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'ambient': '15840.j', 'ambient_counter': 10, 'ambient_order': 15840, 'ambient_tex': '\\SL(2,11):D_6', 'central': False, 'central_factor': False, 'centralizer_order': 12, 'characteristic': False, 'core_order': 6, 'counter': 113, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '15840.j.220.p1.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': None, 'minimal': False, 'minimal_normal': False, 'nilpotent': False, 'normal': False, 'old_label': '220.p1.a1', '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': 220, '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': '72.26', 'subgroup_hash': 26, 'subgroup_order': 72, 'subgroup_tex': 'C_3^2:Q_8', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '15840.j', 'aut_centralizer_order': None, 'aut_label': '220.p1', 'aut_quo_index': None, 'aut_stab_index': None, 'aut_weyl_group': None, 'aut_weyl_index': None, 'centralizer': '1320.a1.a1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['4.a1.a1', '110.h1.a1', '110.j1.a1', '110.j1.b1'], 'contains': ['440.g1.a1', '440.h1.a1', '440.h1.a2', '660.ba1.a1', '660.bc1.a1'], 'core': '2640.a1.a1', 'coset_action_label': None, 'count': 55, 'diagramx': [8490, -1, 3947, -1, 4439, -1, 5613, -1], 'generators': [28153846240628575, 5707748328679427, 32261473703376547, 34137651804035287, 41772741070013040], 'label': '15840.j.220.p1.a1', 'mobius_quo': None, 'mobius_sub': -2, 'normal_closure': '4.a1.a1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '55.b1.a1', 'old_label': '220.p1.a1', 'projective_image': None, 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '220.p1.a1', 'subgroup_fusion': None, 'weyl_group': '24.14'}
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gps_groups • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '12.5', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 12, 'aut_gen_orders': [2, 2, 2, 2, 2, 3], 'aut_gens': [[1, 6], [1, 30], [5, 6], [5, 42], [19, 42], [37, 6], [25, 6]], 'aut_group': '96.209', 'aut_hash': 209, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 96, 'aut_permdeg': 9, 'aut_perms': [5160, 120, 16, 121, 7, 45360], 'aut_phi_ratio': 4.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [3, 1, 2, 1], [3, 2, 1, 1], [3, 2, 2, 1], [4, 2, 1, 1], [4, 6, 2, 1], [6, 1, 2, 1], [6, 2, 1, 1], [6, 2, 2, 1], [12, 2, 2, 2], [12, 2, 4, 1], [12, 6, 4, 1]], 'aut_supersolvable': True, 'aut_tex': 'D_4\\times D_6', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 2, 'autcent_group': '8.5', 'autcent_hash': 5, 'autcent_nilpotent': True, 'autcent_order': 8, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^3', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 6, 'autcentquo_group': '12.4', 'autcentquo_hash': 4, 'autcentquo_nilpotent': False, 'autcentquo_order': 12, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'D_6', 'cc_stats': [[1, 1, 1], [2, 1, 1], [3, 1, 2], [3, 2, 3], [4, 2, 1], [4, 6, 2], [6, 1, 2], [6, 2, 3], [12, 2, 8], [12, 6, 4]], 'center_label': '6.2', 'center_order': 6, 'central_product': True, 'central_quotient': '12.4', 'commutator_count': 1, 'commutator_label': '6.2', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '3.1', '3.1'], 'composition_length': 5, 'conjugacy_classes_known': True, 'counter': 26, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['24.4', 1], ['3.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [3, 1, 2, 1], [3, 2, 1, 1], [3, 2, 2, 1], [4, 2, 1, 1], [4, 6, 1, 2], [6, 1, 2, 1], [6, 2, 1, 1], [6, 2, 2, 1], [12, 2, 2, 2], [12, 2, 4, 1], [12, 6, 2, 2]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 12, 'exponent': 12, 'exponents_of_order': [3, 2], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [[2, 0, 4]], 'familial': False, 'frattini_label': '2.1', 'frattini_quotient': '36.12', 'hash': 26, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 6, 'inner_gen_orders': [2, 6], 'inner_gens': [[1, 66], [13, 6]], 'inner_hash': 4, 'inner_nilpotent': False, 'inner_order': 12, 'inner_split': True, 'inner_tex': 'D_6', 'inner_used': [1, 2], 'irrC_degree': 2, 'irrQ_degree': 8, 'irrQ_dim': 8, 'irrR_degree': 4, 'irrep_stats': [[1, 12], [2, 15]], 'label': '72.26', 'linC_count': 4, 'linC_degree': 2, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 6, 'linQ_degree_count': 16, 'linQ_dim': 6, 'linQ_dim_count': 2, 'linR_count': 2, 'linR_degree': 4, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C3^2:Q8', 'ngens': 5, 'nilpotency_class': -1, 'nilpotent': False, 'normal_counts': [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': 14, 'number_conjugacy_classes': 27, 'number_divisions': 16, 'number_normal_subgroups': 18, 'number_subgroup_autclasses': 23, 'number_subgroup_classes': 27, 'number_subgroups': 42, 'old_label': None, 'order': 72, 'order_factorization_type': 32, 'order_stats': [[1, 1], [2, 1], [3, 8], [4, 14], [6, 8], [12, 40]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 2, 'outer_gen_orders': [2, 2, 2], 'outer_gen_pows': [18, 0, 0], 'outer_gens': [[19, 6], [5, 6], [1, 42]], 'outer_group': '8.5', 'outer_hash': 5, 'outer_nilpotent': True, 'outer_order': 8, 'outer_permdeg': 6, 'outer_perms': [1, 6, 120], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^3', 'pc_rank': 2, 'perfect': False, 'permutation_degree': 14, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 3], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 7], [4, 4], [8, 1]], 'representations': {'PC': {'code': 75140255264546281, 'gens': [1, 3], 'pres': [5, -2, -3, -2, -2, -3, 10, 186, 992, 42, 1203, 58, 1204]}, 'GLZ': {'b': 3, 'd': 6, 'gens': [41715828716014388, 58415888739750341]}, 'GLFp': {'d': 2, 'p': 13, 'gens': [15381, 6592, 221]}, 'Perm': {'d': 14, 'gens': [6314119039, 12593, 13971242880, 18619, 20122058880]}}, 'schur_multiplier': [], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 6], 'solvability_type': 6, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_3^2:Q_8', 'transitive_degree': 24, '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': '4.2', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 660, 'aut_gen_orders': [10, 10, 22, 30, 12], 'aut_gens': [[38116481125442430, 44734985446128011, 3867401643996858, 16407051026406918, 12415902285472309, 41772741070013040], [7570747234513837, 44734985446128011, 3867401643996858, 16407051026406918, 20381396155039640, 41772741070013040], [4428455046269754, 5809717384314194, 19369420335591231, 38314925911036591, 10787558627489347, 41772741070013040], [20298186770848793, 5809717384314194, 10842363402478390, 38314925911036591, 24013981109774853, 41772741070013040], [25318357896232412, 5809717384314194, 10842363402478390, 16407051026406918, 19782347835316445, 41772741070013040], [19274507415220467, 5809717384314194, 44087814079863762, 16407051026406918, 24947885402068161, 41772741070013040]], 'aut_group': None, 'aut_hash': 5245041372911580188, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 63360, 'aut_permdeg': 122, 'aut_perms': [38018668124936870644538841841801358389753544288735650037600304403684288164691108996152136774745765153018135729115120725929994482072807909909829247808153925325497576321372544407868853841850731544562418039, 94839013064012133968143008563247218538329212626697535603553481113016396865846749623278433527222277479764951972280977012834655798517171577543533112404779644984857588158868512560951834839139058215798487588, 40942852713954962431690116322698580763138019245796363811148686493058542599286665845286122203169589712823844442012974019501978317814427381980470965241881427843147823125573394316919138256402684641999960706, 45608143476680022603956517836714720710719920304790988427309738161281255767900864772590173465336313783917334048886346309401043329443690887448006194502046448369410605143177422381132275107231574834535358666, 33444315514519069623382808006564138322979893583116501862436789047328479702224724137025101695678185486431696069455652785338525203975770557229260365978388398882961682802562640465512618649210084696648176889], 'aut_phi_ratio': 16.5, 'aut_solvable': False, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 110, 1, 1], [2, 330, 2, 1], [3, 2, 1, 1], [3, 110, 1, 1], [3, 220, 1, 1], [4, 2, 1, 1], [4, 6, 2, 1], [4, 110, 1, 1], [5, 132, 1, 2], [6, 2, 1, 1], [6, 110, 1, 1], [6, 220, 1, 3], [6, 220, 2, 1], [6, 660, 2, 1], [10, 132, 1, 2], [11, 60, 2, 1], [12, 2, 2, 1], [12, 110, 1, 2], [12, 220, 1, 4], [12, 220, 2, 1], [12, 660, 2, 1], [15, 264, 1, 2], [20, 264, 1, 2], [20, 792, 2, 2], [22, 60, 2, 1], [30, 264, 1, 2], [33, 120, 2, 1], [44, 120, 2, 1], [44, 360, 4, 1], [60, 264, 2, 2], [66, 120, 2, 1], [132, 120, 4, 1]], 'aut_supersolvable': False, 'aut_tex': '(S_3\\times D_4).\\PSL(2,11).C_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': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 660, 'autcentquo_group': None, 'autcentquo_hash': 2842205202387851101, 'autcentquo_nilpotent': False, 'autcentquo_order': 15840, 'autcentquo_solvable': False, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'C_2\\times \\PSL(2,11).C_2\\times S_3', 'cc_stats': [[1, 1, 1], [2, 1, 1], [2, 110, 1], [2, 330, 2], [3, 2, 1], [3, 110, 1], [3, 220, 1], [4, 2, 1], [4, 6, 2], [4, 110, 1], [5, 132, 2], [6, 2, 1], [6, 110, 1], [6, 220, 5], [6, 660, 2], [10, 132, 2], [11, 60, 2], [12, 2, 2], [12, 110, 2], [12, 220, 6], [12, 660, 2], [15, 264, 2], [20, 264, 2], [20, 792, 4], [22, 60, 2], [30, 264, 2], [33, 120, 2], [44, 120, 2], [44, 360, 4], [60, 264, 4], [66, 120, 2], [132, 120, 4]], 'center_label': '2.1', 'center_order': 2, 'central_product': True, 'central_quotient': None, 'commutator_count': 1, 'commutator_label': '3960.a', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '3.1', '660.13'], 'composition_length': 5, 'conjugacy_classes_known': True, 'counter': 10, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 110, 1, 1], [2, 330, 1, 2], [3, 2, 1, 1], [3, 110, 1, 1], [3, 220, 1, 1], [4, 2, 1, 1], [4, 6, 1, 2], [4, 110, 1, 1], [5, 132, 2, 1], [6, 2, 1, 1], [6, 110, 1, 1], [6, 220, 1, 5], [6, 660, 1, 2], [10, 132, 2, 1], [11, 60, 2, 1], [12, 2, 2, 1], [12, 110, 2, 1], [12, 220, 1, 2], [12, 220, 2, 2], [12, 660, 1, 2], [15, 264, 2, 1], [20, 264, 2, 1], [20, 792, 2, 2], [22, 60, 2, 1], [30, 264, 2, 1], [33, 120, 2, 1], [44, 120, 2, 1], [44, 360, 2, 2], [60, 264, 4, 1], [66, 120, 2, 1], [132, 120, 4, 1]], 'element_repr_type': 'GLFp', 'elementary': 1, 'eulerian_function': 762, 'exponent': 660, 'exponents_of_order': [5, 2, 1, 1], 'factors_of_aut_order': [2, 3, 5, 11], 'factors_of_order': [2, 3, 5, 11], 'faithful_reps': [[12, 0, 4], [20, 1, 6], [24, 1, 4]], 'familial': False, 'frattini_label': '2.1', 'frattini_quotient': None, 'hash': 6544422013302709301, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 330, 'inner_gen_orders': [6, 2, 2, 3, 6, 1], 'inner_gens': [[38116481125442430, 44734985446128011, 5732743741830854, 16407051026406918, 28934439925485275, 41772741070013040], [38116481125442430, 44734985446128011, 692737359847830, 16407051026406918, 38125662136079015, 41772741070013040], [18542802577580899, 5809717384314194, 3867401643996858, 38314925911036591, 23577295682729774, 41772741070013040], [38116481125442430, 44734985446128011, 6422340100995270, 16407051026406918, 23577295682729774, 41772741070013040], [20298186770848793, 5809717384314194, 44777410439028178, 38314925911036591, 12415902285472309, 41772741070013040], [38116481125442430, 44734985446128011, 3867401643996858, 16407051026406918, 12415902285472309, 41772741070013040]], 'inner_hash': 3267698946426799616, 'inner_nilpotent': False, 'inner_order': 7920, 'inner_split': True, 'inner_tex': 'D_6.\\PSL(2,11)', 'inner_used': [1, 2, 3, 5], 'irrC_degree': 12, 'irrQ_degree': 40, 'irrQ_dim': 40, 'irrR_degree': 20, 'irrep_stats': [[1, 4], [2, 2], [5, 8], [10, 12], [11, 4], [12, 14], [20, 13], [22, 2], [24, 10]], 'label': '15840.j', 'linC_count': 4, 'linC_degree': 12, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 22, 'linQ_degree_count': 2, 'linQ_dim': 22, 'linQ_dim_count': 2, 'linR_count': 6, 'linR_degree': 20, 'maximal_subgroups_known': True, 'metabelian': False, 'metacyclic': False, 'monomial': False, 'name': 'SL(2,11):D6', '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, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 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': 47, 'number_characteristic_subgroups': 12, 'number_conjugacy_classes': 69, 'number_divisions': 45, 'number_normal_subgroups': 16, 'number_subgroup_autclasses': 222, 'number_subgroup_classes': 369, 'number_subgroups': 52694, 'old_label': None, 'order': 15840, 'order_factorization_type': 321, 'order_stats': [[1, 1], [2, 771], [3, 332], [4, 124], [5, 264], [6, 2532], [10, 264], [11, 120], [12, 2864], [15, 528], [20, 3696], [22, 120], [30, 528], [33, 240], [44, 1680], [60, 1056], [66, 240], [132, 480]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 2, 'outer_gen_orders': [2, 2, 2], 'outer_gen_pows': [44734985446128011, 24041890071333445, 4177274107001304], 'outer_gens': [[38116481125442430, 44734985446128011, 19369420335591231, 16407051026406918, 10010015578310341, 41772741070013040], [30762498942946520, 44734985446128011, 3867401643996858, 16407051026406918, 26666891503484643, 41772741070013040], [32001900272348356, 44734985446128011, 3867401643996858, 38314925911036591, 23577295682729774, 41772741070013040]], 'outer_group': '8.5', 'outer_hash': 5, 'outer_nilpotent': True, 'outer_order': 8, 'outer_permdeg': 6, 'outer_perms': [1, 6, 120], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^3', 'pc_rank': None, 'perfect': False, 'permutation_degree': 99, 'pgroup': 0, 'primary_abelian_invariants': [2, 2], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 2], [10, 12], [11, 4], [20, 7], [22, 2], [24, 5], [40, 4], [48, 4], [96, 1]], 'representations': {'GLFp': {'d': 4, 'p': 11, 'gens': [38116481125442430, 44734985446128011, 3867401643996858, 16407051026406918, 12415902285472309, 41772741070013040]}, 'Perm': {'d': 99, 'gens': [9912865607985640210688520757425304283492982083754023567244735069094478857986069652355961642120309514402261393361406239247541186341642226021584966589264346, 19433004882673905101903528632851543957025990188019212483564743183433504399685088339219746870212422516281845835140143925221811805398164200134150728425248245, 28962946864976286730148801058775127556870198413268303696586292272706905783613929772171297175218599800721072690217053340462549742494521723898726280023237787, 38487134623566855944630839265132729035740675974226625717699498233743448741707086799521405066833142785982112928948003231442553999187490051950933893989851144, 48011221021123261975104790839429425162880081167135078852282277906293783291735306514905760462018307555396052604395609809103647008256905353983850263490295364, 28962946864976286730148801058775127556870198413268303696586292272706905783613929772171297175218599800721072690217053340462549742494521723898726280023237784]}}, 'schur_multiplier': [2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2], 'solvability_type': 13, 'solvable': False, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': '\\SL(2,11):D_6', 'transitive_degree': 288, 'wreath_data': None, 'wreath_product': False}
