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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': False, 'ambient': '600.178', 'ambient_counter': 178, 'ambient_order': 600, 'ambient_tex': 'C_5^2:S_4', 'central': False, 'central_factor': False, 'centralizer_order': 25, 'characteristic': False, 'core_order': 60, 'counter': 20, 'cyclic': False, 'direct': False, 'hall': 0, 'label': '600.178.10.a1.f1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': False, 'minimal': False, 'minimal_normal': False, 'nilpotent': False, 'normal': True, 'old_label': '10.a1.f1', 'outer_equivalence': False, 'perfect': False, 'proper': True, 'quotient': '10.1', 'quotient_Agroup': True, 'quotient_abelian': False, 'quotient_cyclic': False, 'quotient_hash': 1, 'quotient_metabelian': True, 'quotient_nilpotent': False, 'quotient_order': 10, 'quotient_simple': False, 'quotient_solvable': True, 'quotient_supersolvable': True, 'quotient_tex': 'D_5', 'simple': False, 'solvable': True, 'special_labels': [], 'split': True, 'standard_generators': False, 'stem': False, 'subgroup': '60.9', 'subgroup_hash': 9, 'subgroup_order': 60, 'subgroup_tex': 'C_5\\times A_4', 'supersolvable': False, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '600.178', 'aut_centralizer_order': 500, 'aut_label': '10.a1', 'aut_quo_index': 1, 'aut_stab_index': 6, 'aut_weyl_group': '96.186', 'aut_weyl_index': 3000, 'centralizer': '24.a1.a1', 'complements': ['60.b1.a1', '60.b1.b1', '60.b1.d1', '60.b1.e1', '60.b1.c1'], 'conjugacy_class_count': 1, 'contained_in': ['2.a1.a1', '5.a1.f1'], 'contains': ['30.a1.f1', '40.a1.f1', '50.a1.a1'], 'core': '10.a1.f1', 'coset_action_label': None, 'count': 1, 'diagramx': [3589, 2225, 1180, 6308, 3497, 2867, 5958, 4900], 'generators': [2, 330, 300, 492], 'label': '600.178.10.a1.f1', 'mobius_quo': 0, 'mobius_sub': 5, 'normal_closure': '10.a1.f1', 'normal_contained_in': ['2.a1.a1'], 'normal_contains': ['30.a1.f1', '50.a1.a1'], 'normalizer': '1.a1.a1', 'old_label': '10.a1.f1', 'projective_image': '600.178', 'quotient_action_image': '2.1', 'quotient_action_kernel': '5.1', 'quotient_action_kernel_order': 5, 'quotient_fusion': None, 'short_label': '10.a1.f1', 'subgroup_fusion': None, 'weyl_group': '24.12'}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '15.1', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 3, 'aut_exponent': 12, 'aut_gen_orders': [2, 2, 4, 4, 2, 6], 'aut_gens': [[1, 3, 6], [1, 3, 54], [34, 3, 54], [4, 3, 18], [1, 3, 42], [35, 33, 54], [31, 30, 57]], 'aut_group': '96.186', 'aut_hash': 186, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 96, 'aut_permdeg': 8, 'aut_perms': [16, 5176, 16689, 18, 15960, 10936], 'aut_phi_ratio': 6.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 3, 1, 1], [3, 4, 2, 1], [5, 1, 4, 1], [10, 3, 4, 1], [15, 4, 8, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_4\\times S_4', 'autcent_abelian': True, 'autcent_cyclic': True, 'autcent_exponent': 4, 'autcent_group': '4.1', 'autcent_hash': 1, 'autcent_nilpotent': True, 'autcent_order': 4, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_4', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 12, 'autcentquo_group': '24.12', 'autcentquo_hash': 12, 'autcentquo_nilpotent': False, 'autcentquo_order': 24, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'S_4', 'cc_stats': [[1, 1, 1], [2, 3, 1], [3, 4, 2], [5, 1, 4], [10, 3, 4], [15, 4, 8]], 'center_label': '5.1', 'center_order': 5, 'central_product': True, 'central_quotient': '12.3', 'commutator_count': 1, 'commutator_label': '4.2', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '3.1', '5.1'], 'composition_length': 4, 'conjugacy_classes_known': True, 'counter': 9, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['12.3', 1], ['5.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 3, 1, 1], [3, 4, 2, 1], [5, 1, 4, 1], [10, 3, 4, 1], [15, 4, 8, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 24, 'exponent': 30, 'exponents_of_order': [2, 1, 1], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3, 5], 'faithful_reps': [[3, 0, 4]], 'familial': False, 'frattini_label': '1.1', 'frattini_quotient': '60.9', 'hash': 9, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 6, 'inner_gen_orders': [3, 2, 2], 'inner_gens': [[1, 30, 9], [34, 3, 6], [4, 3, 6]], 'inner_hash': 3, 'inner_nilpotent': False, 'inner_order': 12, 'inner_split': True, 'inner_tex': 'A_4', 'inner_used': [1, 2], 'irrC_degree': 3, 'irrQ_degree': 12, 'irrQ_dim': 12, 'irrR_degree': 6, 'irrep_stats': [[1, 15], [3, 5]], 'label': '60.9', 'linC_count': 4, 'linC_degree': 3, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 7, 'linQ_degree_count': 1, 'linQ_dim': 7, 'linQ_dim_count': 1, 'linR_count': 6, 'linR_degree': 5, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'C5*A4', 'ngens': 4, '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': 6, 'number_characteristic_subgroups': 6, 'number_conjugacy_classes': 20, 'number_divisions': 6, 'number_normal_subgroups': 6, 'number_subgroup_autclasses': 10, 'number_subgroup_classes': 10, 'number_subgroups': 20, 'old_label': None, 'order': 60, 'order_factorization_type': 222, 'order_stats': [[1, 1], [2, 3], [3, 8], [5, 4], [10, 12], [15, 32]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 4, 'outer_gen_orders': [2, 4], 'outer_gen_pows': [0, 0], 'outer_gens': [[2, 30, 27], [1, 3, 18]], 'outer_group': '8.2', 'outer_hash': 2, 'outer_nilpotent': True, 'outer_order': 8, 'outer_permdeg': 6, 'outer_perms': [120, 9], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2\\times C_4', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 9, 'pgroup': 0, 'primary_abelian_invariants': [3, 5], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 1], [2, 1], [3, 1], [4, 1], [8, 1], [12, 1]], 'representations': {'PC': {'code': 4852423747, 'gens': [1, 2, 3], 'pres': [4, -3, -2, 2, -5, 241, 110, 34]}, 'GLFp': {'d': 3, 'p': 11, 'gens': [656967451, 1691722594, 857494092, 1907820797]}, 'Perm': {'d': 9, 'gens': [5760, 33, 41040, 90720]}}, 'schur_multiplier': [2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [15], 'solvability_type': 8, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': 'C_5\\times A_4', 'transitive_degree': 20, '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': '2.1', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 3, 'aut_exponent': 120, 'aut_gen_orders': [5, 2, 3, 5, 2, 4, 2], 'aut_gens': [[1, 2, 6, 60], [493, 2, 6, 60], [3, 4, 6, 90], [157, 2, 174, 312], [505, 2, 6, 60], [1, 302, 6, 60], [385, 2, 390, 228], [1, 34, 336, 390]], 'aut_group': '288000.i', 'aut_hash': 9054531536665690799, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 288000, 'aut_permdeg': 150, 'aut_perms': [7288427423545544419359782463218305298427908409657834052743479159721332122514073682641872231669609188271609939129842139198638649655801822242936500881555902397822074989755310489924676254024200810023896243872367463486708419442729122632360080563414853546989230134444, 46516980889861091459522595979487926777005791395725302526716364628667030823755590679429758403276252290603303051986568974919806230298241355243057176339884559074729210054619999319123557441930454478546900446886315238003878628324080984015996497914930681675276757979780, 52788432408953306100877706364260732991270478425194569153748008940470939900763269513294760980613255576776454686626956124971725929739846325975264862878332725435893909668645663164771292609072524272777113304956202522101448145653256968083774275834185323548873214202726, 25167206777475741077570155549316665096971473341684862016382740531542057599084440571685923535065601714519866762003740717368068241935447814798481237875140002047520632625204119898202022745603068205944462308152410103183333949953824451919204105428144761845865519090409, 25566987720023625097239041891271863083530712060592224362670923752220879216787409318532933245006740439169377325138079748922059568903995811795761448685132796763095526600779279453496762291145514388301810813974111300335026802943553050867680365722234744985175540082291, 51482403005261233702019627273804287014928962868016797884352032206112639340519739085390297649126095136102393538282956559572468877567910066747960821208622950834096485873103015205470675840275287939915804148842902175584941382700737395964917113349839185375846632942, 52977112306402066928456538086872100992395156070857818222966466682488522843619367187605254399769354643715945356948528222965994011034923708427936911072957898858544019957882307492046169731376278289446619668699957117311261517720144619564147980644924030902837054079936], 'aut_phi_ratio': 1800.0, 'aut_solvable': False, 'aut_stats': [[1, 1, 1, 1], [2, 3, 1, 1], [2, 150, 1, 1], [3, 8, 1, 1], [4, 150, 1, 1], [5, 2, 12, 1], [10, 6, 12, 1], [15, 8, 24, 1]], 'aut_supersolvable': False, 'aut_tex': '(S_4\\times C_5^2):\\GL(2,5)', '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': 120, 'autcentquo_group': '288000.i', 'autcentquo_hash': 9054531536665690799, 'autcentquo_nilpotent': False, 'autcentquo_order': 288000, 'autcentquo_solvable': False, 'autcentquo_supersolvable': False, 'autcentquo_tex': '(S_4\\times C_5^2):\\GL(2,5)', 'cc_stats': [[1, 1, 1], [2, 3, 1], [2, 150, 1], [3, 8, 1], [4, 150, 1], [5, 2, 12], [10, 6, 12], [15, 8, 24]], 'center_label': '1.1', 'center_order': 1, 'central_product': False, 'central_quotient': '600.178', 'commutator_count': 1, 'commutator_label': '300.42', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '3.1', '5.1', '5.1'], 'composition_length': 6, 'conjugacy_classes_known': True, 'counter': 178, 'cyclic': False, 'derived_length': 3, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 3, 1, 1], [2, 150, 1, 1], [3, 8, 1, 1], [4, 150, 1, 1], [5, 2, 2, 6], [10, 6, 2, 6], [15, 8, 4, 6]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 420, 'exponent': 60, 'exponents_of_order': [3, 2, 1], 'factors_of_aut_order': [2, 3, 5], 'factors_of_order': [2, 3, 5], 'faithful_reps': [], 'familial': False, 'frattini_label': '1.1', 'frattini_quotient': '600.178', 'hash': 178, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 60, 'inner_gen_orders': [2, 3, 10, 10], 'inner_gens': [[1, 4, 324, 270], [5, 2, 336, 90], [343, 332, 6, 60], [451, 32, 6, 60]], 'inner_hash': 178, 'inner_nilpotent': False, 'inner_order': 600, 'inner_split': True, 'inner_tex': 'C_5^2:S_4', 'inner_used': [1, 2, 3, 4], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 2], [2, 37], [3, 2], [6, 12]], 'label': '600.178', 'linC_count': 1080, 'linC_degree': 7, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 11, 'linQ_degree_count': 30, 'linQ_dim': 11, 'linQ_dim_count': 30, 'linR_count': 1080, 'linR_degree': 7, 'maximal_subgroups_known': True, 'metabelian': False, 'metacyclic': False, 'monomial': True, 'name': 'C5^2:S4', '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': 8, 'number_characteristic_subgroups': 7, 'number_conjugacy_classes': 53, 'number_divisions': 23, 'number_normal_subgroups': 25, 'number_subgroup_autclasses': 33, 'number_subgroup_classes': 88, 'number_subgroups': 1200, 'old_label': None, 'order': 600, 'order_factorization_type': 321, 'order_stats': [[1, 1], [2, 153], [3, 8], [4, 150], [5, 24], [10, 72], [15, 192]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 120, 'outer_gen_orders': [4, 3], 'outer_gen_pows': [0, 1], 'outer_gens': [[1, 2, 390, 228], [1, 4, 552, 258]], 'outer_group': '480.218', 'outer_hash': 218, 'outer_nilpotent': False, 'outer_order': 480, 'outer_permdeg': 24, 'outer_perms': [100394386315720751646004, 338763832937352652069381], 'outer_solvable': False, 'outer_supersolvable': False, 'outer_tex': '\\GL(2,5)', 'pc_rank': 4, 'perfect': False, 'permutation_degree': 14, 'pgroup': 0, 'primary_abelian_invariants': [2], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 2], [2, 1], [3, 2], [4, 6], [8, 6], [12, 6]], 'representations': {'PC': {'code': 5058985512374097289171958323167539, 'gens': [1, 2, 3, 5], 'pres': [6, 2, 3, 2, 5, 2, 5, 49, 5834, 3032, 50, 1155, 8104, 1360, 88, 17285]}, 'GLZN': {'d': 2, 'p': 100, 'gens': [81208021, 1002001, 51500051, 46057754, 1505001, 51752576]}, 'Perm': {'d': 14, 'gens': [482635562, 3, 7188652800, 7188703320, 7, 16]}}, 'schur_multiplier': [10], '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_5^2:S_4', 'transitive_degree': 100, 'wreath_data': None, 'wreath_product': False}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': False, 'abelian_quotient': '2.1', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 20, 'aut_gen_orders': [4, 5], 'aut_gens': [[1, 2], [3, 6], [5, 2]], 'aut_group': '20.3', 'aut_hash': 3, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 20, 'aut_permdeg': 5, 'aut_perms': [17, 89], 'aut_phi_ratio': 5.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 5, 1, 1], [5, 2, 2, 1]], 'aut_supersolvable': True, 'aut_tex': 'F_5', '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': 20, 'autcentquo_group': '20.3', 'autcentquo_hash': 3, 'autcentquo_nilpotent': False, 'autcentquo_order': 20, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'F_5', 'cc_stats': [[1, 1, 1], [2, 5, 1], [5, 2, 2]], 'center_label': '1.1', 'center_order': 1, 'central_product': False, 'central_quotient': '10.1', 'commutator_count': 1, 'commutator_label': '5.1', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '5.1'], 'composition_length': 2, 'conjugacy_classes_known': True, 'counter': 1, 'cyclic': False, 'derived_length': 2, 'dihedral': True, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 5, 1, 1], [5, 2, 2, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 3, 'exponent': 10, 'exponents_of_order': [1, 1], 'factors_of_aut_order': [2, 5], 'factors_of_order': [2, 5], 'faithful_reps': [[2, 1, 2]], 'familial': True, 'frattini_label': '1.1', 'frattini_quotient': '10.1', 'hash': 1, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 10, 'inner_gen_orders': [2, 5], 'inner_gens': [[1, 8], [5, 2]], 'inner_hash': 1, 'inner_nilpotent': False, 'inner_order': 10, 'inner_split': False, 'inner_tex': 'D_5', 'inner_used': [1, 2], 'irrC_degree': 2, 'irrQ_degree': 4, 'irrQ_dim': 4, 'irrR_degree': 2, 'irrep_stats': [[1, 2], [2, 2]], 'label': '10.1', 'linC_count': 2, 'linC_degree': 2, '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': 'D5', 'ngens': 2, 'nilpotency_class': -1, 'nilpotent': False, 'normal_counts': [0, 0, 0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 3, 'number_characteristic_subgroups': 3, 'number_conjugacy_classes': 4, 'number_divisions': 3, 'number_normal_subgroups': 3, 'number_subgroup_autclasses': 4, 'number_subgroup_classes': 4, 'number_subgroups': 8, 'old_label': None, 'order': 10, 'order_factorization_type': 11, 'order_stats': [[1, 1], [2, 5], [5, 4]], 'outer_abelian': True, 'outer_cyclic': True, 'outer_equivalence': False, 'outer_exponent': 2, 'outer_gen_orders': [2], 'outer_gen_pows': [1], 'outer_gens': [[1, 6]], 'outer_group': '2.1', 'outer_hash': 1, 'outer_nilpotent': True, 'outer_order': 2, 'outer_permdeg': 2, 'outer_perms': [1], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2', 'pc_rank': 2, 'perfect': False, 'permutation_degree': 5, 'pgroup': 0, 'primary_abelian_invariants': [2], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 2], [4, 1]], 'representations': {'PC': {'code': 227, 'gens': [1, 2], 'pres': [2, -2, -5, 33]}, 'GLZ': {'b': 3, 'd': 4, 'gens': [23707240, 26329372]}, 'Lie': [{'d': 2, 'q': 4, 'gens': [81, 94], 'family': 'CSOMinus'}], 'GLFp': {'d': 2, 'p': 5, 'gens': [131, 129]}, 'Perm': {'d': 5, 'gens': [23, 96]}}, 'schur_multiplier': [], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2], 'solvability_type': 6, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'D_5', 'transitive_degree': 5, 'wreath_data': None, 'wreath_product': False}
