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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': False, 'ambient': '13200.f', 'ambient_counter': 6, 'ambient_order': 13200, 'ambient_tex': '\\SL(2,11):C_{10}', 'central': False, 'central_factor': False, 'centralizer_order': 20, 'characteristic': False, 'core_order': 20, 'counter': 7, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '13200.f.12.a1.a1', 'maximal': True, 'maximal_normal': False, 'metabelian': True, 'metacyclic': None, 'minimal': False, 'minimal_normal': False, 'nilpotent': False, 'normal': False, 'old_label': '12.a1.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': 12, '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': '1100.19', 'subgroup_hash': 19, 'subgroup_order': 1100, 'subgroup_tex': 'C_{220}:C_5', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '13200.f', 'aut_centralizer_order': None, 'aut_label': '12.a1', 'aut_quo_index': None, 'aut_stab_index': None, 'aut_weyl_group': None, 'aut_weyl_index': None, 'centralizer': '660.a1.a1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['1.a1.a1'], 'contains': ['24.a1.a1', '60.a1.a1', '60.b1.a1', '60.c1.a1', '60.c1.a2', '60.c1.b1', '60.c1.b2', '132.b1.a1'], 'core': '660.a1.a1', 'coset_action_label': None, 'count': 12, 'diagramx': [1013, -1, 8158, -1, 1726, -1, 1626, -1], 'generators': [29060178607192161, 16709096428005216, 1204643848318542, 29240918749009128, 11619123519371993], 'label': '13200.f.12.a1.a1', 'mobius_quo': None, 'mobius_sub': -1, 'normal_closure': '1.a1.a1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '12.a1.a1', 'old_label': '12.a1.a1', 'projective_image': '660.13', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '12.a1.a1', 'subgroup_fusion': None, 'weyl_group': '55.1'}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '100.8', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 220, 'aut_gen_orders': [4, 10, 110], 'aut_gens': [[1, 5], [221, 985], [1, 505], [121, 555]], 'aut_group': '4400.r', 'aut_hash': 3690388632023574097, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 4400, 'aut_permdeg': 57, 'aut_perms': [552177193327368596654017085942357817688391572855886588656427788613773164687, 331744024647865077884773496468222514952908633936920622364143262719468981721, 33453142282050453399316860010159861191778300937354515526373447781095490124912], 'aut_phi_ratio': 11.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [4, 1, 2, 1], [5, 1, 4, 1], [5, 11, 5, 4], [10, 1, 4, 1], [10, 11, 5, 4], [11, 5, 2, 1], [20, 1, 8, 1], [20, 11, 10, 4], [22, 5, 2, 1], [44, 5, 4, 1], [55, 5, 8, 1], [110, 5, 8, 1], [220, 5, 16, 1]], 'aut_supersolvable': True, 'aut_tex': 'D_{110}:C_{20}', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 20, 'autcent_group': '40.12', 'autcent_hash': 12, 'autcent_nilpotent': False, 'autcent_order': 40, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2\\times F_5', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 110, 'autcentquo_group': '110.1', 'autcentquo_hash': 1, 'autcentquo_nilpotent': False, 'autcentquo_order': 110, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'F_{11}', 'cc_stats': [[1, 1, 1], [2, 1, 1], [4, 1, 2], [5, 1, 4], [5, 11, 20], [10, 1, 4], [10, 11, 20], [11, 5, 2], [20, 1, 8], [20, 11, 40], [22, 5, 2], [44, 5, 4], [55, 5, 8], [110, 5, 8], [220, 5, 16]], 'center_label': '20.2', 'center_order': 20, 'central_product': True, 'central_quotient': '55.1', 'commutator_count': 1, 'commutator_label': '11.1', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '5.1', '5.1', '11.1'], 'composition_length': 5, 'conjugacy_classes_known': True, 'counter': 19, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['4.1', 1], ['5.1', 1], ['55.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [4, 1, 2, 1], [5, 1, 4, 1], [5, 11, 4, 5], [10, 1, 4, 1], [10, 11, 4, 5], [11, 5, 2, 1], [20, 1, 8, 1], [20, 11, 8, 5], [22, 5, 2, 1], [44, 5, 4, 1], [55, 5, 8, 1], [110, 5, 8, 1], [220, 5, 16, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 144, 'exponent': 220, 'exponents_of_order': [2, 2, 1], 'factors_of_aut_order': [2, 5, 11], 'factors_of_order': [2, 5, 11], 'faithful_reps': [[5, 0, 16]], 'familial': False, 'frattini_label': '2.1', 'frattini_quotient': '550.10', 'hash': 19, 'hyperelementary': 5, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 55, 'inner_gen_orders': [5, 11], 'inner_gens': [[1, 405], [701, 5]], 'inner_hash': 1, 'inner_nilpotent': False, 'inner_order': 55, 'inner_split': True, 'inner_tex': 'C_{11}:C_5', 'inner_used': [1, 2], 'irrC_degree': 5, 'irrQ_degree': 80, 'irrQ_dim': 80, 'irrR_degree': 10, 'irrep_stats': [[1, 100], [5, 40]], 'label': '1100.19', 'linC_count': 16, 'linC_degree': 5, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 16, 'linQ_degree_count': 20, 'linQ_dim': 16, 'linQ_dim_count': 20, 'linR_count': 8, 'linR_degree': 10, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C220:C5', 'ngens': 5, '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': 24, 'number_characteristic_subgroups': 15, 'number_conjugacy_classes': 140, 'number_divisions': 27, 'number_normal_subgroups': 30, 'number_subgroup_autclasses': 24, 'number_subgroup_classes': 48, 'number_subgroups': 228, 'old_label': None, 'order': 1100, 'order_factorization_type': 222, 'order_stats': [[1, 1], [2, 1], [4, 2], [5, 224], [10, 224], [11, 10], [20, 448], [22, 10], [44, 20], [55, 40], [110, 40], [220, 80]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 20, 'outer_gen_orders': [2, 4, 10], 'outer_gen_pows': [0, 0, 0], 'outer_gens': [[1, 655], [441, 765], [221, 305]], 'outer_group': '80.50', 'outer_hash': 50, 'outer_nilpotent': False, 'outer_order': 80, 'outer_permdeg': 9, 'outer_perms': [7, 5896, 122536], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^2\\times F_5', 'pc_rank': 2, 'perfect': False, 'permutation_degree': 20, 'pgroup': 0, 'primary_abelian_invariants': [4, 5, 5], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 2], [2, 1], [4, 12], [8, 6], [10, 2], [20, 1], [40, 2], [80, 1]], 'representations': {'PC': {'code': 8720789124096363978777430039, 'gens': [1, 2], 'pres': [5, -5, -2, -2, -5, -11, 4051, 26, 12152, 42, 10403, 118, 10004]}, 'GLFq': {'d': 2, 'q': 121, 'gens': [15680518, 143403964, 15944058, 83263414, 17715620]}, 'Perm': {'d': 20, 'gens': [9, 7159866360647160, 96480, 16, 135229490543347200]}}, 'schur_multiplier': [5], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [5, 20], 'solvability_type': 6, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_{220}:C_5', 'transitive_degree': 220, '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': '10.2', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 1, 'aut_exponent': 660, 'aut_gen_orders': [5, 20, 20, 4], 'aut_gens': [[12531822321003912, 41464684206115003, 11619123519371993, 33418192856010432, 22490751747208555], [12531822321003912, 43327743730865604, 11619123519371993, 33418192856010432, 24624238455994713], [20886370535006520, 23200846966214846, 37293136249742816, 25063644642007824, 17300916687584358], [37595466963011736, 38976194801611340, 18646723749506926, 8354548214002608, 3087320187979837], [20886370535006520, 20714951285005102, 13251566580699389, 25063644642007824, 5763220669462131]], 'aut_group': None, 'aut_hash': 659480313428745573, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 10560, 'aut_permdeg': 116, 'aut_perms': [3194156700897740723371623325499630721483029152536576679269536160855311435948781177751293215593778397817915614467694402947588820374178690783219246245043190110413878921181288757528, 22511156894193493512373275951646814407570921144361069332521345757554882481391276055393668162482232814870237396184862316807130448926000452982360547646981034406870322875368839168249747710, 292576305144496382309800584797861240913342167225904624598864478109046156085044383471768191451240477958430937725428228962568694743494053336096027763717431143116685164906607825720614228852110, 292531880498102255736608290879343104776737516614317705351525408468667006947841236793759571515728009862412437663240965826863597034392505786297940499442922506608984872941015103389040115403379], 'aut_phi_ratio': 3.3, 'aut_solvable': False, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 110, 1, 1], [3, 110, 1, 1], [4, 1, 2, 1], [4, 110, 1, 1], [5, 1, 4, 1], [5, 132, 1, 2], [5, 132, 4, 2], [6, 110, 1, 1], [6, 110, 2, 1], [10, 1, 4, 1], [10, 110, 4, 1], [10, 132, 1, 2], [10, 132, 4, 2], [11, 60, 2, 1], [12, 110, 1, 2], [12, 110, 2, 1], [15, 110, 4, 1], [20, 1, 8, 1], [20, 110, 4, 1], [20, 132, 2, 2], [20, 132, 8, 2], [22, 60, 2, 1], [30, 110, 4, 1], [30, 110, 8, 1], [44, 60, 4, 1], [55, 60, 8, 1], [60, 110, 4, 2], [60, 110, 8, 1], [110, 60, 8, 1], [220, 60, 16, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_2\\times C_4\\times \\PSL(2,11).C_2', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 4, 'autcent_group': '8.2', 'autcent_hash': 2, 'autcent_nilpotent': True, 'autcent_order': 8, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2\\times C_4', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 660, 'autcentquo_group': '1320.133', 'autcentquo_hash': 133, 'autcentquo_nilpotent': False, 'autcentquo_order': 1320, 'autcentquo_solvable': False, 'autcentquo_supersolvable': False, 'autcentquo_tex': '\\PGL(2,11)', 'cc_stats': [[1, 1, 1], [2, 1, 1], [2, 110, 1], [3, 110, 1], [4, 1, 2], [4, 110, 1], [5, 1, 4], [5, 132, 10], [6, 110, 3], [10, 1, 4], [10, 110, 4], [10, 132, 10], [11, 60, 2], [12, 110, 4], [15, 110, 4], [20, 1, 8], [20, 110, 4], [20, 132, 20], [22, 60, 2], [30, 110, 12], [44, 60, 4], [55, 60, 8], [60, 110, 16], [110, 60, 8], [220, 60, 16]], 'center_label': '20.2', 'center_order': 20, 'central_product': True, 'central_quotient': '660.13', 'commutator_count': 1, 'commutator_label': '1320.13', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '5.1', '660.13'], 'composition_length': 4, 'conjugacy_classes_known': True, 'counter': 6, 'cyclic': False, 'derived_length': 1, 'dihedral': False, 'direct_factorization': [['2640.l', 1], ['5.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 110, 1, 1], [3, 110, 1, 1], [4, 1, 2, 1], [4, 110, 1, 1], [5, 1, 4, 1], [5, 132, 2, 1], [5, 132, 4, 2], [6, 110, 1, 1], [6, 110, 2, 1], [10, 1, 4, 1], [10, 110, 4, 1], [10, 132, 2, 1], [10, 132, 4, 2], [11, 60, 2, 1], [12, 110, 2, 2], [15, 110, 4, 1], [20, 1, 8, 1], [20, 110, 4, 1], [20, 132, 4, 1], [20, 132, 8, 2], [22, 60, 2, 1], [30, 110, 4, 1], [30, 110, 8, 1], [44, 60, 4, 1], [55, 60, 8, 1], [60, 110, 8, 2], [110, 60, 8, 1], [220, 60, 16, 1]], 'element_repr_type': 'GLFp', 'elementary': 1, 'eulerian_function': 9144, 'exponent': 660, 'exponents_of_order': [4, 2, 1, 1], 'factors_of_aut_order': [2, 3, 5, 11], 'factors_of_order': [2, 3, 5, 11], 'faithful_reps': [[6, 0, 16], [10, 0, 24], [12, 0, 16]], 'familial': False, 'frattini_label': '2.1', 'frattini_quotient': None, 'hash': 2506923444263895416, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 330, 'inner_gen_orders': [1, 2, 1, 1, 3], 'inner_gens': [[12531822321003912, 41464684206115003, 11619123519371993, 33418192856010432, 22490751747208555], [12531822321003912, 41464684206115003, 11619123519371993, 33418192856010432, 9597536389346543], [12531822321003912, 41464684206115003, 11619123519371993, 33418192856010432, 22490751747208555], [12531822321003912, 41464684206115003, 11619123519371993, 33418192856010432, 22490751747208555], [12531822321003912, 6492813895957181, 11619123519371993, 33418192856010432, 22490751747208555]], 'inner_hash': 13, 'inner_nilpotent': False, 'inner_order': 660, 'inner_split': True, 'inner_tex': '\\PSL(2,11)', 'inner_used': [2, 5], 'irrC_degree': 6, 'irrQ_degree': 80, 'irrQ_dim': 80, 'irrR_degree': 12, 'irrep_stats': [[1, 10], [5, 20], [6, 20], [10, 50], [11, 10], [12, 40]], 'label': '13200.f', 'linC_count': 16, 'linC_degree': 6, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 24, 'linQ_degree_count': 2, 'linQ_dim': 24, 'linQ_dim_count': 2, 'linR_count': 8, 'linR_degree': 12, 'maximal_subgroups_known': True, 'metabelian': False, 'metacyclic': False, 'monomial': False, 'name': 'SL(2,11):C10', 'ngens': 5, '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], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 40, 'number_characteristic_subgroups': 10, 'number_conjugacy_classes': 150, 'number_divisions': 35, 'number_normal_subgroups': 10, 'number_subgroup_autclasses': 94, 'number_subgroup_classes': 122, 'number_subgroups': 6084, 'old_label': None, 'order': 13200, 'order_factorization_type': 321, 'order_stats': [[1, 1], [2, 111], [3, 110], [4, 112], [5, 1324], [6, 330], [10, 1764], [11, 120], [12, 440], [15, 440], [20, 3088], [22, 120], [30, 1320], [44, 240], [55, 480], [60, 1760], [110, 480], [220, 960]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 4, 'outer_gen_orders': [2, 2, 4], 'outer_gen_pows': [4177274107001304, 10357425199337165, 4177274107001304], 'outer_gens': [[12531822321003912, 9045469760383817, 38925579311070212, 33418192856010432, 22490751747208555], [12531822321003912, 44017340070541639, 11619123519371993, 33418192856010432, 42532629880179343], [20886370535006520, 6802945752681387, 37293136249742816, 25063644642007824, 39800857355970325]], 'outer_group': '16.10', 'outer_hash': 10, 'outer_nilpotent': True, 'outer_order': 16, 'outer_permdeg': 8, 'outer_perms': [5040, 120, 9], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^2\\times C_4', 'pc_rank': None, 'perfect': False, 'permutation_degree': 53, 'pgroup': 0, 'primary_abelian_invariants': [2, 5], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 2], [4, 2], [10, 6], [11, 2], [20, 1], [24, 3], [40, 7], [44, 2], [48, 5], [80, 1], [96, 3], [160, 1]], 'representations': {'GLFp': {'d': 4, 'p': 11, 'gens': [12531822321003912, 41464684206115003, 11619123519371993, 33418192856010432, 22490751747208555]}, 'Perm': {'d': 53, 'gens': [82304843882576109505520288306399373534789183417785172520939754165793, 33, 167523928873291142461325549870881476168953888619050054297545846272064, 249917571529954138829421813737149937941237191291150576021731579579584, 332157813609048095638643826871184318824766784793775515639033756403793]}}, 'schur_multiplier': [], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [10], 'solvability_type': 13, 'solvable': False, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': '\\SL(2,11):C_{10}', 'transitive_degree': 240, 'wreath_data': None, 'wreath_product': False}
