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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': False, 'ambient': '4400.m', 'ambient_counter': 13, 'ambient_order': 4400, 'ambient_tex': 'C_{44}.C_{10}^2', 'central': False, 'central_factor': True, 'centralizer_order': 40, 'characteristic': True, 'core_order': 1100, 'counter': 5, 'cyclic': False, 'direct': False, 'hall': 0, 'label': '4400.m.4.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': None, 'minimal': False, 'minimal_normal': False, 'nilpotent': False, 'normal': True, 'old_label': '4.a1', 'outer_equivalence': True, 'perfect': False, 'proper': True, 'quotient': '4.2', 'quotient_Agroup': True, 'quotient_abelian': True, 'quotient_cyclic': False, 'quotient_hash': 2, 'quotient_metabelian': True, 'quotient_nilpotent': True, 'quotient_order': 4, 'quotient_simple': False, 'quotient_solvable': True, 'quotient_supersolvable': True, 'quotient_tex': 'C_2^2', 'simple': False, 'solvable': True, 'special_labels': [], 'split': False, 'standard_generators': False, 'stem': False, 'subgroup': '1100.29', 'subgroup_hash': 29, 'subgroup_order': 1100, 'subgroup_tex': 'C_{10}\\times F_{11}', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '4400.m', 'aut_centralizer_order': 24, 'aut_label': '4.a1', 'aut_quo_index': 1, 'aut_stab_index': 1, 'aut_weyl_group': '4400.r', 'aut_weyl_index': 24, 'centralizer': '110.a1', 'complements': [], 'conjugacy_class_count': 1, 'contained_in': ['2.a1'], 'contains': ['8.a1', '8.b1', '20.a1', '20.b1', '44.a1'], 'core': '4.a1', 'coset_action_label': None, 'count': 1, 'diagramx': [2169, 2518, 1508, 1227], 'generators': [10, 4, 880, 2200, 400], 'label': '4400.m.4.a1', 'mobius_quo': 0, 'mobius_sub': 2, 'normal_closure': '4.a1', 'normal_contained_in': ['2.a1'], 'normal_contains': ['8.a1', '8.b1', '20.a1', '20.b1'], 'normalizer': '1.a1', 'old_label': '4.a1', 'projective_image': '440.42', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '4.a1', 'subgroup_fusion': None, 'weyl_group': '110.1'}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '100.16', '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, 10], [221, 870], [1, 1010], [1071, 10]], 'aut_group': '4400.r', 'aut_hash': 3690388632023574097, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 4400, 'aut_permdeg': 77, 'aut_perms': [56883759698427363657421140742347031429101440190046605383223984815895734387032106449957147462527776529098702985233, 29956808623520794514028071565247402522112806851483006125820260451801484110597682345036015190324403568587385192560, 92157247970159676524817938842705522413666998993160816869970806325543758955058982550685979128966382436715094427168], 'aut_phi_ratio': 11.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 11, 2, 1], [5, 1, 4, 1], [5, 11, 5, 4], [10, 1, 4, 1], [10, 11, 5, 4], [10, 11, 8, 1], [10, 11, 10, 4], [11, 10, 1, 1], [22, 10, 1, 1], [55, 10, 4, 1], [110, 10, 4, 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], [2, 11, 2], [5, 1, 4], [5, 11, 20], [10, 1, 4], [10, 11, 68], [11, 10, 1], [22, 10, 1], [55, 10, 4], [110, 10, 4]], 'center_label': '10.2', 'center_order': 10, 'central_product': True, 'central_quotient': '110.1', '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': 29, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['110.1', 1], ['2.1', 1], ['5.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 11, 1, 2], [5, 1, 4, 1], [5, 11, 4, 5], [10, 1, 4, 1], [10, 11, 4, 17], [11, 10, 1, 1], [22, 10, 1, 1], [55, 10, 4, 1], [110, 10, 4, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 72, 'exponent': 110, 'exponents_of_order': [2, 2, 1], 'factors_of_aut_order': [2, 5, 11], 'factors_of_order': [2, 5, 11], 'faithful_reps': [[10, 0, 4]], 'familial': False, 'frattini_label': '1.1', 'frattini_quotient': '1100.29', 'hash': 29, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 110, 'inner_gen_orders': [10, 11], 'inner_gens': [[1, 510], [601, 10]], 'inner_hash': 1, 'inner_nilpotent': False, 'inner_order': 110, 'inner_split': True, 'inner_tex': 'F_{11}', 'inner_used': [1, 2], 'irrC_degree': 10, 'irrQ_degree': 40, 'irrQ_dim': 40, 'irrR_degree': 20, 'irrep_stats': [[1, 100], [10, 10]], 'label': '1100.29', 'linC_count': 4, 'linC_degree': 10, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 14, 'linQ_degree_count': 30, 'linQ_dim': 14, 'linQ_dim_count': 30, 'linR_count': 60, 'linR_degree': 12, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C10*F11', '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': 22, 'number_characteristic_subgroups': 13, 'number_conjugacy_classes': 110, 'number_divisions': 32, 'number_normal_subgroups': 44, 'number_subgroup_autclasses': 32, 'number_subgroup_classes': 80, 'number_subgroups': 440, 'old_label': None, 'order': 1100, 'order_factorization_type': 222, 'order_stats': [[1, 1], [2, 23], [5, 224], [10, 752], [11, 10], [22, 10], [55, 40], [110, 40]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 20, 'outer_gen_orders': [4, 10], 'outer_gen_pows': [0, 0], 'outer_gens': [[221, 430], [771, 210]], 'outer_group': '40.12', 'outer_hash': 12, 'outer_nilpotent': False, 'outer_order': 40, 'outer_permdeg': 7, 'outer_perms': [247, 1729], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2\\times F_5', 'pc_rank': 2, 'perfect': False, 'permutation_degree': 18, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 5, 5], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [4, 24], [10, 2], [40, 2]], 'representations': {'PC': {'code': 14057822293040673023954469066903039, 'gens': [1, 3], 'pres': [5, -2, -5, -2, -5, -11, 10, 7652, 5332, 42, 20403, 3208, 118, 17504, 6259]}, 'GLFp': {'d': 2, 'p': 11, 'gens': [1343, 7988, 2663]}, 'Perm': {'d': 18, 'gens': [21115987966081, 1, 46057571844480, 870, 418750544136960]}}, 'schur_multiplier': [10], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [10, 10], 'solvability_type': 6, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_{10}\\times F_{11}', 'transitive_degree': 110, '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': '200.52', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 3, 'aut_exponent': 660, 'aut_gen_orders': [10, 60, 12, 20, 20], 'aut_gens': [[1, 2, 20], [4316, 3122, 2685], [4396, 2602, 3105], [196, 2722, 1825], [2421, 642, 420], [1241, 3522, 2825]], 'aut_group': None, 'aut_hash': 333570381098051054, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 105600, 'aut_permdeg': 83, 'aut_perms': [1555750564830194189365818167316049118515730820254172870430299209378207112214587798433352009503664159661588417888550266258535, 30925874656572050026632831560543697614536016798283942523179543606857669881882668975401193508990631331796641210710672180435612, 16016153995587373590234130532119699956256783529628351478298656723848150907274674305151549848114365394944983733702028537211995, 35250534075927931843308016573561266339361049966756948979592167107945321095774225314123686903477870341728651171110467231148838, 18526419365673994712313357694661675149996324968304186846271297412930531710982138630026035546497560961994497085219536489711342], 'aut_phi_ratio': 66.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 11, 2, 1], [4, 2, 3, 1], [4, 22, 3, 1], [5, 1, 4, 1], [5, 11, 5, 4], [10, 1, 4, 1], [10, 11, 5, 4], [10, 11, 8, 1], [10, 11, 10, 4], [11, 10, 1, 1], [20, 2, 12, 1], [20, 22, 12, 1], [20, 22, 15, 8], [22, 10, 1, 1], [44, 20, 3, 1], [55, 10, 4, 1], [110, 10, 4, 1], [220, 20, 12, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_2^2.C_{165}.C_{10}.C_2^4', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 20, 'autcent_group': '160.236', 'autcent_hash': 236, 'autcent_nilpotent': False, 'autcent_order': 160, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^3\\times F_5', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 330, 'autcentquo_group': '660.15', 'autcentquo_hash': 15, 'autcentquo_nilpotent': False, 'autcentquo_order': 660, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'S_3\\times F_{11}', 'cc_stats': [[1, 1, 1], [2, 1, 1], [2, 11, 2], [4, 2, 3], [4, 22, 3], [5, 1, 4], [5, 11, 20], [10, 1, 4], [10, 11, 68], [11, 10, 1], [20, 2, 12], [20, 22, 132], [22, 10, 1], [44, 20, 3], [55, 10, 4], [110, 10, 4], [220, 20, 12]], 'center_label': '10.2', 'center_order': 10, 'central_product': True, 'central_quotient': '440.42', 'commutator_count': 1, 'commutator_label': '22.2', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '5.1', '5.1', '11.1'], 'composition_length': 7, 'conjugacy_classes_known': True, 'counter': 13, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['110.1', 1], ['5.1', 1], ['8.4', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 11, 1, 2], [4, 2, 1, 3], [4, 22, 1, 3], [5, 1, 4, 1], [5, 11, 4, 5], [10, 1, 4, 1], [10, 11, 4, 17], [11, 10, 1, 1], [20, 2, 4, 3], [20, 22, 4, 33], [22, 10, 1, 1], [44, 20, 1, 3], [55, 10, 4, 1], [110, 10, 4, 1], [220, 20, 4, 3]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 249984, 'exponent': 220, 'exponents_of_order': [4, 2, 1], 'factors_of_aut_order': [2, 3, 5, 11], 'factors_of_order': [2, 5, 11], 'faithful_reps': [[20, 0, 4]], 'familial': False, 'frattini_label': '2.1', 'frattini_quotient': '2200.a', 'hash': 7359625474499212845, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 110, 'inner_gen_orders': [10, 10, 22], 'inner_gens': [[1, 2002, 1820], [2401, 2, 2020], [2601, 2402, 20]], 'inner_hash': 42, 'inner_nilpotent': False, 'inner_order': 440, 'inner_split': True, 'inner_tex': 'C_2^2\\times F_{11}', 'inner_used': [1, 2, 3], 'irrC_degree': 20, 'irrQ_degree': 80, 'irrQ_dim': 80, 'irrR_degree': 40, 'irrep_stats': [[1, 200], [2, 50], [10, 20], [20, 5]], 'label': '4400.m', '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': True, 'metacyclic': False, 'monomial': True, 'name': 'C44.C10^2', '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], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 36, 'number_characteristic_subgroups': 21, 'number_conjugacy_classes': 275, 'number_divisions': 80, 'number_normal_subgroups': 164, 'number_subgroup_autclasses': 80, 'number_subgroup_classes': 304, 'number_subgroups': 1704, 'old_label': None, 'order': 4400, 'order_factorization_type': 321, 'order_stats': [[1, 1], [2, 23], [4, 72], [5, 224], [10, 752], [11, 10], [20, 2928], [22, 10], [44, 60], [55, 40], [110, 40], [220, 240]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': True, 'outer_exponent': 60, 'outer_gen_orders': [2, 4, 30], 'outer_gen_pows': [0, 0, 0], 'outer_gens': [[1356, 2, 285], [876, 882, 2525], [476, 442, 3585]], 'outer_group': '240.195', 'outer_hash': 195, 'outer_nilpotent': False, 'outer_order': 240, 'outer_permdeg': 10, 'outer_perms': [7, 46087, 1099489], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'D_6\\times F_5', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 24, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 2, 5, 5], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 8], [2, 2], [4, 48], [8, 12], [10, 4], [20, 1], [40, 4], [80, 1]], 'representations': {'PC': {'code': '1539018528973074079978991569647978877460577145669761487615340476114120171470030039', 'gens': [1, 2, 4], 'pres': [7, -2, -2, -5, -2, -2, -5, -11, 27244, 28029, 36, 33686, 50963, 28290, 11357, 80, 127404, 70711, 28368, 102, 120965, 77292, 21859, 250, 58806, 19613, 19620]}, 'GLFp': {'d': 4, 'p': 11, 'gens': [12531822321003912, 31318850038632571, 39096446745583924, 41217174402924011, 29417756511587215, 33418192856010432, 23726635956891084]}, 'Perm': {'d': 24, 'gens': [33, 29482547546685500467233, 4603563859131472435264, 486985713, 52013621325460976143264, 80933046670022808522960, 109189379508851625984090]}}, 'schur_multiplier': [2, 10], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 10, 10], 'solvability_type': 7, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_{44}.C_{10}^2', 'transitive_degree': 440, 'wreath_data': None, 'wreath_product': False}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': True, 'abelian_quotient': '4.2', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 6, 'aut_gen_orders': [2, 3], 'aut_gens': [[1, 2], [3, 2], [2, 3]], 'aut_group': '6.1', 'aut_hash': 1, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 6, 'aut_permdeg': 3, 'aut_perms': [1, 4], 'aut_phi_ratio': 3.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 3, 1]], 'aut_supersolvable': True, 'aut_tex': 'S_3', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 6, 'autcent_group': '6.1', 'autcent_hash': 1, 'autcent_nilpotent': False, 'autcent_order': 6, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'S_3', '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, 3]], 'center_label': '4.2', 'center_order': 4, 'central_product': True, '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'], 'composition_length': 2, 'conjugacy_classes_known': True, 'counter': 2, 'cyclic': False, 'derived_length': 1, 'dihedral': True, 'direct_factorization': [['2.1', 2]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3]], 'element_repr_type': 'PC', 'elementary': 2, 'eulerian_function': 1, 'exponent': 2, 'exponents_of_order': [2], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2], 'faithful_reps': [], 'familial': True, 'frattini_label': '1.1', 'frattini_quotient': '4.2', 'hash': 2, 'hyperelementary': 2, 'inner_abelian': True, 'inner_cyclic': True, 'inner_exponent': 1, 'inner_gen_orders': [1, 1], 'inner_gens': [[1, 2], [1, 2]], 'inner_hash': 1, 'inner_nilpotent': True, 'inner_order': 1, 'inner_split': True, 'inner_tex': 'C_1', 'inner_used': [], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 4]], 'label': '4.2', 'linC_count': 3, 'linC_degree': 2, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 2, 'linQ_degree_count': 3, 'linQ_dim': 2, 'linQ_dim_count': 3, 'linR_count': 3, 'linR_degree': 2, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C2^2', 'ngens': 2, 'nilpotency_class': 1, 'nilpotent': True, 'normal_counts': [0, 0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 2, 'number_characteristic_subgroups': 2, 'number_conjugacy_classes': 4, 'number_divisions': 4, 'number_normal_subgroups': 5, 'number_subgroup_autclasses': 3, 'number_subgroup_classes': 5, 'number_subgroups': 5, 'old_label': None, 'order': 4, 'order_factorization_type': 2, 'order_stats': [[1, 1], [2, 3]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 6, 'outer_gen_orders': [2, 3], 'outer_gen_pows': [0, 0], 'outer_gens': [[3, 2], [2, 3]], 'outer_group': '6.1', 'outer_hash': 1, 'outer_nilpotent': False, 'outer_order': 6, 'outer_permdeg': 3, 'outer_perms': [1, 4], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'S_3', 'pc_rank': 2, 'perfect': False, 'permutation_degree': 4, 'pgroup': 2, 'primary_abelian_invariants': [2, 2], 'quasisimple': False, 'rank': 2, 'rational': True, 'rational_characters_known': True, 'ratrep_stats': [[1, 4]], 'representations': {'PC': {'code': 0, 'gens': [1, 2], 'pres': [2, -2, 2]}, 'GLZ': {'b': 3, 'd': 2, 'gens': [14, 12]}, 'GLFp': {'d': 2, 'p': 3, 'gens': [55, 56]}, 'Perm': {'d': 4, 'gens': [6, 1]}}, 'schur_multiplier': [2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2], 'solvability_type': 1, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_2^2', 'transitive_degree': 4, 'wreath_data': None, 'wreath_product': False}
