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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': False, 'ambient': '990.9', 'ambient_counter': 9, 'ambient_order': 990, 'ambient_tex': 'C_5\\times D_{99}', 'central': False, 'central_factor': True, 'centralizer_order': 5, 'characteristic': True, 'core_order': 198, 'counter': 4, 'cyclic': False, 'direct': True, 'hall': 66, 'label': '990.9.5.a1.a1', 'maximal': True, 'maximal_normal': True, 'metabelian': True, 'metacyclic': True, 'minimal': False, 'minimal_normal': False, 'nilpotent': False, 'normal': True, 'old_label': '5.a1.a1', 'outer_equivalence': False, 'perfect': False, 'proper': True, 'quotient': '5.1', 'quotient_Agroup': True, 'quotient_abelian': True, 'quotient_cyclic': True, 'quotient_hash': 1, 'quotient_metabelian': True, 'quotient_nilpotent': True, 'quotient_order': 5, 'quotient_simple': True, 'quotient_solvable': True, 'quotient_supersolvable': True, 'quotient_tex': 'C_5', 'simple': False, 'solvable': True, 'special_labels': [], 'split': True, 'standard_generators': False, 'stem': False, 'subgroup': '198.3', 'subgroup_hash': 3, 'subgroup_order': 198, 'subgroup_tex': 'D_{99}', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '990.9', 'aut_centralizer_order': 4, 'aut_label': '5.a1', 'aut_quo_index': 1, 'aut_stab_index': 1, 'aut_weyl_group': '5940.d', 'aut_weyl_index': 4, 'centralizer': '198.a1.a1', 'complements': ['198.a1.a1'], 'conjugacy_class_count': 1, 'contained_in': ['1.a1.a1'], 'contains': ['10.a1.a1', '15.a1.a1', '55.a1.a1'], 'core': '5.a1.a1', 'coset_action_label': None, 'count': 1, 'diagramx': [3724, 9009, 7895, 839, 2835, 1081, 8855, 8919], 'generators': [1, 90, 330, 440], 'label': '990.9.5.a1.a1', 'mobius_quo': 0, 'mobius_sub': -1, 'normal_closure': '5.a1.a1', 'normal_contained_in': ['1.a1.a1'], 'normal_contains': ['10.a1.a1'], 'normalizer': '1.a1.a1', 'old_label': '5.a1.a1', 'projective_image': '990.9', 'quotient_action_image': '1.1', 'quotient_action_kernel': '5.1', 'quotient_action_kernel_order': 5, 'quotient_fusion': None, 'short_label': '5.a1.a1', 'subgroup_fusion': None, 'weyl_group': '198.3'}
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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': 990, 'aut_gen_orders': [6, 10, 99], 'aut_gens': [[1, 2], [1, 64], [1, 38], [51, 2]], 'aut_group': '5940.d', 'aut_hash': 4717361004838527357, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 5940, 'aut_permdeg': 20, 'aut_perms': [21048432566543308, 14315224873420800, 1014091472311194428], 'aut_phi_ratio': 99.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 99, 1, 1], [3, 2, 1, 1], [9, 2, 3, 1], [11, 2, 5, 1], [33, 2, 10, 1], [99, 2, 30, 1]], 'aut_supersolvable': True, 'aut_tex': 'D_{99}:C_{30}', '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': 990, 'autcentquo_group': '5940.d', 'autcentquo_hash': 4717361004838527357, 'autcentquo_nilpotent': False, 'autcentquo_order': 5940, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'D_{99}:C_{30}', 'cc_stats': [[1, 1, 1], [2, 99, 1], [3, 2, 1], [9, 2, 3], [11, 2, 5], [33, 2, 10], [99, 2, 30]], 'center_label': '1.1', 'center_order': 1, 'central_product': False, 'central_quotient': '198.3', 'commutator_count': 1, 'commutator_label': '99.1', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '3.1', '3.1', '11.1'], 'composition_length': 4, 'conjugacy_classes_known': True, 'counter': 3, 'cyclic': False, 'derived_length': 2, 'dihedral': True, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 99, 1, 1], [3, 2, 1, 1], [9, 2, 3, 1], [11, 2, 5, 1], [33, 2, 10, 1], [99, 2, 30, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 3, 'exponent': 198, 'exponents_of_order': [2, 1, 1], 'factors_of_aut_order': [2, 3, 5, 11], 'factors_of_order': [2, 3, 11], 'faithful_reps': [[2, 1, 30]], 'familial': True, 'frattini_label': '3.1', 'frattini_quotient': '66.3', 'hash': 3, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 198, 'inner_gen_orders': [2, 99], 'inner_gens': [[1, 196], [5, 2]], 'inner_hash': 3, 'inner_nilpotent': False, 'inner_order': 198, 'inner_split': True, 'inner_tex': 'D_{99}', 'inner_used': [1, 2], 'irrC_degree': 2, 'irrQ_degree': 60, 'irrQ_dim': 60, 'irrR_degree': 2, 'irrep_stats': [[1, 2], [2, 49]], 'label': '198.3', 'linC_count': 30, 'linC_degree': 2, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 16, 'linQ_degree_count': 1, 'linQ_dim': 16, 'linQ_dim_count': 1, 'linR_count': 30, 'linR_degree': 2, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'D99', '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': 7, 'number_characteristic_subgroups': 7, 'number_conjugacy_classes': 51, 'number_divisions': 7, 'number_normal_subgroups': 7, 'number_subgroup_autclasses': 12, 'number_subgroup_classes': 12, 'number_subgroups': 162, 'old_label': None, 'order': 198, 'order_factorization_type': 222, 'order_stats': [[1, 1], [2, 99], [3, 2], [9, 6], [11, 10], [33, 20], [99, 60]], 'outer_abelian': True, 'outer_cyclic': True, 'outer_equivalence': False, 'outer_exponent': 30, 'outer_gen_orders': [30], 'outer_gen_pows': [0], 'outer_gens': [[89, 172]], 'outer_group': '30.4', 'outer_hash': 4, 'outer_nilpotent': True, 'outer_order': 30, 'outer_permdeg': 10, 'outer_perms': [373024], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_{30}', 'pc_rank': 2, 'perfect': False, 'permutation_degree': 20, 'pgroup': 0, 'primary_abelian_invariants': [2], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 2], [2, 1], [6, 1], [10, 1], [20, 1], [60, 1]], 'representations': {'PC': {'code': 71670483175020119, 'gens': [1, 2], 'pres': [4, -2, -3, -3, -11, 1569, 29, 2306, 46, 2883]}, 'GLFp': {'d': 2, 'p': 197, 'gens': [1498493109, 88607853]}, 'Perm': {'d': 20, 'gens': [6423384156550615, 57658, 134491868718278400, 104227]}}, '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_{99}', 'transitive_degree': 99, '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': '10.2', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 1980, 'aut_gen_orders': [18, 30, 60, 45], 'aut_gens': [[1, 2], [671, 218], [471, 562], [841, 914], [401, 182]], 'aut_group': None, 'aut_hash': 6501222207799632290, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 23760, 'aut_permdeg': 103, 'aut_perms': [58631780448728065287135228909534274338409798673011026543416759075534902456662803777181290515976861792570234865415247978235479098926548290375645927878615435047513723, 76561370777285245072544813815123372773626907103710683701335794978445628549769858082222325024101717326925507596171645302954785166414958455927924112982278599767168131, 14101066196705344742100923373916347772354365109125628855734875161079372521550808390690223098453199046786612478048376822780009755977199836882391408362392464213245269, 64305900509757953735241874522962313660235937779076417639806044445189586630770446550298997201902468272863703519044884027207003659714448864637385769951924724028934411], 'aut_phi_ratio': 99.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 99, 1, 1], [3, 2, 1, 1], [5, 1, 4, 1], [9, 2, 3, 1], [10, 99, 4, 1], [11, 2, 5, 1], [15, 2, 4, 1], [33, 2, 10, 1], [45, 2, 12, 1], [55, 2, 20, 1], [99, 2, 30, 1], [165, 2, 40, 1], [495, 2, 120, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_{99}.C_{60}.C_2^2', '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': 990, 'autcentquo_group': '5940.d', 'autcentquo_hash': 4717361004838527357, 'autcentquo_nilpotent': False, 'autcentquo_order': 5940, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'D_{99}:C_{30}', 'cc_stats': [[1, 1, 1], [2, 99, 1], [3, 2, 1], [5, 1, 4], [9, 2, 3], [10, 99, 4], [11, 2, 5], [15, 2, 4], [33, 2, 10], [45, 2, 12], [55, 2, 20], [99, 2, 30], [165, 2, 40], [495, 2, 120]], 'center_label': '5.1', 'center_order': 5, 'central_product': True, 'central_quotient': '198.3', 'commutator_count': 1, 'commutator_label': '99.1', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '3.1', '3.1', '5.1', '11.1'], 'composition_length': 5, 'conjugacy_classes_known': True, 'counter': 9, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['198.3', 1], ['5.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 99, 1, 1], [3, 2, 1, 1], [5, 1, 4, 1], [9, 2, 3, 1], [10, 99, 4, 1], [11, 2, 5, 1], [15, 2, 4, 1], [33, 2, 10, 1], [45, 2, 12, 1], [55, 2, 20, 1], [99, 2, 30, 1], [165, 2, 40, 1], [495, 2, 120, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 18, 'exponent': 990, 'exponents_of_order': [2, 1, 1, 1], 'factors_of_aut_order': [2, 3, 5, 11], 'factors_of_order': [2, 3, 5, 11], 'faithful_reps': [[2, 0, 120]], 'familial': False, 'frattini_label': '3.1', 'frattini_quotient': '330.9', 'hash': 9, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 198, 'inner_gen_orders': [2, 99], 'inner_gens': [[1, 592], [401, 2]], 'inner_hash': 3, 'inner_nilpotent': False, 'inner_order': 198, 'inner_split': True, 'inner_tex': 'D_{99}', 'inner_used': [1, 2], 'irrC_degree': 2, 'irrQ_degree': 240, 'irrQ_dim': 240, 'irrR_degree': 4, 'irrep_stats': [[1, 10], [2, 245]], 'label': '990.9', 'linC_count': 120, 'linC_degree': 2, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 20, 'linQ_degree_count': 2, 'linQ_dim': 20, 'linQ_dim_count': 2, 'linR_count': 180, 'linR_degree': 4, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C5*D99', '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], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 14, 'number_characteristic_subgroups': 14, 'number_conjugacy_classes': 255, 'number_divisions': 14, 'number_normal_subgroups': 14, 'number_subgroup_autclasses': 24, 'number_subgroup_classes': 24, 'number_subgroups': 324, 'old_label': None, 'order': 990, 'order_factorization_type': 222, 'order_stats': [[1, 1], [2, 99], [3, 2], [5, 4], [9, 6], [10, 396], [11, 10], [15, 8], [33, 20], [45, 24], [55, 40], [99, 60], [165, 80], [495, 240]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 60, 'outer_gen_orders': [2, 60], 'outer_gen_pows': [968, 0], 'outer_gens': [[771, 812], [1, 346]], 'outer_group': '120.31', 'outer_hash': 31, 'outer_nilpotent': True, 'outer_order': 120, 'outer_permdeg': 14, 'outer_perms': [6267300480, 127013793], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2\\times C_{60}', 'pc_rank': 2, 'perfect': False, 'permutation_degree': 25, 'pgroup': 0, 'primary_abelian_invariants': [2, 5], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 2], [2, 1], [4, 2], [6, 1], [8, 1], [10, 1], [20, 1], [24, 1], [40, 1], [60, 1], [80, 1], [240, 1]], 'representations': {'PC': {'code': 13889073669778879356953401139, 'gens': [1, 2], 'pres': [5, -2, -3, -3, -5, -11, 5921, 36, 11792, 57, 7563, 118, 22504]}, 'GLFp': {'d': 2, 'p': 991, 'gens': [781513543924, 793192451170, 983072]}, 'Perm': {'d': 25, 'gens': [56207700030206140784047, 1401558681600, 701376831873275830272000, 4364893, 1348964242702978277376000]}}, 'schur_multiplier': [], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [10], 'solvability_type': 6, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_5\\times D_{99}', 'transitive_degree': 495, 'wreath_data': None, 'wreath_product': False}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': True, 'abelian_quotient': '5.1', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': True, 'aut_cyclic': True, 'aut_derived_length': 1, 'aut_exponent': 4, 'aut_gen_orders': [4], 'aut_gens': [[1], [2]], 'aut_group': '4.1', 'aut_hash': 1, 'aut_nilpotency_class': 1, 'aut_nilpotent': True, 'aut_order': 4, 'aut_permdeg': 4, 'aut_perms': [9], 'aut_phi_ratio': 1.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [5, 1, 4, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_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': 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], [5, 1, 4]], 'center_label': '5.1', 'center_order': 5, 'central_product': False, 'central_quotient': '1.1', 'commutator_count': 0, 'commutator_label': '1.1', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['5.1'], 'composition_length': 1, 'conjugacy_classes_known': True, 'counter': 1, 'cyclic': True, 'derived_length': 1, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [5, 1, 4, 1]], 'element_repr_type': 'PC', 'elementary': 5, 'eulerian_function': 1, 'exponent': 5, 'exponents_of_order': [1], 'factors_of_aut_order': [2], 'factors_of_order': [5], 'faithful_reps': [[1, 0, 4]], 'familial': True, 'frattini_label': '1.1', 'frattini_quotient': '5.1', 'hash': 1, 'hyperelementary': 5, 'inner_abelian': True, 'inner_cyclic': True, 'inner_exponent': 1, 'inner_gen_orders': [1], 'inner_gens': [[1]], 'inner_hash': 1, 'inner_nilpotent': True, 'inner_order': 1, 'inner_split': True, 'inner_tex': 'C_1', 'inner_used': [], 'irrC_degree': 1, 'irrQ_degree': 4, 'irrQ_dim': 4, 'irrR_degree': 2, 'irrep_stats': [[1, 5]], 'label': '5.1', 'linC_count': 4, 'linC_degree': 1, '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': 'C5', 'ngens': 1, 'nilpotency_class': 1, 'nilpotent': True, 'normal_counts': [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': 5, 'number_divisions': 2, 'number_normal_subgroups': 2, 'number_subgroup_autclasses': 2, 'number_subgroup_classes': 2, 'number_subgroups': 2, 'old_label': None, 'order': 5, 'order_factorization_type': 1, 'order_stats': [[1, 1], [5, 4]], 'outer_abelian': True, 'outer_cyclic': True, 'outer_equivalence': False, 'outer_exponent': 4, 'outer_gen_orders': [4], 'outer_gen_pows': [0], 'outer_gens': [[2]], 'outer_group': '4.1', 'outer_hash': 1, 'outer_nilpotent': True, 'outer_order': 4, 'outer_permdeg': 4, 'outer_perms': [9], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_4', 'pc_rank': 1, 'perfect': False, 'permutation_degree': 5, 'pgroup': 5, 'primary_abelian_invariants': [5], 'quasisimple': False, 'rank': 1, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 1], [4, 1]], 'representations': {'PC': {'code': 0, 'gens': [1], 'pres': [1, -5]}, 'GLZ': {'b': 3, 'd': 4, 'gens': [26329372]}, 'Lie': [{'d': 1, 'q': 5, 'gens': [33], 'family': 'ASigmaL'}], 'GLFp': {'d': 2, 'p': 5, 'gens': [131]}, 'Perm': {'d': 5, 'gens': [96]}}, 'schur_multiplier': [], 'semidirect_product': False, 'simple': True, 'smith_abelian_invariants': [5], 'solvability_type': 0, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_5', 'transitive_degree': 5, 'wreath_data': None, 'wreath_product': False}
