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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': True, 'ambient': '832.230', 'ambient_counter': 230, 'ambient_order': 832, 'ambient_tex': 'D_{26}.\\OD_{16}', 'central': False, 'central_factor': False, 'centralizer_order': 64, 'characteristic': False, 'core_order': 1, 'counter': 114, 'cyclic': True, 'direct': None, 'hall': 0, 'label': '832.230.416.d1.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': True, 'minimal_normal': False, 'nilpotent': True, 'normal': False, 'old_label': '416.d1.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': 416, 'quotient_simple': None, 'quotient_solvable': None, 'quotient_supersolvable': None, 'quotient_tex': None, 'simple': True, 'solvable': True, 'special_labels': [], 'split': None, 'standard_generators': False, 'stem': False, 'subgroup': '2.1', 'subgroup_hash': 1, 'subgroup_order': 2, 'subgroup_tex': 'C_2', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '832.230', 'aut_centralizer_order': 384, 'aut_label': '416.d1', 'aut_quo_index': None, 'aut_stab_index': 26, 'aut_weyl_group': '1.1', 'aut_weyl_index': 9984, 'centralizer': '13.a1.a1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['32.d1.a1', '208.d1.a1', '208.h1.a1', '208.i1.a1'], 'contains': ['832.a1.a1'], 'core': '832.a1.a1', 'coset_action_label': None, 'count': 13, 'diagramx': [1685, -1, 2578, -1, 2823, -1, 3001, -1], 'generators': [454], 'label': '832.230.416.d1.a1', 'mobius_quo': None, 'mobius_sub': 0, 'normal_closure': '32.d1.a1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '13.a1.a1', 'old_label': '416.d1.a1', 'projective_image': '832.230', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '416.d1.a1', 'subgroup_fusion': None, 'weyl_group': '1.1'}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': True, 'abelian_quotient': '2.1', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': True, 'aut_cyclic': True, 'aut_derived_length': 0, 'aut_exponent': 1, 'aut_gen_orders': [], 'aut_gens': [[1]], 'aut_group': '1.1', 'aut_hash': 1, 'aut_nilpotency_class': 0, 'aut_nilpotent': True, 'aut_order': 1, 'aut_permdeg': 1, 'aut_perms': [], 'aut_phi_ratio': 1.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_1', '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': 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, 1]], 'center_label': '2.1', 'center_order': 2, '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': ['2.1'], 'composition_length': 1, 'conjugacy_classes_known': True, 'counter': 1, 'cyclic': True, 'derived_length': 1, 'dihedral': True, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1]], 'element_repr_type': 'PC', 'elementary': 2, 'eulerian_function': 1, 'exponent': 2, 'exponents_of_order': [1], 'factors_of_aut_order': [], 'factors_of_order': [2], 'faithful_reps': [[1, 1, 1]], 'familial': True, 'frattini_label': '1.1', 'frattini_quotient': '2.1', 'hash': 1, 'hyperelementary': 2, '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': 1, 'irrQ_dim': 1, 'irrR_degree': 1, 'irrep_stats': [[1, 2]], 'label': '2.1', 'linC_count': 1, 'linC_degree': 1, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 1, 'linQ_degree_count': 1, 'linQ_dim': 1, 'linQ_dim_count': 1, 'linR_count': 1, 'linR_degree': 1, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C2', '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': 2, 'number_divisions': 2, 'number_normal_subgroups': 2, 'number_subgroup_autclasses': 2, 'number_subgroup_classes': 2, 'number_subgroups': 2, 'old_label': None, 'order': 2, 'order_factorization_type': 1, 'order_stats': [[1, 1], [2, 1]], 'outer_abelian': True, 'outer_cyclic': True, 'outer_equivalence': False, 'outer_exponent': 1, 'outer_gen_orders': [], 'outer_gen_pows': [], 'outer_gens': [], 'outer_group': '1.1', 'outer_hash': 1, 'outer_nilpotent': True, 'outer_order': 1, 'outer_permdeg': 1, 'outer_perms': [], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_1', 'pc_rank': 1, 'perfect': False, 'permutation_degree': 2, 'pgroup': 2, 'primary_abelian_invariants': [2], 'quasisimple': False, 'rank': 1, 'rational': True, 'rational_characters_known': True, 'ratrep_stats': [[1, 2]], 'representations': {'PC': {'code': 0, 'gens': [1], 'pres': [1, -2]}, 'GLZ': {'b': 3, 'd': 2, 'gens': [12]}, 'Lie': [{'d': 2, 'q': 2, 'gens': [6], 'family': 'CSOPlus'}, {'d': 2, 'q': 2, 'gens': [6], 'family': 'COPlus'}, {'d': 1, 'q': 2, 'gens': [1], 'family': 'AGL'}, {'d': 1, 'q': 2, 'gens': [1], 'family': 'AGammaL'}, {'d': 1, 'q': 2, 'gens': [1], 'family': 'ASigmaL'}], 'GLFp': {'d': 2, 'p': 2, 'gens': [11]}, 'Perm': {'d': 2, 'gens': [1]}}, 'schur_multiplier': [], 'semidirect_product': False, 'simple': True, 'smith_abelian_invariants': [2], 'solvability_type': 0, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_2', 'transitive_degree': 2, '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': '32.3', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 156, 'aut_gen_orders': [6, 6, 4, 12, 6, 12, 2], 'aut_gens': [[1, 4, 8], [657, 4, 184], [97, 4, 396], [165, 4, 792], [533, 4, 780], [769, 4, 764], [753, 4, 744], [225, 4, 408]], 'aut_group': None, 'aut_hash': 8707403541713051872, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 9984, 'aut_permdeg': 112, 'aut_perms': [148968553413263001104669729442471201413592077058526686608916920263808878458768866903334668766593056417329557494213053758187136921917621450142539132281297924280004717390284074572977518, 164929720540772897658792861954620659642171013658881373452472813168016521637067190243889161788984664639689834116917360227285357679383116402686950082365935577775192686271186268168588964, 155793835390016600141014030903475782950518177557896375467891183983631423270192106052542097199925683299506570829193377367051794302460749810460718114737109840844130247386124518092720874, 128970710481314623913787441263239967677185278176377649571934416227793623631893049346272976964609506715326353899503118234403105755527993670280418265296068520350585520945724287742665308, 85847537825573044279763208670188159180040438047843909602658398429991970221684892024733027494633609984631735373266633599008210169665734034377554846631536902227973581479755692140315434, 144478735678365966626934791143230004321124089839295688329247275526193655552354693783090567945489935253425660454098488704125988345793794289735103028050253245596498383576239867230936691, 122460314186937301601865433238234618579097706319382923601733052303473866793204998348287858954906402749325683326671908907089616422547089988135242329150965994702835375305264289443143198], 'aut_phi_ratio': 26.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [2, 13, 2, 2], [4, 1, 2, 2], [4, 13, 2, 2], [4, 26, 4, 2], [8, 2, 4, 1], [8, 26, 4, 3], [13, 4, 3, 1], [26, 4, 3, 3], [52, 4, 6, 2], [104, 4, 24, 1]], 'aut_supersolvable': True, 'aut_tex': '(C_2\\times D_{13}:C_3).C_2^6', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 4, 'autcent_group': '64.261', 'autcent_hash': 261, 'autcent_nilpotent': True, 'autcent_order': 64, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'D_4\\times C_2^3', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 156, 'autcentquo_group': '156.7', 'autcentquo_hash': 7, 'autcentquo_nilpotent': False, 'autcentquo_order': 156, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'F_{13}', 'cc_stats': [[1, 1, 1], [2, 1, 3], [2, 13, 4], [4, 1, 4], [4, 13, 4], [4, 26, 8], [8, 2, 4], [8, 26, 12], [13, 4, 3], [26, 4, 9], [52, 4, 12], [104, 4, 24]], 'center_label': '8.2', 'center_order': 8, 'central_product': False, 'central_quotient': '104.12', 'commutator_count': 1, 'commutator_label': '26.2', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '2.1', '2.1', '13.1'], 'composition_length': 7, 'conjugacy_classes_known': True, 'counter': 230, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [2, 13, 1, 4], [4, 1, 2, 2], [4, 13, 2, 2], [4, 26, 2, 4], [8, 2, 4, 1], [8, 26, 4, 3], [13, 4, 3, 1], [26, 4, 3, 3], [52, 4, 6, 2], [104, 4, 24, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 24, 'exponent': 104, 'exponents_of_order': [6, 1], 'factors_of_aut_order': [2, 3, 13], 'factors_of_order': [2, 13], 'faithful_reps': [], 'familial': False, 'frattini_label': '8.2', 'frattini_quotient': '104.12', 'hash': 230, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 52, 'inner_gen_orders': [4, 1, 26], 'inner_gens': [[1, 4, 460], [1, 4, 8], [389, 4, 8]], 'inner_hash': 12, 'inner_nilpotent': False, 'inner_order': 104, 'inner_split': True, 'inner_tex': 'C_{26}:C_4', 'inner_used': [1, 3], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 32], [2, 8], [4, 48]], 'label': '832.230', 'linC_count': 384, 'linC_degree': 5, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 18, 'linQ_degree_count': 48, 'linQ_dim': 18, 'linQ_dim_count': 36, 'linR_count': 48, 'linR_degree': 6, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'D26.OD16', 'ngens': 7, 'nilpotency_class': -1, 'nilpotent': False, 'normal_counts': [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': 23, 'number_characteristic_subgroups': 34, 'number_conjugacy_classes': 88, 'number_divisions': 27, 'number_normal_subgroups': 50, 'number_subgroup_autclasses': 88, 'number_subgroup_classes': 118, 'number_subgroups': 946, 'old_label': None, 'order': 832, 'order_factorization_type': 51, 'order_stats': [[1, 1], [2, 55], [4, 264], [8, 320], [13, 12], [26, 36], [52, 48], [104, 96]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 12, 'outer_gen_orders': [2, 2, 2, 12], 'outer_gen_pows': [0, 0, 0, 0], 'outer_gens': [[417, 4, 424], [1, 4, 204], [1, 4, 408], [209, 4, 780]], 'outer_group': '96.221', 'outer_hash': 221, 'outer_nilpotent': True, 'outer_order': 96, 'outer_permdeg': 11, 'outer_perms': [2424, 744, 766080, 4033684], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_{12}:C_2^3', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 25, 'pgroup': 0, 'primary_abelian_invariants': [4, 8], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 10], [4, 6], [12, 2], [24, 3], [48, 2]], 'representations': {'PC': {'code': 7004825012609263011183427708564855919005866442763, 'gens': [1, 3, 4], 'pres': [7, -2, -2, -2, 2, -2, -2, -13, 14, 12883, 2810, 80, 2804, 7011, 102, 6725, 16812, 124, 15686, 18829]}, 'Perm': {'d': 25, 'gens': [27085524463079670033771, 80209890, 55085801063538531704164, 127391646, 127370880, 17764, 702508280364956153088000]}}, 'schur_multiplier': [2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [4, 8], 'solvability_type': 7, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'D_{26}.\\OD_{16}', 'transitive_degree': 208, 'wreath_data': None, 'wreath_product': False}
