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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': True, 'ambient': '448.1213', 'ambient_counter': 1213, 'ambient_order': 448, 'ambient_tex': '\\SD_{16}:D_{14}', 'central': False, 'central_factor': False, 'centralizer_order': 112, 'characteristic': False, 'core_order': 1, 'counter': 255, 'cyclic': True, 'direct': None, 'hall': 0, 'label': '448.1213.224.c1.b1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': True, 'minimal_normal': False, 'nilpotent': True, 'normal': False, 'old_label': '224.c1.b1', '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': 224, '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': '448.1213', 'aut_centralizer_order': None, 'aut_label': '224.c1', 'aut_quo_index': None, 'aut_stab_index': None, 'aut_weyl_group': None, 'aut_weyl_index': None, 'centralizer': '4.p1.a1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['32.c1.b1', '112.d1.b1', '112.j1.a1', '112.j1.b1'], 'contains': ['448.a1.a1'], 'core': '448.a1.a1', 'coset_action_label': None, 'count': 4, 'diagramx': None, 'generators': [3], 'label': '448.1213.224.c1.b1', 'mobius_quo': None, 'mobius_sub': None, 'normal_closure': '56.b1.b1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '4.p1.a1', 'old_label': '224.c1.b1', 'projective_image': None, 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '224.c1.b1', 'subgroup_fusion': None, 'weyl_group': None}
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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': '16.14', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 84, 'aut_gen_orders': [2, 12, 2, 6, 12, 14, 4, 12], 'aut_gens': [[1, 2, 4, 8], [3, 2, 230, 328], [1, 226, 102, 248], [115, 226, 260, 328], [113, 2, 70, 26], [113, 2, 390, 72], [227, 2, 324, 344], [113, 2, 68, 330], [225, 226, 70, 360]], 'aut_group': None, 'aut_hash': 8598609004174151551, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 21504, 'aut_permdeg': 44, 'aut_perms': [889658661130361708796866284383900238721311659549598727, 57008927448338046238284817069294364051304830956584499, 1323908876264596401024635744064012028728210127331746995, 806002296932791481443446308197826245748433628343324194, 809126126234024175128747343651020393248149583690368338, 479020569830682308638973764380445105714879991415582264, 797849378983640456836837249908965339345177638870322434, 2257602086510776411419588758291004030988078305202890184], 'aut_phi_ratio': 112.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 1, 2, 1], [2, 4, 2, 1], [2, 14, 2, 1], [4, 2, 1, 2], [4, 4, 2, 1], [4, 14, 2, 1], [4, 28, 2, 2], [7, 2, 3, 1], [8, 4, 2, 1], [8, 28, 2, 1], [14, 2, 3, 1], [14, 2, 6, 1], [14, 8, 6, 1], [28, 4, 3, 2], [28, 8, 6, 1], [56, 4, 12, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_7.(C_2^4\\times C_6).C_2^5', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 4, 'autcent_group': '128.1578', 'autcent_hash': 1578, 'autcent_nilpotent': True, 'autcent_order': 128, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^3\\wr C_2', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 42, 'autcentquo_group': '168.47', 'autcentquo_hash': 47, 'autcentquo_nilpotent': False, 'autcentquo_order': 168, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'C_2^2\\times F_7', 'cc_stats': [[1, 1, 1], [2, 1, 3], [2, 4, 2], [2, 14, 2], [4, 2, 2], [4, 4, 2], [4, 14, 2], [4, 28, 4], [7, 2, 3], [8, 4, 2], [8, 28, 2], [14, 2, 9], [14, 8, 6], [28, 4, 6], [28, 8, 6], [56, 4, 12]], 'center_label': '4.2', 'center_order': 4, 'central_product': True, 'central_quotient': '112.31', 'commutator_count': 1, 'commutator_label': '28.2', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '2.1', '2.1', '7.1'], 'composition_length': 7, 'conjugacy_classes_known': True, 'counter': 1213, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['2.1', 1], ['224.110', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [2, 4, 1, 2], [2, 14, 1, 2], [4, 2, 1, 2], [4, 4, 1, 2], [4, 14, 1, 2], [4, 28, 1, 4], [7, 2, 3, 1], [8, 4, 1, 2], [8, 28, 1, 2], [14, 2, 3, 3], [14, 8, 3, 2], [28, 4, 3, 2], [28, 8, 3, 2], [56, 4, 6, 2]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 574560, 'exponent': 56, 'exponents_of_order': [6, 1], 'factors_of_aut_order': [2, 3, 7], 'factors_of_order': [2, 7], 'faithful_reps': [], 'familial': False, 'frattini_label': '4.1', 'frattini_quotient': '112.42', 'hash': 1213, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 28, 'inner_gen_orders': [2, 1, 2, 28], 'inner_gens': [[1, 2, 228, 344], [1, 2, 4, 8], [225, 2, 4, 104], [113, 2, 356, 8]], 'inner_hash': 31, 'inner_nilpotent': False, 'inner_order': 112, 'inner_split': True, 'inner_tex': 'D_4\\times D_7', 'inner_used': [1, 3, 4], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 16], [2, 28], [4, 20]], 'label': '448.1213', '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': 'SD16:D14', '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': 21, 'number_characteristic_subgroups': 33, 'number_conjugacy_classes': 64, 'number_divisions': 34, 'number_normal_subgroups': 103, 'number_subgroup_autclasses': 136, 'number_subgroup_classes': 258, 'number_subgroups': 1156, 'old_label': None, 'order': 448, 'order_factorization_type': 51, 'order_stats': [[1, 1], [2, 39], [4, 152], [7, 6], [8, 64], [14, 66], [28, 72], [56, 48]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 12, 'outer_gen_orders': [2, 2, 2, 2, 12], 'outer_gen_pows': [0, 0, 0, 0, 0], 'outer_gens': [[3, 2, 4, 10], [3, 2, 4, 8], [3, 2, 6, 234], [1, 2, 228, 8], [227, 226, 4, 42]], 'outer_group': '192.1410', 'outer_hash': 1410, 'outer_nilpotent': True, 'outer_order': 192, 'outer_permdeg': 13, 'outer_perms': [1037877864, 41064, 24, 482631144, 1041468028], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^5:C_6', 'pc_rank': 4, 'perfect': False, 'permutation_degree': 25, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 2, 2], 'quasisimple': False, 'rank': 4, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 16], [2, 4], [4, 2], [6, 8], [12, 2], [24, 2]], 'representations': {'PC': {'code': 136931637622278297732535226723446453253, 'gens': [1, 2, 3, 4], 'pres': [7, -2, -2, -2, -2, -2, -2, -7, 4790, 9635, 745, 80, 8404, 1838, 102, 4387, 124, 4724]}, 'Perm': {'d': 25, 'gens': [6798636541085957573167, 1, 706294213562692347264000, 1344776913906383647468801, 1999773221030994941280000, 2644161760785465806054400, 50646]}}, 'schur_multiplier': [2, 2, 2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 2, 2], 'solvability_type': 7, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': '\\SD_{16}:D_{14}', 'transitive_degree': 224, 'wreath_data': None, 'wreath_product': False}
