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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': True, 'ambient': '256.12509', 'ambient_counter': 12509, 'ambient_order': 256, 'ambient_tex': 'C_4^2.C_2^4', 'central': False, 'central_factor': False, 'centralizer_order': 32, 'characteristic': False, 'core_order': 8, 'counter': 171, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '256.12509.16.u1.b1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': False, 'old_label': '16.u1.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': 16, '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': '16.2', 'subgroup_hash': 2, 'subgroup_order': 16, 'subgroup_tex': 'C_4^2', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '256.12509', 'aut_centralizer_order': None, 'aut_label': '16.u1', 'aut_quo_index': None, 'aut_stab_index': None, 'aut_weyl_group': None, 'aut_weyl_index': None, 'centralizer': '8.bl1.b1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['8.e1.b1', '8.t1.a1', '8.bi1.a1', '8.bl1.b1', '8.bl2.b1', '8.bm1.b1', '8.bm2.b1'], 'contains': ['32.b1.a1', '32.i1.b1', '32.j1.b1'], 'core': '32.b1.a1', 'coset_action_label': None, 'count': 2, 'diagramx': [6509, -1, 4018, -1, 6534, -1, 4027, -1], 'generators': [3, 64], 'label': '256.12509.16.u1.b1', 'mobius_quo': None, 'mobius_sub': 0, 'normal_closure': '8.e1.b1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '2.e1.a1', 'old_label': '16.u1.b1', 'projective_image': None, 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '16.u1.b1', 'subgroup_fusion': None, 'weyl_group': None}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': True, 'abelian_quotient': '16.2', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 3, 'aut_exponent': 12, 'aut_gen_orders': [2, 2, 6, 2, 2], 'aut_gens': [[1, 4], [3, 5], [3, 12], [14, 13], [9, 6], [3, 14]], 'aut_group': '96.195', 'aut_hash': 195, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 96, 'aut_permdeg': 8, 'aut_perms': [134, 16, 1447, 11520, 5160], 'aut_phi_ratio': 12.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 3, 1], [4, 1, 12, 1]], 'aut_supersolvable': False, 'aut_tex': '\\GL(2,\\mathbb{Z}/4)', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 12, 'autcent_group': '96.195', 'autcent_hash': 195, 'autcent_nilpotent': False, 'autcent_order': 96, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': False, 'autcent_tex': '\\GL(2,\\mathbb{Z}/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], [2, 1, 3], [4, 1, 12]], 'center_label': '16.2', 'center_order': 16, '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', '2.1', '2.1'], 'composition_length': 4, 'conjugacy_classes_known': True, 'counter': 2, 'cyclic': False, 'derived_length': 1, 'dihedral': False, 'direct_factorization': [['4.1', 2]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [4, 1, 2, 6]], 'element_repr_type': 'PC', 'elementary': 2, 'eulerian_function': 1, 'exponent': 4, 'exponents_of_order': [4], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2], 'faithful_reps': [], 'familial': False, 'frattini_label': '4.2', '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, 4], [1, 4]], '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, 16]], 'label': '16.2', 'linC_count': 48, 'linC_degree': 2, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 4, 'linQ_degree_count': 12, 'linQ_dim': 4, 'linQ_dim_count': 12, 'linR_count': 12, 'linR_degree': 4, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C4^2', 'ngens': 2, 'nilpotency_class': 1, 'nilpotent': True, 'normal_counts': [0, 0, 0, 0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 3, 'number_characteristic_subgroups': 3, 'number_conjugacy_classes': 16, 'number_divisions': 10, 'number_normal_subgroups': 15, 'number_subgroup_autclasses': 6, 'number_subgroup_classes': 15, 'number_subgroups': 15, 'old_label': None, 'order': 16, 'order_factorization_type': 3, 'order_stats': [[1, 1], [2, 3], [4, 12]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 12, 'outer_gen_orders': [2, 2, 6, 2, 2], 'outer_gen_pows': [0, 0, 0, 0, 0], 'outer_gens': [[3, 5], [3, 12], [14, 13], [9, 6], [3, 14]], 'outer_group': '96.195', 'outer_hash': 195, 'outer_nilpotent': False, 'outer_order': 96, 'outer_permdeg': 8, 'outer_perms': [134, 16, 1447, 11520, 5160], 'outer_solvable': True, 'outer_supersolvable': False, 'outer_tex': '\\GL(2,\\mathbb{Z}/4)', 'pc_rank': 2, 'perfect': False, 'permutation_degree': 8, 'pgroup': 2, 'primary_abelian_invariants': [4, 4], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 6]], 'representations': {'PC': {'code': 10245, 'gens': [1, 3], 'pres': [4, 2, 2, 2, 2, 8, 34]}, 'GLZ': {'b': 3, 'd': 4, 'gens': [16917782, 35931238]}, 'GLFp': {'d': 2, 'p': 5, 'gens': [251, 377]}, 'Perm': {'d': 8, 'gens': [16560, 22, 5160, 7]}}, 'schur_multiplier': [4], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [4, 4], 'solvability_type': 1, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_4^2', 'transitive_degree': 16, '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': 4, 'aut_gen_orders': [2, 2, 2, 4, 4, 4, 4, 4], 'aut_gens': [[1, 2, 4, 32], [129, 10, 156, 112], [17, 154, 28, 112], [17, 130, 148, 240], [145, 67, 141, 104], [145, 138, 220, 32], [17, 154, 4, 32], [129, 2, 220, 176], [145, 203, 93, 232]], 'aut_group': None, 'aut_hash': 5881906593575344611, 'aut_nilpotency_class': 2, 'aut_nilpotent': True, 'aut_order': 8192, 'aut_permdeg': 48, 'aut_perms': [7937101482942227503746333454431119931530246126323833099964128, 2384688650769100178979655843873248212040347293521878064929248, 5291856004995458059690824267973006971725641817112821681620165, 2620527253998221229142994783952371532031054653888748300981711, 2443087770591139662453800790717113970304064376594149019194229, 11623881476522574833791081074607517298695116311658470558713161, 3964409943419644123646862046440451104424186199587400949230782, 6644892343978855621545741384409573172618034901587468819417851], 'aut_phi_ratio': 64.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [2, 4, 1, 1], [2, 16, 1, 1], [4, 2, 2, 2], [4, 4, 1, 4], [4, 8, 2, 2], [4, 16, 1, 3], [8, 4, 4, 1], [8, 8, 2, 1], [8, 8, 4, 2], [8, 16, 2, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_2^9.C_2^4', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 2, 'autcent_group': '256.56092', 'autcent_hash': 56092, 'autcent_nilpotent': True, 'autcent_order': 256, 'autcent_solvable': True, 'autcent_split': False, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^8', 'autcentquo_abelian': True, 'autcentquo_cyclic': False, 'autcentquo_exponent': 2, 'autcentquo_group': '32.51', 'autcentquo_hash': 51, 'autcentquo_nilpotent': True, 'autcentquo_order': 32, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'C_2^5', 'cc_stats': [[1, 1, 1], [2, 1, 3], [2, 4, 1], [2, 16, 1], [4, 2, 4], [4, 4, 4], [4, 8, 4], [4, 16, 3], [8, 4, 4], [8, 8, 10], [8, 16, 2]], 'center_label': '4.2', 'center_order': 4, 'central_product': False, 'central_quotient': '64.203', 'commutator_count': 1, 'commutator_label': '16.2', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '2.1', '2.1', '2.1', '2.1'], 'composition_length': 8, 'conjugacy_classes_known': True, 'counter': 12509, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [2, 4, 1, 1], [2, 16, 1, 1], [4, 2, 1, 4], [4, 4, 1, 4], [4, 8, 1, 4], [4, 16, 1, 3], [8, 4, 2, 2], [8, 8, 1, 2], [8, 8, 2, 4], [8, 16, 1, 2]], 'element_repr_type': 'PC', 'elementary': 2, 'eulerian_function': 161280, 'exponent': 8, 'exponents_of_order': [8], 'factors_of_aut_order': [2], 'factors_of_order': [2], 'faithful_reps': [], 'familial': False, 'frattini_label': '16.2', 'frattini_quotient': '16.14', 'hash': 12509, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 4, 'inner_gen_orders': [2, 2, 4, 4], 'inner_gens': [[1, 130, 132, 48], [129, 2, 140, 32], [129, 154, 4, 96], [17, 2, 196, 32]], 'inner_hash': 203, 'inner_nilpotent': True, 'inner_order': 64, 'inner_split': True, 'inner_tex': 'C_2^3:D_4', 'inner_used': [1, 2, 3, 4], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 16], [2, 12], [4, 8], [8, 1]], 'label': '256.12509', 'linC_count': 16, 'linC_degree': 8, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 16, 'linQ_degree_count': 8, 'linQ_dim': 16, 'linQ_dim_count': 4, 'linR_count': 4, 'linR_degree': 16, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'C4^2.C2^4', 'ngens': 4, 'nilpotency_class': 3, 'nilpotent': True, 'normal_counts': [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': 47, 'number_conjugacy_classes': 37, 'number_divisions': 31, 'number_normal_subgroups': 105, 'number_subgroup_autclasses': 205, 'number_subgroup_classes': 291, 'number_subgroups': 727, 'old_label': None, 'order': 256, 'order_factorization_type': 7, 'order_stats': [[1, 1], [2, 23], [4, 104], [8, 128]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 4, 'outer_gen_orders': [2, 2, 2, 2, 2, 4], 'outer_gen_pows': [0, 0, 0, 0, 0, 0], 'outer_gens': [[1, 2, 4, 112], [129, 130, 4, 32], [17, 2, 4, 32], [1, 2, 20, 32], [1, 130, 4, 32], [17, 195, 21, 184]], 'outer_group': '128.1009', 'outer_hash': 1009, 'outer_nilpotent': True, 'outer_order': 128, 'outer_permdeg': 14, 'outer_perms': [14377673521, 19239897655, 21196264375, 288, 21196264608, 32627710206], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^5:C_4', 'pc_rank': 4, 'perfect': False, 'permutation_degree': 32, 'pgroup': 2, 'primary_abelian_invariants': [2, 2, 2, 2], 'quasisimple': False, 'rank': 4, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 16], [2, 8], [4, 2], [8, 5]], 'representations': {'PC': {'code': 10718209585077369834943883663686814190, 'gens': [1, 2, 3, 6], 'pres': [8, 2, 2, 2, 2, 2, 2, 2, 2, 2081, 137, 3170, 1690, 66, 395, 91, 2309, 1173, 141, 2710, 166]}, 'Perm': {'d': 32, 'gens': [9620189608523220657277537279019806, 17268312130521262350485277730445136, 25207874400137674715523130071627653, 34538973774836392403927035693471009, 43051836646078837836722321827071649, 5794634545962, 51550024053101171754201062105088000, 4315567359649]}}, 'schur_multiplier': [2, 2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 2, 2], 'solvability_type': 4, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_4^2.C_2^4', 'transitive_degree': 128, 'wreath_data': None, 'wreath_product': False}
