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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': False, 'ambient': '784.61', 'ambient_counter': 61, 'ambient_order': 784, 'ambient_tex': 'C_{14}.D_{28}', 'central': False, 'central_factor': False, 'centralizer_order': 4, 'characteristic': False, 'core_order': 2, 'counter': 38, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '784.61.28.d1.b2', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': None, 'minimal': False, 'minimal_normal': False, 'nilpotent': False, 'normal': False, 'old_label': '28.d1.b2', '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': 28, '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': '28.3', 'subgroup_hash': 3, 'subgroup_order': 28, 'subgroup_tex': 'D_{14}', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '784.61', 'aut_centralizer_order': 4, 'aut_label': '28.d1', 'aut_quo_index': None, 'aut_stab_index': 168, 'aut_weyl_group': '84.7', 'aut_weyl_index': 672, 'centralizer': '196.a1.a1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['4.c1.a1', '14.c1.b1'], 'contains': ['56.d1.b1', '56.g1.b2', '56.g1.e2', '196.b1.a1'], 'core': '392.a1.a1', 'coset_action_label': None, 'count': 14, 'diagramx': [1511, -1, 5808, -1, 2953, -1, 7584, -1], 'generators': [395, 568, 392], 'label': '784.61.28.d1.b2', 'mobius_quo': None, 'mobius_sub': 0, 'normal_closure': '2.a1.a1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '14.c1.b1', 'old_label': '28.d1.b2', 'projective_image': '392.21', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '28.d1.b2', 'subgroup_fusion': None, 'weyl_group': '14.1'}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '4.2', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 42, 'aut_gen_orders': [6, 14], 'aut_gens': [[1, 2], [1, 6], [11, 2]], 'aut_group': '84.7', 'aut_hash': 7, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 84, 'aut_permdeg': 9, 'aut_perms': [6055, 203047], 'aut_phi_ratio': 7.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 7, 2, 1], [7, 2, 3, 1], [14, 2, 3, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_2\\times F_7', 'autcent_abelian': True, 'autcent_cyclic': True, 'autcent_exponent': 2, 'autcent_group': '2.1', 'autcent_hash': 1, 'autcent_nilpotent': True, 'autcent_order': 2, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 42, 'autcentquo_group': '42.1', 'autcentquo_hash': 1, 'autcentquo_nilpotent': False, 'autcentquo_order': 42, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'F_7', 'cc_stats': [[1, 1, 1], [2, 1, 1], [2, 7, 2], [7, 2, 3], [14, 2, 3]], 'center_label': '2.1', 'center_order': 2, 'central_product': True, 'central_quotient': '14.1', 'commutator_count': 1, 'commutator_label': '7.1', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '7.1'], 'composition_length': 3, 'conjugacy_classes_known': True, 'counter': 3, 'cyclic': False, 'derived_length': 2, 'dihedral': True, 'direct_factorization': [['14.1', 1], ['2.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 7, 1, 2], [7, 2, 3, 1], [14, 2, 3, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 3, 'exponent': 14, 'exponents_of_order': [2, 1], 'factors_of_aut_order': [2, 3, 7], 'factors_of_order': [2, 7], 'faithful_reps': [[2, 1, 3]], 'familial': True, 'frattini_label': '1.1', 'frattini_quotient': '28.3', 'hash': 3, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 14, 'inner_gen_orders': [2, 7], 'inner_gens': [[1, 26], [5, 2]], 'inner_hash': 1, 'inner_nilpotent': False, 'inner_order': 14, 'inner_split': True, 'inner_tex': 'D_7', 'inner_used': [1, 2], 'irrC_degree': 2, 'irrQ_degree': 6, 'irrQ_dim': 6, 'irrR_degree': 2, 'irrep_stats': [[1, 4], [2, 6]], 'label': '28.3', 'linC_count': 3, 'linC_degree': 2, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 6, 'linQ_degree_count': 1, 'linQ_dim': 6, 'linQ_dim_count': 1, 'linR_count': 3, 'linR_degree': 2, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'D14', 'ngens': 3, 'nilpotency_class': -1, 'nilpotent': False, 'normal_counts': [0, 0, 0, 0, 0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 5, 'number_characteristic_subgroups': 5, 'number_conjugacy_classes': 10, 'number_divisions': 6, 'number_normal_subgroups': 7, 'number_subgroup_autclasses': 8, 'number_subgroup_classes': 10, 'number_subgroups': 28, 'old_label': None, 'order': 28, 'order_factorization_type': 22, 'order_stats': [[1, 1], [2, 15], [7, 6], [14, 6]], 'outer_abelian': True, 'outer_cyclic': True, 'outer_equivalence': False, 'outer_exponent': 6, 'outer_gen_orders': [6], 'outer_gen_pows': [0], 'outer_gens': [[15, 10]], 'outer_group': '6.2', 'outer_hash': 2, 'outer_nilpotent': True, 'outer_order': 6, 'outer_permdeg': 5, 'outer_perms': [28], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_6', 'pc_rank': 2, 'perfect': False, 'permutation_degree': 9, 'pgroup': 0, 'primary_abelian_invariants': [2, 2], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [6, 2]], 'representations': {'PC': {'code': 28266197, 'gens': [1, 2], 'pres': [3, -2, -2, -7, 157, 16, 218]}, 'GLZ': {'b': 3, 'd': 6, 'gens': [26761470676238614, 122882691976452762]}, 'GLFp': {'d': 2, 'p': 7, 'gens': [351, 2059, 2064]}, 'Perm': {'d': 9, 'gens': [5166, 1, 50646]}}, 'schur_multiplier': [2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2], 'solvability_type': 6, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'D_{14}', 'transitive_degree': 14, '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': '8.2', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 3, 'aut_exponent': 84, 'aut_gen_orders': [6, 12, 6, 6, 42], 'aut_gens': [[1, 2, 56], [417, 498, 168], [294, 261, 20], [441, 402, 616], [37, 286, 616], [17, 290, 56]], 'aut_group': None, 'aut_hash': 1520373495714903548, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 56448, 'aut_permdeg': 56, 'aut_perms': [161762553786710758117390235289347109014566925694975222727392218660486329460, 324067325609204538456165273261961327751836150773063614330626977645338260363, 72889628226951902902658418967103738842455766455310233011549365166066450168, 45373710044838608684471416573195609846450378417878583251925030643800798053, 382754636916805046304444686568473585363613621058946001398122800046780434002], 'aut_phi_ratio': 168.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 1, 2, 1], [2, 98, 2, 1], [4, 14, 4, 1], [7, 2, 6, 1], [7, 4, 9, 1], [14, 2, 6, 3], [14, 4, 9, 1], [14, 4, 18, 1], [28, 14, 24, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_7:D_7:C_3.C_2.C_6.C_2^4', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 2, 'autcent_group': '16.14', 'autcent_hash': 14, 'autcent_nilpotent': True, 'autcent_order': 16, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^4', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 84, 'autcentquo_group': '3528.br', 'autcentquo_hash': 519392570028930314, 'autcentquo_nilpotent': False, 'autcentquo_order': 3528, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'F_7\\wr C_2', 'cc_stats': [[1, 1, 1], [2, 1, 3], [2, 98, 2], [4, 14, 4], [7, 2, 6], [7, 4, 9], [14, 2, 18], [14, 4, 27], [28, 14, 24]], 'center_label': '4.2', 'center_order': 4, 'central_product': False, 'central_quotient': '196.9', 'commutator_count': 1, 'commutator_label': '98.5', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '7.1', '7.1'], 'composition_length': 6, 'conjugacy_classes_known': True, 'counter': 61, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [2, 98, 1, 2], [4, 14, 2, 2], [7, 2, 3, 2], [7, 4, 3, 3], [14, 2, 3, 6], [14, 4, 3, 9], [28, 14, 12, 2]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 3, 'exponent': 28, 'exponents_of_order': [4, 2], 'factors_of_aut_order': [2, 3, 7], 'factors_of_order': [2, 7], 'faithful_reps': [], 'familial': False, 'frattini_label': '4.2', 'frattini_quotient': '196.9', 'hash': 61, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 14, 'inner_gen_orders': [2, 14, 7], 'inner_gens': [[1, 446, 56], [397, 2, 728], [1, 114, 56]], 'inner_hash': 9, 'inner_nilpotent': False, 'inner_order': 196, 'inner_split': True, 'inner_tex': 'D_7^2', 'inner_used': [1, 2, 3], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 8], [2, 50], [4, 36]], 'label': '784.61', 'linC_count': 216, 'linC_degree': 4, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 14, 'linQ_degree_count': 24, 'linQ_dim': 14, 'linQ_dim_count': 24, 'linR_count': 36, 'linR_degree': 4, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'C14.D28', 'ngens': 6, 'nilpotency_class': -1, 'nilpotent': False, 'normal_counts': [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': 13, 'number_characteristic_subgroups': 8, 'number_conjugacy_classes': 94, 'number_divisions': 30, 'number_normal_subgroups': 28, 'number_subgroup_autclasses': 45, 'number_subgroup_classes': 116, 'number_subgroups': 1376, 'old_label': None, 'order': 784, 'order_factorization_type': 32, 'order_stats': [[1, 1], [2, 199], [4, 56], [7, 48], [14, 144], [28, 336]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 12, 'outer_gen_orders': [2, 4, 6, 6], 'outer_gen_pows': [0, 0, 0, 0], 'outer_gens': [[421, 30, 56], [14, 225, 52], [393, 426, 616], [421, 410, 616]], 'outer_group': '288.724', 'outer_hash': 724, 'outer_nilpotent': False, 'outer_order': 288, 'outer_permdeg': 14, 'outer_perms': [39916800, 13452729360, 747, 403224], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_6^2:D_4', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 22, 'pgroup': 0, 'primary_abelian_invariants': [2, 4], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 4], [6, 4], [12, 18]], 'representations': {'PC': {'code': 34968744627080427978299469610080377, 'gens': [1, 2, 5], 'pres': [6, -2, -2, -2, -7, -2, -7, 2352, 5353, 31, 938, 50, 1155, 10930, 88, 12107]}, 'Perm': {'d': 22, 'gens': [99632700849, 2439325479507840007, 274467916816, 16, 55969571858264064000, 4364760]}}, 'schur_multiplier': [2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 4], 'solvability_type': 7, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_{14}.D_{28}', 'transitive_degree': 56, 'wreath_data': None, 'wreath_product': False}
