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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': False, 'ambient': '1944.3893', 'ambient_counter': 3893, 'ambient_order': 1944, 'ambient_tex': 'C_3^3:F_9', 'central': False, 'central_factor': False, 'centralizer_order': 1, 'characteristic': False, 'core_order': 18, 'counter': 25, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '1944.3893.27.b1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': False, 'minimal': False, 'minimal_normal': False, 'nilpotent': False, 'normal': False, 'old_label': '27.b1', 'outer_equivalence': True, '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': 27, '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': '72.39', 'subgroup_hash': 39, 'subgroup_order': 72, 'subgroup_tex': 'F_9', 'supersolvable': False, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '1944.3893', 'aut_centralizer_order': 16, 'aut_label': '27.b1', 'aut_quo_index': None, 'aut_stab_index': 27, 'aut_weyl_group': '144.182', 'aut_weyl_index': 432, 'centralizer': '1944.a1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['3.a1', '9.b1'], 'contains': ['54.c1', '243.a1'], 'core': '108.a1', 'coset_action_label': None, 'count': 27, 'diagramx': [9693, -1, 853, -1], 'generators': [19342, 275331, 357088511410993, 147591, 2789705331707], 'label': '1944.3893.27.b1', 'mobius_quo': None, 'mobius_sub': 1, 'normal_closure': '1.a1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '27.b1', 'old_label': '27.b1', 'projective_image': '1944.3893', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '27.b1', 'subgroup_fusion': None, 'weyl_group': '72.39'}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '8.1', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 3, 'aut_exponent': 24, 'aut_gen_orders': [8, 2, 4, 2, 3, 3], 'aut_gens': [[1, 8, 24], [1, 56, 32], [3, 24, 8], [1, 24, 16], [1, 16, 48], [25, 8, 24], [33, 8, 24]], 'aut_group': '144.182', 'aut_hash': 182, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 144, 'aut_permdeg': 9, 'aut_perms': [46597, 12269, 94968, 321832, 99115, 51868], 'aut_phi_ratio': 6.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 9, 1, 1], [3, 8, 1, 1], [4, 9, 2, 1], [8, 9, 2, 2]], 'aut_supersolvable': False, 'aut_tex': 'F_9:C_2', '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': 24, 'autcentquo_group': '144.182', 'autcentquo_hash': 182, 'autcentquo_nilpotent': False, 'autcentquo_order': 144, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'F_9:C_2', 'cc_stats': [[1, 1, 1], [2, 9, 1], [3, 8, 1], [4, 9, 2], [8, 9, 4]], 'center_label': '1.1', 'center_order': 1, 'central_product': False, 'central_quotient': '72.39', 'commutator_count': 1, 'commutator_label': '9.2', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '3.1', '3.1'], 'composition_length': 5, 'conjugacy_classes_known': True, 'counter': 39, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 9, 1, 1], [3, 8, 1, 1], [4, 9, 2, 1], [8, 9, 4, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 24, 'exponent': 24, 'exponents_of_order': [3, 2], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [[8, 1, 1]], 'familial': True, 'frattini_label': '1.1', 'frattini_quotient': '72.39', 'hash': 39, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 24, 'inner_gen_orders': [8, 3, 3], 'inner_gens': [[1, 32, 40], [49, 8, 24], [9, 8, 24]], 'inner_hash': 39, 'inner_nilpotent': False, 'inner_order': 72, 'inner_split': True, 'inner_tex': 'F_9', 'inner_used': [1, 2], 'irrC_degree': 8, 'irrQ_degree': 8, 'irrQ_dim': 8, 'irrR_degree': 8, 'irrep_stats': [[1, 8], [8, 1]], 'label': '72.39', 'linC_count': 1, 'linC_degree': 8, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 8, 'linQ_degree_count': 1, 'linQ_dim': 8, 'linQ_dim_count': 1, 'linR_count': 1, 'linR_degree': 8, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'F9', 'ngens': 5, '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': 6, 'number_characteristic_subgroups': 5, 'number_conjugacy_classes': 9, 'number_divisions': 5, 'number_normal_subgroups': 5, 'number_subgroup_autclasses': 10, 'number_subgroup_classes': 10, 'number_subgroups': 48, 'old_label': None, 'order': 72, 'order_factorization_type': 32, 'order_stats': [[1, 1], [2, 9], [3, 8], [4, 18], [8, 36]], 'outer_abelian': True, 'outer_cyclic': True, 'outer_equivalence': False, 'outer_exponent': 2, 'outer_gen_orders': [2], 'outer_gen_pows': [0], 'outer_gens': [[3, 24, 8]], 'outer_group': '2.1', 'outer_hash': 1, 'outer_nilpotent': True, 'outer_order': 2, 'outer_permdeg': 2, 'outer_perms': [1], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 9, 'pgroup': 0, 'primary_abelian_invariants': [8], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 2], [2, 1], [4, 1], [8, 1]], 'representations': {'PC': {'code': 5483238280211454051, 'gens': [1, 4, 5], 'pres': [5, -2, -2, -2, -3, 3, 10, 26, 643, 488, 93, 1004, 109, 314]}, 'Lie': [{'d': 1, 'q': 9, 'gens': [62619, 5913], 'family': 'AGL'}], 'GLFp': {'d': 3, 'p': 3, 'gens': [11017, 8101, 10267, 8183, 7433]}, 'Perm': {'d': 9, 'gens': [23802, 37817, 5515, 276480, 80884]}}, 'schur_multiplier': [], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [8], 'solvability_type': 8, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': 'F_9', 'transitive_degree': 9, 'wreath_data': None, 'wreath_product': False}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '8.1', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 3, 'aut_exponent': 24, 'aut_gen_orders': [12, 24, 8, 12, 6], 'aut_gens': [[357088511410993, 2789705331707, 19342, 57983, 104624, 43333837747200, 756010141286400, 43545600], [380713864785518, 795159421163666, 154160, 336331, 104624, 1485698665651200, 735000173107200, 43545600], [1133934344322937, 732478229938354, 351059, 104624, 147591, 756010141286400, 1443759680563200, 43545600], [377924202841393, 753220435981307, 19342, 336331, 239460, 756010141286400, 1443759680563200, 79833600], [1866051363255177, 732478229866169, 314348, 239460, 147591, 85005060940800, 799076217139200, 43545600], [1781139671338377, 1509230577437369, 314348, 166201, 239460, 1443759680563200, 1506440871936000, 43545600]], 'aut_group': None, 'aut_hash': 7049098469290830887, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 62208, 'aut_permdeg': 486, 'aut_perms': 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'aut_phi_ratio': 96.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 9, 1, 1], [3, 2, 1, 1], [3, 4, 2, 1], [3, 4, 4, 1], [3, 8, 1, 1], [3, 8, 2, 1], [3, 8, 8, 1], [3, 8, 16, 1], [4, 81, 2, 1], [6, 18, 1, 1], [6, 36, 2, 1], [6, 36, 4, 1], [8, 243, 2, 2], [12, 162, 2, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_3^5.C_4^2.C_2^4', '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': 24, 'autcentquo_group': None, 'autcentquo_hash': 7049098469290830887, 'autcentquo_nilpotent': False, 'autcentquo_order': 62208, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'C_3^5.C_4^2.C_2^4', 'cc_stats': [[1, 1, 1], [2, 9, 1], [3, 2, 1], [3, 4, 6], [3, 8, 27], [4, 81, 2], [6, 18, 1], [6, 36, 6], [8, 243, 4], [12, 162, 2]], 'center_label': '1.1', 'center_order': 1, 'central_product': False, 'central_quotient': '1944.3893', 'commutator_count': 1, 'commutator_label': '243.67', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '3.1', '3.1', '3.1', '3.1', '3.1'], 'composition_length': 8, 'conjugacy_classes_known': True, 'counter': 3893, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 9, 1, 1], [3, 2, 1, 1], [3, 4, 1, 2], [3, 4, 2, 2], [3, 8, 1, 1], [3, 8, 2, 13], [4, 81, 2, 1], [6, 18, 1, 1], [6, 36, 1, 2], [6, 36, 2, 2], [8, 243, 4, 1], [12, 162, 2, 1]], 'element_repr_type': 'Perm', 'elementary': 1, 'eulerian_function': 24, 'exponent': 24, 'exponents_of_order': [5, 3], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [[8, 0, 16]], 'familial': False, 'frattini_label': '1.1', 'frattini_quotient': '1944.3893', 'hash': 3893, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 24, 'inner_gen_orders': [8, 4, 2, 3, 3, 3, 3, 3], 'inner_gens': [[357088511410993, 2789705331707, 19342, 104624, 147591, 1443759680563200, 1485698665651200, 79833600], [357088511410993, 2789705331707, 19342, 147591, 275331, 85005060940800, 1506440871936000, 43545600], [357088511410993, 2789705331707, 19342, 166201, 297796, 43333837747200, 756010141286400, 43545600], [357088511535304, 2789705406210, 169567, 57983, 104624, 43333837747200, 756010141286400, 43545600], [357088511668799, 2789705476682, 314348, 57983, 104624, 43333837747200, 756010141286400, 43545600], [1112830890802993, 87794766272507, 19342, 57983, 104624, 43333837747200, 756010141286400, 43545600], [421164555442993, 1509230577267707, 19342, 57983, 104624, 43333837747200, 756010141286400, 43545600], [357088591244593, 2789705331707, 19342, 57983, 104624, 43333837747200, 756010141286400, 43545600]], 'inner_hash': 3893, 'inner_nilpotent': False, 'inner_order': 1944, 'inner_split': True, 'inner_tex': 'C_3^3:F_9', 'inner_used': [1, 4, 6, 8], 'irrC_degree': 8, 'irrQ_degree': 16, 'irrQ_dim': 16, 'irrR_degree': 16, 'irrep_stats': [[1, 8], [2, 4], [4, 12], [8, 27]], 'label': '1944.3893', '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': 'C3^3:F9', 'ngens': 8, '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': 16, 'number_characteristic_subgroups': 15, 'number_conjugacy_classes': 51, 'number_divisions': 29, 'number_normal_subgroups': 15, 'number_subgroup_autclasses': 112, 'number_subgroup_classes': 440, 'number_subgroups': 4120, 'old_label': None, 'order': 1944, 'order_factorization_type': 33, 'order_stats': [[1, 1], [2, 9], [3, 242], [4, 162], [6, 234], [8, 972], [12, 324]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': True, 'outer_exponent': 8, 'outer_gen_orders': [2, 2, 8], 'outer_gen_pows': [19342, 19342, 0], 'outer_gens': [[357088511410993, 2789705331707, 19342, 336331, 239460, 43333837747200, 756010141286400, 79833600], [359610454856728, 2789705356603, 19342, 104624, 166201, 1443759680563200, 1506440871936000, 43545600], [357088511410993, 2789705331707, 19342, 104624, 147591, 1485698665651200, 85005060940800, 43545600]], 'outer_group': '32.40', 'outer_hash': 40, 'outer_nilpotent': True, 'outer_order': 32, 'outer_permdeg': 10, 'outer_perms': [1, 767647, 1272600], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2\\times \\SD_{16}', 'pc_rank': 6, 'perfect': False, 'permutation_degree': 18, 'pgroup': 0, 'primary_abelian_invariants': [8], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 2], [2, 3], [4, 6], [8, 5], [16, 13]], 'representations': {'PC': {'code': 25742640805760433951273635006530035508372537098502943, 'gens': [1, 4, 5, 6, 7, 8], 'pres': [8, -2, -2, -2, -3, -3, 3, 3, 3, 16, 41, 515, 19204, 972, 31109, 32845, 1749, 14790, 12110, 87559, 46095, 20759]}, 'Perm': {'d': 18, 'gens': [357088511410993, 2789705331707, 19342, 57983, 104624, 43333837747200, 756010141286400, 43545600]}}, 'schur_multiplier': [3], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [8], 'solvability_type': 8, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': 'C_3^3:F_9', 'transitive_degree': 24, 'wreath_data': None, 'wreath_product': False}
