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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': True, 'ambient': '729.355', 'ambient_counter': 355, 'ambient_order': 729, 'ambient_tex': 'C_3^3.C_3^3', 'central': False, 'central_factor': False, 'centralizer_order': 243, 'characteristic': False, 'core_order': 3, 'counter': 95, 'cyclic': True, 'direct': None, 'hall': 0, 'label': '729.355.81.g1.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': False, 'old_label': '81.g1.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': 81, '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': '9.1', 'subgroup_hash': 1, 'subgroup_order': 9, 'subgroup_tex': 'C_9', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '729.355', 'aut_centralizer_order': 1458, 'aut_label': '81.g1', 'aut_quo_index': None, 'aut_stab_index': 3, 'aut_weyl_group': '3.1', 'aut_weyl_index': 4374, 'centralizer': '3.b1.a1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['27.b1.a1', '27.m1.a1', '27.n1.a1', '27.n1.a2', '27.n1.a3'], 'contains': ['243.a1.a1'], 'core': '243.a1.a1', 'coset_action_label': None, 'count': 3, 'diagramx': [1545, -1, 4178, -1, 1544, -1, 4109, -1], 'generators': [9], 'label': '729.355.81.g1.a1', 'mobius_quo': None, 'mobius_sub': 0, 'normal_closure': '27.b1.a1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '3.b1.a1', 'old_label': '81.g1.a1', 'projective_image': '243.58', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '81.g1.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': '9.1', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': True, 'aut_cyclic': True, 'aut_derived_length': 1, 'aut_exponent': 6, 'aut_gen_orders': [6], 'aut_gens': [[1], [2]], 'aut_group': '6.2', 'aut_hash': 2, 'aut_nilpotency_class': 1, 'aut_nilpotent': True, 'aut_order': 6, 'aut_permdeg': 5, 'aut_perms': [27], 'aut_phi_ratio': 1.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [3, 1, 2, 1], [9, 1, 6, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_6', 'autcent_abelian': True, 'autcent_cyclic': True, 'autcent_exponent': 6, 'autcent_group': '6.2', 'autcent_hash': 2, 'autcent_nilpotent': True, 'autcent_order': 6, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_6', '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], [3, 1, 2], [9, 1, 6]], 'center_label': '9.1', 'center_order': 9, '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': ['3.1', '3.1'], 'composition_length': 2, 'conjugacy_classes_known': True, 'counter': 1, 'cyclic': True, 'derived_length': 1, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [3, 1, 2, 1], [9, 1, 6, 1]], 'element_repr_type': 'PC', 'elementary': 3, 'eulerian_function': 1, 'exponent': 9, 'exponents_of_order': [2], 'factors_of_aut_order': [2, 3], 'factors_of_order': [3], 'faithful_reps': [[1, 0, 6]], 'familial': True, 'frattini_label': '3.1', 'frattini_quotient': '3.1', 'hash': 1, 'hyperelementary': 3, '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': 6, 'irrQ_dim': 6, 'irrR_degree': 2, 'irrep_stats': [[1, 9]], 'label': '9.1', 'linC_count': 6, 'linC_degree': 1, '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': 'C9', 'ngens': 1, 'nilpotency_class': 1, 'nilpotent': True, 'normal_counts': [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': 9, 'number_divisions': 3, 'number_normal_subgroups': 3, 'number_subgroup_autclasses': 3, 'number_subgroup_classes': 3, 'number_subgroups': 3, 'old_label': None, 'order': 9, 'order_factorization_type': 2, 'order_stats': [[1, 1], [3, 2], [9, 6]], 'outer_abelian': True, 'outer_cyclic': True, 'outer_equivalence': False, 'outer_exponent': 6, 'outer_gen_orders': [6], 'outer_gen_pows': [0], 'outer_gens': [[2]], 'outer_group': '6.2', 'outer_hash': 2, 'outer_nilpotent': True, 'outer_order': 6, 'outer_permdeg': 5, 'outer_perms': [27], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_6', 'pc_rank': 1, 'perfect': False, 'permutation_degree': 9, 'pgroup': 3, 'primary_abelian_invariants': [9], 'quasisimple': False, 'rank': 1, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 1], [2, 1], [6, 1]], 'representations': {'PC': {'code': 5, 'gens': [1], 'pres': [2, -3, -3, 6]}, 'GLZ': {'b': 3, 'd': 6, 'gens': [41907234971425459]}, 'GLFp': {'d': 2, 'p': 17, 'gens': [78572]}, 'Perm': {'d': 9, 'gens': [357120, 80884]}}, 'schur_multiplier': [], 'semidirect_product': False, 'simple': False, 'smith_abelian_invariants': [9], 'solvability_type': 0, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_9', 'transitive_degree': 9, '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': '27.5', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 6, 'aut_gen_orders': [6, 6, 6, 6, 3, 3], 'aut_gens': [[1, 3, 9, 81], [32, 6, 549, 354], [59, 519, 252, 351], [59, 303, 549, 87], [491, 6, 36, 624], [34, 516, 9, 597], [4, 300, 36, 114]], 'aut_group': None, 'aut_hash': 545426196672643097, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 13122, 'aut_permdeg': 63, 'aut_perms': [223892941276936202605427202200837499465178720908896068040974533208572677945439127645327, 31977694508668918481864960088567763538054276300875766625504815972930585139811362729938, 223892941276936307008881517755341823532041599965140759439374976859671540952642393340879, 127935741115422915416683968216764153974006725539381303750583658173905768309123487496256, 164639992466873735842671590045764756701861137287501268745220165451545424, 127935741115422979097998682374305646934676150622792903391438100250471953083201833344338], 'aut_phi_ratio': 27.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [3, 1, 1, 8], [3, 3, 2, 3], [3, 9, 6, 1], [3, 27, 2, 2], [9, 3, 3, 2], [9, 9, 2, 2], [9, 9, 3, 6], [9, 9, 6, 2], [9, 27, 2, 4]], 'aut_supersolvable': True, 'aut_tex': 'C_3^4\\times C_3^3:S_3', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 3, 'autcent_group': '729.504', 'autcent_hash': 504, 'autcent_nilpotent': True, 'autcent_order': 729, 'autcent_solvable': True, 'autcent_split': False, 'autcent_supersolvable': True, 'autcent_tex': 'C_3^6', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 6, 'autcentquo_group': '18.3', 'autcentquo_hash': 3, 'autcentquo_nilpotent': False, 'autcentquo_order': 18, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'C_3\\times S_3', 'cc_stats': [[1, 1, 1], [3, 1, 8], [3, 3, 6], [3, 9, 6], [3, 27, 4], [9, 3, 6], [9, 9, 34], [9, 27, 8]], 'center_label': '9.2', 'center_order': 9, 'central_product': False, 'central_quotient': '81.12', 'commutator_count': 1, 'commutator_label': '27.5', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['3.1', '3.1', '3.1', '3.1', '3.1', '3.1'], 'composition_length': 6, 'conjugacy_classes_known': True, 'counter': 355, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [3, 1, 2, 4], [3, 3, 2, 3], [3, 9, 2, 3], [3, 27, 2, 2], [9, 3, 6, 1], [9, 9, 2, 5], [9, 9, 6, 4], [9, 27, 2, 4]], 'element_repr_type': 'PC', 'elementary': 3, 'eulerian_function': 16848, 'exponent': 9, 'exponents_of_order': [6], 'factors_of_aut_order': [2, 3], 'factors_of_order': [3], 'faithful_reps': [], 'familial': False, 'frattini_label': '27.5', 'frattini_quotient': '27.5', 'hash': 355, 'hyperelementary': 3, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 3, 'inner_gen_orders': [3, 3, 3, 3], 'inner_gens': [[1, 516, 9, 84], [298, 3, 9, 81], [1, 3, 9, 324], [7, 3, 495, 81]], 'inner_hash': 12, 'inner_nilpotent': True, 'inner_order': 81, 'inner_split': True, 'inner_tex': 'C_3\\times \\He_3', 'inner_used': [1, 2, 3, 4], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 27], [3, 42], [9, 4]], 'label': '729.355', 'linC_count': 324, 'linC_degree': 6, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 24, 'linQ_degree_count': 15, 'linQ_dim': 24, 'linQ_dim_count': 15, 'linR_count': 81, 'linR_degree': 12, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'C3^3.C3^3', 'ngens': 3, 'nilpotency_class': 3, 'nilpotent': True, 'normal_counts': [0, 0, 0, 0, 0, 0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 31, 'number_characteristic_subgroups': 20, 'number_conjugacy_classes': 73, 'number_divisions': 27, 'number_normal_subgroups': 43, 'number_subgroup_autclasses': 81, 'number_subgroup_classes': 134, 'number_subgroups': 562, 'old_label': None, 'order': 729, 'order_factorization_type': 3, 'order_stats': [[1, 1], [3, 188], [9, 540]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 6, 'outer_gen_orders': [3, 3, 6, 3], 'outer_gen_pows': [0, 0, 0, 0], 'outer_gens': [[1, 3, 495, 81], [1, 3, 9, 594], [2, 519, 306, 81], [244, 3, 9, 81]], 'outer_group': '162.51', 'outer_hash': 51, 'outer_nilpotent': False, 'outer_order': 162, 'outer_permdeg': 12, 'outer_perms': [244, 144, 171002160, 134305920], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'S_3\\times C_3^3', 'pc_rank': 4, 'perfect': False, 'permutation_degree': 36, 'pgroup': 3, 'primary_abelian_invariants': [3, 3, 3], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 1], [2, 13], [6, 6], [18, 7]], 'representations': {'PC': {'code': 329060390198608180, 'gens': [1, 2, 3, 5], 'pres': [6, 3, 3, 3, 3, 3, 3, 6193, 68, 2524, 1096, 118]}, 'Perm': {'d': 36, 'gens': [12183800246267215569781660475425318897668, 22822425369578395314170292761614048377401, 31095063165338359457866671016434907814400, 43732644043298441989459486545617138646913, 53840356007958409139381551062772296921600, 64772813369522083038401720571569143891200]}}, 'schur_multiplier': [3, 3, 3], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [3, 3, 3], 'solvability_type': 4, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_3^3.C_3^3', 'transitive_degree': 243, 'wreath_data': None, 'wreath_product': False}
