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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': True, 'ambient': '729.497', 'ambient_counter': 497, 'ambient_order': 729, 'ambient_tex': 'C_3^4\\times C_9', 'central': True, 'central_factor': False, 'centralizer_order': 729, 'characteristic': False, 'core_order': 3, 'counter': 13, 'cyclic': True, 'direct': True, 'hall': 0, 'label': '729.497.243.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': True, 'minimal_normal': True, 'nilpotent': True, 'normal': True, 'old_label': '243.a1', 'outer_equivalence': True, 'perfect': False, 'proper': True, 'quotient': '243.61', 'quotient_Agroup': True, 'quotient_abelian': True, 'quotient_cyclic': False, 'quotient_hash': 61, 'quotient_metabelian': True, 'quotient_nilpotent': True, 'quotient_order': 243, 'quotient_simple': False, 'quotient_solvable': True, 'quotient_supersolvable': True, 'quotient_tex': 'C_3^3\\times C_9', 'simple': True, 'solvable': True, 'special_labels': [], 'split': True, 'standard_generators': False, 'stem': False, 'subgroup': '3.1', 'subgroup_hash': 1, 'subgroup_order': 3, 'subgroup_tex': 'C_3', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '729.497', 'aut_centralizer_order': 3979430208, 'aut_label': '243.a1', 'aut_quo_index': 1, 'aut_stab_index': 120, 'aut_weyl_group': '2.1', 'aut_weyl_index': 477531624960, 'centralizer': '1.a1', 'complements': ['3.a1'], 'conjugacy_class_count': 120, 'contained_in': ['81.a1', '81.b1'], 'contains': ['729.a1'], 'core': '243.a1', 'coset_action_label': None, 'count': 120, 'diagramx': [4550, 4550, 3987, 3987], 'generators': [9], 'label': '729.497.243.a1', 'mobius_quo': -1, 'mobius_sub': 0, 'normal_closure': '243.a1', 'normal_contained_in': ['81.a1', '81.b1'], 'normal_contains': ['729.a1'], 'normalizer': '1.a1', 'old_label': '243.a1', 'projective_image': '243.61', 'quotient_action_image': '1.1', 'quotient_action_kernel': '243.61', 'quotient_action_kernel_order': 243, 'quotient_fusion': None, 'short_label': '243.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': '3.1', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': True, 'aut_cyclic': True, 'aut_derived_length': 1, 'aut_exponent': 2, 'aut_gen_orders': [2], 'aut_gens': [[1], [2]], 'aut_group': '2.1', 'aut_hash': 1, 'aut_nilpotency_class': 1, 'aut_nilpotent': True, 'aut_order': 2, 'aut_permdeg': 2, 'aut_perms': [1], 'aut_phi_ratio': 1.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [3, 1, 2, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_2', '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': 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]], 'center_label': '3.1', 'center_order': 3, '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'], 'composition_length': 1, '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]], 'element_repr_type': 'PC', 'elementary': 3, 'eulerian_function': 1, 'exponent': 3, 'exponents_of_order': [1], 'factors_of_aut_order': [2], 'factors_of_order': [3], 'faithful_reps': [[1, 0, 2]], 'familial': True, 'frattini_label': '1.1', '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': 2, 'irrQ_dim': 2, 'irrR_degree': 2, 'irrep_stats': [[1, 3]], 'label': '3.1', 'linC_count': 2, 'linC_degree': 1, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 2, 'linQ_degree_count': 1, 'linQ_dim': 2, 'linQ_dim_count': 1, 'linR_count': 1, 'linR_degree': 2, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C3', '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': 3, 'number_divisions': 2, 'number_normal_subgroups': 2, 'number_subgroup_autclasses': 2, 'number_subgroup_classes': 2, 'number_subgroups': 2, 'old_label': None, 'order': 3, 'order_factorization_type': 1, 'order_stats': [[1, 1], [3, 2]], 'outer_abelian': True, 'outer_cyclic': True, 'outer_equivalence': False, 'outer_exponent': 2, 'outer_gen_orders': [2], 'outer_gen_pows': [0], 'outer_gens': [[2]], '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': 1, 'perfect': False, 'permutation_degree': 3, 'pgroup': 3, 'primary_abelian_invariants': [3], 'quasisimple': False, 'rank': 1, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 1], [2, 1]], 'representations': {'PC': {'code': 0, 'gens': [1], 'pres': [1, -3]}, 'GLZ': {'b': 3, 'd': 2, 'gens': [7]}, 'Lie': [{'d': 1, 'q': 3, 'gens': [3], 'family': 'ASigmaL'}], 'GLFp': {'d': 2, 'p': 2, 'gens': [7]}, 'Perm': {'d': 3, 'gens': [4]}}, 'schur_multiplier': [], 'semidirect_product': False, 'simple': True, 'smith_abelian_invariants': [3], 'solvability_type': 0, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_3', 'transitive_degree': 3, 'wreath_data': None, 'wreath_product': False}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': True, 'abelian_quotient': '729.497', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 1, 'aut_exponent': 9360, 'aut_gen_orders': [78, 6, 24], 'aut_gens': [[1, 3, 9, 27, 81], [535, 313, 18, 281, 146], [271, 73, 318, 527, 681], [11, 64, 51, 9, 464]], 'aut_group': None, 'aut_hash': 8072738789780754013, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 955063249920, 'aut_permdeg': 486, 'aut_perms': 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'aut_phi_ratio': 1965150720.0, 'aut_solvable': False, 'aut_stats': [[1, 1, 1, 1], [3, 1, 2, 1], [3, 1, 240, 1], [9, 1, 486, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_3^5.C_3^4.C_2^2.\\PSL(4,3).C_2', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 9360, 'autcent_group': None, 'autcent_hash': 8072738789780754013, 'autcent_nilpotent': False, 'autcent_order': 955063249920, 'autcent_solvable': False, 'autcent_split': True, 'autcent_supersolvable': False, 'autcent_tex': 'C_3^5.C_3^4.C_2^2.\\PSL(4,3).C_2', '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, 242], [9, 1, 486]], 'center_label': '729.497', 'center_order': 729, 'central_product': True, 'central_quotient': '1.1', 'commutator_count': 0, 'commutator_label': '1.1', 'complements_known': True, 'complete': False, 'complex_characters_known': False, 'composition_factors': ['3.1', '3.1', '3.1', '3.1', '3.1', '3.1'], 'composition_length': 6, 'conjugacy_classes_known': True, 'counter': 497, 'cyclic': False, 'derived_length': 1, 'dihedral': False, 'direct_factorization': [['3.1', 4], ['9.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [3, 1, 2, 121], [9, 1, 6, 81]], 'element_repr_type': 'PC', 'elementary': 3, 'eulerian_function': 121, 'exponent': 9, 'exponents_of_order': [6], 'factors_of_aut_order': [2, 3, 5, 13], 'factors_of_order': [3], 'faithful_reps': [], 'familial': False, 'frattini_label': '3.1', 'frattini_quotient': '243.67', 'hash': 497, 'hyperelementary': 3, 'inner_abelian': True, 'inner_cyclic': True, 'inner_exponent': 1, 'inner_gen_orders': [1, 1, 1, 1, 1], 'inner_gens': [[1, 3, 9, 27, 81], [1, 3, 9, 27, 81], [1, 3, 9, 27, 81], [1, 3, 9, 27, 81], [1, 3, 9, 27, 81]], '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': None, 'irrep_stats': [[1, 729]], 'label': '729.497', 'linC_count': None, 'linC_degree': 5, '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': 'C3^4*C9', 'ngens': 5, 'nilpotency_class': 1, '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': 4, 'number_characteristic_subgroups': 4, 'number_conjugacy_classes': 729, 'number_divisions': 203, 'number_normal_subgroups': 5116, 'number_subgroup_autclasses': 15, 'number_subgroup_classes': 5116, 'number_subgroups': 5116, 'old_label': None, 'order': 729, 'order_factorization_type': 3, 'order_stats': [[1, 1], [3, 242], [9, 486]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': True, 'outer_exponent': 9360, 'outer_gen_orders': [78, 80, 240], 'outer_gen_pows': [0, 0, 0], 'outer_gens': [[511, 51, 60, 32, 701], [24, 77, 28, 294, 728], [537, 303, 26, 551, 458]], 'outer_group': None, 'outer_hash': 8072738789780754013, 'outer_nilpotent': False, 'outer_order': 955063249920, 'outer_permdeg': 486, 'outer_perms': 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'outer_solvable': False, 'outer_supersolvable': False, 'outer_tex': 'C_3^5.C_3^4.C_2^2.\\PSL(4,3).C_2', 'pc_rank': 5, 'perfect': False, 'permutation_degree': 21, 'pgroup': 3, 'primary_abelian_invariants': [3, 3, 3, 3, 9], 'quasisimple': False, 'rank': 5, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 1], [2, 121], [6, 81]], 'representations': {'PC': {'code': 1048592, 'gens': [1, 2, 3, 4, 5], 'pres': [6, -3, 3, 3, 3, 3, -3, 118]}, 'GLZN': {'d': 2, 'p': 54, 'gens': [157483, 158437, 3046285, 5949307, 1637677, 2991835]}, 'Perm': {'d': 21, 'gens': [357120, 4865804016353280000, 711374856192000, 174356582400, 79833600, 80884]}}, 'schur_multiplier': [3, 3, 3, 3, 3, 3, 3, 3, 3, 3], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [3, 3, 3, 3, 9], 'solvability_type': 2, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_3^4\\times C_9', 'transitive_degree': 729, 'wreath_data': None, 'wreath_product': False}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': True, 'abelian_quotient': '243.61', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 1, 'aut_exponent': 936, 'aut_gen_orders': [3, 3, 13, 3, 6, 3, 3, 3, 3, 2], 'aut_gens': [[1, 3, 9, 27], [1, 3, 9, 203], [82, 165, 90, 189], [16, 88, 82, 45], [1, 3, 9, 200], [163, 3, 9, 71], [163, 3, 171, 27], [1, 3, 9, 112], [1, 3, 9, 189], [163, 84, 90, 189], [86, 3, 94, 27]], 'aut_group': '49128768.a', 'aut_hash': 6829789377147908262, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 49128768, 'aut_permdeg': 83, 'aut_perms': [1908345068815238937354193556108802943040289821207358369935475573264099584066390679277349104248117090040053137229561290129040, 10241132782696443443314086248258889552096493367769102167529388866584357555382939086758781906325662260631567071779659212225560, 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'243.61', 'center_order': 243, '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': ['3.1', '3.1', '3.1', '3.1', '3.1'], 'composition_length': 5, 'conjugacy_classes_known': True, 'counter': 61, 'cyclic': False, 'derived_length': 1, 'dihedral': False, 'direct_factorization': [['3.1', 3], ['9.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [3, 1, 2, 40], [9, 1, 6, 27]], 'element_repr_type': 'PC', 'elementary': 3, 'eulerian_function': 40, 'exponent': 9, 'exponents_of_order': [5], 'factors_of_aut_order': [2, 3, 13], 'factors_of_order': [3], 'faithful_reps': [], 'familial': False, 'frattini_label': '3.1', 'frattini_quotient': '81.15', 'hash': 61, 'hyperelementary': 3, 'inner_abelian': True, 'inner_cyclic': True, 'inner_exponent': 1, 'inner_gen_orders': [1, 1, 1, 1], 'inner_gens': [[1, 3, 9, 27], [1, 3, 9, 27], [1, 3, 9, 27], [1, 3, 9, 27]], '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, 243]], 'label': '243.61', 'linC_count': None, 'linC_degree': 4, '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*C9', 'ngens': 4, 'nilpotency_class': 1, 'nilpotent': True, 'normal_counts': [0, 0, 0, 0, 0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 4, 'number_characteristic_subgroups': 4, 'number_conjugacy_classes': 243, 'number_divisions': 68, 'number_normal_subgroups': 396, 'number_subgroup_autclasses': 12, 'number_subgroup_classes': 396, 'number_subgroups': 396, 'old_label': None, 'order': 243, 'order_factorization_type': 3, 'order_stats': [[1, 1], [3, 80], [9, 162]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': True, 'outer_exponent': 936, 'outer_gen_orders': [3, 3, 13, 3, 6, 3, 3, 3, 3, 2], 'outer_gen_pows': [0, 0, 0, 0, 0, 0, 0, 0, 0, 0], 'outer_gens': [[1, 3, 9, 203], [82, 165, 90, 189], [16, 88, 82, 45], [1, 3, 9, 200], [163, 3, 9, 71], [163, 3, 171, 27], [1, 3, 9, 112], [1, 3, 9, 189], [163, 84, 90, 189], [86, 3, 94, 27]], 'outer_group': '49128768.a', 'outer_hash': 6829789377147908262, 'outer_nilpotent': False, 'outer_order': 49128768, 'outer_permdeg': 83, 'outer_perms': [1908345068815238937354193556108802943040289821207358369935475573264099584066390679277349104248117090040053137229561290129040, 10241132782696443443314086248258889552096493367769102167529388866584357555382939086758781906325662260631567071779659212225560, 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268866802959050458010886843533925454282204483500860054481959047180565072333878513949201756680960901618043809561139011657808], 'outer_solvable': False, 'outer_supersolvable': False, 'outer_tex': 'C_3^4.(C_2\\times C_3^3:\\GL(3,3))', 'pc_rank': 4, 'perfect': False, 'permutation_degree': 18, 'pgroup': 3, 'primary_abelian_invariants': [3, 3, 3, 9], 'quasisimple': False, 'rank': 4, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 1], [2, 40], [6, 27]], 'representations': {'PC': {'code': 16392, 'gens': [1, 2, 3, 4], 'pres': [5, -3, 3, 3, 3, -3, 78]}, 'GLZN': {'d': 2, 'p': 54, 'gens': [157483, 158437, 211285, 7272037, 2991835]}, 'GLZq': {'d': 2, 'q': 27, 'gens': [196840, 19687, 203896, 19693, 19927]}, 'Perm': {'d': 18, 'gens': [357120, 711374856192000, 174356582400, 79833600, 80884]}}, 'schur_multiplier': [3, 3, 3, 3, 3, 3], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [3, 3, 3, 9], 'solvability_type': 2, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_3^3\\times C_9', 'transitive_degree': 243, 'wreath_data': None, 'wreath_product': False}
