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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': False, 'ambient': '600.157', 'ambient_counter': 157, 'ambient_order': 600, 'ambient_tex': 'D_5^2.C_6', 'central': False, 'central_factor': False, 'centralizer_order': 3, 'characteristic': True, 'core_order': 300, 'counter': 2, 'cyclic': False, 'direct': False, 'hall': 0, 'label': '600.157.2.a1.a1', 'maximal': True, 'maximal_normal': True, 'metabelian': True, 'metacyclic': False, 'minimal': False, 'minimal_normal': False, 'nilpotent': False, 'normal': True, 'old_label': '2.a1.a1', 'outer_equivalence': False, 'perfect': False, 'proper': True, 'quotient': '2.1', 'quotient_Agroup': True, 'quotient_abelian': True, 'quotient_cyclic': True, 'quotient_hash': 1, 'quotient_metabelian': True, 'quotient_nilpotent': True, 'quotient_order': 2, 'quotient_simple': True, 'quotient_solvable': True, 'quotient_supersolvable': True, 'quotient_tex': 'C_2', 'simple': False, 'solvable': True, 'special_labels': [], 'split': False, 'standard_generators': False, 'stem': False, 'subgroup': '300.36', 'subgroup_hash': 36, 'subgroup_order': 300, 'subgroup_tex': 'C_3\\times D_5^2', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '600.157', 'aut_centralizer_order': 1, 'aut_label': '2.a1', 'aut_quo_index': 1, 'aut_stab_index': 1, 'aut_weyl_group': '1600.10254', 'aut_weyl_index': 1, 'centralizer': '200.a1.a1', 'complements': [], 'conjugacy_class_count': 1, 'contained_in': ['1.a1.a1'], 'contains': ['4.a1.a1', '4.a1.b1', '4.b1.a1', '6.a1.a1', '10.a1.a1', '10.a1.b1'], 'core': '2.a1.a1', 'coset_action_label': None, 'count': 1, 'diagramx': [3592, 3149, 3673, 5208, 3257, 1493, 3676, 4700], 'generators': [2, 400, 8, 20, 120], 'label': '600.157.2.a1.a1', 'mobius_quo': 0, 'mobius_sub': -1, 'normal_closure': '2.a1.a1', 'normal_contained_in': ['1.a1.a1'], 'normal_contains': ['4.a1.a1', '4.a1.b1', '4.b1.a1', '6.a1.a1'], 'normalizer': '1.a1.a1', 'old_label': '2.a1.a1', 'projective_image': '200.42', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '2.a1.a1', 'subgroup_fusion': None, 'weyl_group': '200.42'}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '12.5', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 3, 'aut_exponent': 40, 'aut_gen_orders': [4, 5, 8, 4, 2, 10, 5, 2], 'aut_gens': [[1, 2, 20], [17, 254, 140], [9, 242, 20], [190, 245, 108], [10, 73, 216], [1, 2, 220], [13, 2, 80], [9, 62, 20], [9, 138, 80]], 'aut_group': '1600.10254', 'aut_hash': 10254, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 1600, 'aut_permdeg': 12, 'aut_perms': [125002707, 288787367, 347754344, 81134427, 26070, 218633902, 316406170, 286490529], 'aut_phi_ratio': 20.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 5, 2, 1], [2, 25, 1, 1], [3, 1, 2, 1], [5, 2, 4, 1], [5, 4, 4, 1], [6, 5, 4, 1], [6, 25, 2, 1], [10, 10, 4, 1], [15, 2, 8, 1], [15, 4, 8, 1], [30, 10, 8, 1]], 'aut_supersolvable': False, 'aut_tex': 'F_5^2:C_2^2', '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': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 40, 'autcentquo_group': '800.1191', 'autcentquo_hash': 1191, 'autcentquo_nilpotent': False, 'autcentquo_order': 800, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'F_5\\wr C_2', 'cc_stats': [[1, 1, 1], [2, 5, 2], [2, 25, 1], [3, 1, 2], [5, 2, 4], [5, 4, 4], [6, 5, 4], [6, 25, 2], [10, 10, 4], [15, 2, 8], [15, 4, 8], [30, 10, 8]], 'center_label': '3.1', 'center_order': 3, 'central_product': True, 'central_quotient': '100.13', 'commutator_count': 1, 'commutator_label': '25.2', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '3.1', '5.1', '5.1'], 'composition_length': 5, 'conjugacy_classes_known': True, 'counter': 36, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['10.1', 2], ['3.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 5, 1, 2], [2, 25, 1, 1], [3, 1, 2, 1], [5, 2, 2, 2], [5, 4, 2, 2], [6, 5, 2, 2], [6, 25, 2, 1], [10, 10, 2, 2], [15, 2, 4, 2], [15, 4, 4, 2], [30, 10, 4, 2]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 12, 'exponent': 30, 'exponents_of_order': [2, 2, 1], 'factors_of_aut_order': [2, 5], 'factors_of_order': [2, 3, 5], 'faithful_reps': [[4, 0, 8]], 'familial': False, 'frattini_label': '1.1', 'frattini_quotient': '300.36', 'hash': 36, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 10, 'inner_gen_orders': [2, 10, 5], 'inner_gens': [[1, 18, 20], [5, 2, 80], [1, 242, 20]], 'inner_hash': 13, 'inner_nilpotent': False, 'inner_order': 100, 'inner_split': False, 'inner_tex': 'D_5^2', 'inner_used': [1, 2, 3], 'irrC_degree': 4, 'irrQ_degree': 16, 'irrQ_dim': 16, 'irrR_degree': 8, 'irrep_stats': [[1, 12], [2, 24], [4, 12]], 'label': '300.36', 'linC_count': 136, 'linC_degree': 4, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 10, 'linQ_degree_count': 24, 'linQ_dim': 10, 'linQ_dim_count': 24, 'linR_count': 112, 'linR_degree': 6, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'C3*D5^2', 'ngens': 5, '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], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 12, 'number_characteristic_subgroups': 8, 'number_conjugacy_classes': 48, 'number_divisions': 20, 'number_normal_subgroups': 20, 'number_subgroup_autclasses': 30, 'number_subgroup_classes': 48, 'number_subgroups': 248, 'old_label': None, 'order': 300, 'order_factorization_type': 222, 'order_stats': [[1, 1], [2, 35], [3, 2], [5, 24], [6, 70], [10, 40], [15, 48], [30, 80]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 4, 'outer_gen_orders': [2, 2, 2, 2], 'outer_gen_pows': [11, 10, 11, 11], 'outer_gens': [[10, 181, 112], [1, 2, 160], [1, 14, 160], [1, 6, 260]], 'outer_group': '16.11', 'outer_hash': 11, 'outer_nilpotent': True, 'outer_order': 16, 'outer_permdeg': 6, 'outer_perms': [24, 289, 415, 414], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2\\times D_4', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 13, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 3], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 4], [4, 4], [8, 6], [16, 2]], 'representations': {'PC': {'code': 122462275009292855351, 'gens': [1, 2, 4], 'pres': [5, -2, -2, -5, -3, -5, 181, 26, 242, 808, 78, 3009]}, 'GLFq': {'d': 4, 'q': 4, 'gens': [2210795970, 2160064514, 2149582850, 32591745, 3856979952]}, 'Perm': {'d': 13, 'gens': [40279680, 7, 5760, 37, 559198080]}}, 'schur_multiplier': [2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 6], 'solvability_type': 7, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_3\\times D_5^2', 'transitive_degree': 30, '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': '24.9', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 3, 'aut_exponent': 40, 'aut_gen_orders': [4, 5, 8, 4, 2, 10, 5, 2], 'aut_gens': [[1, 4, 40], [265, 148, 280], [145, 364, 40], [369, 246, 432], [481, 502, 408], [1, 4, 440], [481, 276, 40], [137, 364, 40], [153, 396, 160]], 'aut_group': '1600.10254', 'aut_hash': 10254, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 1600, 'aut_permdeg': 37, 'aut_perms': [3238262482707351533797839011838221359110145, 11600654198913531160649487977849218582916290, 4241628021758069895306228955652874937150401, 8240951998411539081021698694448651078819331, 9333755695734919995571200, 8500901416650935786869085294790108092499064, 9963292272173982211537425289246229004115497, 13352570873795207331426896862794804661925234], 'aut_phi_ratio': 10.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 5, 2, 1], [2, 25, 1, 1], [3, 1, 2, 1], [4, 25, 1, 2], [4, 25, 2, 1], [5, 4, 2, 1], [5, 8, 2, 1], [6, 5, 4, 1], [6, 25, 2, 1], [10, 20, 2, 1], [12, 25, 2, 2], [12, 25, 4, 1], [15, 4, 4, 1], [15, 8, 4, 1], [30, 20, 4, 1]], 'aut_supersolvable': False, 'aut_tex': 'F_5^2:C_2^2', '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': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 40, 'autcentquo_group': '800.1191', 'autcentquo_hash': 1191, 'autcentquo_nilpotent': False, 'autcentquo_order': 800, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'F_5\\wr C_2', 'cc_stats': [[1, 1, 1], [2, 5, 2], [2, 25, 1], [3, 1, 2], [4, 25, 4], [5, 4, 2], [5, 8, 2], [6, 5, 4], [6, 25, 2], [10, 20, 2], [12, 25, 8], [15, 4, 4], [15, 8, 4], [30, 20, 4]], 'center_label': '3.1', 'center_order': 3, 'central_product': True, 'central_quotient': '200.42', 'commutator_count': 1, 'commutator_label': '25.2', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '3.1', '5.1', '5.1'], 'composition_length': 6, 'conjugacy_classes_known': True, 'counter': 157, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['200.42', 1], ['3.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 5, 1, 2], [2, 25, 1, 1], [3, 1, 2, 1], [4, 25, 2, 2], [5, 4, 1, 2], [5, 8, 1, 2], [6, 5, 2, 2], [6, 25, 2, 1], [10, 20, 1, 2], [12, 25, 4, 2], [15, 4, 2, 2], [15, 8, 2, 2], [30, 20, 2, 2]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 48, 'exponent': 60, 'exponents_of_order': [3, 2, 1], 'factors_of_aut_order': [2, 5], 'factors_of_order': [2, 3, 5], 'faithful_reps': [[8, 0, 4]], 'familial': False, 'frattini_label': '1.1', 'frattini_quotient': '600.157', 'hash': 157, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 20, 'inner_gen_orders': [4, 10, 5], 'inner_gens': [[1, 28, 280], [17, 4, 160], [361, 484, 40]], 'inner_hash': 42, 'inner_nilpotent': False, 'inner_order': 200, 'inner_split': False, 'inner_tex': 'D_5:F_5', 'inner_used': [1, 2, 3], 'irrC_degree': 8, 'irrQ_degree': 16, 'irrQ_dim': 16, 'irrR_degree': 16, 'irrep_stats': [[1, 24], [4, 12], [8, 6]], 'label': '600.157', 'linC_count': 36, 'linC_degree': 8, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 10, 'linQ_degree_count': 24, 'linQ_dim': 10, 'linQ_dim_count': 24, 'linR_count': 48, 'linR_degree': 10, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'D5^2.C6', 'ngens': 6, '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': 18, 'number_characteristic_subgroups': 14, 'number_conjugacy_classes': 42, 'number_divisions': 24, 'number_normal_subgroups': 26, 'number_subgroup_autclasses': 50, 'number_subgroup_classes': 76, 'number_subgroups': 504, 'old_label': None, 'order': 600, 'order_factorization_type': 321, 'order_stats': [[1, 1], [2, 35], [3, 2], [4, 100], [5, 24], [6, 70], [10, 40], [12, 200], [15, 48], [30, 80]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 2, 'outer_gen_orders': [2, 2, 2], 'outer_gen_pows': [22, 2, 0], 'outer_gens': [[1, 12, 40], [1, 12, 320], [1, 142, 408]], 'outer_group': '8.5', 'outer_hash': 5, 'outer_nilpotent': True, 'outer_order': 8, 'outer_permdeg': 6, 'outer_perms': [1, 6, 120], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^3', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 13, 'pgroup': 0, 'primary_abelian_invariants': [2, 4, 3], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 6], [4, 6], [8, 6], [16, 2]], 'representations': {'PC': {'code': 28063286802408873369098244177140948766904375, 'gens': [1, 3, 5], 'pres': [6, -2, -2, -2, -5, -3, -5, 12, 506, 332, 50, 387, 393, 8404, 2410, 1216, 118, 8645, 8651, 4337]}, 'Perm': {'d': 13, 'gens': [43908489, 16, 5760, 87091216, 46, 606009600]}}, 'schur_multiplier': [2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 12], 'solvability_type': 7, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'D_5^2.C_6', 'transitive_degree': 30, 'wreath_data': None, 'wreath_product': False}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': True, 'abelian_quotient': '2.1', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': True, 'aut_cyclic': True, 'aut_derived_length': 0, 'aut_exponent': 1, 'aut_gen_orders': [], 'aut_gens': [[1]], 'aut_group': '1.1', 'aut_hash': 1, 'aut_nilpotency_class': 0, 'aut_nilpotent': True, 'aut_order': 1, 'aut_permdeg': 1, 'aut_perms': [], 'aut_phi_ratio': 1.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_1', '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': 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, 1]], 'center_label': '2.1', 'center_order': 2, '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': ['2.1'], 'composition_length': 1, 'conjugacy_classes_known': True, 'counter': 1, 'cyclic': True, 'derived_length': 1, 'dihedral': True, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1]], 'element_repr_type': 'PC', 'elementary': 2, 'eulerian_function': 1, 'exponent': 2, 'exponents_of_order': [1], 'factors_of_aut_order': [], 'factors_of_order': [2], 'faithful_reps': [[1, 1, 1]], 'familial': True, 'frattini_label': '1.1', 'frattini_quotient': '2.1', 'hash': 1, 'hyperelementary': 2, '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': 1, 'irrQ_dim': 1, 'irrR_degree': 1, 'irrep_stats': [[1, 2]], 'label': '2.1', 'linC_count': 1, 'linC_degree': 1, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 1, 'linQ_degree_count': 1, 'linQ_dim': 1, 'linQ_dim_count': 1, 'linR_count': 1, 'linR_degree': 1, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C2', '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': 2, 'number_divisions': 2, 'number_normal_subgroups': 2, 'number_subgroup_autclasses': 2, 'number_subgroup_classes': 2, 'number_subgroups': 2, 'old_label': None, 'order': 2, 'order_factorization_type': 1, 'order_stats': [[1, 1], [2, 1]], 'outer_abelian': True, 'outer_cyclic': True, 'outer_equivalence': False, 'outer_exponent': 1, 'outer_gen_orders': [], 'outer_gen_pows': [], 'outer_gens': [], 'outer_group': '1.1', 'outer_hash': 1, 'outer_nilpotent': True, 'outer_order': 1, 'outer_permdeg': 1, 'outer_perms': [], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_1', 'pc_rank': 1, 'perfect': False, 'permutation_degree': 2, 'pgroup': 2, 'primary_abelian_invariants': [2], 'quasisimple': False, 'rank': 1, 'rational': True, 'rational_characters_known': True, 'ratrep_stats': [[1, 2]], 'representations': {'PC': {'code': 0, 'gens': [1], 'pres': [1, -2]}, 'GLZ': {'b': 3, 'd': 2, 'gens': [12]}, 'Lie': [{'d': 2, 'q': 2, 'gens': [6], 'family': 'CSOPlus'}, {'d': 2, 'q': 2, 'gens': [6], 'family': 'COPlus'}, {'d': 1, 'q': 2, 'gens': [1], 'family': 'AGL'}, {'d': 1, 'q': 2, 'gens': [1], 'family': 'AGammaL'}, {'d': 1, 'q': 2, 'gens': [1], 'family': 'ASigmaL'}], 'GLFp': {'d': 2, 'p': 2, 'gens': [11]}, 'Perm': {'d': 2, 'gens': [1]}}, 'schur_multiplier': [], 'semidirect_product': False, 'simple': True, 'smith_abelian_invariants': [2], 'solvability_type': 0, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_2', 'transitive_degree': 2, 'wreath_data': None, 'wreath_product': False}
