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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': False, 'ambient': '2880.gn', 'ambient_counter': 170, 'ambient_order': 2880, 'ambient_tex': 'C_2^4:\\GL(2,4)', 'central': False, 'central_factor': False, 'centralizer_order': 12, 'characteristic': False, 'core_order': 1, 'counter': 236, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '2880.gn.240.bd1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': False, 'minimal': False, 'minimal_normal': False, 'nilpotent': False, 'normal': False, 'old_label': '240.bd1', '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': 240, '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': '12.3', 'subgroup_hash': 3, 'subgroup_order': 12, 'subgroup_tex': 'A_4', 'supersolvable': False, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '2880.gn', 'aut_centralizer_order': 18, 'aut_label': '240.bd1', 'aut_quo_index': None, 'aut_stab_index': 40, 'aut_weyl_group': '24.12', 'aut_weyl_index': 720, 'centralizer': '240.j1', 'complements': None, 'conjugacy_class_count': 2, 'contained_in': ['60.x1', '80.e1', '120.n1'], 'contains': ['720.z1', '960.c1'], 'core': '2880.a1', 'coset_action_label': None, 'count': 40, 'diagramx': [5420, -1, 4913, -1], 'generators': [1484179200, 1963543680, 1005575056], 'label': '2880.gn.240.bd1', 'mobius_quo': None, 'mobius_sub': None, 'normal_closure': '4.a1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '20.d1', 'old_label': '240.bd1', 'projective_image': '2880.gn', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '240.bd1', 'subgroup_fusion': None, 'weyl_group': '12.3'}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '3.1', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 3, 'aut_exponent': 12, 'aut_gen_orders': [2, 3, 2, 2], 'aut_gens': [[4, 16, 7], [11, 16, 23], [19, 23, 16], [8, 16, 7], [15, 16, 7]], 'aut_group': '24.12', 'aut_hash': 12, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 24, 'aut_permdeg': 4, 'aut_perms': [2, 4, 16, 7], 'aut_phi_ratio': 6.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 3, 1, 1], [3, 4, 2, 1]], 'aut_supersolvable': False, 'aut_tex': 'S_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': 12, 'autcentquo_group': '24.12', 'autcentquo_hash': 12, 'autcentquo_nilpotent': False, 'autcentquo_order': 24, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'S_4', 'cc_stats': [[1, 1, 1], [2, 3, 1], [3, 4, 2]], 'center_label': '1.1', 'center_order': 1, 'central_product': False, 'central_quotient': '12.3', 'commutator_count': 1, 'commutator_label': '4.2', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '3.1'], 'composition_length': 3, 'conjugacy_classes_known': True, 'counter': 3, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 3, 1, 1], [3, 4, 2, 1]], 'element_repr_type': 'Perm', 'elementary': 1, 'eulerian_function': 4, 'exponent': 6, 'exponents_of_order': [2, 1], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [[3, 1, 1]], 'familial': True, 'frattini_label': '1.1', 'frattini_quotient': '12.3', 'hash': 3, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 6, 'inner_gen_orders': [3, 2, 2], 'inner_gens': [[4, 7, 23], [19, 16, 7], [15, 16, 7]], 'inner_hash': 3, 'inner_nilpotent': False, 'inner_order': 12, 'inner_split': True, 'inner_tex': 'A_4', 'inner_used': [1, 2], 'irrC_degree': 3, 'irrQ_degree': 3, 'irrQ_dim': 3, 'irrR_degree': 3, 'irrep_stats': [[1, 3], [3, 1]], 'label': '12.3', 'linC_count': 1, 'linC_degree': 3, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 3, 'linQ_degree_count': 1, 'linQ_dim': 3, 'linQ_dim_count': 1, 'linR_count': 1, 'linR_degree': 3, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'A4', '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': 3, 'number_characteristic_subgroups': 3, 'number_conjugacy_classes': 4, 'number_divisions': 3, 'number_normal_subgroups': 3, 'number_subgroup_autclasses': 5, 'number_subgroup_classes': 5, 'number_subgroups': 10, 'old_label': None, 'order': 12, 'order_factorization_type': 22, 'order_stats': [[1, 1], [2, 3], [3, 8]], 'outer_abelian': True, 'outer_cyclic': True, 'outer_equivalence': False, 'outer_exponent': 2, 'outer_gen_orders': [2], 'outer_gen_pows': [0], 'outer_gens': [[3, 7, 16]], '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': 4, 'pgroup': 0, 'primary_abelian_invariants': [3], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 1], [2, 1], [3, 1]], 'representations': {'PC': {'code': 9572, 'gens': [1, 2, 3], 'pres': [3, -3, -2, 2, 55, 29]}, 'GLZ': {'b': 3, 'd': 3, 'gens': [4828, 4708]}, 'Lie': [{'d': 2, 'q': 3, 'gens': [31, 21], 'family': 'PSL'}, {'d': 2, 'q': 3, 'gens': [362, 766], 'family': 'PSU'}, {'d': 3, 'q': 3, 'gens': [10315, 1638], 'family': 'Omega'}, {'d': 3, 'q': 3, 'gens': [10315, 1638], 'family': 'POmega'}, {'d': 2, 'q': 3, 'gens': [31, 21], 'family': 'PSigmaL'}, {'d': 1, 'q': 4, 'gens': [3, 7], 'family': 'AGL'}], 'GLFp': {'d': 3, 'p': 2, 'gens': [458, 314, 124]}, 'Perm': {'d': 4, 'gens': [4, 16, 7]}}, 'schur_multiplier': [2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [3], 'solvability_type': 8, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': 'A_4', 'transitive_degree': 4, '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': '12.5', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 3, 'aut_exponent': 60, 'aut_gen_orders': [2, 2, 2, 2, 6, 3, 3, 12], 'aut_gens': [[121, 34696, 5041, 1480947982, 482630407, 127, 120], [121, 34696, 5041, 1480952903, 482635566, 5046, 120], [121, 39616, 5041, 1480952902, 482630407, 127, 120], [5160, 17158, 5041, 1480930432, 482635440, 5166, 126], [121, 34577, 5041, 1480947863, 482630407, 127, 120], [5041, 39622, 5160, 1476961216, 479369520, 5047, 7], [121, 34702, 5041, 1480952903, 482635566, 5040, 126], [5160, 39623, 121, 1480947976, 482630520, 7, 127], [121, 17273, 5160, 79850879, 486622206, 127, 1]], 'aut_group': '17280.bd', 'aut_hash': 456236538737532445, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 17280, 'aut_permdeg': 26, 'aut_perms': [14478346316280, 611974989316170741876785, 26927487852072165279719, 243456023430396555166081, 347443423247591566413535034, 17093695052286, 411288879116256716484, 257614205581872841253811917], 'aut_phi_ratio': 22.5, 'aut_solvable': False, 'aut_stats': [[1, 1, 1, 1], [2, 1, 3, 1], [2, 3, 1, 1], [2, 3, 3, 1], [2, 15, 1, 1], [2, 15, 3, 1], [2, 45, 1, 1], [2, 45, 3, 1], [3, 4, 2, 1], [3, 20, 1, 1], [3, 80, 2, 1], [5, 12, 2, 1], [6, 4, 6, 1], [6, 20, 3, 1], [6, 60, 1, 1], [6, 60, 2, 1], [6, 60, 3, 1], [6, 60, 6, 1], [6, 80, 6, 1], [10, 12, 6, 1], [10, 36, 2, 1], [10, 36, 6, 1], [15, 48, 4, 1], [30, 48, 12, 1]], 'aut_supersolvable': False, 'aut_tex': 'S_3\\times S_4\\times S_5', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 6, 'autcent_group': '6.1', 'autcent_hash': 1, 'autcent_nilpotent': False, 'autcent_order': 6, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'S_3', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 60, 'autcentquo_group': '2880.dv', 'autcentquo_hash': 409072078557532110, 'autcentquo_nilpotent': False, 'autcentquo_order': 2880, 'autcentquo_solvable': False, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'S_4\\times S_5', 'cc_stats': [[1, 1, 1], [2, 1, 3], [2, 3, 4], [2, 15, 4], [2, 45, 4], [3, 4, 2], [3, 20, 1], [3, 80, 2], [5, 12, 2], [6, 4, 6], [6, 20, 3], [6, 60, 12], [6, 80, 6], [10, 12, 6], [10, 36, 8], [15, 48, 4], [30, 48, 12]], 'center_label': '4.2', 'center_order': 4, 'central_product': True, 'central_quotient': '720.768', 'commutator_count': 1, 'commutator_label': '240.190', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '3.1', '60.5'], 'composition_length': 6, 'conjugacy_classes_known': True, 'counter': 170, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['12.3', 1], ['2.1', 2], ['60.5', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [2, 3, 1, 4], [2, 15, 1, 4], [2, 45, 1, 4], [3, 4, 2, 1], [3, 20, 1, 1], [3, 80, 2, 1], [5, 12, 2, 1], [6, 4, 2, 3], [6, 20, 1, 3], [6, 60, 1, 4], [6, 60, 2, 4], [6, 80, 2, 3], [10, 12, 2, 3], [10, 36, 2, 4], [15, 48, 4, 1], [30, 48, 4, 3]], 'element_repr_type': 'Perm', 'elementary': 1, 'eulerian_function': None, 'exponent': 30, 'exponents_of_order': [6, 2, 1], 'factors_of_aut_order': [2, 3, 5], 'factors_of_order': [2, 3, 5], 'faithful_reps': [], 'familial': False, 'frattini_label': '1.1', 'frattini_quotient': '2880.gn', 'hash': 5094570882729245637, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 30, 'inner_gen_orders': [2, 3, 2, 3, 2, 2, 2], 'inner_gens': [[121, 39616, 5041, 1480952902, 482630407, 127, 120], [5160, 34696, 121, 1480947982, 482630526, 5166, 5040], [121, 34577, 5041, 1480947863, 482630407, 127, 120], [5160, 34696, 121, 1480947982, 87091326, 5166, 5040], [121, 39737, 5041, 47577023, 482630407, 127, 120], [121, 39616, 5041, 1480952902, 482630407, 127, 120], [121, 34577, 5041, 1480947863, 482630407, 127, 120]], 'inner_hash': 768, 'inner_nilpotent': False, 'inner_order': 720, 'inner_split': True, 'inner_tex': 'A_4\\times A_5', 'inner_used': [1, 2, 4, 5], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 12], [3, 28], [4, 12], [5, 12], [9, 8], [12, 4], [15, 4]], 'label': '2880.gn', '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': False, 'metacyclic': False, 'monomial': False, 'name': 'C2^4:GL(2,4)', 'ngens': 7, '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, 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': 24, 'number_characteristic_subgroups': 12, 'number_conjugacy_classes': 80, 'number_divisions': 48, 'number_normal_subgroups': 30, 'number_subgroup_autclasses': 339, 'number_subgroup_classes': 921, 'number_subgroups': 24598, 'old_label': None, 'order': 2880, 'order_factorization_type': 321, 'order_stats': [[1, 1], [2, 255], [3, 188], [5, 24], [6, 1284], [10, 360], [15, 192], [30, 576]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': True, 'outer_exponent': 6, 'outer_gen_orders': [2, 2, 6], 'outer_gen_pows': [0, 0, 0], 'outer_gens': [[5160, 17152, 5041, 1480930438, 482635446, 5166, 120], [121, 39623, 5041, 1480947982, 482630407, 5040, 5047], [121, 34702, 5041, 4031303, 1520472366, 5040, 126]], 'outer_group': '24.14', 'outer_hash': 14, 'outer_nilpotent': False, 'outer_order': 24, 'outer_permdeg': 7, 'outer_perms': [7, 136, 856], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2\\times D_6', 'pc_rank': None, '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], [3, 4], [4, 4], [5, 4], [6, 4], [8, 4], [10, 4], [12, 8], [15, 4], [18, 4]], 'representations': {'Perm': {'d': 13, 'gens': [121, 34696, 5041, 1480947982, 482630407, 127, 120]}}, 'schur_multiplier': [2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 6], 'solvability_type': 13, 'solvable': False, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': 'C_2^4:\\GL(2,4)', 'transitive_degree': 60, 'wreath_data': None, 'wreath_product': False}
