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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': True, 'ambient': '240.204', 'ambient_counter': 204, 'ambient_order': 240, 'ambient_tex': 'C_2^4:C_{15}', 'central': False, 'central_factor': False, 'centralizer_order': 80, 'characteristic': False, 'core_order': 4, 'counter': 51, 'cyclic': False, 'direct': False, 'hall': 0, 'label': '240.204.60.a1.e1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': False, 'minimal_normal': True, 'nilpotent': True, 'normal': True, 'old_label': '60.a1.e1', 'outer_equivalence': False, 'perfect': False, 'proper': True, 'quotient': '60.9', 'quotient_Agroup': True, 'quotient_abelian': False, 'quotient_cyclic': False, 'quotient_hash': 9, 'quotient_metabelian': True, 'quotient_nilpotent': False, 'quotient_order': 60, 'quotient_simple': False, 'quotient_solvable': True, 'quotient_supersolvable': False, 'quotient_tex': 'C_5\\times A_4', 'simple': False, 'solvable': True, 'special_labels': [], 'split': True, 'standard_generators': False, 'stem': False, 'subgroup': '4.2', 'subgroup_hash': 2, 'subgroup_order': 4, 'subgroup_tex': 'C_2^2', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '240.204', 'aut_centralizer_order': 768, 'aut_label': '60.a1', 'aut_quo_index': 1, 'aut_stab_index': 5, 'aut_weyl_group': '6.1', 'aut_weyl_index': 3840, 'centralizer': '3.a1.a1', 'complements': ['4.a1.c1', '4.a1.a1', '4.a1.b1', '4.a1.d1'], 'conjugacy_class_count': 1, 'contained_in': ['12.a1.e1', '20.a1.e1', '30.a1.a1'], 'contains': ['120.a1.e1'], 'core': '60.a1.e1', 'coset_action_label': None, 'count': 1, 'diagramx': [5898, 6146, 7327, 5742, 3772, 4275, 1587, 2729], 'generators': [17556, 52028], 'label': '240.204.60.a1.e1', 'mobius_quo': -1, 'mobius_sub': -4, 'normal_closure': '60.a1.e1', 'normal_contained_in': ['12.a1.e1', '15.a1.a1'], 'normal_contains': ['240.a1.a1'], 'normalizer': '1.a1.a1', 'old_label': '60.a1.e1', 'projective_image': '240.204', 'quotient_action_image': '3.1', 'quotient_action_kernel': '20.5', 'quotient_action_kernel_order': 20, 'quotient_fusion': None, 'short_label': '60.a1.e1', 'subgroup_fusion': None, 'weyl_group': '3.1'}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': True, 'abelian_quotient': '4.2', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 6, 'aut_gen_orders': [2, 3], 'aut_gens': [[1, 2], [3, 2], [2, 3]], 'aut_group': '6.1', 'aut_hash': 1, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 6, 'aut_permdeg': 3, 'aut_perms': [1, 4], 'aut_phi_ratio': 3.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 3, 1]], 'aut_supersolvable': True, 'aut_tex': 'S_3', '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': 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, 3]], 'center_label': '4.2', 'center_order': 4, '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': ['2.1', '2.1'], 'composition_length': 2, 'conjugacy_classes_known': True, 'counter': 2, 'cyclic': False, 'derived_length': 1, 'dihedral': True, 'direct_factorization': [['2.1', 2]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3]], 'element_repr_type': 'PC', 'elementary': 2, 'eulerian_function': 1, 'exponent': 2, 'exponents_of_order': [2], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2], 'faithful_reps': [], 'familial': True, 'frattini_label': '1.1', 'frattini_quotient': '4.2', 'hash': 2, 'hyperelementary': 2, 'inner_abelian': True, 'inner_cyclic': True, 'inner_exponent': 1, 'inner_gen_orders': [1, 1], 'inner_gens': [[1, 2], [1, 2]], '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, 4]], 'label': '4.2', 'linC_count': 3, 'linC_degree': 2, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 2, 'linQ_degree_count': 3, 'linQ_dim': 2, 'linQ_dim_count': 3, 'linR_count': 3, 'linR_degree': 2, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C2^2', 'ngens': 2, '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': 2, 'number_characteristic_subgroups': 2, 'number_conjugacy_classes': 4, 'number_divisions': 4, 'number_normal_subgroups': 5, 'number_subgroup_autclasses': 3, 'number_subgroup_classes': 5, 'number_subgroups': 5, 'old_label': None, 'order': 4, 'order_factorization_type': 2, 'order_stats': [[1, 1], [2, 3]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 6, 'outer_gen_orders': [2, 3], 'outer_gen_pows': [0, 0], 'outer_gens': [[3, 2], [2, 3]], 'outer_group': '6.1', 'outer_hash': 1, 'outer_nilpotent': False, 'outer_order': 6, 'outer_permdeg': 3, 'outer_perms': [1, 4], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'S_3', 'pc_rank': 2, 'perfect': False, 'permutation_degree': 4, 'pgroup': 2, 'primary_abelian_invariants': [2, 2], 'quasisimple': False, 'rank': 2, 'rational': True, 'rational_characters_known': True, 'ratrep_stats': [[1, 4]], 'representations': {'PC': {'code': 0, 'gens': [1, 2], 'pres': [2, -2, 2]}, 'GLZ': {'b': 3, 'd': 2, 'gens': [14, 12]}, 'GLFp': {'d': 2, 'p': 3, 'gens': [55, 56]}, 'Perm': {'d': 4, 'gens': [6, 1]}}, 'schur_multiplier': [2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2], 'solvability_type': 1, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_2^2', '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': '15.1', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 120, 'aut_gen_orders': [2, 2, 3, 3, 2, 6, 12, 2], 'aut_gens': [[40110, 63055, 62623, 49164, 23685, 62255], [62792, 63055, 62623, 49164, 23685, 62255], [51259, 63055, 62623, 49164, 23685, 62255], [40110, 40873, 12194, 49164, 13667, 39625], [40110, 60558, 51692, 49164, 23685, 31879], [4870, 63055, 62623, 49164, 23685, 62255], [49722, 25382, 36440, 40970, 38041, 60110], [40110, 31207, 60110, 32776, 12787, 25078], [61321, 63055, 62623, 49164, 23685, 62255]], 'aut_group': '23040.es', 'aut_hash': 4311431573477723808, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 23040, 'aut_permdeg': 20, 'aut_perms': [874314345654974160, 1000424634086158560, 164733774136455000, 70456306672402680, 405852046407329880, 1655084741027202120, 1431364260697792698, 743223441383016480], 'aut_phi_ratio': 360.0, 'aut_solvable': False, 'aut_stats': [[1, 1, 1, 1], [2, 3, 5, 1], [3, 16, 2, 1], [5, 1, 4, 1], [10, 3, 20, 1], [15, 16, 8, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_4\\times \\AGammaL(2,4)', 'autcent_abelian': True, 'autcent_cyclic': True, 'autcent_exponent': 4, 'autcent_group': '4.1', 'autcent_hash': 1, 'autcent_nilpotent': True, 'autcent_order': 4, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_4', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 120, 'autcentquo_group': '5760.c', 'autcentquo_hash': 699824500729758013, 'autcentquo_nilpotent': False, 'autcentquo_order': 5760, 'autcentquo_solvable': False, 'autcentquo_supersolvable': False, 'autcentquo_tex': '\\AGammaL(2,4)', 'cc_stats': [[1, 1, 1], [2, 3, 5], [3, 16, 2], [5, 1, 4], [10, 3, 20], [15, 16, 8]], 'center_label': '5.1', 'center_order': 5, 'central_product': True, 'central_quotient': '48.50', 'commutator_count': 1, 'commutator_label': '16.14', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '3.1', '5.1'], 'composition_length': 6, 'conjugacy_classes_known': True, 'counter': 204, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['48.50', 1], ['5.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 3, 1, 5], [3, 16, 2, 1], [5, 1, 4, 1], [10, 3, 4, 5], [15, 16, 8, 1]], 'element_repr_type': 'GLFq', 'elementary': 1, 'eulerian_function': 403, 'exponent': 30, 'exponents_of_order': [4, 1, 1], 'factors_of_aut_order': [2, 3, 5], 'factors_of_order': [2, 3, 5], 'faithful_reps': [], 'familial': False, 'frattini_label': '1.1', 'frattini_quotient': '240.204', 'hash': 204, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 6, 'inner_gen_orders': [3, 2, 2, 1, 2, 2], 'inner_gens': [[40110, 53757, 12194, 49164, 4817, 59886], [23890, 63055, 62623, 49164, 23685, 62255], [13492, 63055, 62623, 49164, 23685, 62255], [40110, 63055, 62623, 49164, 23685, 62255], [18323, 63055, 62623, 49164, 23685, 62255], [54010, 63055, 62623, 49164, 23685, 62255]], 'inner_hash': 50, 'inner_nilpotent': False, 'inner_order': 48, 'inner_split': True, 'inner_tex': 'C_2^2:A_4', 'inner_used': [1, 2, 3], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 15], [3, 25]], 'label': '240.204', 'linC_count': 240, 'linC_degree': 6, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 10, 'linQ_degree_count': 10, 'linQ_dim': 10, 'linQ_dim_count': 10, 'linR_count': 60, 'linR_degree': 8, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'C2^4:C15', '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], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 6, 'number_characteristic_subgroups': 6, 'number_conjugacy_classes': 40, 'number_divisions': 14, 'number_normal_subgroups': 16, 'number_subgroup_autclasses': 18, 'number_subgroup_classes': 68, 'number_subgroups': 208, 'old_label': None, 'order': 240, 'order_factorization_type': 311, 'order_stats': [[1, 1], [2, 15], [3, 32], [5, 4], [10, 60], [15, 128]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 60, 'outer_gen_orders': [3, 4, 2, 4], 'outer_gen_pows': [4097, 4097, 61956, 4097], 'outer_gens': [[24609, 17556, 62623, 49164, 40313, 59886], [19858, 41690, 59326, 40970, 21861, 55229], [18323, 25382, 6961, 40970, 21861, 12787], [40110, 63055, 5697, 32776, 60110, 34744]], 'outer_group': '480.943', 'outer_hash': 943, 'outer_nilpotent': False, 'outer_order': 480, 'outer_permdeg': 9, 'outer_perms': [840, 121816, 40456, 18], 'outer_solvable': False, 'outer_supersolvable': False, 'outer_tex': 'C_4\\times S_5', 'pc_rank': 5, 'perfect': False, 'permutation_degree': 13, 'pgroup': 0, 'primary_abelian_invariants': [3, 5], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 1], [2, 1], [3, 5], [4, 1], [8, 1], [12, 5]], 'representations': {'PC': {'code': 143516016090544014339, 'gens': [1, 2, 3, 4, 5], 'pres': [6, -3, -2, 2, -2, 2, -5, 73, 164, 2883, 1084, 88]}, 'GLFq': {'d': 2, 'q': 16, 'gens': [40110, 63055, 62623, 49164, 23685, 62255]}, 'Perm': {'d': 13, 'gens': [43551360, 33, 482630400, 1037836800, 1037877840, 1520557920]}}, 'schur_multiplier': [2, 2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [15], 'solvability_type': 8, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': 'C_2^4:C_{15}', 'transitive_degree': 60, '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': '15.1', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 3, 'aut_exponent': 12, 'aut_gen_orders': [2, 2, 4, 4, 2, 6], 'aut_gens': [[1, 3, 6], [1, 3, 54], [34, 3, 54], [4, 3, 18], [1, 3, 42], [35, 33, 54], [31, 30, 57]], 'aut_group': '96.186', 'aut_hash': 186, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 96, 'aut_permdeg': 8, 'aut_perms': [16, 5176, 16689, 18, 15960, 10936], 'aut_phi_ratio': 6.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 3, 1, 1], [3, 4, 2, 1], [5, 1, 4, 1], [10, 3, 4, 1], [15, 4, 8, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_4\\times S_4', 'autcent_abelian': True, 'autcent_cyclic': True, 'autcent_exponent': 4, 'autcent_group': '4.1', 'autcent_hash': 1, 'autcent_nilpotent': True, 'autcent_order': 4, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_4', '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], [5, 1, 4], [10, 3, 4], [15, 4, 8]], 'center_label': '5.1', 'center_order': 5, 'central_product': True, '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', '5.1'], 'composition_length': 4, 'conjugacy_classes_known': True, 'counter': 9, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['12.3', 1], ['5.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 3, 1, 1], [3, 4, 2, 1], [5, 1, 4, 1], [10, 3, 4, 1], [15, 4, 8, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 24, 'exponent': 30, 'exponents_of_order': [2, 1, 1], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3, 5], 'faithful_reps': [[3, 0, 4]], 'familial': False, 'frattini_label': '1.1', 'frattini_quotient': '60.9', 'hash': 9, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 6, 'inner_gen_orders': [3, 2, 2], 'inner_gens': [[1, 30, 9], [34, 3, 6], [4, 3, 6]], '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': 12, 'irrQ_dim': 12, 'irrR_degree': 6, 'irrep_stats': [[1, 15], [3, 5]], 'label': '60.9', 'linC_count': 4, 'linC_degree': 3, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 7, 'linQ_degree_count': 1, 'linQ_dim': 7, 'linQ_dim_count': 1, 'linR_count': 6, 'linR_degree': 5, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'C5*A4', 'ngens': 4, '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': 6, 'number_conjugacy_classes': 20, 'number_divisions': 6, 'number_normal_subgroups': 6, 'number_subgroup_autclasses': 10, 'number_subgroup_classes': 10, 'number_subgroups': 20, 'old_label': None, 'order': 60, 'order_factorization_type': 222, 'order_stats': [[1, 1], [2, 3], [3, 8], [5, 4], [10, 12], [15, 32]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 4, 'outer_gen_orders': [2, 4], 'outer_gen_pows': [0, 0], 'outer_gens': [[2, 30, 27], [1, 3, 18]], 'outer_group': '8.2', 'outer_hash': 2, 'outer_nilpotent': True, 'outer_order': 8, 'outer_permdeg': 6, 'outer_perms': [120, 9], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2\\times C_4', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 9, 'pgroup': 0, 'primary_abelian_invariants': [3, 5], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 1], [2, 1], [3, 1], [4, 1], [8, 1], [12, 1]], 'representations': {'PC': {'code': 4852423747, 'gens': [1, 2, 3], 'pres': [4, -3, -2, 2, -5, 241, 110, 34]}, 'GLFp': {'d': 3, 'p': 11, 'gens': [656967451, 1691722594, 857494092, 1907820797]}, 'Perm': {'d': 9, 'gens': [5760, 33, 41040, 90720]}}, 'schur_multiplier': [2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [15], 'solvability_type': 8, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': 'C_5\\times A_4', 'transitive_degree': 20, 'wreath_data': None, 'wreath_product': False}
