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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': True, 'ambient': '480.1097', 'ambient_counter': 1097, 'ambient_order': 480, 'ambient_tex': 'D_{12}:D_{10}', 'central': False, 'central_factor': False, 'centralizer_order': 16, 'characteristic': False, 'core_order': 4, 'counter': 291, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '480.1097.30.h1.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': False, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': False, 'old_label': '30.h1.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': 30, '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': '16.10', 'subgroup_hash': 10, 'subgroup_order': 16, 'subgroup_tex': 'C_2^2\\times C_4', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '480.1097', 'aut_centralizer_order': 32, 'aut_label': '30.h1', 'aut_quo_index': None, 'aut_stab_index': 15, 'aut_weyl_group': '8.5', 'aut_weyl_index': 480, 'centralizer': '30.h1.a1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['6.h1.a1', '10.h1.a1', '15.a1.a1'], 'contains': ['60.h1.a1', '60.j1.a1', '60.k1.a1', '60.m1.a1', '60.n1.a1', '60.u1.a1', '60.v1.a1'], 'core': '120.b1.a1', 'coset_action_label': None, 'count': 15, 'diagramx': [8335, -1, 9741, -1, 1736, -1, 9808, -1], 'generators': [1, 4, 120], 'label': '480.1097.30.h1.a1', 'mobius_quo': None, 'mobius_sub': -1, 'normal_closure': '2.h1.a1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '15.a1.a1', 'old_label': '30.h1.a1', 'projective_image': '240.202', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '30.h1.a1', 'subgroup_fusion': None, 'weyl_group': '2.1'}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': True, 'abelian_quotient': '16.10', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 4, 'aut_exponent': 12, 'aut_gen_orders': [2, 3, 2, 2, 2, 2, 2], 'aut_gens': [[1, 2, 4], [10, 9, 7], [10, 3, 15], [1, 2, 5], [1, 2, 14], [9, 2, 14], [1, 10, 13], [1, 2, 12]], 'aut_group': '192.1493', 'aut_hash': 1493, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 192, 'aut_permdeg': 8, 'aut_perms': [17764, 31287, 5329, 40319, 37965, 21769, 12316], 'aut_phi_ratio': 24.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 1, 6, 1], [4, 1, 8, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_2^3:S_4', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 12, 'autcent_group': '192.1493', 'autcent_hash': 1493, 'autcent_nilpotent': False, 'autcent_order': 192, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': False, 'autcent_tex': 'C_2^3:S_4', '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, 7], [4, 1, 8]], 'center_label': '16.10', 'center_order': 16, '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', '2.1', '2.1'], 'composition_length': 4, 'conjugacy_classes_known': True, 'counter': 10, 'cyclic': False, 'derived_length': 1, 'dihedral': False, 'direct_factorization': [['2.1', 2], ['4.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 7], [4, 1, 2, 4]], 'element_repr_type': 'PC', 'elementary': 2, 'eulerian_function': 7, 'exponent': 4, 'exponents_of_order': [4], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2], 'faithful_reps': [], 'familial': False, 'frattini_label': '2.1', 'frattini_quotient': '8.5', 'hash': 10, 'hyperelementary': 2, 'inner_abelian': True, 'inner_cyclic': True, 'inner_exponent': 1, 'inner_gen_orders': [1, 1, 1], 'inner_gens': [[1, 2, 4], [1, 2, 4], [1, 2, 4]], '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, 16]], 'label': '16.10', 'linC_count': 224, 'linC_degree': 3, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 4, 'linQ_degree_count': 48, 'linQ_dim': 4, 'linQ_dim_count': 48, 'linR_count': 48, 'linR_degree': 4, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'C2^2*C4', 'ngens': 3, 'nilpotency_class': 1, 'nilpotent': True, 'normal_counts': [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': 16, 'number_divisions': 12, 'number_normal_subgroups': 27, 'number_subgroup_autclasses': 9, 'number_subgroup_classes': 27, 'number_subgroups': 27, 'old_label': None, 'order': 16, 'order_factorization_type': 3, 'order_stats': [[1, 1], [2, 7], [4, 8]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 12, 'outer_gen_orders': [2, 3, 2, 2, 2, 2, 2], 'outer_gen_pows': [0, 0, 0, 0, 0, 0, 0], 'outer_gens': [[10, 9, 7], [10, 3, 15], [1, 2, 5], [1, 2, 14], [9, 2, 14], [1, 10, 13], [1, 2, 12]], 'outer_group': '192.1493', 'outer_hash': 1493, 'outer_nilpotent': False, 'outer_order': 192, 'outer_permdeg': 8, 'outer_perms': [17764, 31287, 5329, 40319, 37965, 21769, 12316], 'outer_solvable': True, 'outer_supersolvable': False, 'outer_tex': 'C_2^3:S_4', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 8, 'pgroup': 2, 'primary_abelian_invariants': [2, 2, 4], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 8], [2, 4]], 'representations': {'PC': {'code': 516, 'gens': [1, 2, 3], 'pres': [4, -2, 2, 2, -2, 34]}, 'GLZ': {'b': 3, 'd': 4, 'gens': [7233746, 7115326, 7115160]}, 'GLFp': {'d': 3, 'p': 5, 'gens': [1173753, 1832002, 438254, 1565004]}, 'Perm': {'d': 8, 'gens': [22, 5040, 120, 7]}}, 'schur_multiplier': [2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 4], 'solvability_type': 2, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_2^2\\times C_4', 'transitive_degree': 16, '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': '16.14', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 60, 'aut_gen_orders': [12, 4, 4, 10, 20, 12, 4], 'aut_gens': [[1, 2, 4, 8], [321, 242, 468, 344], [241, 122, 244, 136], [241, 122, 244, 8], [241, 2, 388, 328], [401, 362, 308, 328], [401, 362, 68, 344], [1, 122, 148, 344]], 'aut_group': None, 'aut_hash': 2590250131169074209, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 3840, 'aut_permdeg': 20, 'aut_perms': [1443698682922712859, 2396863850796989291, 13168375807915442, 1054849567346512607, 712765677513853206, 1411588745921329904, 1359821827288367693], 'aut_phi_ratio': 30.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 2, 2, 1], [2, 3, 2, 1], [2, 5, 2, 1], [2, 6, 2, 1], [2, 10, 2, 1], [2, 15, 2, 1], [2, 30, 2, 1], [3, 2, 1, 1], [4, 2, 1, 1], [4, 6, 1, 1], [4, 10, 1, 1], [4, 30, 1, 1], [5, 2, 2, 1], [6, 2, 1, 1], [6, 4, 2, 1], [6, 10, 2, 1], [6, 20, 2, 1], [10, 2, 2, 1], [10, 4, 4, 1], [10, 6, 4, 1], [10, 12, 4, 1], [12, 4, 1, 1], [12, 20, 1, 1], [15, 4, 2, 1], [20, 4, 2, 1], [20, 12, 2, 1], [30, 4, 2, 1], [30, 8, 4, 1], [60, 8, 2, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_{15}:(C_2^2.C_2^6)', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 2, 'autcent_group': '16.14', 'autcent_hash': 14, 'autcent_nilpotent': True, 'autcent_order': 16, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^4', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 60, 'autcentquo_group': '240.195', 'autcentquo_hash': 195, 'autcentquo_nilpotent': False, 'autcentquo_order': 240, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'D_6\\times F_5', 'cc_stats': [[1, 1, 1], [2, 1, 1], [2, 2, 2], [2, 3, 2], [2, 5, 2], [2, 6, 2], [2, 10, 2], [2, 15, 2], [2, 30, 2], [3, 2, 1], [4, 2, 1], [4, 6, 1], [4, 10, 1], [4, 30, 1], [5, 2, 2], [6, 2, 1], [6, 4, 2], [6, 10, 2], [6, 20, 2], [10, 2, 2], [10, 4, 4], [10, 6, 4], [10, 12, 4], [12, 4, 1], [12, 20, 1], [15, 4, 2], [20, 4, 2], [20, 12, 2], [30, 4, 2], [30, 8, 4], [60, 8, 2]], 'center_label': '2.1', 'center_order': 2, 'central_product': True, 'central_quotient': '240.202', 'commutator_count': 1, 'commutator_label': '30.4', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '2.1', '3.1', '5.1'], 'composition_length': 7, 'conjugacy_classes_known': True, 'counter': 1097, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['10.1', 1], ['6.1', 1], ['8.3', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 2, 1, 2], [2, 3, 1, 2], [2, 5, 1, 2], [2, 6, 1, 2], [2, 10, 1, 2], [2, 15, 1, 2], [2, 30, 1, 2], [3, 2, 1, 1], [4, 2, 1, 1], [4, 6, 1, 1], [4, 10, 1, 1], [4, 30, 1, 1], [5, 2, 2, 1], [6, 2, 1, 1], [6, 4, 1, 2], [6, 10, 1, 2], [6, 20, 1, 2], [10, 2, 2, 1], [10, 4, 2, 2], [10, 6, 2, 2], [10, 12, 2, 2], [12, 4, 1, 1], [12, 20, 1, 1], [15, 4, 2, 1], [20, 4, 2, 1], [20, 12, 2, 1], [30, 4, 2, 1], [30, 8, 2, 2], [60, 8, 2, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 4062240, 'exponent': 60, 'exponents_of_order': [5, 1, 1], 'factors_of_aut_order': [2, 3, 5], 'factors_of_order': [2, 3, 5], 'faithful_reps': [[8, 1, 2]], 'familial': False, 'frattini_label': '2.1', 'frattini_quotient': '240.202', 'hash': 1097, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 30, 'inner_gen_orders': [2, 2, 2, 30], 'inner_gens': [[1, 2, 4, 328], [1, 2, 4, 248], [1, 2, 4, 232], [161, 242, 260, 8]], 'inner_hash': 202, 'inner_nilpotent': False, 'inner_order': 240, 'inner_split': False, 'inner_tex': 'D_6\\times D_{10}', 'inner_used': [1, 2, 3, 4], 'irrC_degree': 8, 'irrQ_degree': 16, 'irrQ_dim': 16, 'irrR_degree': 8, 'irrep_stats': [[1, 16], [2, 28], [4, 14], [8, 2]], 'label': '480.1097', 'linC_count': 608, 'linC_degree': 6, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 8, 'linQ_degree_count': 272, 'linQ_dim': 8, 'linQ_dim_count': 272, 'linR_count': 608, 'linR_degree': 6, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'D12:D10', '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], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 31, 'number_characteristic_subgroups': 50, 'number_conjugacy_classes': 60, 'number_divisions': 45, 'number_normal_subgroups': 122, 'number_subgroup_autclasses': 252, 'number_subgroup_classes': 472, 'number_subgroups': 2892, 'old_label': None, 'order': 480, 'order_factorization_type': 311, 'order_stats': [[1, 1], [2, 143], [3, 2], [4, 48], [5, 4], [6, 70], [10, 92], [12, 24], [15, 8], [20, 32], [30, 40], [60, 16]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 2, 'outer_gen_orders': [2, 2, 2, 2], 'outer_gen_pows': [0, 0, 120, 5], 'outer_gens': [[241, 2, 4, 8], [1, 2, 244, 8], [1, 122, 4, 8], [1, 2, 4, 104]], 'outer_group': '16.14', 'outer_hash': 14, 'outer_nilpotent': True, 'outer_order': 16, 'outer_permdeg': 8, 'outer_perms': [5040, 120, 6, 1], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^4', 'pc_rank': 4, 'perfect': False, 'permutation_degree': 12, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 2, 2], 'quasisimple': False, 'rank': 4, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 16], [2, 12], [4, 10], [8, 6], [16, 1]], 'representations': {'PC': {'code': 4310008185727079791829190034761456680967, 'gens': [1, 2, 3, 4], 'pres': [7, -2, -2, -2, -2, -2, -3, -5, 9187, 3482, 1641, 80, 6164, 4078, 102, 14789, 4723, 166, 4724]}, 'GLZN': {'d': 2, 'p': 30, 'gens': [185256, 252609, 40966, 297011, 27181, 252309, 27649]}, 'Perm': {'d': 12, 'gens': [7, 720, 40279680, 7620480, 87091200, 5760, 37]}}, 'schur_multiplier': [2, 2, 2, 2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 2, 2], 'solvability_type': 7, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'D_{12}:D_{10}', 'transitive_degree': 60, 'wreath_data': None, 'wreath_product': False}
