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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': True, 'ambient': '2800.h', 'ambient_counter': 8, 'ambient_order': 2800, 'ambient_tex': 'C_{280}.C_{10}', 'central': False, 'central_factor': False, 'centralizer_order': 1400, 'characteristic': True, 'core_order': 8, 'counter': 76, 'cyclic': True, 'direct': False, 'hall': 0, 'label': '2800.h.350.a1.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': True, 'old_label': '350.a1.a1', 'outer_equivalence': False, 'perfect': False, 'proper': True, 'quotient': '350.7', 'quotient_Agroup': True, 'quotient_abelian': False, 'quotient_cyclic': False, 'quotient_hash': 7, 'quotient_metabelian': True, 'quotient_nilpotent': False, 'quotient_order': 350, 'quotient_simple': False, 'quotient_solvable': True, 'quotient_supersolvable': True, 'quotient_tex': 'C_5\\times D_{35}', 'simple': False, 'solvable': True, 'special_labels': [], 'split': False, 'standard_generators': False, 'stem': False, 'subgroup': '8.1', 'subgroup_hash': 1, 'subgroup_order': 8, 'subgroup_tex': 'C_8', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '2800.h', 'aut_centralizer_order': 26880, 'aut_label': '350.a1', 'aut_quo_index': 1, 'aut_stab_index': 1, 'aut_weyl_group': '4.2', 'aut_weyl_index': 26880, 'centralizer': '2.b1.a1', 'complements': [], 'conjugacy_class_count': 1, 'contained_in': ['50.a1.a1', '70.a1.a1', '70.b1.a1', '70.e1.a1', '70.e1.b1', '175.a1.a1'], 'contains': ['700.a1.a1'], 'core': '350.a1.a1', 'coset_action_label': None, 'count': 1, 'diagramx': [1396, 2200, 8873, 1143, 7764, 8660, 8596, 8655], 'generators': [350], 'label': '2800.h.350.a1.a1', 'mobius_quo': 0, 'mobius_sub': 35, 'normal_closure': '350.a1.a1', 'normal_contained_in': ['50.a1.a1', '70.a1.a1', '70.b1.a1'], 'normal_contains': ['700.a1.a1'], 'normalizer': '1.a1.a1', 'old_label': '350.a1.a1', 'projective_image': '1400.98', 'quotient_action_image': '2.1', 'quotient_action_kernel': '175.2', 'quotient_action_kernel_order': 175, 'quotient_fusion': None, 'short_label': '350.a1.a1', 'subgroup_fusion': None, 'weyl_group': '2.1'}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': True, 'abelian_quotient': '8.1', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': True, 'aut_cyclic': False, 'aut_derived_length': 1, 'aut_exponent': 2, 'aut_gen_orders': [2, 2], 'aut_gens': [[1], [5], [3]], 'aut_group': '4.2', 'aut_hash': 2, 'aut_nilpotency_class': 1, 'aut_nilpotent': True, 'aut_order': 4, 'aut_permdeg': 4, 'aut_perms': [1, 6], 'aut_phi_ratio': 1.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [4, 1, 2, 1], [8, 1, 4, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_2^2', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 2, 'autcent_group': '4.2', 'autcent_hash': 2, 'autcent_nilpotent': True, 'autcent_order': 4, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^2', '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], [2, 1, 1], [4, 1, 2], [8, 1, 4]], 'center_label': '8.1', 'center_order': 8, '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', '2.1', '2.1'], 'composition_length': 3, 'conjugacy_classes_known': True, 'counter': 1, 'cyclic': True, 'derived_length': 1, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [4, 1, 2, 1], [8, 1, 4, 1]], 'element_repr_type': 'PC', 'elementary': 2, 'eulerian_function': 1, 'exponent': 8, 'exponents_of_order': [3], 'factors_of_aut_order': [2], 'factors_of_order': [2], 'faithful_reps': [[1, 0, 4]], 'familial': True, 'frattini_label': '4.1', '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': 4, 'irrQ_dim': 4, 'irrR_degree': 2, 'irrep_stats': [[1, 8]], 'label': '8.1', 'linC_count': 4, 'linC_degree': 1, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 4, 'linQ_degree_count': 1, 'linQ_dim': 4, 'linQ_dim_count': 1, 'linR_count': 2, 'linR_degree': 2, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C8', 'ngens': 1, 'nilpotency_class': 1, 'nilpotent': True, 'normal_counts': [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': 8, 'number_divisions': 4, 'number_normal_subgroups': 4, 'number_subgroup_autclasses': 4, 'number_subgroup_classes': 4, 'number_subgroups': 4, 'old_label': None, 'order': 8, 'order_factorization_type': 3, 'order_stats': [[1, 1], [2, 1], [4, 2], [8, 4]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 2, 'outer_gen_orders': [2, 2], 'outer_gen_pows': [0, 0], 'outer_gens': [[5], [3]], 'outer_group': '4.2', 'outer_hash': 2, 'outer_nilpotent': True, 'outer_order': 4, 'outer_permdeg': 4, 'outer_perms': [1, 6], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^2', 'pc_rank': 1, 'perfect': False, 'permutation_degree': 8, 'pgroup': 2, 'primary_abelian_invariants': [8], 'quasisimple': False, 'rank': 1, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 2], [2, 1], [4, 1]], 'representations': {'PC': {'code': 323, 'gens': [1], 'pres': [3, -2, -2, -2, 6, 16]}, 'GLZ': {'b': 3, 'd': 4, 'gens': [26315074]}, 'GLFp': {'d': 2, 'p': 3, 'gens': [77]}, 'Perm': {'d': 8, 'gens': [40176, 16582, 5167]}}, 'schur_multiplier': [], 'semidirect_product': False, 'simple': False, 'smith_abelian_invariants': [8], 'solvability_type': 0, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_8', 'transitive_degree': 8, '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': '20.5', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 840, 'aut_gen_orders': [24, 12, 12, 12, 12, 12, 4], 'aut_gens': [[1, 10], [1253, 730], [191, 2630], [2601, 370], [1847, 2630], [1383, 2690], [1887, 90], [1927, 2090]], 'aut_group': None, 'aut_hash': 2535493821222932349, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 107520, 'aut_permdeg': 284, 'aut_perms': 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[140, 2, 24, 2], [140, 2, 96, 1], [280, 2, 48, 2], [280, 2, 192, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_{140}.C_6.C_2.C_2^6', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 4, 'autcent_group': '16.10', 'autcent_hash': 10, 'autcent_nilpotent': True, 'autcent_order': 16, 'autcent_solvable': True, 'autcent_split': False, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^2\\times C_4', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 420, 'autcentquo_group': None, 'autcentquo_hash': 1492037419303921345, 'autcentquo_nilpotent': False, 'autcentquo_order': 6720, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'C_{70}.(C_2^3\\times C_{12})', 'cc_stats': [[1, 1, 1], [2, 1, 1], [4, 2, 1], [4, 140, 2], [5, 1, 4], [5, 2, 10], [7, 2, 3], [8, 2, 2], [10, 1, 4], [10, 2, 10], [14, 2, 3], [20, 2, 24], [20, 140, 8], [28, 2, 6], [35, 2, 72], [40, 2, 48], [56, 2, 12], [70, 2, 72], [140, 2, 144], [280, 2, 288]], 'center_label': '10.2', 'center_order': 10, 'central_product': True, 'central_quotient': '280.26', 'commutator_count': 1, 'commutator_label': '140.4', 'complements_known': True, 'complete': False, 'complex_characters_known': False, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '5.1', '5.1', '7.1'], 'composition_length': 7, 'conjugacy_classes_known': True, 'counter': 8, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['5.1', 1], ['560.68', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [4, 2, 1, 1], [4, 140, 1, 2], [5, 1, 4, 1], [5, 2, 2, 1], [5, 2, 4, 2], [7, 2, 3, 1], [8, 2, 2, 1], [10, 1, 4, 1], [10, 2, 2, 1], [10, 2, 4, 2], [14, 2, 3, 1], [20, 2, 4, 2], [20, 2, 8, 2], [20, 140, 4, 2], [28, 2, 6, 1], [35, 2, 12, 2], [35, 2, 24, 2], [40, 2, 8, 2], [40, 2, 16, 2], [56, 2, 12, 1], [70, 2, 12, 2], [70, 2, 24, 2], [140, 2, 24, 2], [140, 2, 48, 2], [280, 2, 48, 2], [280, 2, 96, 2]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 18, 'exponent': 280, 'exponents_of_order': [4, 2, 1], 'factors_of_aut_order': [2, 3, 5, 7], 'factors_of_order': [2, 5, 7], 'faithful_reps': [[2, 0, 192]], 'familial': False, 'frattini_label': '4.1', 'frattini_quotient': '700.33', 'hash': 4146598981154431277, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 140, 'inner_gen_orders': [2, 140], 'inner_gens': [[1, 2790], [21, 10]], 'inner_hash': 26, 'inner_nilpotent': False, 'inner_order': 280, 'inner_split': False, 'inner_tex': 'D_{140}', 'inner_used': [1, 2], 'irrC_degree': 2, 'irrQ_degree': 192, 'irrQ_dim': 192, 'irrR_degree': None, 'irrep_stats': [[1, 20], [2, 695]], 'label': '2800.h', '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': True, 'metacyclic': True, 'monomial': True, 'name': 'C280.C10', '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], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 34, 'number_characteristic_subgroups': 34, 'number_conjugacy_classes': 715, 'number_divisions': 44, 'number_normal_subgroups': 38, 'number_subgroup_autclasses': 64, 'number_subgroup_classes': 88, 'number_subgroups': 736, 'old_label': None, 'order': 2800, 'order_factorization_type': 321, 'order_stats': [[1, 1], [2, 1], [4, 282], [5, 24], [7, 6], [8, 4], [10, 24], [14, 6], [20, 1168], [28, 12], [35, 144], [40, 96], [56, 24], [70, 144], [140, 288], [280, 576]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 12, 'outer_gen_orders': [2, 2, 2, 4, 12], 'outer_gen_pows': [0, 0, 350, 0, 0], 'outer_gens': [[1, 710], [1, 1410], [351, 10], [1, 1130], [1757, 2430]], 'outer_group': '384.19878', 'outer_hash': 19878, 'outer_nilpotent': True, 'outer_order': 384, 'outer_permdeg': 17, 'outer_perms': [486985704, 20922789888000, 87178291200, 4032000, 21010458068907], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^3\\times C_4\\times C_{12}', 'pc_rank': 2, 'perfect': False, 'permutation_degree': 33, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 5], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 1], [4, 7], [6, 2], [8, 6], [12, 1], [16, 4], [24, 5], [32, 2], [48, 6], [96, 4], [192, 2]], 'representations': {'PC': {'code': '5575144021664765978760299061298352131462519764407', 'gens': [1, 3], 'pres': [7, -2, -5, -2, -2, -2, -5, -7, 14, 9808, 58592, 58, 77843, 80, 96604, 102, 114245, 250, 117606]}, 'GLFp': {'d': 2, 'p': 281, 'gens': [1908171527, 2085675857, 114929]}, 'Perm': {'d': 33, 'gens': [280408000847069544612145368990841440, 44635285094441, 552036980173586265260141429650007588]}}, 'schur_multiplier': [], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 10], 'solvability_type': 6, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_{280}.C_{10}', 'transitive_degree': 560, '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': '10.2', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 420, 'aut_gen_orders': [60, 30, 28, 12], 'aut_gens': [[1, 10], [327, 110], [311, 260], [151, 220], [223, 230]], 'aut_group': None, 'aut_hash': 6689779008057805882, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 3360, 'aut_permdeg': 39, 'aut_perms': [12276634611259940266075176873975331271664283490, 19221019525894338726674953538768658820018229872, 19779225335434016187892073765821497684583046432, 1165022133730743137378744179309508987377555714], 'aut_phi_ratio': 28.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 35, 1, 1], [5, 1, 4, 1], [5, 2, 2, 1], [5, 2, 8, 1], [7, 2, 3, 1], [10, 35, 4, 1], [35, 2, 12, 2], [35, 2, 48, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_4\\times F_5\\times F_7', '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': 420, 'autcentquo_group': '840.139', 'autcentquo_hash': 139, 'autcentquo_nilpotent': False, 'autcentquo_order': 840, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'F_5\\times F_7', 'cc_stats': [[1, 1, 1], [2, 35, 1], [5, 1, 4], [5, 2, 10], [7, 2, 3], [10, 35, 4], [35, 2, 72]], 'center_label': '5.1', 'center_order': 5, 'central_product': True, 'central_quotient': '70.3', 'commutator_count': 1, 'commutator_label': '35.1', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '5.1', '5.1', '7.1'], 'composition_length': 4, 'conjugacy_classes_known': True, 'counter': 7, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['5.1', 1], ['70.3', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 35, 1, 1], [5, 1, 4, 1], [5, 2, 2, 1], [5, 2, 4, 2], [7, 2, 3, 1], [10, 35, 4, 1], [35, 2, 12, 2], [35, 2, 24, 2]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 18, 'exponent': 70, 'exponents_of_order': [2, 1, 1], 'factors_of_aut_order': [2, 3, 5, 7], 'factors_of_order': [2, 5, 7], 'faithful_reps': [[2, 0, 48]], 'familial': False, 'frattini_label': '1.1', 'frattini_quotient': '350.7', 'hash': 7, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 70, 'inner_gen_orders': [2, 35], 'inner_gens': [[1, 340], [21, 10]], 'inner_hash': 3, 'inner_nilpotent': False, 'inner_order': 70, 'inner_split': True, 'inner_tex': 'D_{35}', 'inner_used': [1, 2], 'irrC_degree': 2, 'irrQ_degree': 48, 'irrQ_dim': 48, 'irrR_degree': 4, 'irrep_stats': [[1, 10], [2, 85]], 'label': '350.7', 'linC_count': 48, 'linC_degree': 2, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 14, 'linQ_degree_count': 4, 'linQ_dim': 14, 'linQ_dim_count': 4, 'linR_count': 72, 'linR_degree': 4, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C5*D35', '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': 10, 'number_characteristic_subgroups': 10, 'number_conjugacy_classes': 95, 'number_divisions': 12, 'number_normal_subgroups': 10, 'number_subgroup_autclasses': 18, 'number_subgroup_classes': 20, 'number_subgroups': 112, 'old_label': None, 'order': 350, 'order_factorization_type': 222, 'order_stats': [[1, 1], [2, 35], [5, 24], [7, 6], [10, 140], [35, 144]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 12, 'outer_gen_orders': [4, 12], 'outer_gen_pows': [0, 0], 'outer_gens': [[1, 80], [3, 160]], 'outer_group': '48.20', 'outer_hash': 20, 'outer_nilpotent': True, 'outer_order': 48, 'outer_permdeg': 11, 'outer_perms': [864, 4032003], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_4\\times C_{12}', 'pc_rank': 2, 'perfect': False, 'permutation_degree': 17, 'pgroup': 0, 'primary_abelian_invariants': [2, 5], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 2], [4, 3], [6, 1], [8, 2], [24, 2], [48, 2]], 'representations': {'PC': {'code': 144443913776339, 'gens': [1, 3], 'pres': [4, -2, -5, -5, -7, 8, 4082, 70, 4803]}, 'GLFp': {'d': 2, 'p': 71, 'gens': [19327195, 10379468, 5112]}, 'Perm': {'d': 17, 'gens': [21115948009087, 46032173550720, 66718256814720, 973]}}, 'schur_multiplier': [], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [10], 'solvability_type': 6, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_5\\times D_{35}', 'transitive_degree': 70, 'wreath_data': None, 'wreath_product': False}
