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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': True, 'ambient': '232.3', 'ambient_counter': 3, 'ambient_order': 232, 'ambient_tex': 'C_{29}:C_8', 'central': False, 'central_factor': False, 'centralizer_order': 8, 'characteristic': False, 'core_order': 2, 'counter': 5, 'cyclic': True, 'direct': None, 'hall': 2, 'label': '232.3.29.a1.a1', 'maximal': True, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': False, 'old_label': '29.a1.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': 29, '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': '8.1', 'subgroup_hash': 1, 'subgroup_order': 8, 'subgroup_tex': 'C_8', 'supersolvable': True, 'sylow': 2}
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gps_subgroup_data • Show schema
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{'ambient': '232.3', 'aut_centralizer_order': 28, 'aut_label': '29.a1', 'aut_quo_index': None, 'aut_stab_index': 29, 'aut_weyl_group': '2.1', 'aut_weyl_index': 812, 'centralizer': '29.a1.a1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['1.a1.a1'], 'contains': ['58.a1.a1'], 'core': '116.a1.a1', 'coset_action_label': None, 'count': 29, 'diagramx': [1238, -1, 1238, -1, 1556, -1, 1556, -1], 'generators': [1], 'label': '232.3.29.a1.a1', 'mobius_quo': None, 'mobius_sub': -1, 'normal_closure': '1.a1.a1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '29.a1.a1', 'old_label': '29.a1.a1', 'projective_image': '116.3', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '29.a1.a1', 'subgroup_fusion': None, 'weyl_group': '1.1'}
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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': True, 'Zgroup': True, 'abelian': False, 'abelian_quotient': '8.1', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 812, 'aut_gen_orders': [28, 58], 'aut_gens': [[1, 8], [5, 16], [229, 8]], 'aut_group': '1624.43', 'aut_hash': 43, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 1624, 'aut_permdeg': 31, 'aut_perms': [18316338061114844722254956024454, 531117947303972730619403121641041], 'aut_phi_ratio': 14.5, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [4, 29, 1, 2], [8, 29, 2, 2], [29, 4, 7, 1], [58, 4, 7, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_2\\times F_{29}', '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': 812, 'autcentquo_group': '812.7', 'autcentquo_hash': 7, 'autcentquo_nilpotent': False, 'autcentquo_order': 812, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'F_{29}', 'cc_stats': [[1, 1, 1], [2, 1, 1], [4, 29, 2], [8, 29, 4], [29, 4, 7], [58, 4, 7]], 'center_label': '2.1', 'center_order': 2, 'central_product': False, 'central_quotient': '116.3', 'commutator_count': 1, 'commutator_label': '29.1', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '29.1'], 'composition_length': 4, '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, 1, 1, 1], [4, 29, 2, 1], [8, 29, 4, 1], [29, 4, 7, 1], [58, 4, 7, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 24, 'exponent': 232, 'exponents_of_order': [3, 1], 'factors_of_aut_order': [2, 7, 29], 'factors_of_order': [2, 29], 'faithful_reps': [[4, -1, 7]], 'familial': False, 'frattini_label': '2.1', 'frattini_quotient': '116.3', 'hash': 3, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 116, 'inner_gen_orders': [4, 29], 'inner_gens': [[1, 96], [145, 8]], 'inner_hash': 3, 'inner_nilpotent': False, 'inner_order': 116, 'inner_split': True, 'inner_tex': 'C_{29}:C_4', 'inner_used': [1, 2], 'irrC_degree': 4, 'irrQ_degree': 28, 'irrQ_dim': 56, 'irrR_degree': 8, 'irrep_stats': [[1, 8], [4, 14]], 'label': '232.3', 'linC_count': 7, 'linC_degree': 4, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 28, 'linQ_degree_count': 1, 'linQ_dim': 32, 'linQ_dim_count': 1, 'linR_count': 14, 'linR_degree': 6, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C29:C8', 'ngens': 4, 'nilpotency_class': -1, 'nilpotent': False, 'normal_counts': [0, 0, 0, 0, 0, 0, 0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 8, 'number_characteristic_subgroups': 6, 'number_conjugacy_classes': 22, 'number_divisions': 6, 'number_normal_subgroups': 6, 'number_subgroup_autclasses': 8, 'number_subgroup_classes': 8, 'number_subgroups': 64, 'old_label': None, 'order': 232, 'order_factorization_type': 31, 'order_stats': [[1, 1], [2, 1], [4, 58], [8, 116], [29, 28], [58, 28]], 'outer_abelian': True, 'outer_cyclic': True, 'outer_equivalence': False, 'outer_exponent': 14, 'outer_gen_orders': [14], 'outer_gen_pows': [2], 'outer_gens': [[5, 144]], 'outer_group': '14.2', 'outer_hash': 2, 'outer_nilpotent': True, 'outer_order': 14, 'outer_permdeg': 9, 'outer_perms': [43776], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_{14}', 'pc_rank': 2, 'perfect': False, 'permutation_degree': 37, 'pgroup': 0, 'primary_abelian_invariants': [8], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 2], [2, 1], [4, 1], [28, 2]], 'representations': {'PC': {'code': 106291983751627547, 'gens': [1, 4], 'pres': [4, -2, -2, -2, -29, 8, 21, 1539, 1799]}, 'GLFq': {'d': 2, 'q': 841, 'gens': [10174237197, 16655053016, 430473849674, 129746059462]}, 'Perm': {'d': 37, 'gens': [10646538139028000373722798092792667850537, 21553048506737100747439045268983004746648, 18498, 404175060581506640821862170738864023398400]}}, 'schur_multiplier': [], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [8], 'solvability_type': 6, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_{29}:C_8', 'transitive_degree': 232, 'wreath_data': None, 'wreath_product': False}
