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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': True, 'ambient': '1350.41', 'ambient_counter': 41, 'ambient_order': 1350, 'ambient_tex': 'C_{15}^2.S_3', 'central': False, 'central_factor': False, 'centralizer_order': 27, 'characteristic': False, 'core_order': 3, 'counter': 36, 'cyclic': True, 'direct': None, 'hall': 0, 'label': '1350.41.150.b1.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': False, 'old_label': '150.b1.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': 150, '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': '9.1', 'subgroup_hash': 1, 'subgroup_order': 9, 'subgroup_tex': 'C_9', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '1350.41', 'aut_centralizer_order': 72, 'aut_label': '150.b1', 'aut_quo_index': None, 'aut_stab_index': 25, 'aut_weyl_group': '6.2', 'aut_weyl_index': 1800, 'centralizer': '50.a1.a1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['6.b1.a1', '50.a1.a1', '75.b1.a1'], 'contains': ['450.b1.a1'], 'core': '450.b1.a1', 'coset_action_label': None, 'count': 25, 'diagramx': [708, -1, 8987, -1, 980, -1, 986, -1], 'generators': [8], 'label': '1350.41.150.b1.a1', 'mobius_quo': None, 'mobius_sub': -1, 'normal_closure': '6.b1.a1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '25.a1.a1', 'old_label': '150.b1.a1', 'projective_image': '1350.41', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '150.b1.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': '9.1', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': True, 'aut_cyclic': True, 'aut_derived_length': 1, 'aut_exponent': 6, 'aut_gen_orders': [6], 'aut_gens': [[1], [2]], 'aut_group': '6.2', 'aut_hash': 2, 'aut_nilpotency_class': 1, 'aut_nilpotent': True, 'aut_order': 6, 'aut_permdeg': 5, 'aut_perms': [27], 'aut_phi_ratio': 1.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [3, 1, 2, 1], [9, 1, 6, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_6', 'autcent_abelian': True, 'autcent_cyclic': True, 'autcent_exponent': 6, 'autcent_group': '6.2', 'autcent_hash': 2, 'autcent_nilpotent': True, 'autcent_order': 6, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_6', '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], [3, 1, 2], [9, 1, 6]], 'center_label': '9.1', 'center_order': 9, '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': ['3.1', '3.1'], 'composition_length': 2, 'conjugacy_classes_known': True, 'counter': 1, 'cyclic': True, 'derived_length': 1, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [3, 1, 2, 1], [9, 1, 6, 1]], 'element_repr_type': 'PC', 'elementary': 3, 'eulerian_function': 1, 'exponent': 9, 'exponents_of_order': [2], 'factors_of_aut_order': [2, 3], 'factors_of_order': [3], 'faithful_reps': [[1, 0, 6]], 'familial': True, 'frattini_label': '3.1', 'frattini_quotient': '3.1', 'hash': 1, 'hyperelementary': 3, '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': 6, 'irrQ_dim': 6, 'irrR_degree': 2, 'irrep_stats': [[1, 9]], 'label': '9.1', 'linC_count': 6, 'linC_degree': 1, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 6, 'linQ_degree_count': 1, 'linQ_dim': 6, 'linQ_dim_count': 1, 'linR_count': 3, 'linR_degree': 2, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C9', 'ngens': 1, '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': 3, 'number_characteristic_subgroups': 3, 'number_conjugacy_classes': 9, 'number_divisions': 3, 'number_normal_subgroups': 3, 'number_subgroup_autclasses': 3, 'number_subgroup_classes': 3, 'number_subgroups': 3, 'old_label': None, 'order': 9, 'order_factorization_type': 2, 'order_stats': [[1, 1], [3, 2], [9, 6]], 'outer_abelian': True, 'outer_cyclic': True, 'outer_equivalence': False, 'outer_exponent': 6, 'outer_gen_orders': [6], 'outer_gen_pows': [0], 'outer_gens': [[2]], 'outer_group': '6.2', 'outer_hash': 2, 'outer_nilpotent': True, 'outer_order': 6, 'outer_permdeg': 5, 'outer_perms': [27], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_6', 'pc_rank': 1, 'perfect': False, 'permutation_degree': 9, 'pgroup': 3, 'primary_abelian_invariants': [9], 'quasisimple': False, 'rank': 1, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 1], [2, 1], [6, 1]], 'representations': {'PC': {'code': 5, 'gens': [1], 'pres': [2, -3, -3, 6]}, 'GLZ': {'b': 3, 'd': 6, 'gens': [41907234971425459]}, 'GLFp': {'d': 2, 'p': 17, 'gens': [78572]}, 'Perm': {'d': 9, 'gens': [357120, 80884]}}, 'schur_multiplier': [], 'semidirect_product': False, 'simple': False, 'smith_abelian_invariants': [9], 'solvability_type': 0, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_9', 'transitive_degree': 9, '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': '6.2', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 3, 'aut_exponent': 180, 'aut_gen_orders': [36, 12, 18, 30], 'aut_gens': [[1, 2, 18, 90], [305, 278, 288, 198], [73, 1144, 36, 792], [881, 824, 576, 1296], [85, 314, 18, 990]], 'aut_group': None, 'aut_hash': 106406116256727718, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 10800, 'aut_permdeg': 47, 'aut_perms': [86651058368638026046283518960489069968330873510942592604880, 206089238202062876161870084447779885557147713819038680760438, 97286679572383692931064340704701306132789295204064231518960, 90839569153188758152570305185423397832529412800416861286649], 'aut_phi_ratio': 30.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 45, 1, 1], [3, 1, 2, 1], [3, 2, 1, 1], [3, 2, 2, 1], [5, 3, 4, 1], [5, 6, 2, 1], [6, 45, 2, 1], [9, 50, 3, 1], [9, 50, 6, 1], [10, 45, 4, 1], [15, 3, 8, 1], [15, 6, 4, 3], [15, 6, 8, 2], [30, 45, 8, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_5^2:C_9.C_6.C_2^3', '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': 180, 'autcentquo_group': '5400.w', 'autcentquo_hash': 1369849614865178234, 'autcentquo_nilpotent': False, 'autcentquo_order': 5400, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'C_{15}^2.(C_4\\times S_3)', 'cc_stats': [[1, 1, 1], [2, 45, 1], [3, 1, 2], [3, 2, 3], [5, 3, 4], [5, 6, 2], [6, 45, 2], [9, 50, 9], [10, 45, 4], [15, 3, 8], [15, 6, 28], [30, 45, 8]], 'center_label': '3.1', 'center_order': 3, 'central_product': True, 'central_quotient': '450.11', 'commutator_count': 2, 'commutator_label': '225.3', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '3.1', '3.1', '3.1', '5.1', '5.1'], 'composition_length': 6, 'conjugacy_classes_known': True, 'counter': 41, 'cyclic': False, 'derived_length': 3, 'dihedral': False, 'direct_factorization': [['3.1', 1], ['450.11', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 45, 1, 1], [3, 1, 2, 1], [3, 2, 1, 1], [3, 2, 2, 1], [5, 3, 4, 1], [5, 6, 2, 1], [6, 45, 2, 1], [9, 50, 3, 1], [9, 50, 6, 1], [10, 45, 4, 1], [15, 3, 8, 1], [15, 6, 4, 3], [15, 6, 8, 2], [30, 45, 8, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 72, 'exponent': 90, 'exponents_of_order': [3, 2, 1], 'factors_of_aut_order': [2, 3, 5], 'factors_of_order': [2, 3, 5], 'faithful_reps': [[6, 0, 16]], 'familial': False, 'frattini_label': '3.1', 'frattini_quotient': '450.20', 'hash': 41, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 90, 'inner_gen_orders': [2, 9, 5, 5], 'inner_gens': [[1, 16, 72, 144], [5, 2, 864, 144], [37, 596, 18, 90], [37, 38, 18, 90]], 'inner_hash': 11, 'inner_nilpotent': False, 'inner_order': 450, 'inner_split': True, 'inner_tex': 'C_5^2:D_9', 'inner_used': [1, 2, 3], 'irrC_degree': 6, 'irrQ_degree': 48, 'irrQ_dim': 48, 'irrR_degree': 12, 'irrep_stats': [[1, 6], [2, 12], [3, 24], [6, 30]], 'label': '1350.41', 'linC_count': 192, 'linC_degree': 5, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 20, 'linQ_degree_count': 6, 'linQ_dim': 20, 'linQ_dim_count': 6, 'linR_count': 32, 'linR_degree': 8, 'maximal_subgroups_known': True, 'metabelian': False, 'metacyclic': False, 'monomial': True, 'name': 'C15^2.S3', '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, 0, 0, 0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 18, 'number_characteristic_subgroups': 12, 'number_conjugacy_classes': 72, 'number_divisions': 18, 'number_normal_subgroups': 12, 'number_subgroup_autclasses': 46, 'number_subgroup_classes': 46, 'number_subgroups': 492, 'old_label': None, 'order': 1350, 'order_factorization_type': 321, 'order_stats': [[1, 1], [2, 45], [3, 8], [5, 24], [6, 90], [9, 450], [10, 180], [15, 192], [30, 360]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 12, 'outer_gen_orders': [2, 12], 'outer_gen_pows': [6, 0], 'outer_gens': [[7, 2, 72, 1260], [1, 8, 36, 180]], 'outer_group': '24.9', 'outer_hash': 9, 'outer_nilpotent': True, 'outer_order': 24, 'outer_permdeg': 9, 'outer_perms': [40320, 2163], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2\\times C_{12}', 'pc_rank': 4, 'perfect': False, 'permutation_degree': 27, 'pgroup': 0, 'primary_abelian_invariants': [2, 3], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 2], [2, 3], [4, 1], [6, 1], [12, 4], [24, 5], [48, 2]], 'representations': {'PC': {'code': 3840222315474576447459146927530951254391, 'gens': [1, 2, 4, 5], 'pres': [6, 2, 3, 3, 5, 3, 5, 193, 43, 218, 1731, 10377, 4324, 2170, 118, 12317, 6167]}, 'Perm': {'d': 27, 'gens': [31647428478114742152777600, 3, 435559488968024445075894240, 95943600, 855061339457165861872051200, 47263223632421202729523200]}}, 'schur_multiplier': [], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [6], 'solvability_type': 11, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': 'C_{15}^2.S_3', 'transitive_degree': 90, 'wreath_data': None, 'wreath_product': False}
