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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': False, 'ambient': '910.6', 'ambient_counter': 6, 'ambient_order': 910, 'ambient_tex': 'C_{13}\\times D_{35}', 'central': False, 'central_factor': False, 'centralizer_order': 13, 'characteristic': True, 'core_order': 910, 'counter': 1, 'cyclic': False, 'direct': True, 'hall': 910, 'label': '910.6.1.a1.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': False, 'minimal_normal': False, 'nilpotent': False, 'normal': True, 'old_label': '1.a1.a1', 'outer_equivalence': False, 'perfect': False, 'proper': False, 'quotient': '1.1', 'quotient_Agroup': True, 'quotient_abelian': True, 'quotient_cyclic': True, 'quotient_hash': 1, 'quotient_metabelian': True, 'quotient_nilpotent': True, 'quotient_order': 1, 'quotient_simple': False, 'quotient_solvable': True, 'quotient_supersolvable': True, 'quotient_tex': 'C_1', 'simple': False, 'solvable': True, 'special_labels': ['R', 'L0', 'D0', 'C0'], 'split': True, 'standard_generators': False, 'stem': False, 'subgroup': '910.6', 'subgroup_hash': 6, 'subgroup_order': 910, 'subgroup_tex': 'C_{13}\\times D_{35}', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '910.6', 'aut_centralizer_order': None, 'aut_label': '1.a1', 'aut_quo_index': None, 'aut_stab_index': None, 'aut_weyl_group': None, 'aut_weyl_index': None, 'centralizer': '70.a1.a1', 'complements': ['910.a1.a1'], 'conjugacy_class_count': 1, 'contained_in': [], 'contains': ['2.a1.a1', '5.a1.a1', '7.a1.a1', '13.a1.a1'], 'core': '1.a1.a1', 'coset_action_label': None, 'count': 1, 'diagramx': [5464, 3196, 6620, 3196, 5080, 6574, 4803, 6574], 'generators': [1, 182, 780, 70], 'label': '910.6.1.a1.a1', 'mobius_quo': 0, 'mobius_sub': 1, 'normal_closure': '1.a1.a1', 'normal_contained_in': [], 'normal_contains': ['2.a1.a1', '13.a1.a1'], 'normalizer': '1.a1.a1', 'old_label': '1.a1.a1', 'projective_image': '70.3', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '1.a1.a1', 'subgroup_fusion': None, 'weyl_group': '70.3'}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': False, 'abelian_quotient': '26.2', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 420, 'aut_gen_orders': [30, 4, 12, 12], 'aut_gens': [[1, 2], [833, 292], [365, 726], [365, 794], [391, 816]], 'aut_group': None, 'aut_hash': 1353831758173859334, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 10080, 'aut_permdeg': 47, 'aut_perms': [198710101440593196243218347790811720298878256370630268737019, 140267350450781368047622894677561570976535544194091094500653, 139188430599882027313978426784561684589484624043701455244064, 188253137915665707653624865308908802773522479547081256014097], 'aut_phi_ratio': 35.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 35, 1, 1], [5, 2, 2, 1], [7, 2, 3, 1], [13, 1, 12, 1], [26, 35, 12, 1], [35, 2, 12, 1], [65, 2, 24, 1], [91, 2, 36, 1], [455, 2, 144, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_{12}\\times F_5\\times F_7', 'autcent_abelian': True, 'autcent_cyclic': True, 'autcent_exponent': 12, 'autcent_group': '12.2', 'autcent_hash': 2, 'autcent_nilpotent': True, 'autcent_order': 12, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_{12}', '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, 2, 2], [7, 2, 3], [13, 1, 12], [26, 35, 12], [35, 2, 12], [65, 2, 24], [91, 2, 36], [455, 2, 144]], 'center_label': '13.1', 'center_order': 13, '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', '7.1', '13.1'], 'composition_length': 4, 'conjugacy_classes_known': True, 'counter': 6, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['13.1', 1], ['70.3', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 35, 1, 1], [5, 2, 2, 1], [7, 2, 3, 1], [13, 1, 12, 1], [26, 35, 12, 1], [35, 2, 12, 1], [65, 2, 24, 1], [91, 2, 36, 1], [455, 2, 144, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 42, 'exponent': 910, 'exponents_of_order': [1, 1, 1, 1], 'factors_of_aut_order': [2, 3, 5, 7], 'factors_of_order': [2, 5, 7, 13], 'faithful_reps': [[2, 0, 144]], 'familial': False, 'frattini_label': '1.1', 'frattini_quotient': '910.6', 'hash': 6, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 70, 'inner_gen_orders': [2, 35], 'inner_gens': [[1, 418], [495, 2]], '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': 288, 'irrQ_dim': 288, 'irrR_degree': 4, 'irrep_stats': [[1, 26], [2, 221]], 'label': '910.6', 'linC_count': 144, 'linC_degree': 2, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 22, 'linQ_degree_count': 2, 'linQ_dim': 22, 'linQ_dim_count': 2, 'linR_count': 216, 'linR_degree': 4, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C13*D35', 'ngens': 4, 'nilpotency_class': -1, 'nilpotent': False, 'normal_counts': [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': 10, 'number_characteristic_subgroups': 10, 'number_conjugacy_classes': 247, 'number_divisions': 10, 'number_normal_subgroups': 10, 'number_subgroup_autclasses': 16, 'number_subgroup_classes': 16, 'number_subgroups': 104, 'old_label': None, 'order': 910, 'order_factorization_type': 11, 'order_stats': [[1, 1], [2, 35], [5, 4], [7, 6], [13, 12], [26, 420], [35, 24], [65, 48], [91, 72], [455, 288]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 12, 'outer_gen_orders': [12, 12], 'outer_gen_pows': [0, 0], 'outer_gens': [[1, 786], [1, 172]], 'outer_group': '144.101', 'outer_hash': 101, 'outer_nilpotent': True, 'outer_order': 144, 'outer_permdeg': 14, 'outer_perms': [408384, 6745939203], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_{12}^2', 'pc_rank': 2, 'perfect': False, 'permutation_degree': 25, 'pgroup': 0, 'primary_abelian_invariants': [2, 13], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 2], [4, 1], [6, 1], [12, 2], [24, 1], [48, 1], [72, 1], [288, 1]], 'representations': {'PC': {'code': 547166386175049719, 'gens': [1, 2], 'pres': [4, -2, -5, -7, -13, 3345, 45, 3242, 94]}, 'GLFp': {'d': 2, 'p': 911, 'gens': [155747955098, 569311697591, 830832]}, 'Perm': {'d': 25, 'gens': [25903229347167063091200, 522956313, 397620186163200, 672254738740453220352000]}}, 'schur_multiplier': [], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [26], 'solvability_type': 6, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_{13}\\times D_{35}', 'transitive_degree': 455, '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': '26.2', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 420, 'aut_gen_orders': [30, 4, 12, 12], 'aut_gens': [[1, 2], [833, 292], [365, 726], [365, 794], [391, 816]], 'aut_group': None, 'aut_hash': 1353831758173859334, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 10080, 'aut_permdeg': 47, 'aut_perms': [198710101440593196243218347790811720298878256370630268737019, 140267350450781368047622894677561570976535544194091094500653, 139188430599882027313978426784561684589484624043701455244064, 188253137915665707653624865308908802773522479547081256014097], 'aut_phi_ratio': 35.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 35, 1, 1], [5, 2, 2, 1], [7, 2, 3, 1], [13, 1, 12, 1], [26, 35, 12, 1], [35, 2, 12, 1], [65, 2, 24, 1], [91, 2, 36, 1], [455, 2, 144, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_{12}\\times F_5\\times F_7', 'autcent_abelian': True, 'autcent_cyclic': True, 'autcent_exponent': 12, 'autcent_group': '12.2', 'autcent_hash': 2, 'autcent_nilpotent': True, 'autcent_order': 12, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_{12}', '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, 2, 2], [7, 2, 3], [13, 1, 12], [26, 35, 12], [35, 2, 12], [65, 2, 24], [91, 2, 36], [455, 2, 144]], 'center_label': '13.1', 'center_order': 13, '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', '7.1', '13.1'], 'composition_length': 4, 'conjugacy_classes_known': True, 'counter': 6, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['13.1', 1], ['70.3', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 35, 1, 1], [5, 2, 2, 1], [7, 2, 3, 1], [13, 1, 12, 1], [26, 35, 12, 1], [35, 2, 12, 1], [65, 2, 24, 1], [91, 2, 36, 1], [455, 2, 144, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 42, 'exponent': 910, 'exponents_of_order': [1, 1, 1, 1], 'factors_of_aut_order': [2, 3, 5, 7], 'factors_of_order': [2, 5, 7, 13], 'faithful_reps': [[2, 0, 144]], 'familial': False, 'frattini_label': '1.1', 'frattini_quotient': '910.6', 'hash': 6, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 70, 'inner_gen_orders': [2, 35], 'inner_gens': [[1, 418], [495, 2]], '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': 288, 'irrQ_dim': 288, 'irrR_degree': 4, 'irrep_stats': [[1, 26], [2, 221]], 'label': '910.6', 'linC_count': 144, 'linC_degree': 2, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 22, 'linQ_degree_count': 2, 'linQ_dim': 22, 'linQ_dim_count': 2, 'linR_count': 216, 'linR_degree': 4, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C13*D35', 'ngens': 4, 'nilpotency_class': -1, 'nilpotent': False, 'normal_counts': [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': 10, 'number_characteristic_subgroups': 10, 'number_conjugacy_classes': 247, 'number_divisions': 10, 'number_normal_subgroups': 10, 'number_subgroup_autclasses': 16, 'number_subgroup_classes': 16, 'number_subgroups': 104, 'old_label': None, 'order': 910, 'order_factorization_type': 11, 'order_stats': [[1, 1], [2, 35], [5, 4], [7, 6], [13, 12], [26, 420], [35, 24], [65, 48], [91, 72], [455, 288]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 12, 'outer_gen_orders': [12, 12], 'outer_gen_pows': [0, 0], 'outer_gens': [[1, 786], [1, 172]], 'outer_group': '144.101', 'outer_hash': 101, 'outer_nilpotent': True, 'outer_order': 144, 'outer_permdeg': 14, 'outer_perms': [408384, 6745939203], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_{12}^2', 'pc_rank': 2, 'perfect': False, 'permutation_degree': 25, 'pgroup': 0, 'primary_abelian_invariants': [2, 13], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 2], [4, 1], [6, 1], [12, 2], [24, 1], [48, 1], [72, 1], [288, 1]], 'representations': {'PC': {'code': 547166386175049719, 'gens': [1, 2], 'pres': [4, -2, -5, -7, -13, 3345, 45, 3242, 94]}, 'GLFp': {'d': 2, 'p': 911, 'gens': [155747955098, 569311697591, 830832]}, 'Perm': {'d': 25, 'gens': [25903229347167063091200, 522956313, 397620186163200, 672254738740453220352000]}}, 'schur_multiplier': [], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [26], 'solvability_type': 6, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_{13}\\times D_{35}', 'transitive_degree': 455, 'wreath_data': None, 'wreath_product': False}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': True, 'abelian_quotient': '1.1', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': True, 'aut_cyclic': True, 'aut_derived_length': 0, 'aut_exponent': 1, 'aut_gen_orders': [], 'aut_gens': [], 'aut_group': '1.1', 'aut_hash': 1, 'aut_nilpotency_class': 0, 'aut_nilpotent': True, 'aut_order': 1, 'aut_permdeg': 1, 'aut_perms': [], 'aut_phi_ratio': 1.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_1', 'autcent_abelian': True, 'autcent_cyclic': True, 'autcent_exponent': 1, 'autcent_group': '1.1', 'autcent_hash': 1, 'autcent_nilpotent': True, 'autcent_order': 1, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_1', '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]], 'center_label': '1.1', 'center_order': 1, 'central_product': False, 'central_quotient': '1.1', 'commutator_count': 0, 'commutator_label': '1.1', 'complements_known': True, 'complete': True, 'complex_characters_known': True, 'composition_factors': [], 'composition_length': 0, 'conjugacy_classes_known': True, 'counter': 1, 'cyclic': True, 'derived_length': 0, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 1, 'exponent': 1, 'exponents_of_order': [], 'factors_of_aut_order': [], 'factors_of_order': [], 'faithful_reps': [[1, 1, 1]], 'familial': True, 'frattini_label': '1.1', 'frattini_quotient': '1.1', 'hash': 1, 'hyperelementary': 1, 'inner_abelian': True, 'inner_cyclic': True, 'inner_exponent': 1, 'inner_gen_orders': [], 'inner_gens': [], '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, 1]], 'label': '1.1', 'linC_count': 1, 'linC_degree': 0, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 0, 'linQ_degree_count': 1, 'linQ_dim': 0, 'linQ_dim_count': 1, 'linR_count': 1, 'linR_degree': 0, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C1', 'ngens': 0, 'nilpotency_class': 0, 'nilpotent': True, 'normal_counts': [1], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 1, 'number_characteristic_subgroups': 1, 'number_conjugacy_classes': 1, 'number_divisions': 1, 'number_normal_subgroups': 1, 'number_subgroup_autclasses': 1, 'number_subgroup_classes': 1, 'number_subgroups': 1, 'old_label': None, 'order': 1, 'order_factorization_type': 0, 'order_stats': [[1, 1]], 'outer_abelian': True, 'outer_cyclic': True, 'outer_equivalence': False, 'outer_exponent': 1, 'outer_gen_orders': [], 'outer_gen_pows': [], 'outer_gens': [], 'outer_group': '1.1', 'outer_hash': 1, 'outer_nilpotent': True, 'outer_order': 1, 'outer_permdeg': 1, 'outer_perms': [], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_1', 'pc_rank': 0, 'perfect': True, 'permutation_degree': 1, 'pgroup': 1, 'primary_abelian_invariants': [], 'quasisimple': False, 'rank': 0, 'rational': True, 'rational_characters_known': True, 'ratrep_stats': [[1, 1]], 'representations': {'PC': {'code': 0, 'gens': [], 'pres': []}}, 'schur_multiplier': [], 'semidirect_product': False, 'simple': False, 'smith_abelian_invariants': [], 'solvability_type': 0, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_1', 'transitive_degree': 1, 'wreath_data': None, 'wreath_product': False}
