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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': True, 'ambient': '5400.q', 'ambient_counter': 17, 'ambient_order': 5400, 'ambient_tex': 'C_{15}^2:(C_2\\times D_6)', 'central': False, 'central_factor': False, 'centralizer_order': 20, 'characteristic': False, 'core_order': 1, 'counter': 264, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '5400.q.270.b1.b1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': False, 'old_label': '270.b1.b1', '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': 270, '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': '20.5', 'subgroup_hash': 5, 'subgroup_order': 20, 'subgroup_tex': 'C_2\\times C_{10}', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '5400.q', 'aut_centralizer_order': None, 'aut_label': '270.b1', 'aut_quo_index': None, 'aut_stab_index': None, 'aut_weyl_group': None, 'aut_weyl_index': None, 'centralizer': '270.b1.b1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['54.b1.b1', '90.d1.b1', '90.e1.b1', '135.a1.b1'], 'contains': ['540.a1.b1', '540.c1.b1', '540.e1.b1', '1350.b1.b1'], 'core': '5400.a1.a1', 'coset_action_label': None, 'count': 135, 'diagramx': [4916, -1, 5117, -1, 6491, -1, 3192, -1], 'generators': [1801, 2232, 12], 'label': '5400.q.270.b1.b1', 'mobius_quo': None, 'mobius_sub': 0, 'normal_closure': '2.b1.b1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '135.a1.b1', 'old_label': '270.b1.b1', 'projective_image': '5400.q', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '270.b1.b1', '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': '20.5', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 12, 'aut_gen_orders': [2, 12], 'aut_gens': [[1, 2], [10, 13], [10, 15]], 'aut_group': '24.5', 'aut_hash': 5, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 24, 'aut_permdeg': 7, 'aut_perms': [127, 857], 'aut_phi_ratio': 3.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 3, 1], [5, 1, 4, 1], [10, 1, 12, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_4\\times S_3', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 12, 'autcent_group': '24.5', 'autcent_hash': 5, 'autcent_nilpotent': False, 'autcent_order': 24, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_4\\times S_3', '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, 3], [5, 1, 4], [10, 1, 12]], 'center_label': '20.5', 'center_order': 20, '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', '5.1'], 'composition_length': 3, 'conjugacy_classes_known': True, 'counter': 5, 'cyclic': False, 'derived_length': 1, 'dihedral': False, 'direct_factorization': [['2.1', 2], ['5.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [5, 1, 4, 1], [10, 1, 4, 3]], 'element_repr_type': 'PC', 'elementary': 2, 'eulerian_function': 6, 'exponent': 10, 'exponents_of_order': [2, 1], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 5], 'faithful_reps': [], 'familial': False, 'frattini_label': '1.1', 'frattini_quotient': '20.5', 'hash': 5, 'hyperelementary': 2, 'inner_abelian': True, 'inner_cyclic': True, 'inner_exponent': 1, 'inner_gen_orders': [1, 1], 'inner_gens': [[1, 2], [1, 2]], '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, 20]], 'label': '20.5', 'linC_count': 72, 'linC_degree': 2, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 5, 'linQ_degree_count': 6, 'linQ_dim': 5, 'linQ_dim_count': 6, 'linR_count': 12, 'linR_degree': 3, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C2*C10', 'ngens': 3, 'nilpotency_class': 1, 'nilpotent': True, 'normal_counts': [0, 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': 20, 'number_divisions': 8, 'number_normal_subgroups': 10, 'number_subgroup_autclasses': 6, 'number_subgroup_classes': 10, 'number_subgroups': 10, 'old_label': None, 'order': 20, 'order_factorization_type': 22, 'order_stats': [[1, 1], [2, 3], [5, 4], [10, 12]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 12, 'outer_gen_orders': [2, 12], 'outer_gen_pows': [0, 0], 'outer_gens': [[10, 13], [10, 15]], 'outer_group': '24.5', 'outer_hash': 5, 'outer_nilpotent': False, 'outer_order': 24, 'outer_permdeg': 7, 'outer_perms': [127, 857], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_4\\times S_3', 'pc_rank': 2, 'perfect': False, 'permutation_degree': 9, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 5], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [4, 4]], 'representations': {'PC': {'code': 2179, 'gens': [1, 2], 'pres': [3, -2, -2, -5, 16]}, 'GLZ': {'b': 3, 'd': 5, 'gens': [141602720900, 706303489327]}, 'GLFp': {'d': 2, 'p': 11, 'gens': [13320, 4001]}, 'Perm': {'d': 9, 'gens': [40320, 720, 96]}}, 'schur_multiplier': [2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 10], 'solvability_type': 1, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_2\\times C_{10}', 'transitive_degree': 20, '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': '8.5', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 3, 'aut_exponent': 120, 'aut_gen_orders': [12, 24, 12, 12], 'aut_gens': [[1, 2, 4, 24, 360], [281, 250, 5276, 3552, 840], [5130, 5329, 1924, 1176, 4872], [3761, 4866, 3748, 2400, 4704], [4689, 3706, 1324, 2208, 1824]], 'aut_group': None, 'aut_hash': 7509908978670635385, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 21600, 'aut_permdeg': 99, 'aut_perms': [468643248296311546907098131181129810862167156002129258346069595807777651414243918068855878869239229525948336797888664353217757476067296946057395903011537236, 646858487552754921344605030948085459321743594208415625630457868892710840208293734215129807327986457065841148908476621565972730904709881481080822413390761247, 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'autcent_tex': 'C_1', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 120, 'autcentquo_group': None, 'autcentquo_hash': 7509908978670635385, 'autcentquo_nilpotent': False, 'autcentquo_order': 21600, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'C_5^2.\\He_3.C_4.C_2^3', 'cc_stats': [[1, 1, 1], [2, 9, 1], [2, 25, 1], [2, 45, 4], [2, 225, 1], [3, 2, 1], [3, 6, 1], [3, 150, 1], [3, 300, 1], [5, 6, 4], [6, 50, 1], [6, 90, 4], [6, 150, 2], [6, 300, 1], [6, 450, 2], [10, 54, 4], [10, 90, 8], [15, 12, 8], [15, 24, 4], [30, 180, 8]], 'center_label': '1.1', 'center_order': 1, 'central_product': False, 'central_quotient': '5400.q', 'commutator_count': 1, 'commutator_label': '675.12', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '3.1', '3.1', '3.1', '5.1', '5.1'], 'composition_length': 8, 'conjugacy_classes_known': True, 'counter': 17, 'cyclic': False, 'derived_length': 3, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 9, 1, 1], [2, 25, 1, 1], [2, 45, 1, 4], [2, 225, 1, 1], [3, 2, 1, 1], [3, 6, 1, 1], [3, 150, 1, 1], [3, 300, 1, 1], [5, 6, 2, 2], [6, 50, 1, 1], [6, 90, 1, 4], [6, 150, 1, 2], [6, 300, 1, 1], [6, 450, 1, 2], [10, 54, 2, 2], [10, 90, 2, 4], [15, 12, 2, 4], [15, 24, 2, 2], [30, 180, 2, 4]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 1886976, 'exponent': 30, 'exponents_of_order': [3, 3, 2], 'factors_of_aut_order': [2, 3, 5], 'factors_of_order': [2, 3, 5], 'faithful_reps': [[12, 1, 8], [24, 1, 4]], 'familial': False, 'frattini_label': '3.1', 'frattini_quotient': '1800.575', 'hash': 4786563836012664664, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 30, 'inner_gen_orders': [2, 10, 6, 15, 15], 'inner_gens': [[1, 210, 2060, 4560, 1872], [2409, 2, 4052, 24, 5352], [3609, 2698, 4, 4656, 1968], [1225, 2, 1132, 24, 360], [4249, 770, 4156, 24, 360]], 'inner_hash': 4786563836012664664, 'inner_nilpotent': False, 'inner_order': 5400, 'inner_split': False, 'inner_tex': 'C_{15}^2:(C_2\\times D_6)', 'inner_used': [1, 2, 3], 'irrC_degree': 12, 'irrQ_degree': 24, 'irrQ_dim': 24, 'irrR_degree': 12, 'irrep_stats': [[1, 8], [2, 8], [4, 2], [6, 20], [12, 16], [24, 4]], 'label': '5400.q', 'linC_count': 72, 'linC_degree': 12, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 18, 'linQ_degree_count': 32, 'linQ_dim': 18, 'linQ_dim_count': 32, 'linR_count': 72, 'linR_degree': 12, 'maximal_subgroups_known': True, 'metabelian': False, 'metacyclic': False, 'monomial': True, 'name': 'C15^2:(C2*D6)', 'ngens': 8, '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, 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': 27, 'number_characteristic_subgroups': 24, 'number_conjugacy_classes': 58, 'number_divisions': 40, 'number_normal_subgroups': 34, 'number_subgroup_autclasses': 253, 'number_subgroup_classes': 382, 'number_subgroups': 17960, 'old_label': None, 'order': 5400, 'order_factorization_type': 321, 'order_stats': [[1, 1], [2, 439], [3, 458], [5, 24], [6, 1910], [10, 936], [15, 192], [30, 1440]], 'outer_abelian': True, 'outer_cyclic': True, 'outer_equivalence': False, 'outer_exponent': 4, 'outer_gen_orders': [4], 'outer_gen_pows': [2411], 'outer_gens': [[3450, 1289, 20, 2424, 1488]], 'outer_group': '4.1', 'outer_hash': 1, 'outer_nilpotent': True, 'outer_order': 4, 'outer_permdeg': 4, 'outer_perms': [17], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_4', 'pc_rank': 5, 'perfect': False, 'permutation_degree': 24, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 2], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 8], [2, 8], [4, 2], [6, 4], [12, 8], [24, 8], [48, 2]], 'representations': {'PC': {'code': '1401720944733762749398017699641141431646380288460945042217301025960558010733611925391170607299035511', 'gens': [1, 2, 3, 5, 7], 'pres': [8, 2, 2, 2, 3, 3, 5, 3, 5, 3361, 1161, 49442, 48634, 66, 8195, 52491, 182404, 46580, 6628, 156, 103685, 29397, 19469, 104838, 149870, 27574, 12294, 222, 13831, 145167, 2327, 36319]}, 'Perm': {'d': 24, 'gens': [28161232801305787013443, 55292692370478153345213, 4654263295722373817069]}}, 'schur_multiplier': [2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 2], 'solvability_type': 11, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': 'C_{15}^2:(C_2\\times D_6)', 'transitive_degree': 45, 'wreath_data': None, 'wreath_product': False}
