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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': True, 'ambient': '3440.a', 'ambient_counter': 1, 'ambient_order': 3440, 'ambient_tex': 'D_8\\times C_{215}', 'central': False, 'central_factor': False, 'centralizer_order': 1720, 'characteristic': True, 'core_order': 20, 'counter': 27, 'cyclic': True, 'direct': False, 'hall': 0, 'label': '3440.a.172.a1.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': True, 'old_label': '172.a1.a1', 'outer_equivalence': False, 'perfect': False, 'proper': True, 'quotient': '172.4', 'quotient_Agroup': True, 'quotient_abelian': True, 'quotient_cyclic': False, 'quotient_hash': 4, 'quotient_metabelian': True, 'quotient_nilpotent': True, 'quotient_order': 172, 'quotient_simple': False, 'quotient_solvable': True, 'quotient_supersolvable': True, 'quotient_tex': 'C_2\\times C_{86}', 'simple': False, 'solvable': True, 'special_labels': [], 'split': False, 'standard_generators': False, 'stem': False, 'subgroup': '20.2', 'subgroup_hash': 2, 'subgroup_order': 20, 'subgroup_tex': 'C_{20}', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '3440.a', 'aut_centralizer_order': 672, 'aut_label': '172.a1', 'aut_quo_index': 3, 'aut_stab_index': 1, 'aut_weyl_group': '8.2', 'aut_weyl_index': 672, 'centralizer': '2.b1.a1', 'complements': [], 'conjugacy_class_count': 1, 'contained_in': ['4.a1.a1', '86.a1.a1', '86.a1.b1', '86.b1.a1'], 'contains': ['344.a1.a1', '860.a1.a1'], 'core': '172.a1.a1', 'coset_action_label': None, 'count': 1, 'diagramx': [7324, 3736, 3233, 6028, 7077, 3594, 4724, 2954], 'generators': [860, 1720, 1376], 'label': '3440.a.172.a1.a1', 'mobius_quo': 0, 'mobius_sub': -2, 'normal_closure': '172.a1.a1', 'normal_contained_in': ['4.a1.a1', '86.a1.a1', '86.a1.b1', '86.b1.a1'], 'normal_contains': ['344.a1.a1', '860.a1.a1'], 'normalizer': '1.a1.a1', 'old_label': '172.a1.a1', 'projective_image': '344.9', 'quotient_action_image': '2.1', 'quotient_action_kernel': '86.2', 'quotient_action_kernel_order': 86, 'quotient_fusion': None, 'short_label': '172.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': '20.2', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': True, 'aut_cyclic': False, 'aut_derived_length': 1, 'aut_exponent': 4, 'aut_gen_orders': [2, 4], 'aut_gens': [[1], [11], [17]], 'aut_group': '8.2', 'aut_hash': 2, 'aut_nilpotency_class': 1, 'aut_nilpotent': True, 'aut_order': 8, 'aut_permdeg': 6, 'aut_perms': [120, 9], 'aut_phi_ratio': 1.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [4, 1, 2, 1], [5, 1, 4, 1], [10, 1, 4, 1], [20, 1, 8, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_2\\times C_4', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 4, 'autcent_group': '8.2', 'autcent_hash': 2, 'autcent_nilpotent': True, 'autcent_order': 8, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2\\times C_4', '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], [5, 1, 4], [10, 1, 4], [20, 1, 8]], 'center_label': '20.2', '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': 2, 'cyclic': True, 'derived_length': 1, 'dihedral': False, 'direct_factorization': [['4.1', 1], ['5.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [4, 1, 2, 1], [5, 1, 4, 1], [10, 1, 4, 1], [20, 1, 8, 1]], 'element_repr_type': 'PC', 'elementary': 10, 'eulerian_function': 1, 'exponent': 20, 'exponents_of_order': [2, 1], 'factors_of_aut_order': [2], 'factors_of_order': [2, 5], 'faithful_reps': [[1, 0, 8]], 'familial': True, 'frattini_label': '2.1', 'frattini_quotient': '10.2', 'hash': 2, 'hyperelementary': 10, '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': 8, 'irrQ_dim': 8, 'irrR_degree': 2, 'irrep_stats': [[1, 20]], 'label': '20.2', 'linC_count': 8, 'linC_degree': 1, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 6, 'linQ_degree_count': 2, 'linQ_dim': 6, 'linQ_dim_count': 2, 'linR_count': 4, 'linR_degree': 2, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C20', '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': 6, 'number_characteristic_subgroups': 6, 'number_conjugacy_classes': 20, 'number_divisions': 6, 'number_normal_subgroups': 6, 'number_subgroup_autclasses': 6, 'number_subgroup_classes': 6, 'number_subgroups': 6, 'old_label': None, 'order': 20, 'order_factorization_type': 22, 'order_stats': [[1, 1], [2, 1], [4, 2], [5, 4], [10, 4], [20, 8]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 4, 'outer_gen_orders': [2, 4], 'outer_gen_pows': [0, 0], 'outer_gens': [[11], [17]], 'outer_group': '8.2', 'outer_hash': 2, 'outer_nilpotent': True, 'outer_order': 8, 'outer_permdeg': 6, 'outer_perms': [120, 9], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2\\times C_4', 'pc_rank': 1, 'perfect': False, 'permutation_degree': 9, 'pgroup': 0, 'primary_abelian_invariants': [4, 5], 'quasisimple': False, 'rank': 1, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 2], [2, 1], [4, 2], [8, 1]], 'representations': {'PC': {'code': 51395, 'gens': [1], 'pres': [3, -2, -2, -5, 6, 16]}, 'GLZ': {'b': 3, 'd': 6, 'gens': [58438750843733785]}, 'GLFp': {'d': 2, 'p': 5, 'gens': [131, 504, 252]}, 'Perm': {'d': 9, 'gens': [131040, 96, 41040]}}, 'schur_multiplier': [], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [20], 'solvability_type': 0, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_{20}', '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': '860.11', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 168, 'aut_gen_orders': [8, 2, 2, 4, 4, 42], 'aut_gens': [[1, 2], [431, 2], [1, 1722], [1, 862], [861, 2], [1, 2754], [1, 2242]], 'aut_group': None, 'aut_hash': 6966770769406336981, 'aut_nilpotency_class': 3, 'aut_nilpotent': True, 'aut_order': 5376, 'aut_permdeg': 54, 'aut_perms': [459579231522570221689172693150444621650952251430954786461925904, 558890449870141255052508537344023828902955543210857557860940304, 124403404131378705277074714084746530548217363356521258614009104, 260956496977302661005525516850211075472165707277635083894009104, 53852414376475861580288, 90083814753215547094071010943469195448280033383666676473625768512221661], 'aut_phi_ratio': 4.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 4, 2, 1], [4, 2, 1, 1], [5, 1, 4, 1], [8, 2, 2, 1], [10, 1, 4, 1], [10, 4, 8, 1], [20, 2, 4, 1], [40, 2, 8, 1], [43, 1, 42, 1], [86, 1, 42, 1], [86, 4, 84, 1], [172, 2, 42, 1], [215, 1, 168, 1], [344, 2, 84, 1], [430, 1, 168, 1], [430, 4, 336, 1], [860, 2, 168, 1], [1720, 2, 336, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_2\\times C_{84}\\times C_8:C_2^2', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 84, 'autcent_group': '672.1248', 'autcent_hash': 1248, 'autcent_nilpotent': True, 'autcent_order': 672, 'autcent_solvable': True, 'autcent_split': False, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^3\\times C_{84}', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 4, 'autcentquo_group': '8.3', 'autcentquo_hash': 3, 'autcentquo_nilpotent': True, 'autcentquo_order': 8, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'D_4', 'cc_stats': [[1, 1, 1], [2, 1, 1], [2, 4, 2], [4, 2, 1], [5, 1, 4], [8, 2, 2], [10, 1, 4], [10, 4, 8], [20, 2, 4], [40, 2, 8], [43, 1, 42], [86, 1, 42], [86, 4, 84], [172, 2, 42], [215, 1, 168], [344, 2, 84], [430, 1, 168], [430, 4, 336], [860, 2, 168], [1720, 2, 336]], 'center_label': '430.4', 'center_order': 430, 'central_product': True, 'central_quotient': '8.3', 'commutator_count': 1, 'commutator_label': '4.1', 'complements_known': True, 'complete': False, 'complex_characters_known': False, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '5.1', '43.1'], 'composition_length': 6, 'conjugacy_classes_known': True, 'counter': 1, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['16.7', 1], ['43.1', 1], ['5.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 4, 1, 2], [4, 2, 1, 1], [5, 1, 4, 1], [8, 2, 2, 1], [10, 1, 4, 1], [10, 4, 4, 2], [20, 2, 4, 1], [40, 2, 8, 1], [43, 1, 42, 1], [86, 1, 42, 1], [86, 4, 42, 2], [172, 2, 42, 1], [215, 1, 168, 1], [344, 2, 84, 1], [430, 1, 168, 1], [430, 4, 168, 2], [860, 2, 168, 1], [1720, 2, 336, 1]], 'element_repr_type': 'PC', 'elementary': 2, 'eulerian_function': 792, 'exponent': 1720, 'exponents_of_order': [4, 1, 1], 'factors_of_aut_order': [2, 3, 7], 'factors_of_order': [2, 5, 43], 'faithful_reps': [[2, 0, 336]], 'familial': False, 'frattini_label': '4.1', 'frattini_quotient': '860.11', 'hash': 8535201396817996670, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 4, 'inner_gen_orders': [2, 4], 'inner_gens': [[1, 862], [2581, 2]], 'inner_hash': 3, 'inner_nilpotent': True, 'inner_order': 8, 'inner_split': False, 'inner_tex': 'D_4', 'inner_used': [1, 2], 'irrC_degree': 2, 'irrQ_degree': 672, 'irrQ_dim': None, 'irrR_degree': None, 'irrep_stats': [[1, 860], [2, 645]], 'label': '3440.a', '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': 'D8*C215', 'ngens': 6, 'nilpotency_class': 3, 'nilpotent': True, 'normal_counts': [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': 20, 'number_characteristic_subgroups': 20, 'number_conjugacy_classes': 1505, 'number_divisions': 24, 'number_normal_subgroups': 28, 'number_subgroup_autclasses': 32, 'number_subgroup_classes': 44, 'number_subgroups': 76, 'old_label': None, 'order': 3440, 'order_factorization_type': 311, 'order_stats': [[1, 1], [2, 9], [4, 2], [5, 4], [8, 4], [10, 36], [20, 8], [40, 16], [43, 42], [86, 378], [172, 84], [215, 168], [344, 168], [430, 1512], [860, 336], [1720, 672]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 84, 'outer_gen_orders': [2, 2, 2, 84], 'outer_gen_pows': [3438, 0, 4, 4], 'outer_gens': [[431, 2], [1, 1722], [861, 1202], [2151, 3274]], 'outer_group': '672.1248', 'outer_hash': 1248, 'outer_nilpotent': True, 'outer_order': 672, 'outer_permdeg': 20, 'outer_perms': [121645100408832000, 355687428096000, 1321086412800, 122002114192762473], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^3\\times C_{84}', 'pc_rank': 2, 'perfect': False, 'permutation_degree': 56, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 5, 43], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': False, 'ratrep_stats': [[1, 4], [2, 1], [4, 5], [8, 1], [16, 1], [42, 4], [84, 1], [168, 5], [336, 1], [672, 1]], 'representations': {'PC': {'code': '893410431492827361160361000669883431', 'gens': [1, 2], 'pres': [6, -2, -2, -2, -2, -5, -43, 10345, 31, 31034, 50, 69, 178]}, 'GLFp': {'d': 2, 'p': 431, 'gens': [21161365840, 34427086131]}, 'Perm': {'d': 56, 'gens': [50785779384340497831356839620572308871767772717203314440847264280420940313, 39021137719137419925676829989424472940979880226731715712778240000000000000]}}, 'schur_multiplier': [2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 430], 'solvability_type': 3, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'D_8\\times C_{215}', 'transitive_degree': 1720, 'wreath_data': None, 'wreath_product': False}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': True, 'abelian_quotient': '172.4', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 42, 'aut_gen_orders': [6, 42], 'aut_gens': [[1, 2], [87, 98], [87, 45]], 'aut_group': '252.42', 'aut_hash': 42, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 252, 'aut_permdeg': 15, 'aut_perms': [367437772800, 275030427031], 'aut_phi_ratio': 3.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 3, 1], [43, 1, 42, 1], [86, 1, 126, 1]], 'aut_supersolvable': True, 'aut_tex': 'S_3\\times C_{42}', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 42, 'autcent_group': '252.42', 'autcent_hash': 42, 'autcent_nilpotent': False, 'autcent_order': 252, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'S_3\\times C_{42}', '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], [43, 1, 42], [86, 1, 126]], 'center_label': '172.4', 'center_order': 172, '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', '43.1'], 'composition_length': 3, 'conjugacy_classes_known': True, 'counter': 4, 'cyclic': False, 'derived_length': 1, 'dihedral': False, 'direct_factorization': [['2.1', 2], ['43.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [43, 1, 42, 1], [86, 1, 42, 3]], 'element_repr_type': 'PC', 'elementary': 2, 'eulerian_function': 44, 'exponent': 86, 'exponents_of_order': [2, 1], 'factors_of_aut_order': [2, 3, 7], 'factors_of_order': [2, 43], 'faithful_reps': [], 'familial': False, 'frattini_label': '1.1', 'frattini_quotient': '172.4', 'hash': 4, '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, 172]], 'label': '172.4', 'linC_count': 5544, 'linC_degree': 2, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 43, 'linQ_degree_count': 6, 'linQ_dim': 43, 'linQ_dim_count': 6, 'linR_count': 126, 'linR_degree': 3, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C2*C86', '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': 172, 'number_divisions': 8, 'number_normal_subgroups': 10, 'number_subgroup_autclasses': 6, 'number_subgroup_classes': 10, 'number_subgroups': 10, 'old_label': None, 'order': 172, 'order_factorization_type': 22, 'order_stats': [[1, 1], [2, 3], [43, 42], [86, 126]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 42, 'outer_gen_orders': [6, 42], 'outer_gen_pows': [0, 0], 'outer_gens': [[87, 98], [87, 45]], 'outer_group': '252.42', 'outer_hash': 42, 'outer_nilpotent': False, 'outer_order': 252, 'outer_permdeg': 15, 'outer_perms': [367437772800, 275030427031], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'S_3\\times C_{42}', 'pc_rank': 2, 'perfect': False, 'permutation_degree': 47, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 43], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [42, 4]], 'representations': {'PC': {'code': 2519033, 'gens': [1, 2], 'pres': [3, -2, -2, -43, 16]}, 'GLFp': {'d': 2, 'p': 173, 'gens': [150153817, 890567496]}, 'Perm': {'d': 47, 'gens': [5502622159812088949850305428800254892961651752960000000000, 2658271574788448768043625811014615890319638528000000000, 59010256945620955738811989462269485937328128000000000]}}, 'schur_multiplier': [2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 86], '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_{86}', 'transitive_degree': 172, 'wreath_data': None, 'wreath_product': False}
