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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': False, 'ambient': '115248.bg', 'ambient_counter': 33, 'ambient_order': 115248, 'ambient_tex': 'D_7\\times C_7^3:S_4', 'central': False, 'central_factor': False, 'centralizer_order': 1, 'characteristic': False, 'core_order': 1, 'counter': 111, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '115248.bg.84.bh1.b1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': False, 'minimal': False, 'minimal_normal': False, 'nilpotent': False, 'normal': False, 'old_label': '84.bh1.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': 84, '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': '1372.31', 'subgroup_hash': 31, 'subgroup_order': 1372, 'subgroup_tex': 'C_7^3:C_4', 'supersolvable': False, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '115248.bg', 'aut_centralizer_order': None, 'aut_label': '84.bh1', 'aut_quo_index': None, 'aut_stab_index': None, 'aut_weyl_group': None, 'aut_weyl_index': None, 'centralizer': '115248.a1.a1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['12.i1.a1', '42.j1.b1'], 'contains': ['168.z1.b1', '588.bu1.a1', '4116.be1.c1'], 'core': '115248.a1.a1', 'coset_action_label': None, 'count': 42, 'diagramx': [9898, -1, 9228, -1, 188, -1, 9409, -1], 'generators': [33769, 16488, 14448, 756, 2352], 'label': '115248.bg.84.bh1.b1', 'mobius_quo': None, 'mobius_sub': 0, 'normal_closure': '2.c1.a1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '42.j1.b1', 'old_label': '84.bh1.b1', 'projective_image': '115248.bg', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '84.bh1.b1', 'subgroup_fusion': None, 'weyl_group': '2744.z'}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '4.1', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 3, 'aut_exponent': 336, 'aut_gen_orders': [48, 42, 48, 12], 'aut_gens': [[1, 4, 28, 196], [861, 20, 196, 224], [1283, 12, 644, 1148], [405, 20, 420, 644], [1363, 12, 1260, 896]], 'aut_group': None, 'aut_hash': 5704297344339064290, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 197568, 'aut_permdeg': 686, 'aut_perms': 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'aut_phi_ratio': 336.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 49, 1, 1], [4, 343, 2, 1], [7, 2, 3, 1], [7, 4, 12, 1], [7, 4, 72, 1], [14, 98, 3, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_7^3.C_8.C_6^2.C_2', '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': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 336, 'autcentquo_group': None, 'autcentquo_hash': 5704297344339064290, 'autcentquo_nilpotent': False, 'autcentquo_order': 197568, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'C_7^3.C_8.C_6^2.C_2', 'cc_stats': [[1, 1, 1], [2, 49, 1], [4, 343, 2], [7, 2, 3], [7, 4, 84], [14, 98, 3]], 'center_label': '1.1', 'center_order': 1, 'central_product': False, 'central_quotient': '1372.31', 'commutator_count': 1, 'commutator_label': '343.5', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '7.1', '7.1', '7.1'], 'composition_length': 5, 'conjugacy_classes_known': True, 'counter': 31, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 49, 1, 1], [4, 343, 2, 1], [7, 2, 3, 1], [7, 4, 3, 4], [7, 4, 6, 12], [14, 98, 3, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 6, 'exponent': 28, 'exponents_of_order': [3, 2], 'factors_of_aut_order': [2, 3, 7], 'factors_of_order': [2, 7], 'faithful_reps': [[4, 0, 72]], 'familial': False, 'frattini_label': '1.1', 'frattini_quotient': '1372.31', 'hash': 31, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 28, 'inner_gen_orders': [4, 7, 7, 7], 'inner_gens': [[1, 24, 644, 1064], [9, 4, 28, 196], [953, 4, 28, 196], [701, 4, 28, 196]], 'inner_hash': 31, 'inner_nilpotent': False, 'inner_order': 1372, 'inner_split': False, 'inner_tex': 'C_7^3:C_4', 'inner_used': [1, 2, 3], 'irrC_degree': 4, 'irrQ_degree': 24, 'irrQ_dim': 24, 'irrR_degree': 8, 'irrep_stats': [[1, 4], [2, 6], [4, 84]], 'label': '1372.31', 'linC_count': 72, 'linC_degree': 4, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 18, 'linQ_degree_count': 8, 'linQ_dim': 18, 'linQ_dim_count': 4, 'linR_count': 36, 'linR_degree': 6, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'C7^3:C4', 'ngens': 5, 'nilpotency_class': -1, 'nilpotent': False, 'normal_counts': [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': 7, 'number_characteristic_subgroups': 7, 'number_conjugacy_classes': 94, 'number_divisions': 21, 'number_normal_subgroups': 7, 'number_subgroup_autclasses': 18, 'number_subgroup_classes': 52, 'number_subgroups': 728, 'old_label': None, 'order': 1372, 'order_factorization_type': 32, 'order_stats': [[1, 1], [2, 49], [4, 686], [7, 342], [14, 294]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 24, 'outer_gen_orders': [6, 24], 'outer_gen_pows': [2, 2], 'outer_gens': [[3, 12, 588, 112], [1, 8, 504, 1232]], 'outer_group': '144.106', 'outer_hash': 106, 'outer_nilpotent': True, 'outer_order': 144, 'outer_permdeg': 14, 'outer_perms': [1078847284, 46118500080], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_3^2\\times D_8', 'pc_rank': 4, 'perfect': False, 'permutation_degree': 21, 'pgroup': 0, 'primary_abelian_invariants': [4], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 2], [2, 1], [6, 2], [12, 4], [24, 12]], 'representations': {'PC': {'code': 3112633987312405066609463, 'gens': [1, 3, 4, 5], 'pres': [5, -2, -2, -7, -7, 7, 10, 362, 12883, 1688, 26604, 14709]}, 'Perm': {'d': 21, 'gens': [2447552358978854527, 27900573853057920, 5245059686973966720, 7819959613217834880, 973]}}, 'schur_multiplier': [7], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [4], 'solvability_type': 8, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': 'C_7^3:C_4', 'transitive_degree': 28, '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': '4.2', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 4, 'aut_exponent': 84, 'aut_gen_orders': [42, 4, 12, 6], 'aut_gens': [[1, 2, 12, 168, 2352, 16464], [75641, 86426, 8688, 25452, 16464, 672], [79741, 32422, 8556, 12348, 2016, 98784], [95985, 75562, 109092, 57288, 65856, 14112], [44065, 58114, 74388, 39564, 672, 82320]], 'aut_group': None, 'aut_hash': 5409013870120294433, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 2074464, 'aut_permdeg': 49, 'aut_perms': [376762426489910281438409422993128720816034257781019552250536147, 496917807121179171468003852382289237688155082067063963823853195, 73737223738794316053009751302970448599108295316881618416497150, 323232204136004497116876748466108666815919630240014240454638263], 'aut_phi_ratio': 63.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 7, 1, 1], [2, 42, 1, 1], [2, 147, 1, 1], [2, 294, 1, 1], [2, 1029, 1, 1], [3, 392, 1, 1], [4, 2058, 1, 1], [4, 14406, 1, 1], [6, 2744, 1, 1], [7, 2, 3, 1], [7, 4, 6, 1], [7, 6, 3, 1], [7, 8, 18, 1], [7, 12, 3, 1], [7, 12, 6, 2], [7, 12, 9, 1], [7, 24, 2, 1], [7, 24, 3, 1], [7, 24, 9, 1], [7, 24, 18, 2], [7, 48, 6, 1], [7, 48, 9, 1], [14, 28, 6, 1], [14, 42, 3, 1], [14, 42, 6, 1], [14, 84, 3, 3], [14, 84, 6, 5], [14, 84, 18, 1], [14, 168, 2, 1], [14, 168, 3, 1], [14, 168, 9, 1], [14, 168, 18, 3], [14, 294, 3, 2], [14, 294, 6, 1], [14, 588, 3, 1], [14, 588, 6, 3], [14, 588, 9, 1], [14, 2058, 3, 1], [21, 392, 6, 1], [21, 784, 3, 1], [21, 784, 18, 1], [28, 4116, 3, 1], [42, 2744, 6, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_7^3.(C_7\\times A_4).C_6^2.C_2', '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': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 84, 'autcentquo_group': None, 'autcentquo_hash': 5409013870120294433, 'autcentquo_nilpotent': False, 'autcentquo_order': 2074464, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'C_7^3.(C_7\\times A_4).C_6^2.C_2', 'cc_stats': [[1, 1, 1], [2, 7, 1], [2, 42, 1], [2, 147, 1], [2, 294, 1], [2, 1029, 1], [3, 392, 1], [4, 2058, 1], [4, 14406, 1], [6, 2744, 1], [7, 2, 3], [7, 4, 6], [7, 6, 3], [7, 8, 18], [7, 12, 24], [7, 24, 50], [7, 48, 15], [14, 28, 6], [14, 42, 9], [14, 84, 57], [14, 168, 68], [14, 294, 12], [14, 588, 30], [14, 2058, 3], [21, 392, 6], [21, 784, 21], [28, 4116, 3], [42, 2744, 6]], 'center_label': '1.1', 'center_order': 1, 'central_product': True, 'central_quotient': '115248.bg', 'commutator_count': 1, 'commutator_label': '28812.bd', 'complements_known': True, 'complete': False, 'complex_characters_known': False, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '3.1', '7.1', '7.1', '7.1', '7.1'], 'composition_length': 9, 'conjugacy_classes_known': False, 'counter': 33, 'cyclic': False, 'derived_length': 4, 'dihedral': False, 'direct_factorization': [['14.1', 1], ['8232.bt', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 7, 1, 1], [2, 42, 1, 1], [2, 147, 1, 1], [2, 294, 1, 1], [2, 1029, 1, 1], [3, 392, 1, 1], [4, 2058, 1, 1], [4, 14406, 1, 1], [6, 2744, 1, 1], [7, 2, 3, 1], [7, 4, 6, 1], [7, 6, 3, 1], [7, 8, 6, 3], [7, 12, 3, 4], [7, 12, 6, 2], [7, 24, 2, 1], [7, 24, 3, 4], [7, 24, 6, 6], [7, 48, 3, 3], [7, 48, 6, 1], [14, 28, 6, 1], [14, 42, 3, 1], [14, 42, 6, 1], [14, 84, 3, 3], [14, 84, 6, 8], [14, 168, 2, 1], [14, 168, 3, 4], [14, 168, 6, 9], [14, 294, 3, 2], [14, 294, 6, 1], [14, 588, 3, 4], [14, 588, 6, 3], [14, 2058, 3, 1], [21, 392, 6, 1], [21, 784, 3, 1], [21, 784, 6, 3], [28, 4116, 3, 1], [42, 2744, 6, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 1026, 'exponent': 84, 'exponents_of_order': [4, 4, 1], 'factors_of_aut_order': [2, 3, 7], 'factors_of_order': [2, 3, 7], 'faithful_reps': [[8, 0, 36], [12, 0, 18], [12, 1, 18], [16, 0, 18], [24, 0, 72], [24, 1, 18], [48, 0, 6], [48, 1, 9]], 'familial': False, 'frattini_label': '1.1', 'frattini_quotient': '115248.bg', 'hash': 3341210359253535008, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 84, 'inner_gen_orders': [14, 42, 14, 14, 7, 7], 'inner_gens': [[1, 102370, 17964, 42924, 2016, 16464], [88073, 2, 63072, 94668, 32928, 1344], [99817, 93686, 12, 23352, 14112, 16464], [87781, 40742, 110892, 168, 14112, 98784], [2689, 84674, 4716, 4872, 2352, 16464], [1, 17474, 12, 33096, 2352, 16464]], 'inner_hash': 3341210359253535008, 'inner_nilpotent': False, 'inner_order': 115248, 'inner_split': True, 'inner_tex': 'D_7\\times C_7^3:S_4', 'inner_used': [1, 2], 'irrC_degree': 8, 'irrQ_degree': 36, 'irrQ_dim': 36, 'irrR_degree': 12, 'irrep_stats': [[1, 4], [2, 8], [3, 4], [4, 27], [6, 30], [8, 48], [12, 96], [16, 18], [24, 100], [48, 15]], 'label': '115248.bg', 'linC_count': 144, 'linC_degree': 6, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 24, 'linQ_degree_count': 8, 'linQ_dim': 24, 'linQ_dim_count': 8, 'linR_count': 72, 'linR_degree': 8, 'maximal_subgroups_known': True, 'metabelian': False, 'metacyclic': False, 'monomial': None, 'name': 'D7*C7^3:S4', 'ngens': 9, '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, 0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 57, 'number_characteristic_subgroups': 16, 'number_conjugacy_classes': 350, 'number_divisions': 83, 'number_normal_subgroups': 16, 'number_subgroup_autclasses': 527, 'number_subgroup_classes': 733, 'number_subgroups': 193232, 'old_label': None, 'order': 115248, 'order_factorization_type': 321, 'order_stats': [[1, 1], [2, 1519], [3, 392], [4, 16464], [6, 2744], [7, 2400], [14, 44100], [21, 18816], [28, 12348], [42, 16464]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 6, 'outer_gen_orders': [3, 6], 'outer_gen_pows': [98454, 0], 'outer_gens': [[49489, 2, 82380, 168, 2352, 16464], [56533, 48218, 82668, 79464, 7056, 65856]], 'outer_group': '18.5', 'outer_hash': 5, 'outer_nilpotent': True, 'outer_order': 18, 'outer_permdeg': 8, 'outer_perms': [144, 5043], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_3\\times C_6', 'pc_rank': 6, 'perfect': False, 'permutation_degree': 28, 'pgroup': 0, 'primary_abelian_invariants': [2, 2], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': False, 'ratrep_stats': [[1, 4], [2, 2], [3, 4], [6, 2], [12, 1], [18, 6], [24, 4], [36, 12], [48, 10], [72, 19], [96, 3], [144, 15], [288, 1]], 'representations': {'PC': {'code': '543307837735532771015067858902241724184967121635012158936478206041229689504465066671722161147982859538840734000690150032666543202031101126549003587800563291507962292839', 'gens': [1, 2, 4, 6, 8, 9], 'pres': [9, 2, 2, 3, 2, 7, 2, 7, 7, 7, 408240, 1842661, 46, 270110, 480836, 646707, 1135308, 809049, 102, 1488244, 403933, 1217452, 2317901, 2556050, 305069, 105116, 74507, 158, 889062, 74103, 518640, 10617, 145159, 1185424, 6073, 84706, 6100, 54449, 190538, 47681]}, 'Perm': {'d': 28, 'gens': [23052920286645070446581108885, 11744603803563313972885385100]}}, 'schur_multiplier': [2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2], 'solvability_type': 17, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': 'D_7\\times C_7^3:S_4', 'transitive_degree': 42, 'wreath_data': None, 'wreath_product': False}
