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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': True, 'ambient': '1984.320', 'ambient_counter': 320, 'ambient_order': 1984, 'ambient_tex': '(C_2\\times C_{248}):C_4', 'central': False, 'central_factor': False, 'centralizer_order': 992, 'characteristic': True, 'core_order': 4, 'counter': 82, 'cyclic': True, 'direct': False, 'hall': 0, 'label': '1984.320.496.g1.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': True, 'old_label': '496.g1.a1', 'outer_equivalence': False, 'perfect': False, 'proper': True, 'quotient': '496.13', 'quotient_Agroup': False, 'quotient_abelian': False, 'quotient_cyclic': False, 'quotient_hash': 13, 'quotient_metabelian': True, 'quotient_nilpotent': False, 'quotient_order': 496, 'quotient_simple': False, 'quotient_solvable': True, 'quotient_supersolvable': True, 'quotient_tex': 'D_{62}:C_4', 'simple': False, 'solvable': True, 'special_labels': [], 'split': False, 'standard_generators': False, 'stem': False, 'subgroup': '4.1', 'subgroup_hash': 1, 'subgroup_order': 4, 'subgroup_tex': 'C_4', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '1984.320', 'aut_centralizer_order': 119040, 'aut_label': '496.g1', 'aut_quo_index': 1, 'aut_stab_index': 1, 'aut_weyl_group': '2.1', 'aut_weyl_index': 119040, 'centralizer': '2.c1.a1', 'complements': [], 'conjugacy_class_count': 1, 'contained_in': ['16.g1.a1', '248.c1.a1', '248.e1.a1', '248.f1.a1', '248.j1.a1', '248.j1.b1'], 'contains': ['992.a1.a1'], 'core': '496.g1.a1', 'coset_action_label': None, 'count': 1, 'diagramx': [6659, 2756, 6142, 3848, 5710, 6037, 5920, 4257], 'generators': [1018], 'label': '1984.320.496.g1.a1', 'mobius_quo': 0, 'mobius_sub': 0, 'normal_closure': '496.g1.a1', 'normal_contained_in': ['16.g1.a1', '248.e1.a1', '248.f1.a1', '248.c1.a1'], 'normal_contains': ['992.a1.a1'], 'normalizer': '1.a1.a1', 'old_label': '496.g1.a1', 'projective_image': '992.37', 'quotient_action_image': '2.1', 'quotient_action_kernel': '248.11', 'quotient_action_kernel_order': 248, 'quotient_fusion': None, 'short_label': '496.g1.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': '4.1', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': True, 'aut_cyclic': True, 'aut_derived_length': 1, 'aut_exponent': 2, 'aut_gen_orders': [2], 'aut_gens': [[1], [3]], 'aut_group': '2.1', 'aut_hash': 1, 'aut_nilpotency_class': 1, 'aut_nilpotent': True, 'aut_order': 2, 'aut_permdeg': 2, 'aut_perms': [1], 'aut_phi_ratio': 1.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [4, 1, 2, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_2', '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': 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]], 'center_label': '4.1', 'center_order': 4, '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': ['2.1', '2.1'], '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], [2, 1, 1, 1], [4, 1, 2, 1]], 'element_repr_type': 'PC', 'elementary': 2, 'eulerian_function': 1, 'exponent': 4, 'exponents_of_order': [2], 'factors_of_aut_order': [2], 'factors_of_order': [2], 'faithful_reps': [[1, 0, 2]], 'familial': True, 'frattini_label': '2.1', 'frattini_quotient': '2.1', 'hash': 1, 'hyperelementary': 2, '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': 2, 'irrQ_dim': 2, 'irrR_degree': 2, 'irrep_stats': [[1, 4]], 'label': '4.1', 'linC_count': 2, 'linC_degree': 1, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 2, 'linQ_degree_count': 1, 'linQ_dim': 2, 'linQ_dim_count': 1, 'linR_count': 1, 'linR_degree': 2, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C4', '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': 4, 'number_divisions': 3, 'number_normal_subgroups': 3, 'number_subgroup_autclasses': 3, 'number_subgroup_classes': 3, 'number_subgroups': 3, 'old_label': None, 'order': 4, 'order_factorization_type': 2, 'order_stats': [[1, 1], [2, 1], [4, 2]], 'outer_abelian': True, 'outer_cyclic': True, 'outer_equivalence': False, 'outer_exponent': 2, 'outer_gen_orders': [2], 'outer_gen_pows': [0], 'outer_gens': [[3]], 'outer_group': '2.1', 'outer_hash': 1, 'outer_nilpotent': True, 'outer_order': 2, 'outer_permdeg': 2, 'outer_perms': [1], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2', 'pc_rank': 1, 'perfect': False, 'permutation_degree': 4, 'pgroup': 2, 'primary_abelian_invariants': [4], 'quasisimple': False, 'rank': 1, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 2], [2, 1]], 'representations': {'PC': {'code': 5, 'gens': [1], 'pres': [2, -2, -2, 4]}, 'GLZ': {'b': 3, 'd': 2, 'gens': [46]}, 'Lie': [{'d': 2, 'q': 3, 'gens': [56, 15], 'family': 'CSOPlus'}], 'GLFp': {'d': 2, 'p': 3, 'gens': [21]}, 'Perm': {'d': 4, 'gens': [22, 7]}}, 'schur_multiplier': [], 'semidirect_product': False, 'simple': False, 'smith_abelian_invariants': [4], 'solvability_type': 0, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_4', 'transitive_degree': 4, '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': '16.2', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 1860, 'aut_gen_orders': [30, 6, 4, 60, 12, 60, 30, 10], 'aut_gens': [[1, 4, 32], [329, 12, 1888], [1691, 30, 816], [209, 1020, 1968], [1403, 12, 1712], [1147, 30, 176], [337, 14, 1696], [945, 4, 1632], [1801, 28, 736]], 'aut_group': None, 'aut_hash': 1488748237624605799, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 238080, 'aut_permdeg': 512, 'aut_perms': 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'aut_phi_ratio': 248.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 2, 1, 3], [4, 1, 2, 1], [4, 2, 1, 3], [4, 124, 4, 1], [8, 4, 4, 1], [8, 124, 4, 1], [31, 2, 15, 1], [62, 2, 15, 1], [62, 2, 30, 1], [62, 4, 15, 2], [124, 2, 30, 2], [124, 4, 15, 2], [248, 4, 120, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_{62}.C_{30}.C_2^6.C_2', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 4, 'autcent_group': '16.11', 'autcent_hash': 11, 'autcent_nilpotent': True, 'autcent_order': 16, 'autcent_solvable': True, 'autcent_split': False, 'autcent_supersolvable': True, 'autcent_tex': 'C_2\\times D_4', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 930, 'autcentquo_group': None, 'autcentquo_hash': 3780555722307572373, 'autcentquo_nilpotent': False, 'autcentquo_order': 14880, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'C_{31}.C_{30}.C_2^4', 'cc_stats': [[1, 1, 1], [2, 1, 1], [2, 2, 3], [4, 1, 2], [4, 2, 3], [4, 124, 4], [8, 4, 4], [8, 124, 4], [31, 2, 15], [62, 2, 45], [62, 4, 30], [124, 2, 60], [124, 4, 30], [248, 4, 120]], 'center_label': '4.1', 'center_order': 4, 'central_product': False, 'central_quotient': '496.18', 'commutator_count': 1, 'commutator_label': '124.4', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '2.1', '2.1', '31.1'], 'composition_length': 7, 'conjugacy_classes_known': True, 'counter': 320, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 2, 1, 3], [4, 1, 2, 1], [4, 2, 1, 3], [4, 124, 2, 2], [8, 4, 2, 2], [8, 124, 2, 2], [31, 2, 15, 1], [62, 2, 15, 1], [62, 2, 30, 1], [62, 4, 15, 2], [124, 2, 30, 2], [124, 4, 15, 2], [248, 4, 60, 2]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 6, 'exponent': 248, 'exponents_of_order': [6, 1], 'factors_of_aut_order': [2, 3, 5, 31], 'factors_of_order': [2, 31], 'faithful_reps': [[4, 0, 60]], 'familial': False, 'frattini_label': '16.10', 'frattini_quotient': '124.3', 'hash': 320, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 124, 'inner_gen_orders': [4, 2, 62], 'inner_gens': [[1, 1012, 1968], [1009, 4, 32], [81, 4, 32]], 'inner_hash': 18, 'inner_nilpotent': False, 'inner_order': 496, 'inner_split': False, 'inner_tex': 'C_{62}.D_4', 'inner_used': [1, 2, 3], 'irrC_degree': 4, 'irrQ_degree': 240, 'irrQ_dim': 240, 'irrR_degree': 8, 'irrep_stats': [[1, 16], [2, 244], [4, 62]], 'label': '1984.320', 'linC_count': 60, 'linC_degree': 4, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 38, 'linQ_degree_count': 4, 'linQ_dim': 38, 'linQ_dim_count': 2, 'linR_count': 30, 'linR_degree': 8, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': '(C2*C248):C4', 'ngens': 7, 'nilpotency_class': -1, 'nilpotent': False, 'normal_counts': [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': 22, 'number_characteristic_subgroups': 39, 'number_conjugacy_classes': 322, 'number_divisions': 26, 'number_normal_subgroups': 47, 'number_subgroup_autclasses': 76, 'number_subgroup_classes': 90, 'number_subgroups': 1068, 'old_label': None, 'order': 1984, 'order_factorization_type': 51, 'order_stats': [[1, 1], [2, 7], [4, 504], [8, 512], [31, 30], [62, 210], [124, 240], [248, 480]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 30, 'outer_gen_orders': [2, 2, 2, 2, 30], 'outer_gen_pows': [0, 0, 1008, 0, 0], 'outer_gens': [[17, 4, 1952], [3, 4, 1968], [9, 4, 1952], [1, 22, 1952], [17, 12, 544]], 'outer_group': '480.1213', 'outer_hash': 1213, 'outer_nilpotent': True, 'outer_order': 480, 'outer_permdeg': 18, 'outer_perms': [362880, 355687428096000, 1307674368000, 6227020800, 39922593], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^4\\times C_{30}', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 47, 'pgroup': 0, 'primary_abelian_invariants': [4, 4], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 10], [8, 1], [30, 4], [60, 6], [240, 1]], 'representations': {'PC': {'code': 22026453158040928671374056942482867030000029, 'gens': [1, 3, 6], 'pres': [7, -2, -2, -2, -2, -2, 2, -31, 14, 21254, 219, 58, 80, 82661, 124, 94086]}, 'Perm': {'d': 47, 'gens': [5877986410790946074842168659865725843463839765351872681647, 11614361037669253074836479625972997686672840636497920000000, 17244220622409087225215154609686307171328149505966080000000, 22494898219429864126965915525151759346482568044216320000000, 27534920775933028697185619692206550728033255964016640000000, 22513506124535629832015374686560547139931076223303680000000, 282958192612797845278006145363293]}}, 'schur_multiplier': [2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [4, 4], 'solvability_type': 7, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': '(C_2\\times C_{248}):C_4', 'transitive_degree': 496, 'wreath_data': None, 'wreath_product': False}
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '8.2', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 930, 'aut_gen_orders': [2, 2, 2, 2, 30, 31], 'aut_gens': [[1, 2, 4], [3, 2, 244], [249, 2, 244], [1, 2, 246], [1, 2, 492], [1, 2, 292], [17, 2, 4]], 'aut_group': None, 'aut_hash': 3780555722307572373, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 14880, 'aut_permdeg': 128, 'aut_perms': [212535904754050021820250728584129068943161719065482573873866470722528519958491719463878853322674920743194767812104282494605090537557319519720400579640214601722109973143670452249743681684573044439670192523267879478895, 355254766992662807715426830267175328955586526556413547361110259553894726488363833825724495009774220029359651513261996301745478396336169801140823135524421224606599696726175501694650376002712729026603831066994194687135, 212535904754050021820250728584129068943161719065482573873866470722528519958491719463878853322674920743194767812104283521339805106405187754530433884457440466756425470029602464551991586964916228191692913035267879478439, 355254766992662807715426830267175328955586526556413547361110259553894726488363833825724495009774220029359651513261997670725097821466660780887534211356433667131112230677213171215375660203013492352665571098994194687501, 370726620302777891308616311665875704099753639955138162672579061199498266922548382968355224195857080413706559082092938136184961795059020949753451484297680749424521407215678842461417844020418031558352452414888790041486, 3036571536991761402025023565628831166190670615520904684519790561697467652582233828334907358461356931877577760538021196445998170060964578944183659705512819823311135946413904112267983419132453658879119764938441879777], 'aut_phi_ratio': 62.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [2, 62, 2, 1], [4, 2, 2, 1], [4, 62, 2, 1], [31, 2, 15, 1], [62, 2, 15, 3], [124, 2, 60, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_{31}.C_{30}.C_2^4', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 2, 'autcent_group': '16.14', 'autcent_hash': 14, 'autcent_nilpotent': True, 'autcent_order': 16, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^4', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 930, 'autcentquo_group': '930.1', 'autcentquo_hash': 1, 'autcentquo_nilpotent': False, 'autcentquo_order': 930, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'F_{31}', 'cc_stats': [[1, 1, 1], [2, 1, 3], [2, 62, 2], [4, 2, 2], [4, 62, 2], [31, 2, 15], [62, 2, 45], [124, 2, 60]], 'center_label': '4.2', 'center_order': 4, 'central_product': False, 'central_quotient': '124.3', 'commutator_count': 1, 'commutator_label': '62.2', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '31.1'], 'composition_length': 5, 'conjugacy_classes_known': True, 'counter': 13, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [2, 62, 1, 2], [4, 2, 2, 1], [4, 62, 2, 1], [31, 2, 15, 1], [62, 2, 15, 3], [124, 2, 60, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 6, 'exponent': 124, 'exponents_of_order': [4, 1], 'factors_of_aut_order': [2, 3, 5, 31], 'factors_of_order': [2, 31], 'faithful_reps': [], 'familial': False, 'frattini_label': '4.2', 'frattini_quotient': '124.3', 'hash': 13, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 62, 'inner_gen_orders': [2, 1, 62], 'inner_gens': [[1, 2, 246], [1, 2, 4], [259, 2, 4]], 'inner_hash': 3, 'inner_nilpotent': False, 'inner_order': 124, 'inner_split': True, 'inner_tex': 'D_{62}', 'inner_used': [1, 3], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 8], [2, 122]], 'label': '496.13', 'linC_count': 240, 'linC_degree': 3, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 34, 'linQ_degree_count': 10, 'linQ_dim': 34, 'linQ_dim_count': 10, 'linR_count': 90, 'linR_degree': 4, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'D62:C4', 'ngens': 5, 'nilpotency_class': -1, 'nilpotent': False, 'normal_counts': [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': 12, 'number_characteristic_subgroups': 15, 'number_conjugacy_classes': 130, 'number_divisions': 13, 'number_normal_subgroups': 17, 'number_subgroup_autclasses': 30, 'number_subgroup_classes': 34, 'number_subgroups': 496, 'old_label': None, 'order': 496, 'order_factorization_type': 31, 'order_stats': [[1, 1], [2, 127], [4, 128], [31, 30], [62, 90], [124, 120]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 30, 'outer_gen_orders': [2, 2, 30], 'outer_gen_pows': [0, 0, 0], 'outer_gens': [[3, 2, 244], [249, 2, 244], [3, 2, 44]], 'outer_group': '120.47', 'outer_hash': 47, 'outer_nilpotent': True, 'outer_order': 120, 'outer_permdeg': 14, 'outer_perms': [362880, 6227020800, 39922593], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^2\\times C_{30}', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 39, 'pgroup': 0, 'primary_abelian_invariants': [2, 4], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 4], [30, 2], [60, 3]], 'representations': {'PC': {'code': 700267875683481190800029, 'gens': [1, 2, 3], 'pres': [5, -2, -2, 2, -2, -31, 3692, 42, 9763, 58, 12004]}, 'GLZN': {'d': 2, 'p': 124, 'gens': [120117375, 1907121, 2871531, 234516675, 1914313]}, 'Perm': {'d': 39, 'gens': [13774094930742041323710367962159238592682092, 12460, 5329, 18619, 550570807328085194407427960024392768384990080]}}, 'schur_multiplier': [2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 4], 'solvability_type': 7, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'D_{62}:C_4', 'transitive_degree': 248, 'wreath_data': None, 'wreath_product': False}
