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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': False, 'ambient': '118206.b', 'ambient_counter': 2, 'ambient_order': 118206, 'ambient_tex': 'C_3\\times F_{199}', 'central': False, 'central_factor': False, 'centralizer_order': 3, 'characteristic': True, 'core_order': 39402, 'counter': 3, 'cyclic': False, 'direct': False, 'hall': 0, 'label': '118206.b.3.a1.a1', 'maximal': True, 'maximal_normal': True, 'metabelian': True, 'metacyclic': None, 'minimal': False, 'minimal_normal': False, 'nilpotent': False, 'normal': True, 'old_label': '3.a1.a1', 'outer_equivalence': False, 'perfect': False, 'proper': True, 'quotient': '3.1', 'quotient_Agroup': True, 'quotient_abelian': True, 'quotient_cyclic': True, 'quotient_hash': 1, 'quotient_metabelian': True, 'quotient_nilpotent': True, 'quotient_order': 3, 'quotient_simple': True, 'quotient_solvable': True, 'quotient_supersolvable': True, 'quotient_tex': 'C_3', 'simple': False, 'solvable': True, 'special_labels': [], 'split': False, 'standard_generators': False, 'stem': False, 'subgroup': '39402.b', 'subgroup_hash': 521106686622776765, 'subgroup_order': 39402, 'subgroup_tex': 'C_{597}:C_{66}', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '118206.b', 'aut_centralizer_order': None, 'aut_label': '3.a1', 'aut_quo_index': None, 'aut_stab_index': None, 'aut_weyl_group': None, 'aut_weyl_index': None, 'centralizer': '39402.a1.a1', 'complements': [], 'conjugacy_class_count': 1, 'contained_in': ['1.a1.a1'], 'contains': ['6.a1.a1', '9.a1.a1', '9.b1.a1', '9.b1.b1', '9.c1.a1', '33.a1.a1', '597.a1.a1'], 'core': '3.a1.a1', 'coset_action_label': None, 'count': 1, 'diagramx': [5673, 4227, 5755, 4061, 4442, 4423, 6614, 3437], 'generators': [18, 594, 39402, 132, 99], 'label': '118206.b.3.a1.a1', 'mobius_quo': 0, 'mobius_sub': -1, 'normal_closure': '3.a1.a1', 'normal_contained_in': ['1.a1.a1'], 'normal_contains': ['6.a1.a1', '9.a1.a1', '9.c1.a1', '9.b1.a1', '9.b1.b1', '33.a1.a1'], 'normalizer': '1.a1.a1', 'old_label': '3.a1.a1', 'projective_image': '39402.d', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '3.a1.a1', 'subgroup_fusion': None, 'weyl_group': '39402.d'}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '198.10', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 39402, 'aut_gen_orders': [33, 66, 198], 'aut_gens': [[1, 66], [17029, 32736], [29041, 39072], [36565, 2904]], 'aut_group': None, 'aut_hash': 2208373850004692075, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 236412, 'aut_permdeg': 796, 'aut_perms': 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'aut_phi_ratio': 19.9, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 199, 1, 1], [3, 1, 2, 1], [3, 199, 3, 2], [6, 199, 2, 1], [6, 199, 3, 2], [11, 199, 1, 10], [22, 199, 1, 10], [33, 199, 2, 10], [33, 199, 3, 20], [66, 199, 2, 10], [66, 199, 3, 20], [199, 66, 3, 1], [597, 66, 6, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_{199}:(C_{11}:(C_{18}\\times S_3))', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 6, 'autcent_group': '6.1', 'autcent_hash': 1, 'autcent_nilpotent': False, 'autcent_order': 6, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'S_3', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 39402, 'autcentquo_group': '39402.d', 'autcentquo_hash': 6821401462869295210, 'autcentquo_nilpotent': False, 'autcentquo_order': 39402, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'F_{199}', 'cc_stats': [[1, 1, 1], [2, 199, 1], [3, 1, 2], [3, 199, 6], [6, 199, 8], [11, 199, 10], [22, 199, 10], [33, 199, 80], [66, 199, 80], [199, 66, 3], [597, 66, 6]], 'center_label': '3.1', 'center_order': 3, 'central_product': True, 'central_quotient': '13134.h', 'commutator_count': 1, 'commutator_label': '199.1', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '3.1', '3.1', '11.1', '199.1'], 'composition_length': 5, 'conjugacy_classes_known': True, 'counter': 2, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['13134.h', 1], ['3.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 199, 1, 1], [3, 1, 2, 1], [3, 199, 2, 3], [6, 199, 2, 4], [11, 199, 10, 1], [22, 199, 10, 1], [33, 199, 20, 4], [66, 199, 20, 4], [199, 66, 3, 1], [597, 66, 6, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 2880, 'exponent': 13134, 'exponents_of_order': [2, 1, 1, 1], 'factors_of_aut_order': [2, 3, 11, 199], 'factors_of_order': [2, 3, 11, 199], 'faithful_reps': [[66, 0, 6]], 'familial': False, 'frattini_label': '1.1', 'frattini_quotient': '39402.b', 'hash': 521106686622776765, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 13134, 'inner_gen_orders': [66, 199], 'inner_gens': [[1, 30162], [9307, 66]], 'inner_hash': 18, 'inner_nilpotent': False, 'inner_order': 13134, 'inner_split': False, 'inner_tex': 'C_{199}:C_{66}', 'inner_used': [1, 2], 'irrC_degree': 66, 'irrQ_degree': 396, 'irrQ_dim': 396, 'irrR_degree': 132, 'irrep_stats': [[1, 198], [66, 9]], 'label': '39402.b', 'linC_count': 6, 'linC_degree': 66, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 200, 'linQ_degree_count': 6, 'linQ_dim': 200, 'linQ_dim_count': 6, 'linR_count': 198, 'linR_degree': 68, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C597:C66', 'ngens': 5, '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], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 90, 'number_characteristic_subgroups': 14, 'number_conjugacy_classes': 207, 'number_divisions': 22, 'number_normal_subgroups': 26, 'number_subgroup_autclasses': 32, 'number_subgroup_classes': 48, 'number_subgroups': 4404, 'old_label': None, 'order': 39402, 'order_factorization_type': 222, 'order_stats': [[1, 1], [2, 199], [3, 1196], [6, 1592], [11, 1990], [22, 1990], [33, 15920], [66, 15920], [199, 198], [597, 396]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 6, 'outer_gen_orders': [6, 3], 'outer_gen_pows': [16, 0], 'outer_gens': [[26269, 20724], [13135, 66]], 'outer_group': '18.3', 'outer_hash': 3, 'outer_nilpotent': False, 'outer_order': 18, 'outer_permdeg': 6, 'outer_perms': [305, 466], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_3\\times S_3', 'pc_rank': 2, 'perfect': False, 'permutation_degree': 202, 'pgroup': 0, 'primary_abelian_invariants': [2, 3, 3, 11], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 2], [2, 8], [10, 2], [20, 8], [198, 1], [396, 1]], 'representations': {'PC': {'code': '3475977401404892342291067075012383118162558241527431111', 'gens': [1, 4], 'pres': [5, -2, -3, -11, -3, -199, 10, 36, 603243, 327368, 26633, 78, 292054, 242559, 99839]}, 'GLFp': {'d': 2, 'p': 199, 'gens': [7880799, 1063880905, 725015214]}, 'Perm': {'d': 202, 'gens': [1616928876444447162438635540036043177124476851653520177554972866365538881519883333078681662298242310281474797644482184485568759633593663189259689001201583311090694930075041876923681792959248017588431325039288862599811577487795039208319930994871110730701988909189694996657273845985388552735452775305493431200631284854733446280673670260118814833533590895314986943054739589917443, 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'sylow_subgroups_known': True, 'tex_name': 'C_{597}:C_{66}', 'transitive_degree': 597, 'wreath_data': None, 'wreath_product': False}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '594.20', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 39402, 'aut_gen_orders': [198, 22, 198], 'aut_gens': [[1, 198], [25543, 13464], [33463, 24750], [52471, 54252]], 'aut_group': None, 'aut_hash': 2208373850004692075, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 236412, 'aut_permdeg': 597, 'aut_perms': 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'aut_phi_ratio': 6.633333333333334, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 199, 1, 1], [3, 1, 2, 1], [3, 199, 1, 2], [3, 199, 2, 2], [6, 199, 1, 2], [6, 199, 2, 3], [9, 199, 3, 6], [11, 199, 1, 10], [18, 199, 3, 6], [22, 199, 1, 10], [33, 199, 1, 20], [33, 199, 2, 30], [66, 199, 1, 20], [66, 199, 2, 30], [99, 199, 3, 60], [198, 199, 3, 60], [199, 198, 1, 1], [597, 198, 2, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_{199}:(C_{11}:(C_{18}\\times S_3))', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 6, 'autcent_group': '6.1', 'autcent_hash': 1, 'autcent_nilpotent': False, 'autcent_order': 6, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'S_3', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 39402, 'autcentquo_group': '39402.d', 'autcentquo_hash': 6821401462869295210, 'autcentquo_nilpotent': False, 'autcentquo_order': 39402, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'F_{199}', 'cc_stats': [[1, 1, 1], [2, 199, 1], [3, 1, 2], [3, 199, 6], [6, 199, 8], [9, 199, 18], [11, 199, 10], [18, 199, 18], [22, 199, 10], [33, 199, 80], [66, 199, 80], [99, 199, 180], [198, 199, 180], [199, 198, 1], [597, 198, 2]], 'center_label': '3.1', 'center_order': 3, 'central_product': True, 'central_quotient': '39402.d', 'commutator_count': 1, 'commutator_label': '199.1', 'complements_known': True, 'complete': False, 'complex_characters_known': False, 'composition_factors': ['2.1', '3.1', '3.1', '3.1', '11.1', '199.1'], 'composition_length': 6, 'conjugacy_classes_known': True, 'counter': 2, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['3.1', 1], ['39402.d', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 199, 1, 1], [3, 1, 2, 1], [3, 199, 2, 3], [6, 199, 2, 4], [9, 199, 6, 3], [11, 199, 10, 1], [18, 199, 6, 3], [22, 199, 10, 1], [33, 199, 20, 4], [66, 199, 20, 4], [99, 199, 60, 3], [198, 199, 60, 3], [199, 198, 1, 1], [597, 198, 2, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 25920, 'exponent': 39402, 'exponents_of_order': [3, 1, 1, 1], 'factors_of_aut_order': [2, 3, 11, 199], 'factors_of_order': [2, 3, 11, 199], 'faithful_reps': [[198, 0, 2]], 'familial': False, 'frattini_label': '1.1', 'frattini_quotient': '118206.b', 'hash': 993549773098811332, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 39402, 'inner_gen_orders': [198, 199], 'inner_gens': [[1, 86328], [32077, 198]], 'inner_hash': 6821401462869295210, 'inner_nilpotent': False, 'inner_order': 39402, 'inner_split': True, 'inner_tex': 'F_{199}', 'inner_used': [1, 2], 'irrC_degree': 198, 'irrQ_degree': 396, 'irrQ_dim': 396, 'irrR_degree': None, 'irrep_stats': [[1, 594], [198, 3]], 'label': '118206.b', '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': 'C3*F199', 'ngens': 6, '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], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 266, 'number_characteristic_subgroups': 22, 'number_conjugacy_classes': 597, 'number_divisions': 34, 'number_normal_subgroups': 42, 'number_subgroup_autclasses': 56, 'number_subgroup_classes': 80, 'number_subgroups': 7604, 'old_label': None, 'order': 118206, 'order_factorization_type': 311, 'order_stats': [[1, 1], [2, 199], [3, 1196], [6, 1592], [9, 3582], [11, 1990], [18, 3582], [22, 1990], [33, 15920], [66, 15920], [99, 35820], [198, 35820], [199, 198], [597, 396]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 6, 'outer_gen_orders': [2, 3], 'outer_gen_pows': [0, 0], 'outer_gens': [[78805, 118008], [39403, 198]], 'outer_group': '6.1', 'outer_hash': 1, 'outer_nilpotent': False, 'outer_order': 6, 'outer_permdeg': 3, 'outer_perms': [1, 4], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'S_3', 'pc_rank': 2, 'perfect': False, 'permutation_degree': 202, 'pgroup': 0, 'primary_abelian_invariants': [2, 3, 9, 11], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 2], [2, 8], [6, 6], [10, 2], [20, 8], [60, 6], [198, 1], [396, 1]], 'representations': {'PC': {'code': '238659246580686050565389015627360277025947720797371306756695821870781064794743', 'gens': [1, 5], 'pres': [6, -2, -3, -3, -11, -3, -199, 12, 43, 68, 2589844, 742510, 312856, 20152, 118, 812597, 545303, 417005, 72491]}, 'GLFp': {'d': 2, 'p': 199, 'gens': [7880799, 1008716714, 725015214]}, 'Perm': {'d': 202, 'gens': [1617107125760524307766100365424199373010971604634639225606828417689585168323053353330077072011087644895861673386127722921844440215533147109943204670751965379014642136739140539745050865979379043606840365902069089296329276807637706001055845521129334678412362574408342890869574767270015764787465041502313767900651858237629569988745092874895192355775818958102217826757319360452563, 2409728899142544665694002617428431763699342373216762171439217176948816013893203935440411500988577433714435727611511524343568068758731540559311672140615088929784766255489056102979438836153563905451468293439068281052874898659916019410835019087556304920392130840485924635808963716032116321772421457756199845089703003598915957423708435679300320777307445046041544288858899497114083, 161675261840272413713656168254934723851818547888749537980209141527158143994532142410874015243118211155969703800450240243502807430740906227550703011269515180637950537540327182201746637127738428301787436940258064877062536671335551525418763876116108829399777583208675654656903869536546942154959627862239344794391430764502618263004589118004893647062516949149047611325598654549344480]}}, 'schur_multiplier': [3], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [3, 198], 'solvability_type': 6, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_3\\times F_{199}', 'transitive_degree': 597, 'wreath_data': None, 'wreath_product': False}
-
gps_groups • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': True, 'abelian_quotient': '3.1', '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], [2]], '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], [3, 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], [3, 1, 2]], 'center_label': '3.1', 'center_order': 3, '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': ['3.1'], 'composition_length': 1, 'conjugacy_classes_known': True, 'counter': 1, 'cyclic': True, 'derived_length': 1, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [3, 1, 2, 1]], 'element_repr_type': 'PC', 'elementary': 3, 'eulerian_function': 1, 'exponent': 3, 'exponents_of_order': [1], 'factors_of_aut_order': [2], 'factors_of_order': [3], 'faithful_reps': [[1, 0, 2]], 'familial': True, 'frattini_label': '1.1', 'frattini_quotient': '3.1', 'hash': 1, 'hyperelementary': 3, '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, 3]], 'label': '3.1', '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': 'C3', 'ngens': 1, 'nilpotency_class': 1, 'nilpotent': True, 'normal_counts': [0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 2, 'number_characteristic_subgroups': 2, 'number_conjugacy_classes': 3, 'number_divisions': 2, 'number_normal_subgroups': 2, 'number_subgroup_autclasses': 2, 'number_subgroup_classes': 2, 'number_subgroups': 2, 'old_label': None, 'order': 3, 'order_factorization_type': 1, 'order_stats': [[1, 1], [3, 2]], 'outer_abelian': True, 'outer_cyclic': True, 'outer_equivalence': False, 'outer_exponent': 2, 'outer_gen_orders': [2], 'outer_gen_pows': [0], 'outer_gens': [[2]], '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': 3, 'pgroup': 3, 'primary_abelian_invariants': [3], 'quasisimple': False, 'rank': 1, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 1], [2, 1]], 'representations': {'PC': {'code': 0, 'gens': [1], 'pres': [1, -3]}, 'GLZ': {'b': 3, 'd': 2, 'gens': [7]}, 'Lie': [{'d': 1, 'q': 3, 'gens': [3], 'family': 'ASigmaL'}], 'GLFp': {'d': 2, 'p': 2, 'gens': [7]}, 'Perm': {'d': 3, 'gens': [4]}}, 'schur_multiplier': [], 'semidirect_product': False, 'simple': True, 'smith_abelian_invariants': [3], 'solvability_type': 0, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_3', 'transitive_degree': 3, 'wreath_data': None, 'wreath_product': False}
