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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': False, 'ambient': '768.1088504', 'ambient_counter': 1088504, 'ambient_order': 768, 'ambient_tex': 'C_2\\times A_4:\\OD_{32}', 'central': False, 'central_factor': False, 'centralizer_order': 32, 'characteristic': False, 'core_order': 192, 'counter': 16, 'cyclic': False, 'direct': False, 'hall': 0, 'label': '768.1088504.4.e1.b1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': False, 'minimal': False, 'minimal_normal': False, 'nilpotent': False, 'normal': True, 'old_label': '4.e1.b1', 'outer_equivalence': False, 'perfect': False, 'proper': True, 'quotient': '4.1', 'quotient_Agroup': True, 'quotient_abelian': True, 'quotient_cyclic': True, 'quotient_hash': 1, 'quotient_metabelian': True, 'quotient_nilpotent': True, 'quotient_order': 4, 'quotient_simple': False, 'quotient_solvable': True, 'quotient_supersolvable': True, 'quotient_tex': 'C_4', 'simple': False, 'solvable': True, 'special_labels': [], 'split': False, 'standard_generators': False, 'stem': False, 'subgroup': '192.1010', 'subgroup_hash': 1010, 'subgroup_order': 192, 'subgroup_tex': 'C_2^3:C_{24}', 'supersolvable': False, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '768.1088504', 'aut_centralizer_order': 16, 'aut_label': '4.e1', 'aut_quo_index': 1, 'aut_stab_index': 2, 'aut_weyl_group': '192.1537', 'aut_weyl_index': 32, 'centralizer': '24.b1.a1', 'complements': [], 'conjugacy_class_count': 1, 'contained_in': ['2.a1.a1'], 'contains': ['8.d1.b1', '8.g1.a1', '8.h1.a1', '12.e1.b1', '16.j1.b1'], 'core': '4.e1.b1', 'coset_action_label': None, 'count': 1, 'diagramx': [5116, 812, 4546, 4125, 5718, 4445, 4034, 4016], 'generators': [508447, 573457, 557073, 622611, 557585, 417459, 294921], 'label': '768.1088504.4.e1.b1', 'mobius_quo': 0, 'mobius_sub': 0, 'normal_closure': '4.e1.b1', 'normal_contained_in': ['2.a1.a1'], 'normal_contains': ['8.d1.b1', '8.g1.a1', '8.h1.a1', '12.e1.b1'], 'normalizer': '1.a1.a1', 'old_label': '4.e1.b1', 'projective_image': '96.194', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '4.e1.b1', 'subgroup_fusion': None, 'weyl_group': '24.12'}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '48.23', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 3, 'aut_exponent': 12, 'aut_gen_orders': [2, 2, 2, 2, 2, 3, 2, 2], 'aut_gens': [[24532, 38921, 37001, 12901], [16693, 37001, 38921, 64233], [24532, 38921, 37001, 64233], [24532, 38921, 37001, 5671], [55124, 38921, 37001, 31457], [24532, 38921, 37001, 45677], [24532, 6273, 38921, 12901], [57180, 38921, 37001, 12901], [55260, 38921, 37001, 12901]], 'aut_group': '384.20051', 'aut_hash': 20051, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 384, 'aut_permdeg': 10, 'aut_perms': [40320, 1, 414, 120, 126, 45360, 367920, 806400], 'aut_phi_ratio': 6.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 1, 2, 1], [2, 3, 1, 2], [2, 3, 2, 1], [3, 4, 2, 1], [4, 1, 2, 2], [4, 3, 2, 2], [6, 4, 2, 1], [6, 4, 4, 1], [8, 1, 8, 1], [8, 3, 8, 1], [12, 4, 4, 2], [24, 4, 16, 1]], 'aut_supersolvable': False, 'aut_tex': '\\GL(2,\\mathbb{Z}/4):C_2^2', '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': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2\\times D_4', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 12, 'autcentquo_group': '24.12', 'autcentquo_hash': 12, 'autcentquo_nilpotent': False, 'autcentquo_order': 24, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'S_4', 'cc_stats': [[1, 1, 1], [2, 1, 3], [2, 3, 4], [3, 4, 2], [4, 1, 4], [4, 3, 4], [6, 4, 6], [8, 1, 8], [8, 3, 8], [12, 4, 8], [24, 4, 16]], 'center_label': '16.5', 'center_order': 16, 'central_product': True, 'central_quotient': '12.3', 'commutator_count': 1, 'commutator_label': '4.2', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '2.1', '2.1', '3.1'], 'composition_length': 7, 'conjugacy_classes_known': True, 'counter': 1010, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['12.3', 1], ['2.1', 1], ['8.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [2, 3, 1, 4], [3, 4, 2, 1], [4, 1, 2, 2], [4, 3, 2, 2], [6, 4, 2, 3], [8, 1, 4, 2], [8, 3, 4, 2], [12, 4, 4, 2], [24, 4, 8, 2]], 'element_repr_type': 'GLZq', 'elementary': 1, 'eulerian_function': 24, 'exponent': 24, 'exponents_of_order': [6, 1], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [], 'familial': False, 'frattini_label': '4.1', 'frattini_quotient': '48.49', 'hash': 1010, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 6, 'inner_gen_orders': [3, 2, 2, 1], 'inner_gens': [[24532, 37001, 6273, 12901], [55260, 38921, 37001, 12901], [57180, 38921, 37001, 12901], [24532, 38921, 37001, 12901]], 'inner_hash': 3, 'inner_nilpotent': False, 'inner_order': 12, 'inner_split': True, 'inner_tex': 'A_4', 'inner_used': [1, 2], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 48], [3, 16]], 'label': '192.1010', 'linC_count': 288, 'linC_degree': 4, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 7, 'linQ_degree_count': 4, 'linQ_dim': 7, 'linQ_dim_count': 4, 'linR_count': 24, 'linR_degree': 5, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'C2^3:C24', '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': 18, 'number_characteristic_subgroups': 21, 'number_conjugacy_classes': 64, 'number_divisions': 24, 'number_normal_subgroups': 33, 'number_subgroup_autclasses': 72, 'number_subgroup_classes': 93, 'number_subgroups': 224, 'old_label': None, 'order': 192, 'order_factorization_type': 51, 'order_stats': [[1, 1], [2, 15], [3, 8], [4, 16], [6, 24], [8, 32], [12, 32], [24, 64]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 4, 'outer_gen_orders': [2, 2, 2, 4], 'outer_gen_pows': [4097, 4097, 4097, 4097], 'outer_gens': [[16693, 37001, 38921, 45677], [16693, 37001, 38921, 31457], [16693, 37001, 38921, 5671], [18877, 37001, 38921, 38447]], 'outer_group': '32.46', 'outer_hash': 46, 'outer_nilpotent': True, 'outer_order': 32, 'outer_permdeg': 8, 'outer_perms': [23, 7, 11536, 16583], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^2\\times D_4', 'pc_rank': 4, 'perfect': False, 'permutation_degree': 14, 'pgroup': 0, 'primary_abelian_invariants': [2, 8, 3], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 6], [3, 4], [4, 4], [6, 2], [8, 2], [12, 2]], 'representations': {'PC': {'code': 20691867932091636212357280, 'gens': [1, 3, 4, 5], 'pres': [7, -2, -3, -2, 2, -2, -2, -2, 14, 254, 198, 507, 94, 102, 124]}, 'GLZN': {'d': 2, 'p': 12, 'gens': [12417, 8645, 18685, 2665, 9889, 2043, 1801]}, 'GLZq': {'d': 2, 'q': 16, 'gens': [12901, 36873, 63503, 51027, 12291, 39049, 4225]}, 'Perm': {'d': 14, 'gens': [6314117160, 13982577840, 3, 20761458480, 6314117040, 7, 16]}}, 'schur_multiplier': [2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 24], 'solvability_type': 8, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': 'C_2^3:C_{24}', 'transitive_degree': 48, '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': '32.36', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 3, 'aut_exponent': 12, 'aut_gen_orders': [12, 4, 12, 12, 12, 4, 4], 'aut_gens': [[49681, 299447, 957620, 573457, 557585], [508447, 18336, 433332, 557585, 49665], [1032719, 168763, 141165, 49665, 557585], [508447, 915077, 974516, 49665, 573457], [49681, 20192, 450228, 557585, 49665], [1032719, 581519, 973988, 49665, 573457], [508447, 111101, 664941, 49665, 557585], [1032719, 897765, 681837, 49665, 557585]], 'aut_group': None, 'aut_hash': 3625057183303488335, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 6144, 'aut_permdeg': 192, 'aut_perms': [304170215825141125498351650060121770994619997301352445378259322975349888001776951025299061398553895503471721068186274474755440387144381327013646507349976759914838144186963470239734717528814773268670728705815151351874201877103667825531204526780114954358073982747634721167577084739058003887174644918333983604544608484856565959195206907995792363098406549518578, 55276677921982762997846185304263720345243573683180091184496401878338919607761581566206856658023123897360082885875171849280909773256067652573061394960608897330941744702137666137526478640957505528566824179401155279576485055160916164942542680302248997882701369729592703623323231634138755773136039044903747750691914167876734188450608866018600961227408457976608, 212324832416112070329653132729568087813170878789145978787958504427527056912177613323678112037268767550707977608517496710459393843389083786760048345948765959766515205091862111050106648077052517630338623960385361371570561208913955949758833795497515284084851042226540683337577506058492812623265710807369364069262459015235415414518030032673032349424536260875467, 157996951377493510019053435870975609508804365976241195187747077432622094598235112535251631041990832213183612676980408420048168173168652991575623408621823134362425595409724673051720678793795704859520995571887105089873672553377109781856815408232305187863812670868235563354693598858482260616176372122454123404547148830557470014797694255056055164361823912396800, 143325015352572938061763816185648286948213641658977706454091672234253980084124667760631686832855226654348137616273228683143492516384111986209044222831321018774599068719453045790875896772933358887231308547615633789467840275746360780189934638501584532390886116644071115008517357446630044804577227837240225863455436551462437077956146462562449069248025729139610, 352731483389056514370861034617604725166107112643371034216203048850376255058148281351455534508343432665393791560262349775069697118471827091824377094262505463785651112269788686317986254276729228039638433725944999890083939006394420420034667004367813585140149181113652525947294066673015026749114250428539396277408416046861128628687111851004722352722033188884424, 318299428896529513707849828360789184126542137890191208139354409169047785422065877470648743370021104916125907661463589531747416032710003015108481093200986021447224492787651825835125430993867783091486182418376961593437804822415239428383052461718578741137256343886428066423809284397088581426440357921055221030916941165878646920980405328102972493275686761550623], 'aut_phi_ratio': 24.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 1, 2, 1], [2, 2, 2, 1], [2, 3, 1, 2], [2, 3, 2, 1], [2, 6, 2, 1], [3, 8, 1, 1], [4, 1, 2, 2], [4, 2, 2, 1], [4, 3, 2, 2], [4, 6, 2, 1], [6, 8, 1, 1], [6, 8, 2, 1], [6, 8, 4, 1], [8, 1, 4, 2], [8, 2, 4, 1], [8, 3, 4, 2], [8, 6, 4, 1], [12, 8, 2, 2], [12, 8, 4, 1], [16, 12, 16, 2], [24, 8, 4, 2], [24, 8, 8, 1]], 'aut_supersolvable': False, 'aut_tex': '(C_2^3\\times A_4).C_2^6', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 4, 'autcent_group': '128.2167', 'autcent_hash': 2167, 'autcent_nilpotent': True, 'autcent_order': 128, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_4^2:C_2^3', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 12, 'autcentquo_group': '48.48', 'autcentquo_hash': 48, 'autcentquo_nilpotent': False, 'autcentquo_order': 48, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'C_2\\times S_4', 'cc_stats': [[1, 1, 1], [2, 1, 3], [2, 2, 2], [2, 3, 4], [2, 6, 2], [3, 8, 1], [4, 1, 4], [4, 2, 2], [4, 3, 4], [4, 6, 2], [6, 8, 7], [8, 1, 8], [8, 2, 4], [8, 3, 8], [8, 6, 4], [12, 8, 8], [16, 12, 32], [24, 8, 16]], 'center_label': '16.5', 'center_order': 16, 'central_product': True, 'central_quotient': '48.48', 'commutator_count': 1, 'commutator_label': '24.13', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '2.1', '2.1', '2.1', '2.1', '3.1'], 'composition_length': 9, 'conjugacy_classes_known': True, 'counter': 1088504, 'cyclic': False, 'derived_length': 3, 'dihedral': False, 'direct_factorization': [['2.1', 1], ['384.5680', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [2, 2, 1, 2], [2, 3, 1, 4], [2, 6, 1, 2], [3, 8, 1, 1], [4, 1, 2, 2], [4, 2, 1, 2], [4, 3, 2, 2], [4, 6, 1, 2], [6, 8, 1, 3], [6, 8, 2, 2], [8, 1, 4, 2], [8, 2, 2, 2], [8, 3, 4, 2], [8, 6, 2, 2], [12, 8, 2, 4], [16, 12, 4, 8], [24, 8, 4, 4]], 'element_repr_type': 'GLZq', 'elementary': 1, 'eulerian_function': 20160, 'exponent': 48, 'exponents_of_order': [8, 1], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [], 'familial': False, 'frattini_label': '8.1', 'frattini_quotient': '96.226', 'hash': 1088504, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 12, 'inner_gen_orders': [2, 2, 3, 2, 2], 'inner_gens': [[49681, 824231, 957620, 573457, 557585], [573953, 299447, 681853, 49665, 557585], [49681, 28640, 957620, 557585, 49665], [49681, 823719, 449700, 573457, 557585], [49681, 299447, 433828, 573457, 557585]], 'inner_hash': 48, 'inner_nilpotent': False, 'inner_order': 48, 'inner_split': False, 'inner_tex': 'C_2\\times S_4', 'inner_used': [1, 2, 3, 4], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 32], [2, 40], [3, 32], [6, 8]], 'label': '768.1088504', 'linC_count': 384, 'linC_degree': 5, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 11, 'linQ_degree_count': 8, 'linQ_dim': 11, 'linQ_dim_count': 8, 'linR_count': 48, 'linR_degree': 7, 'maximal_subgroups_known': True, 'metabelian': False, 'metacyclic': False, 'monomial': True, 'name': 'C2*A4:OD32', '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], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 32, 'number_characteristic_subgroups': 43, 'number_conjugacy_classes': 112, 'number_divisions': 50, 'number_normal_subgroups': 101, 'number_subgroup_autclasses': 284, 'number_subgroup_classes': 489, 'number_subgroups': 1626, 'old_label': None, 'order': 768, 'order_factorization_type': 51, 'order_stats': [[1, 1], [2, 31], [3, 8], [4, 32], [6, 56], [8, 64], [12, 64], [16, 384], [24, 128]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 4, 'outer_gen_orders': [2, 2, 2, 2, 2, 4], 'outer_gen_pows': [49681, 49681, 32769, 32769, 32769, 32769], 'outer_gens': [[49681, 37055, 957620, 573457, 557585], [49681, 446515, 957620, 573457, 557585], [49681, 766057, 681853, 49665, 557585], [49681, 914149, 664941, 49665, 557585], [49681, 258169, 957620, 573457, 557585], [508447, 840103, 450228, 573457, 557585]], 'outer_group': '128.2163', 'outer_hash': 2163, 'outer_nilpotent': True, 'outer_order': 128, 'outer_permdeg': 12, 'outer_perms': [87102727, 127387576, 40291223, 11520, 127387560, 303171263], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^4:D_4', 'pc_rank': 5, 'perfect': False, 'permutation_degree': 22, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 8], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 8], [2, 12], [3, 8], [4, 8], [6, 4], [8, 2], [12, 4], [16, 2], [24, 2]], 'representations': {'PC': {'code': '1482085955519320388227563783068344264788363336708', 'gens': [1, 2, 6, 8, 9], 'pres': [9, -2, -2, -2, -2, -2, -2, -3, -2, 2, 325, 46, 74, 102, 4334, 158, 4047, 20752, 916, 709, 1511, 305]}, 'GLZN': {'d': 2, 'p': 68, 'gens': [8020353, 4087629, 2829897, 316745, 16587479, 5236, 10376289, 314465, 18708763]}, 'GLZq': {'d': 2, 'q': 32, 'gens': [32785, 98307, 299447, 294921, 124284, 442909, 33281, 557073, 573953]}, 'Perm': {'d': 22, 'gens': [51347038622219078823, 51212945926204877284, 355461131765, 4397976680734, 5800817303020, 2966658509284, 107425784095838208000, 51212587272118272000, 122000787836928000]}}, 'schur_multiplier': [2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 8], 'solvability_type': 11, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': 'C_2\\times A_4:\\OD_{32}', 'transitive_degree': 96, 'wreath_data': None, 'wreath_product': False}
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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}
