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gps_subgroup_search • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'ambient': '256.10', 'ambient_counter': 10, 'ambient_order': 256, 'ambient_tex': 'C_4^2.D_8', 'central': False, 'central_factor': False, 'centralizer_order': 16, 'characteristic': True, 'core_order': 128, 'counter': 3, 'cyclic': False, 'direct': False, 'hall': 0, 'label': '256.10.2.b1.a1', 'maximal': True, 'maximal_normal': True, 'metabelian': True, 'metacyclic': False, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': True, 'old_label': '2.b1.a1', 'outer_equivalence': False, 'perfect': False, 'proper': True, 'quotient': '2.1', 'quotient_Agroup': True, 'quotient_abelian': True, 'quotient_cyclic': True, 'quotient_hash': 1, 'quotient_metabelian': True, 'quotient_nilpotent': True, 'quotient_order': 2, 'quotient_simple': True, 'quotient_solvable': True, 'quotient_supersolvable': True, 'quotient_tex': 'C_2', 'simple': False, 'solvable': True, 'special_labels': ['C1'], 'split': False, 'standard_generators': False, 'stem': False, 'subgroup': '128.499', 'subgroup_hash': 499, 'subgroup_order': 128, 'subgroup_tex': 'C_4^3.C_2', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '256.10', 'aut_centralizer_order': 16, 'aut_label': '2.b1', 'aut_quo_index': 1, 'aut_stab_index': 1, 'aut_weyl_group': '256.51978', 'aut_weyl_index': 16, 'centralizer': '16.a1.a1', 'complements': [], 'conjugacy_class_count': 1, 'contained_in': ['1.a1.a1'], 'contains': ['4.a1.a1', '4.b1.a1', '4.b1.b1', '4.f1.a1', '4.g1.a1'], 'core': '2.b1.a1', 'coset_action_label': None, 'count': 1, 'diagramx': [5745, 6982, 2205, 3290, 5629, 6780, 2197, 3031], 'generators': [8, 2, 96], 'label': '256.10.2.b1.a1', 'mobius_quo': 0, 'mobius_sub': -1, 'normal_closure': '2.b1.a1', 'normal_contained_in': ['1.a1.a1'], 'normal_contains': ['4.a1.a1', '4.b1.b1', '4.b1.a1'], 'normalizer': '1.a1.a1', 'old_label': '2.b1.a1', 'projective_image': '32.2', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '2.b1.a1', 'subgroup_fusion': None, 'weyl_group': '16.3'}
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gps_groups • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '32.21', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 4, 'aut_gen_orders': [4, 4, 2, 4, 4, 4, 2, 4], 'aut_gens': [[1, 4, 32], [73, 110, 48], [91, 103, 32], [19, 84, 96], [19, 101, 50], [89, 68, 34], [91, 23, 96], [17, 77, 48], [81, 38, 32]], 'aut_group': None, 'aut_hash': 1184556994785001468, 'aut_nilpotency_class': 3, 'aut_nilpotent': True, 'aut_order': 8192, 'aut_permdeg': 64, 'aut_perms': [84265266705985721847532195761042447311022245049485937382530714051095095729731030551893494, 74134686081953047050248015716233693831165729993533843678427823428343958040288185156169504, 120509978517648903330128581294016692621861422543957121988468559183450110286915087547845289, 58150212345087667149002959752254491182900363281556575277629144296788568234110336420202998, 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None, 'autcentquo_nilpotent': None, 'autcentquo_order': 4, 'autcentquo_solvable': None, 'autcentquo_supersolvable': None, 'autcentquo_tex': None, 'cc_stats': [[1, 1, 1], [2, 1, 7], [4, 1, 8], [4, 2, 24], [8, 4, 16]], 'center_label': '16.10', 'center_order': 16, 'central_product': False, 'central_quotient': '8.5', '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', '2.1'], 'composition_length': 7, 'conjugacy_classes_known': True, 'counter': 499, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 7], [4, 1, 2, 4], [4, 2, 1, 4], [4, 2, 2, 10], [8, 4, 2, 8]], 'element_repr_type': 'PC', 'elementary': 2, 'eulerian_function': 84, 'exponent': 8, 'exponents_of_order': [7], 'factors_of_aut_order': [2], 'factors_of_order': [2], 'faithful_reps': [], 'familial': False, 'frattini_label': '16.10', 'frattini_quotient': '8.5', 'hash': 499, 'hyperelementary': 2, 'inner_abelian': True, 'inner_cyclic': False, 'inner_exponent': 2, 'inner_gen_orders': [2, 2, 2], 'inner_gens': [[1, 20, 32], [17, 4, 96], [1, 68, 32]], 'inner_hash': 5, 'inner_nilpotent': True, 'inner_order': 8, 'inner_split': None, 'inner_tex': 'C_2^3', 'inner_used': [1, 2, 3], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 32], [2, 24]], 'label': '128.499', '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': False, 'monomial': True, 'name': 'C4^3.C2', 'ngens': 3, 'nilpotency_class': 2, 'nilpotent': True, 'normal_counts': [0, 0, 0, 0, 0, 0, 0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 13, 'number_characteristic_subgroups': 18, 'number_conjugacy_classes': 56, 'number_divisions': 34, 'number_normal_subgroups': 96, 'number_subgroup_autclasses': 60, 'number_subgroup_classes': 146, 'number_subgroups': 196, 'old_label': None, 'order': 128, 'order_factorization_type': 7, 'order_stats': [[1, 1], [2, 7], [4, 56], [8, 64]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': True, 'outer_exponent': 4, 'outer_gen_orders': [2, 4, 4, 4, 4, 4, 4], 'outer_gen_pows': [1, 0, 0, 0, 0, 0, 0], 'outer_gens': [[81, 109, 48], [89, 55, 48], [89, 36, 32], [83, 31, 32], [11, 45, 32], [83, 124, 50], [3, 38, 34]], 'outer_group': None, 'outer_hash': 2921611870903438452, 'outer_nilpotent': True, 'outer_order': 1024, 'outer_permdeg': 256, 'outer_perms': [736235332398101341907397520719100995390087139222514732794420561568117468815527866893943704173054179958927903126847737344460064901176713225554345709395874772692789630209937996980686744966790945646117066871902211979791110247965745543869757997186126854733912882051681241069312309866131923410827767077900764342410926580727853421690306845341488816276564694990433311853422149224055910326216388086069406250232803333737008110775175624459202089466924129920895131665183804911457739210627550382996614357436447796437092, 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101251840326576017631577223401771431901429646382744885893576693305185097463785133489286955871837196401497466317299705884140540880774581441501752377754710066512693904481642625725527842304565615010759449184858160924367788567419185684611796695463549572085891453749642237264008374845790703506799013402079392747112849411509201210728680288569104214344820924945530740223122617386139991447094072878910244271717300600276608315098927401138297805775871312296274098007018411752513736728322697986269752279505003218684229], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^6.C_2^4', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 16, 'pgroup': 2, 'primary_abelian_invariants': [2, 4, 4], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 8], [2, 16], [4, 10]], 'representations': {'PC': {'code': 4756091619307421869, 'gens': [1, 3, 6], 'pres': [7, 2, 2, 2, 2, 2, 2, 2, 14, 422, 58, 80, 1027, 124]}, 'GLZN': {'d': 2, 'p': 40, 'gens': [1088017, 192087, 576009, 64801, 2171157, 481437, 1344021]}, 'Perm': {'d': 16, 'gens': [1308153376089, 2984786340487, 4209952693080, 1327353713296, 16696, 5517148049280, 1327353713280]}}, 'schur_multiplier': [2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 4, 4], 'solvability_type': 4, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_4^3.C_2', 'transitive_degree': 128, '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': '32.3', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 4, 'aut_gen_orders': [4, 4, 4, 4, 2, 4, 4, 4], 'aut_gens': [[1, 8, 64], [101, 42, 100], [145, 206, 196], [53, 200, 96], [179, 222, 100], [35, 108, 192], [51, 126, 100], [193, 204, 224], [113, 236, 224]], 'aut_group': None, 'aut_hash': 309185806410308548, 'aut_nilpotency_class': 2, 'aut_nilpotent': True, 'aut_order': 4096, 'aut_permdeg': 128, 'aut_perms': [80770401319516950835097450199312845582526684144708097679168545797526079868240059140887420328144168040441639224179481011084740492549657720207501357681391454659410744586033339161858657881736563895216077291171743852333, 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368721883231349794776723264876069927971839303782377782383086626908221321518571069760267855257879238643160527425214763721145040378057458843861791370636047602775289486680880607817504836880586605844079977452937366068981, 315627117510345689337059177542952115010890863014958090073038163478284017671333322003419786598208588976334973990354432072759729914920387539638772597465921623221683886660714091496618269214540124604413253375559793490389, 49926099516117874933175829179008559972008259130422010336975860069555592567955598040017224814755162843022323899193609471698380424524086323637416506688788393881628890411036968632017239004977058538830831897667690769635], 'aut_phi_ratio': 32.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 7], [4, 2, 2, 10], [4, 4, 2, 2], [8, 4, 16, 1], [8, 8, 8, 2]], 'aut_supersolvable': True, 'aut_tex': 'C_2^5.C_2^6.C_2', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 2, 'autcent_group': '64.267', 'autcent_hash': 267, 'autcent_nilpotent': True, 'autcent_order': 64, 'autcent_solvable': True, 'autcent_split': False, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^6', 'autcentquo_abelian': True, 'autcentquo_cyclic': False, 'autcentquo_exponent': 2, 'autcentquo_group': '64.267', 'autcentquo_hash': 267, 'autcentquo_nilpotent': True, 'autcentquo_order': 64, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'C_2^6', 'cc_stats': [[1, 1, 1], [2, 1, 7], [4, 2, 20], [4, 4, 4], [8, 4, 16], [8, 8, 16]], 'center_label': '8.5', 'center_order': 8, 'central_product': False, 'central_quotient': '32.2', 'commutator_count': 1, 'commutator_label': '8.2', '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'], 'composition_length': 8, 'conjugacy_classes_known': True, 'counter': 10, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 7], [4, 2, 1, 4], [4, 2, 2, 8], [4, 4, 1, 2], [4, 4, 2, 1], [8, 4, 4, 4], [8, 8, 2, 4], [8, 8, 4, 2]], 'element_repr_type': 'PC', 'elementary': 2, 'eulerian_function': 6, 'exponent': 8, 'exponents_of_order': [8], 'factors_of_aut_order': [2], 'factors_of_order': [2], 'faithful_reps': [], 'familial': False, 'frattini_label': '64.55', 'frattini_quotient': '4.2', 'hash': 10, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 4, 'inner_gen_orders': [4, 4, 2], 'inner_gens': [[1, 248, 96], [81, 8, 192], [33, 136, 64]], 'inner_hash': 2, 'inner_nilpotent': True, 'inner_order': 32, 'inner_split': True, 'inner_tex': 'C_2.C_4^2', 'inner_used': [1, 2], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 32], [2, 24], [4, 8]], 'label': '256.10', 'linC_count': 2048, 'linC_degree': 7, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 12, 'linQ_degree_count': 128, 'linQ_dim': 12, 'linQ_dim_count': 54, 'linR_count': 16, 'linR_degree': 8, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'C4^2.D8', 'ngens': 2, 'nilpotency_class': 3, 'nilpotent': True, 'normal_counts': [0, 0, 0, 0, 0, 0, 0, 0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 23, 'number_characteristic_subgroups': 49, 'number_conjugacy_classes': 64, 'number_divisions': 33, 'number_normal_subgroups': 73, 'number_subgroup_autclasses': 117, 'number_subgroup_classes': 160, 'number_subgroups': 323, 'old_label': None, 'order': 256, 'order_factorization_type': 7, 'order_stats': [[1, 1], [2, 7], [4, 56], [8, 192]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 4, 'outer_gen_orders': [2, 2, 2, 2, 2, 4], 'outer_gen_pows': [0, 0, 0, 0, 0, 0], 'outer_gens': [[7, 24, 224], [3, 24, 224], [3, 12, 224], [179, 24, 224], [1, 12, 64], [179, 170, 68]], 'outer_group': '128.2163', 'outer_hash': 2163, 'outer_nilpotent': True, 'outer_order': 128, 'outer_permdeg': 12, 'outer_perms': [16, 98155463, 125199496, 283818383, 132452760, 250871183], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^4:D_4', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 24, 'pgroup': 2, 'primary_abelian_invariants': [4, 8], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 10], [4, 16], [8, 3]], 'representations': {'PC': {'code': 165438346102558323643681511347803, 'gens': [1, 4, 7], 'pres': [8, 2, 2, 2, 2, 2, 2, 2, 2, 16, 41, 7939, 2699, 91, 1924, 116, 5382, 1374, 166]}, 'GLZN': {'d': 2, 'p': 48, 'gens': [2764825, 1055911, 111745, 1512989, 3207749, 857333, 4810795, 4534313]}, 'Perm': {'d': 24, 'gens': [28311802671422872189863, 27083199607498106556290, 55188205604114558978309, 82307873133979145022574, 55188205609550831596800, 82307873129612562449764, 17764, 7094612828644]}}, 'schur_multiplier': [2, 4], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [4, 8], 'solvability_type': 4, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_4^2.D_8', 'transitive_degree': 256, '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': '2.1', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': True, 'aut_cyclic': True, 'aut_derived_length': 0, 'aut_exponent': 1, 'aut_gen_orders': [], 'aut_gens': [[1]], 'aut_group': '1.1', 'aut_hash': 1, 'aut_nilpotency_class': 0, 'aut_nilpotent': True, 'aut_order': 1, 'aut_permdeg': 1, 'aut_perms': [], 'aut_phi_ratio': 1.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_1', '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': 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]], 'center_label': '2.1', 'center_order': 2, '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'], 'composition_length': 1, 'conjugacy_classes_known': True, 'counter': 1, 'cyclic': True, 'derived_length': 1, 'dihedral': True, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1]], 'element_repr_type': 'PC', 'elementary': 2, 'eulerian_function': 1, 'exponent': 2, 'exponents_of_order': [1], 'factors_of_aut_order': [], 'factors_of_order': [2], 'faithful_reps': [[1, 1, 1]], 'familial': True, 'frattini_label': '1.1', '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': 1, 'irrQ_dim': 1, 'irrR_degree': 1, 'irrep_stats': [[1, 2]], 'label': '2.1', 'linC_count': 1, 'linC_degree': 1, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 1, 'linQ_degree_count': 1, 'linQ_dim': 1, 'linQ_dim_count': 1, 'linR_count': 1, 'linR_degree': 1, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C2', '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': 2, 'number_divisions': 2, 'number_normal_subgroups': 2, 'number_subgroup_autclasses': 2, 'number_subgroup_classes': 2, 'number_subgroups': 2, 'old_label': None, 'order': 2, 'order_factorization_type': 1, 'order_stats': [[1, 1], [2, 1]], 'outer_abelian': True, 'outer_cyclic': True, 'outer_equivalence': False, 'outer_exponent': 1, 'outer_gen_orders': [], 'outer_gen_pows': [], 'outer_gens': [], 'outer_group': '1.1', 'outer_hash': 1, 'outer_nilpotent': True, 'outer_order': 1, 'outer_permdeg': 1, 'outer_perms': [], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_1', 'pc_rank': 1, 'perfect': False, 'permutation_degree': 2, 'pgroup': 2, 'primary_abelian_invariants': [2], 'quasisimple': False, 'rank': 1, 'rational': True, 'rational_characters_known': True, 'ratrep_stats': [[1, 2]], 'representations': {'PC': {'code': 0, 'gens': [1], 'pres': [1, -2]}, 'GLZ': {'b': 3, 'd': 2, 'gens': [12]}, 'Lie': [{'d': 2, 'q': 2, 'gens': [6], 'family': 'CSOPlus'}, {'d': 2, 'q': 2, 'gens': [6], 'family': 'COPlus'}, {'d': 1, 'q': 2, 'gens': [1], 'family': 'AGL'}, {'d': 1, 'q': 2, 'gens': [1], 'family': 'AGammaL'}, {'d': 1, 'q': 2, 'gens': [1], 'family': 'ASigmaL'}], 'GLFp': {'d': 2, 'p': 2, 'gens': [11]}, 'Perm': {'d': 2, 'gens': [1]}}, 'schur_multiplier': [], 'semidirect_product': False, 'simple': True, 'smith_abelian_invariants': [2], 'solvability_type': 0, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_2', 'transitive_degree': 2, 'wreath_data': None, 'wreath_product': False}
