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gps_subgroup_search • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'ambient': '832.8', 'ambient_counter': 8, 'ambient_order': 832, 'ambient_tex': 'C_{13}:Q_{64}', 'central': False, 'central_factor': False, 'centralizer_order': 2, 'characteristic': False, 'core_order': 32, 'counter': 11, 'cyclic': False, 'direct': None, 'hall': 2, 'label': '832.8.13.a1.a1', 'maximal': True, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': False, 'old_label': '13.a1.a1', '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': 13, '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': '64.54', 'subgroup_hash': 54, 'subgroup_order': 64, 'subgroup_tex': 'Q_{64}', 'supersolvable': True, 'sylow': 2}
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gps_subgroup_data • Show schema
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{'ambient': '832.8', 'aut_centralizer_order': 12, 'aut_label': '13.a1', 'aut_quo_index': None, 'aut_stab_index': 13, 'aut_weyl_group': '512.58302', 'aut_weyl_index': 156, 'centralizer': '416.a1.a1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['1.a1.a1'], 'contains': ['26.a1.a1', '26.b1.a1', '26.b1.b1'], 'core': '26.a1.a1', 'coset_action_label': None, 'count': 13, 'diagramx': [8324, -1, 709, -1, 789, -1, 5804, -1], 'generators': [1, 26], 'label': '832.8.13.a1.a1', 'mobius_quo': None, 'mobius_sub': -1, 'normal_closure': '1.a1.a1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '13.a1.a1', 'old_label': '13.a1.a1', 'projective_image': '416.6', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '13.a1.a1', 'subgroup_fusion': None, 'weyl_group': '32.18'}
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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': 2, 'aut_exponent': 32, 'aut_gen_orders': [2, 8, 32], 'aut_gens': [[1, 2], [31, 62], [21, 6], [51, 34]], 'aut_group': '512.58302', 'aut_hash': 7772715459544409331, 'aut_nilpotency_class': 5, 'aut_nilpotent': True, 'aut_order': 512, 'aut_permdeg': 32, 'aut_perms': [209110625063666949217196394943844509, 166831302240608821283181734550591682, 192584446486825221723391051205525821], 'aut_phi_ratio': 16.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [4, 2, 1, 1], [4, 16, 2, 1], [8, 2, 2, 1], [16, 2, 4, 1], [32, 2, 8, 1]], 'aut_supersolvable': True, 'aut_tex': 'D_{32}:C_8', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 2, 'autcent_group': '4.2', 'autcent_hash': 2, 'autcent_nilpotent': True, 'autcent_order': 4, 'autcent_solvable': True, 'autcent_split': False, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^2', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 16, 'autcentquo_group': '128.913', 'autcentquo_hash': 913, 'autcentquo_nilpotent': True, 'autcentquo_order': 128, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'D_{16}:C_4', 'cc_stats': [[1, 1, 1], [2, 1, 1], [4, 2, 1], [4, 16, 2], [8, 2, 2], [16, 2, 4], [32, 2, 8]], 'center_label': '2.1', 'center_order': 2, 'central_product': False, 'central_quotient': '32.18', 'commutator_count': 1, 'commutator_label': '16.1', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '2.1', '2.1'], 'composition_length': 6, 'conjugacy_classes_known': True, 'counter': 54, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [4, 2, 1, 1], [4, 16, 1, 2], [8, 2, 2, 1], [16, 2, 4, 1], [32, 2, 8, 1]], 'element_repr_type': 'PC', 'elementary': 2, 'eulerian_function': 3, 'exponent': 32, 'exponents_of_order': [6], 'factors_of_aut_order': [2], 'factors_of_order': [2], 'faithful_reps': [[2, -1, 8]], 'familial': True, 'frattini_label': '16.1', 'frattini_quotient': '4.2', 'hash': 54, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 16, 'inner_gen_orders': [2, 16], 'inner_gens': [[1, 62], [5, 2]], 'inner_hash': 18, 'inner_nilpotent': True, 'inner_order': 32, 'inner_split': False, 'inner_tex': 'D_{16}', 'inner_used': [1, 2], 'irrC_degree': 2, 'irrQ_degree': 16, 'irrQ_dim': 32, 'irrR_degree': 4, 'irrep_stats': [[1, 4], [2, 15]], 'label': '64.54', 'linC_count': 8, 'linC_degree': 2, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 16, 'linQ_degree_count': 1, 'linQ_dim': 32, 'linQ_dim_count': 1, 'linR_count': 8, 'linR_degree': 4, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'Q64', 'ngens': 2, 'nilpotency_class': 5, 'nilpotent': True, 'normal_counts': [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': 19, 'number_divisions': 8, 'number_normal_subgroups': 9, 'number_subgroup_autclasses': 11, 'number_subgroup_classes': 15, 'number_subgroups': 37, 'old_label': None, 'order': 64, 'order_factorization_type': 3, 'order_stats': [[1, 1], [2, 1], [4, 34], [8, 4], [16, 8], [32, 16]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 8, 'outer_gen_orders': [2, 8], 'outer_gen_pows': [2, 0], 'outer_gens': [[3, 2], [1, 10]], 'outer_group': '16.5', 'outer_hash': 5, 'outer_nilpotent': True, 'outer_order': 16, 'outer_permdeg': 10, 'outer_perms': [362880, 6057], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2\\times C_8', 'pc_rank': 2, 'perfect': False, 'permutation_degree': 64, 'pgroup': 2, 'primary_abelian_invariants': [2, 2], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 1], [4, 1], [8, 1], [16, 1]], 'representations': {'PC': {'code': 309504273051162771967, 'gens': [1, 2], 'pres': [6, -2, 2, -2, -2, -2, -2, 192, 745, 31, 1082, 50, 1347, 69, 1444, 88]}, 'Lie': [{'d': 2, 'q': 31, 'gens': [896293, 653851, 893760, 405374], 'family': 'CSOMinus'}], 'GLFp': {'d': 2, 'p': 31, 'gens': [893666, 687030]}, 'Perm': {'d': 64, 'gens': [66450134624445933678799560050331607746391532584685435254529204790382042061733888000000000, 34216625329393954566794504615428400130324458319121250657957939996263067413000027270329600, 24175131827739636274127375045840733860693888662420563914564207161518320707982436785337600, 12087040917006882684584735480871444149038188995887599735100184001816510068360415118652856, 6010765313484612917937180565556780809330360821355660294198753718800751569568892186006982, 1983116034341975454789449189498620955194254460449807567074683265838922666583379392681647]}}, 'schur_multiplier': [], 'semidirect_product': False, 'simple': False, 'smith_abelian_invariants': [2, 2], 'solvability_type': 3, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'Q_{64}', 'transitive_degree': 64, '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': 2, 'aut_exponent': 1248, 'aut_gen_orders': [48, 24, 24, 24, 6], 'aut_gens': [[1, 2], [295, 294], [305, 282], [521, 630], [825, 742], [633, 734]], 'aut_group': None, 'aut_hash': 5940659561310699955, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 79872, 'aut_permdeg': 416, 'aut_perms': 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21201969478656878851342293413038559069898040013608280123815221475763782770409285220752797131735345443211099492096198971200145486821994220865720635561335974039744373608917513592200128528320321570651840343124029949256430333024266743321485271262694535315416414302371388627786680404006393099296636428968470623183545610828956106267051068219077148624164492055782205965326027416907095829841376941941594502107645849339175334327817491550207318440908122515819022601618519016441961691205608228853776011521491849074627584536603397330978680134294453454444948738945663451188258544321376825769219181666266099107092927434570020579895487895133258228809480789409985482363771775725521133538466076921736214963093809473735102248942682207685799007582513164676416165963783338385604631811434681490685337474157802484831990475688957607359255351412465086661409302622037445814352292464154476074640261408321926132583141651466984883682691763, 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5867205174196122597307090559697342748403977552827533171952068088771359556553190213150102907374028363924546813593658025876068694099047326871728559141389371234524764229418441383020426799953238160980928382055171030856945117655605615787255043066149372235936857885140865035357802512279923432596425574673278830432013545565142402932094925123754641849484438847717517447425847636691506628714957471683504542759114897105624779048999144093044939766426731470047239264428698606971603653027081972857441238443115435731666023960841023629857735300317627928794698044802716709941107579708543670539227079627486914186948847671109184720131850419590499469592234580513078850013131102247629662898866560194001546718997287678640333373888221182844606749340763048521338116759257715751407555350525478725728661803461460114007753500834737883881433412499121610222745145142887032506722041254652166477191853178109092315147830735700210021994144887], 'aut_phi_ratio': 208.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [4, 2, 1, 1], [4, 208, 2, 1], [8, 2, 2, 1], [13, 2, 6, 1], [16, 2, 4, 1], [26, 2, 6, 1], [32, 2, 8, 1], [52, 2, 12, 1], [104, 2, 24, 1], [208, 2, 48, 1], [416, 2, 96, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_{208}.C_3.C_4^2.C_2^3', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 2, 'autcent_group': '4.2', 'autcent_hash': 2, 'autcent_nilpotent': True, 'autcent_order': 4, 'autcent_solvable': True, 'autcent_split': False, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^2', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 624, 'autcentquo_group': None, 'autcentquo_hash': 2846720442775810740, 'autcentquo_nilpotent': False, 'autcentquo_order': 19968, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'C_{104}.C_6.C_2^4.C_2', 'cc_stats': [[1, 1, 1], [2, 1, 1], [4, 2, 1], [4, 208, 2], [8, 2, 2], [13, 2, 6], [16, 2, 4], [26, 2, 6], [32, 2, 8], [52, 2, 12], [104, 2, 24], [208, 2, 48], [416, 2, 96]], 'center_label': '2.1', 'center_order': 2, 'central_product': False, 'central_quotient': '416.6', 'commutator_count': 1, 'commutator_label': '208.2', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '2.1', '2.1', '13.1'], 'composition_length': 7, 'conjugacy_classes_known': True, 'counter': 8, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [4, 2, 1, 1], [4, 208, 1, 2], [8, 2, 2, 1], [13, 2, 6, 1], [16, 2, 4, 1], [26, 2, 6, 1], [32, 2, 8, 1], [52, 2, 12, 1], [104, 2, 24, 1], [208, 2, 48, 1], [416, 2, 96, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 3, 'exponent': 416, 'exponents_of_order': [6, 1], 'factors_of_aut_order': [2, 3, 13], 'factors_of_order': [2, 13], 'faithful_reps': [[2, -1, 96]], 'familial': False, 'frattini_label': '16.1', 'frattini_quotient': '52.4', 'hash': 8, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 208, 'inner_gen_orders': [2, 208], 'inner_gens': [[1, 830], [5, 2]], 'inner_hash': 6, 'inner_nilpotent': False, 'inner_order': 416, 'inner_split': False, 'inner_tex': 'D_{208}', 'inner_used': [1, 2], 'irrC_degree': 2, 'irrQ_degree': 192, 'irrQ_dim': 384, 'irrR_degree': 4, 'irrep_stats': [[1, 4], [2, 207]], 'label': '832.8', 'linC_count': 96, 'linC_degree': 2, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 28, 'linQ_degree_count': 2, 'linQ_dim': 44, 'linQ_dim_count': 2, 'linR_count': 96, 'linR_degree': 4, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C13:Q64', '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': 13, 'number_characteristic_subgroups': 13, 'number_conjugacy_classes': 211, 'number_divisions': 14, 'number_normal_subgroups': 15, 'number_subgroup_autclasses': 22, 'number_subgroup_classes': 30, 'number_subgroups': 446, 'old_label': None, 'order': 832, 'order_factorization_type': 51, 'order_stats': [[1, 1], [2, 1], [4, 418], [8, 4], [13, 12], [16, 8], [26, 12], [32, 16], [52, 24], [104, 48], [208, 96], [416, 192]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 24, 'outer_gen_orders': [2, 4, 24], 'outer_gen_pows': [26, 0, 0], 'outer_gens': [[27, 258], [1, 770], [417, 202]], 'outer_group': '192.835', 'outer_hash': 835, 'outer_nilpotent': True, 'outer_order': 192, 'outer_permdeg': 17, 'outer_perms': [20922789888000, 187304886960, 187302142323], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2\\times C_4\\times C_{24}', 'pc_rank': 2, 'perfect': False, 'permutation_degree': 77, 'pgroup': 0, 'primary_abelian_invariants': [2, 2], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 1], [4, 1], [8, 1], [12, 2], [16, 1], [24, 1], [48, 1], [96, 1], [192, 1]], 'representations': {'PC': {'code': 7004842521640598333092179609953500601069487341579, 'gens': [1, 2], 'pres': [7, -2, -2, -2, -2, -2, -2, -13, 2912, 11621, 36, 17390, 58, 23075, 80, 28564, 102, 33605, 124, 37638]}, 'Perm': {'d': 77, 'gens': [2012916462237405787325090313710381947170518414653774083352709058707047558227147827810487703921116357919352720047, 3923532776524199585230863962642303070095470296662188000860332634085014639735906313861700991424518423327078144000, 5834189761572771998319707577146687540338811545838753536008087459193944156625985033965235403552152815550113536000, 7744802531930376124421388428069823121581255230724577281722783975409521855541403945248800862813819574770362112000, 9655463791424796606130519865682925179612877594737673981516193118614275998513672973235450734028990344615504435200, 11566103149202927675680463652041288526807971444788451647160552452979411445639160906187065254182427728922041395200, 559571293]}}, 'schur_multiplier': [], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2], 'solvability_type': 6, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_{13}:Q_{64}', 'transitive_degree': 832, 'wreath_data': None, 'wreath_product': False}
