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gps_subgroup_search • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'ambient': '11664.hr', 'ambient_counter': 200, 'ambient_order': 11664, 'ambient_tex': 'C_2\\times \\He_3^2:D_4', 'central': False, 'central_factor': False, 'centralizer_order': 6, 'characteristic': False, 'core_order': 162, 'counter': 14, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '11664.hr.6.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': False, 'metacyclic': False, 'minimal': False, 'minimal_normal': False, 'nilpotent': False, 'normal': False, 'old_label': '6.a1', 'outer_equivalence': True, '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': 6, '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': '1944.2710', 'subgroup_hash': 2710, 'subgroup_order': 1944, 'subgroup_tex': 'C_2\\times C_3^3:S_3^2', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '11664.hr', 'aut_centralizer_order': None, 'aut_label': '6.a1', 'aut_quo_index': None, 'aut_stab_index': None, 'aut_weyl_group': None, 'aut_weyl_index': None, 'centralizer': '1944.a1', 'complements': None, 'conjugacy_class_count': 2, 'contained_in': ['2.a1'], 'contains': ['12.a1', '12.c1', '12.g1', '12.p1', '12.r1', '18.a1', '18.c1', '18.d1', '18.f1'], 'core': '72.a1', 'coset_action_label': None, 'count': 12, 'diagramx': [8648, -1, 8559, -1], 'generators': [2119809122033846544, 2367872583892584, 554460648822350160, 635636139925223071, 2105457054575179680, 1278118975935078150, 67625173003454784, 136007507152018824], 'label': '11664.hr.6.a1', 'mobius_quo': None, 'mobius_sub': 0, 'normal_closure': '2.a1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '6.a1', 'old_label': '6.a1', 'projective_image': '5832.cu', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '6.a1', 'subgroup_fusion': None, 'weyl_group': '324.121'}
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gps_groups • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '8.5', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 3, 'aut_exponent': 6, 'aut_gen_orders': [6, 6, 6, 6, 2, 6], 'aut_gens': [[24577758, 36728797, 2939606, 1017496, 25633100, 28816400, 6810638, 34663142], [7138662, 36728797, 2939606, 30616167, 25945430, 28816400, 10663906, 21642294], [20201582, 8188691, 32864349, 2266061, 25633100, 28816400, 8879620, 34663142], [24577758, 8188691, 2939606, 19088515, 8194004, 28816400, 41094069, 38862198], [43014227, 14725843, 32864349, 40442506, 8194004, 28816400, 30873581, 14111386], [7138662, 25627796, 2939606, 15714108, 29136749, 28816400, 36864279, 29083602], [7138662, 11534524, 2939606, 32415937, 8194004, 28816400, 27280919, 14111386]], 'aut_group': None, 'aut_hash': 8264013052929650798, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 7776, 'aut_permdeg': 72, 'aut_perms': [59551797721467841295332359914818605004394622464876014245872037305394154988567776864382831560736654544840, 44879114161687353345735867789452421498650279771436830672573472386814442960749315236725384552642801700394, 48286843718904780111533301866882441389038884812008993486670615076121533795445717793921239867304168777002, 50225042802464493413505859353407099287870961117547815967014470273515979686442790731226810683209719461885, 60733088966652816343600151314155604205971091265413771127003756716829567299873883898270224240163346725833, 28128091333034457387631006435137011501623047182457571883579221239844126954821371418264753906832284494659], 'aut_phi_ratio': 12.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 9, 2, 1], [2, 27, 2, 2], [3, 1, 2, 1], [3, 2, 1, 1], [3, 2, 2, 1], [3, 6, 1, 2], [3, 6, 2, 1], [3, 12, 1, 2], [3, 12, 2, 1], [3, 18, 3, 1], [3, 36, 3, 1], [6, 1, 2, 1], [6, 2, 1, 1], [6, 2, 2, 1], [6, 6, 1, 2], [6, 6, 2, 1], [6, 9, 4, 1], [6, 12, 1, 2], [6, 12, 2, 1], [6, 18, 3, 1], [6, 27, 4, 2], [6, 36, 3, 1], [6, 54, 2, 3], [6, 54, 4, 2], [6, 54, 6, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_3^3.D_6^2.C_2', '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': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^2', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 6, 'autcentquo_group': '1944.2342', 'autcentquo_hash': 2342, 'autcentquo_nilpotent': False, 'autcentquo_order': 1944, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'C_3^2.S_3^3', 'cc_stats': [[1, 1, 1], [2, 1, 1], [2, 9, 2], [2, 27, 4], [3, 1, 2], [3, 2, 3], [3, 6, 4], [3, 12, 4], [3, 18, 3], [3, 36, 3], [6, 1, 2], [6, 2, 3], [6, 6, 4], [6, 9, 4], [6, 12, 4], [6, 18, 3], [6, 27, 8], [6, 36, 3], [6, 54, 20]], 'center_label': '6.2', 'center_order': 6, 'central_product': True, 'central_quotient': '324.121', 'commutator_count': 1, 'commutator_label': '243.37', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '3.1', '3.1', '3.1', '3.1', '3.1'], 'composition_length': 8, 'conjugacy_classes_known': True, 'counter': 2710, 'cyclic': False, 'derived_length': 3, 'dihedral': False, 'direct_factorization': [['2.1', 1], ['972.455', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 9, 1, 2], [2, 27, 1, 4], [3, 1, 2, 1], [3, 2, 1, 1], [3, 2, 2, 1], [3, 6, 1, 2], [3, 6, 2, 1], [3, 12, 1, 2], [3, 12, 2, 1], [3, 18, 1, 3], [3, 36, 1, 3], [6, 1, 2, 1], [6, 2, 1, 1], [6, 2, 2, 1], [6, 6, 1, 2], [6, 6, 2, 1], [6, 9, 2, 2], [6, 12, 1, 2], [6, 12, 2, 1], [6, 18, 1, 3], [6, 27, 2, 4], [6, 36, 1, 3], [6, 54, 1, 12], [6, 54, 2, 4]], 'element_repr_type': 'GLFp', 'elementary': 1, 'eulerian_function': 163296, 'exponent': 6, 'exponents_of_order': [5, 3], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [[6, 0, 4], [12, 0, 2]], 'familial': False, 'frattini_label': '9.2', 'frattini_quotient': '216.171', 'hash': 2710, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 6, 'inner_gen_orders': [3, 2, 1, 3, 3, 1, 6, 3], 'inner_gens': [[24577758, 11534524, 2939606, 1017496, 25633100, 28816400, 6810638, 34663142], [7138662, 36728797, 2939606, 1017496, 8092874, 28816400, 24131645, 13939616], [24577758, 36728797, 2939606, 1017496, 25633100, 28816400, 6810638, 34663142], [24577758, 36728797, 2939606, 1017496, 8194004, 28816400, 5236001, 38862198], [24577758, 14414350, 2939606, 23121499, 25633100, 28816400, 32325881, 34663142], [24577758, 36728797, 2939606, 1017496, 25633100, 28816400, 6810638, 34663142], [24577758, 11534524, 2939606, 2651191, 8092874, 28816400, 6810638, 38915318], [24577758, 11222302, 2939606, 10990224, 25633100, 28816400, 12657380, 34663142]], 'inner_hash': 121, 'inner_nilpotent': False, 'inner_order': 324, 'inner_split': True, 'inner_tex': 'C_3^2:S_3^2', 'inner_used': [1, 2, 4, 5, 7, 8], 'irrC_degree': 6, 'irrQ_degree': 12, 'irrQ_dim': 12, 'irrR_degree': 12, 'irrep_stats': [[1, 8], [2, 20], [3, 16], [4, 8], [6, 20], [12, 6]], 'label': '1944.2710', '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': False, 'metacyclic': False, 'monomial': True, 'name': 'C2*C3^3:S3^2', 'ngens': 8, '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': 36, 'number_characteristic_subgroups': 30, 'number_conjugacy_classes': 78, 'number_divisions': 60, 'number_normal_subgroups': 69, 'number_subgroup_autclasses': 400, 'number_subgroup_classes': 884, 'number_subgroups': 9908, 'old_label': None, 'order': 1944, 'order_factorization_type': 33, 'order_stats': [[1, 1], [2, 127], [3, 242], [6, 1574]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': True, 'outer_exponent': 6, 'outer_gen_orders': [2, 2, 6], 'outer_gen_pows': [14408200, 14408200, 14408200], 'outer_gens': [[7138662, 36728797, 2939606, 1017496, 8092874, 28816400, 33855322, 13939616], [20201582, 36728797, 32864349, 15714108, 25633100, 28816400, 24131645, 34663142], [24577758, 25627796, 2939606, 18762443, 25633100, 28816400, 6810638, 34663142]], 'outer_group': '24.14', 'outer_hash': 14, 'outer_nilpotent': False, 'outer_order': 24, 'outer_permdeg': 7, 'outer_perms': [23, 127, 847], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2\\times D_6', 'pc_rank': 6, 'perfect': False, 'permutation_degree': 20, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 2], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 8], [2, 20], [4, 8], [6, 12], [12, 10], [24, 2]], 'representations': {'PC': {'code': 153901516914712113722945590587773731510072423608141933, 'gens': [1, 2, 3, 4, 6, 7], 'pres': [8, 2, 3, 3, 2, 3, 3, 2, 3, 65, 2883, 395, 91, 2884, 972, 10373, 605, 5742, 878, 166, 5023, 1383]}, 'GLFp': {'d': 4, 'p': 3, 'gens': [24577758, 36728797, 2939606, 1017496, 25633100, 28816400, 6810638, 34663142]}, 'Perm': {'d': 20, 'gens': [1093228705460575, 13874425523481600, 1, 4478784, 19630994903192190, 135165413461478400, 8891526, 263653312782182400]}}, 'schur_multiplier': [2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 2], 'solvability_type': 10, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_2\\times C_3^3:S_3^2', 'transitive_degree': 36, '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': '8.5', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 4, 'aut_exponent': 12, 'aut_gen_orders': [12, 12, 2, 6, 6, 6, 12], 'aut_gens': [[27895765165899480, 136007507152018824, 264307680506237791], [1776311427424854960, 1604905223641345104, 974048342169275401], [886320566834634768, 1950974782166199846, 278179758442822351], [2079979595265450991, 1464013730410852351, 331180062312839928], [899898749404867086, 1547833750661728440, 1500316259193829609], [1886416192140931441, 1407656805947181385, 1672590101234580774], [432050101798667071, 714161221920937488, 276150336453155545], [918478804111725031, 1490858901327822024, 712926327685239649]], 'aut_group': None, 'aut_hash': 8232036993248398305, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 93312, 'aut_permdeg': 432, 'aut_perms': 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'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 54, 2, 1], [2, 81, 1, 2], [2, 162, 2, 1], [3, 2, 1, 2], [3, 4, 1, 1], [3, 12, 2, 1], [3, 24, 1, 1], [3, 24, 2, 2], [3, 36, 2, 3], [3, 72, 1, 1], [3, 72, 4, 1], [4, 486, 2, 1], [6, 2, 1, 2], [6, 4, 1, 1], [6, 12, 2, 1], [6, 24, 1, 1], [6, 24, 2, 2], [6, 36, 2, 3], [6, 54, 4, 1], [6, 72, 1, 1], [6, 72, 4, 1], [6, 162, 1, 4], [6, 324, 1, 2], [6, 324, 4, 4], [12, 486, 4, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_3^2.C_3^4.D_4.C_2^4', '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': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^2', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 12, 'autcentquo_group': '23328.jh', 'autcentquo_hash': 1424162380239288937, 'autcentquo_nilpotent': False, 'autcentquo_order': 23328, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': '\\He_3^2:D_4:C_2^2', 'cc_stats': [[1, 1, 1], [2, 1, 1], [2, 54, 2], [2, 81, 2], [2, 162, 2], [3, 2, 2], [3, 4, 1], [3, 12, 2], [3, 24, 5], [3, 36, 6], [3, 72, 5], [4, 486, 2], [6, 2, 2], [6, 4, 1], [6, 12, 2], [6, 24, 5], [6, 36, 6], [6, 54, 4], [6, 72, 5], [6, 162, 4], [6, 324, 18], [12, 486, 4]], 'center_label': '2.1', 'center_order': 2, 'central_product': True, 'central_quotient': '5832.cu', 'commutator_count': 1, 'commutator_label': '1458.1576', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '3.1', '3.1', '3.1', '3.1', '3.1', '3.1'], 'composition_length': 10, 'conjugacy_classes_known': True, 'counter': 200, 'cyclic': False, 'derived_length': 4, 'dihedral': False, 'direct_factorization': [['2.1', 1], ['5832.cu', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 54, 1, 2], [2, 81, 1, 2], [2, 162, 1, 2], [3, 2, 1, 2], [3, 4, 1, 1], [3, 12, 1, 2], [3, 24, 1, 3], [3, 24, 2, 1], [3, 36, 1, 6], [3, 72, 1, 5], [4, 486, 1, 2], [6, 2, 1, 2], [6, 4, 1, 1], [6, 12, 1, 2], [6, 24, 1, 3], [6, 24, 2, 1], [6, 36, 1, 6], [6, 54, 2, 2], [6, 72, 1, 5], [6, 162, 1, 4], [6, 324, 1, 18], [12, 486, 2, 2]], 'element_repr_type': 'Perm', 'elementary': 1, 'eulerian_function': 5307120, 'exponent': 12, 'exponents_of_order': [6, 4], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [[12, 1, 2], [24, 0, 2], [24, 1, 2]], 'familial': False, 'frattini_label': '9.2', 'frattini_quotient': '1296.3576', 'hash': 7966846194014259031, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 12, 'inner_gen_orders': [6, 6, 6], 'inner_gens': [[27895765165899480, 700142500597375704, 1017984368595559831], [860123306189940054, 136007507152018824, 278326226637255727], [948272594830612926, 1601933836935890190, 264307680506237791]], 'inner_hash': 4547925521878059488, 'inner_nilpotent': False, 'inner_order': 5832, 'inner_split': True, 'inner_tex': '\\He_3^2:D_4', 'inner_used': [1, 2, 3], 'irrC_degree': 12, 'irrQ_degree': 12, 'irrQ_dim': 12, 'irrR_degree': 12, 'irrep_stats': [[1, 8], [2, 2], [4, 32], [8, 12], [12, 4], [18, 16], [24, 8]], 'label': '11664.hr', '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': False, 'metacyclic': False, 'monomial': True, 'name': 'C2*He3^2:D4', 'ngens': 3, '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, 0, 0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 43, 'number_characteristic_subgroups': 17, 'number_conjugacy_classes': 82, 'number_divisions': 76, 'number_normal_subgroups': 35, 'number_subgroup_autclasses': 1134, 'number_subgroup_classes': 2650, 'number_subgroups': 142235, 'old_label': None, 'order': 11664, 'order_factorization_type': 33, 'order_stats': [[1, 1], [2, 595], [3, 728], [4, 972], [6, 7424], [12, 1944]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': True, 'outer_exponent': 2, 'outer_gen_orders': [2, 2, 2, 2], 'outer_gen_pows': [0, 933612249010302894, 0, 0], 'outer_gens': [[73404667274365135, 1430044278556322880, 635783994312223807], [2126051662664581567, 1387697247690578808, 647919575725471495], [501980557627348615, 1951037551499396430, 611282376164533975], [501980557627348614, 1951037551499396431, 611282376164533974]], 'outer_group': '16.14', 'outer_hash': 14, 'outer_nilpotent': True, 'outer_order': 16, 'outer_permdeg': 8, 'outer_perms': [5040, 120, 6, 1], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^4', 'pc_rank': None, 'perfect': False, 'permutation_degree': 20, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 2], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 8], [2, 2], [4, 32], [8, 12], [12, 4], [18, 8], [24, 4], [36, 4], [48, 2]], 'representations': {'PC': {'code': '1177676132415137680895647952676181094482769089254277256453737649709580178995223691660501028315793149981667270645231585880551472191', 'gens': [1, 2, 4, 6, 7, 8, 9, 10], 'pres': [10, -2, -2, -2, -2, -3, 3, 3, 3, -3, 3, 114880, 45561, 51, 160922, 38902, 438723, 118573, 104103, 113, 511204, 187214, 65224, 8645, 8655, 1465, 10115, 9945, 184806, 1696, 5066, 632327, 158097, 172827, 2467, 699848, 174978, 259209, 64819]}, 'Perm': {'d': 20, 'gens': [27895765165899480, 136007507152018824, 264307680506237791]}}, 'schur_multiplier': [2, 2, 6], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 2], 'solvability_type': 11, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': 'C_2\\times \\He_3^2:D_4', 'transitive_degree': 36, 'wreath_data': None, 'wreath_product': False}
