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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': False, 'ambient': '20736.ex', 'ambient_counter': 128, 'ambient_order': 20736, 'ambient_tex': '(S_3^2\\times A_4^2):C_2^2', 'central': False, 'central_factor': False, 'centralizer_order': 18, 'characteristic': True, 'core_order': 2592, 'counter': 27, 'cyclic': False, 'direct': False, 'hall': 0, 'label': '20736.ex.8.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': False, 'minimal': False, 'minimal_normal': False, 'nilpotent': False, 'normal': True, 'old_label': '8.a1', 'outer_equivalence': True, 'perfect': False, 'proper': True, 'quotient': '8.3', 'quotient_Agroup': False, 'quotient_abelian': False, 'quotient_cyclic': False, 'quotient_hash': 3, 'quotient_metabelian': True, 'quotient_nilpotent': True, 'quotient_order': 8, 'quotient_simple': False, 'quotient_solvable': True, 'quotient_supersolvable': True, 'quotient_tex': 'D_4', 'simple': False, 'solvable': True, 'special_labels': [], 'split': True, 'standard_generators': False, 'stem': False, 'subgroup': None, 'subgroup_hash': 2992611077904635648, 'subgroup_order': 2592, 'subgroup_tex': 'C_2^4.C_3^4.C_2', 'supersolvable': False, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '20736.ex', 'aut_centralizer_order': None, 'aut_label': '8.a1', 'aut_quo_index': None, 'aut_stab_index': None, 'aut_weyl_group': None, 'aut_weyl_index': None, 'centralizer': '1152.A', 'complements': ['2592.A', '2592.B', '2592.C', '2592.D'], 'conjugacy_class_count': 1, 'contained_in': ['4.a1', '4.e1', '4.f1'], 'contains': ['16.a1', '24.a1', '24.b1', '24.u1', '24.v1', '24.w1', '24.x1', '24.y1', '24.z1', '24.ba1', '24.bb1', '24.bc1', '32.b1'], 'core': '8.a1', 'coset_action_label': None, 'count': 1, 'diagramx': [4460, 7518, 3311, 8412], 'generators': [374659087824, 5413204702080, 87181922310, 3024, 19203609600, 19766073600, 13200, 1, 9335309722704], 'label': '20736.ex.8.a1', 'mobius_quo': 0, 'mobius_sub': None, 'normal_closure': '8.a1', 'normal_contained_in': ['4.a1'], 'normal_contains': ['16.a1', '24.a1', '24.b1', '72.b1'], 'normalizer': '1.a1', 'old_label': '8.a1', 'projective_image': '10368.fm', 'quotient_action_image': '8.3', 'quotient_action_kernel': '1.1', 'quotient_action_kernel_order': 1, 'quotient_fusion': None, 'short_label': '8.a1', 'subgroup_fusion': None, 'weyl_group': '1152.157517'}
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label None does not appear in gps_groups
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gps_groups • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '24.15', 'all_subgroups_known': False, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 3, 'aut_exponent': 12, 'aut_gen_orders': [6, 12, 6, 4, 6, 4, 4], 'aut_gens': [[1315425854455, 2890375863361, 1482989316486], [2622621234606, 8213959727407, 7086948788310], [1601824901449, 3152832089280, 3158656666711], [6918826145430, 1850023942807, 1340250105606], [1495051111830, 8395980342006, 2648403486001], [2622621236065, 6801387642727, 6832039739904], [1582222486614, 6801387639150, 7877787700441], [6907283293734, 7877184943944, 2821399273561]], 'aut_group': None, 'aut_hash': 8422280042728975113, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 165888, 'aut_permdeg': 172, 'aut_perms': [5269004349345360718919445140142841178540062189635967320089659734350158774703673879008073855119362371541761082096201865684309312786732325213459322580739818808839909349905270888906138749889101990810836354244563127983618025331341678597961071174662426291477415739380163771385220120954283053369142464821234334567224, 173032968311709546285582212614314793690417433969326251662518165195029293806097543981928381161500059895889091516469704075146444441326641537525882340918593315038067650447271279893376921672020516846619313955174332388415986812846188484199394558410180110913751405962420063255536477942849157259585264633738313619010942, 182171393606653777241996463207662838658750167186604062660970158008194534958376884782920403315141741005993028709652677108919922624866320661549171195963534391904572771398353155930035960406302619127524316035245072806865554128478815070422978001008657090654836363881197600692126536992037888498563010407207350006316385, 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122014765157944968840203992206415077971652709096548282723353240140859711201677696104713631824568225293695895220779527344760572384437911630519662639361239093041708407252669901012724931307797656066195695630449104146794629890134832998891204305356187493772702718588949629870552751128473632698462323648429553196287255], 'aut_phi_ratio': 24.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 6, 1, 2], [2, 6, 2, 1], [2, 9, 1, 4], [2, 18, 4, 1], [2, 54, 1, 2], [2, 54, 2, 1], [2, 72, 2, 1], [2, 81, 1, 2], [3, 4, 1, 2], [3, 8, 2, 1], [3, 16, 2, 1], [3, 16, 4, 1], [3, 32, 1, 1], [3, 32, 2, 1], [3, 64, 2, 3], [3, 128, 1, 1], [4, 216, 2, 2], [4, 648, 2, 1], [6, 4, 1, 2], [6, 8, 2, 1], [6, 12, 2, 3], [6, 16, 2, 1], [6, 16, 4, 1], [6, 24, 1, 2], [6, 24, 2, 2], [6, 24, 8, 1], [6, 32, 1, 1], [6, 32, 2, 1], [6, 36, 1, 4], [6, 36, 4, 1], [6, 48, 4, 2], [6, 48, 8, 1], [6, 64, 2, 3], [6, 72, 2, 2], [6, 72, 8, 1], [6, 96, 2, 2], [6, 96, 4, 2], [6, 108, 2, 1], [6, 128, 1, 1], [6, 144, 2, 3], [6, 144, 8, 1], [6, 192, 4, 2], [6, 216, 2, 2], [6, 288, 1, 2], [6, 288, 4, 1], [6, 576, 4, 1], [12, 432, 2, 1], [12, 864, 4, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_3^2.A_4^2.C_2.C_2^6', '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': '41472.kr', 'autcentquo_hash': 5131433462079402963, 'autcentquo_nilpotent': False, 'autcentquo_order': 41472, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'A_4^2:(C_6^2:D_4)', 'cc_stats': [[1, 1, 1], [2, 1, 1], [2, 6, 4], [2, 9, 4], [2, 18, 4], [2, 54, 4], [2, 72, 2], [2, 81, 2], [3, 4, 2], [3, 8, 2], [3, 16, 6], [3, 32, 3], [3, 64, 6], [3, 128, 1], [4, 216, 4], [4, 648, 2], [6, 4, 2], [6, 8, 2], [6, 12, 6], [6, 16, 6], [6, 24, 14], [6, 32, 3], [6, 36, 8], [6, 48, 16], [6, 64, 6], [6, 72, 12], [6, 96, 12], [6, 108, 2], [6, 128, 1], [6, 144, 14], [6, 192, 8], [6, 216, 4], [6, 288, 6], [6, 576, 4], [12, 432, 2], [12, 864, 4]], 'center_label': '2.1', 'center_order': 2, 'central_product': True, 'central_quotient': '10368.fm', 'commutator_count': 1, 'commutator_label': '864.4714', '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', '3.1', '3.1', '3.1'], 'composition_length': 12, 'conjugacy_classes_known': True, 'counter': 128, 'cyclic': False, 'derived_length': 3, 'dihedral': False, 'direct_factorization': [['10368.fm', 1], ['2.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 6, 1, 4], [2, 9, 1, 4], [2, 18, 1, 4], [2, 54, 1, 4], [2, 72, 1, 2], [2, 81, 1, 2], [3, 4, 1, 2], [3, 8, 2, 1], [3, 16, 2, 3], [3, 32, 1, 1], [3, 32, 2, 1], [3, 64, 2, 3], [3, 128, 1, 1], [4, 216, 1, 4], [4, 648, 1, 2], [6, 4, 1, 2], [6, 8, 2, 1], [6, 12, 1, 6], [6, 16, 2, 3], [6, 24, 1, 2], [6, 24, 2, 6], [6, 32, 1, 1], [6, 32, 2, 1], [6, 36, 1, 8], [6, 48, 2, 8], [6, 64, 2, 3], [6, 72, 2, 6], [6, 96, 2, 6], [6, 108, 1, 2], [6, 128, 1, 1], [6, 144, 1, 2], [6, 144, 2, 6], [6, 192, 2, 4], [6, 216, 2, 2], [6, 288, 1, 2], [6, 288, 2, 2], [6, 576, 2, 2], [12, 432, 1, 2], [12, 864, 2, 2]], 'element_repr_type': 'Perm', 'elementary': 1, 'eulerian_function': None, 'exponent': 12, 'exponents_of_order': [8, 4], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [[12, 0, 8], [12, 1, 4], [24, 0, 2], [24, 1, 1], [36, 1, 4]], 'familial': False, 'frattini_label': '1.1', 'frattini_quotient': '20736.ex', 'hash': 1224363201338985062, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 12, 'inner_gen_orders': [12, 12, 6], 'inner_gens': [[1315425854455, 6801387629311, 7864727644080], [1495490197345, 2890375863361, 8395102157910], [6819153875689, 6569989587607, 1482989316486]], 'inner_hash': 6068583894714021913, 'inner_nilpotent': False, 'inner_order': 10368, 'inner_split': True, 'inner_tex': '(S_3\\times A_4)\\wr C_2', 'inner_used': [1, 2], 'irrC_degree': 12, 'irrQ_degree': 12, 'irrQ_dim': 12, 'irrR_degree': 12, 'irrep_stats': [[1, 24], [2, 30], [4, 48], [6, 24], [8, 6], [9, 8], [12, 24], [18, 2], [24, 6], [36, 8]], 'label': '20736.ex', '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': None, 'name': '(S3^2*A4^2):C2^2', '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, 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': 79, 'number_characteristic_subgroups': 46, 'number_conjugacy_classes': 180, 'number_divisions': 120, 'number_normal_subgroups': 72, 'number_subgroup_autclasses': 4858, 'number_subgroup_classes': 11907, 'number_subgroups': 686252, 'old_label': None, 'order': 20736, 'order_factorization_type': 33, 'order_stats': [[1, 1], [2, 655], [3, 728], [4, 2160], [6, 12872], [12, 4320]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': True, 'outer_exponent': 2, 'outer_gen_orders': [2, 2, 2, 2], 'outer_gen_pows': [25986199680, 0, 0, 0], 'outer_gens': [[1320208612855, 3152832092647, 6571470489984], [1338932857975, 8369115971047, 6558533817984], [1315425854455, 2890375866726, 1482989318785], [1315425854454, 2890375866727, 1482989318784]], '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': 7, 'perfect': False, 'permutation_degree': 16, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 2, 3], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 8], [2, 14], [4, 20], [6, 8], [8, 22], [9, 8], [12, 16], [16, 2], [18, 2], [24, 10], [36, 8], [48, 2]], 'representations': {'PC': {'code': '3235232651243231414839076009409834969907257276217996397060509785667506970388343795869256914563417240481252829065625470129350953201307466957774883498314489324600441990942419163726441912367741928801129556', 'gens': [1, 3, 5, 7, 9, 11, 12], 'pres': [12, 2, 3, 2, 2, 2, 3, 2, 3, 2, 3, 2, 2, 24, 417098, 93434, 98, 180291, 180004, 169216, 198508, 10240, 172, 1168133, 670481, 148061, 24521, 447558, 350802, 98814, 5082, 27774, 14682, 246, 59935, 4651, 1384136, 629876, 362912, 38924, 85592, 37004, 320, 483849, 80673, 34605, 399178, 1026454, 87970, 29770, 2612747, 373271, 18203, 40247]}, 'Perm': {'d': 16, 'gens': [1315425854455, 2890375863361, 1482989316486]}}, 'schur_multiplier': [2, 2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 6], 'solvability_type': 17, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 128, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': '(S_3^2\\times A_4^2):C_2^2', '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': '4.2', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 4, 'aut_gen_orders': [2, 4], 'aut_gens': [[1, 2], [1, 6], [3, 2]], 'aut_group': '8.3', 'aut_hash': 3, 'aut_nilpotency_class': 2, 'aut_nilpotent': True, 'aut_order': 8, 'aut_permdeg': 4, 'aut_perms': [5, 9], 'aut_phi_ratio': 2.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 2, 2, 1], [4, 2, 1, 1]], 'aut_supersolvable': True, 'aut_tex': 'D_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': True, 'autcentquo_cyclic': True, 'autcentquo_exponent': 2, 'autcentquo_group': '2.1', 'autcentquo_hash': 1, 'autcentquo_nilpotent': True, 'autcentquo_order': 2, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'C_2', 'cc_stats': [[1, 1, 1], [2, 1, 1], [2, 2, 2], [4, 2, 1]], 'center_label': '2.1', 'center_order': 2, 'central_product': False, 'central_quotient': '4.2', 'commutator_count': 1, 'commutator_label': '2.1', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1'], 'composition_length': 3, 'conjugacy_classes_known': True, 'counter': 3, 'cyclic': False, 'derived_length': 2, 'dihedral': True, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 2, 1, 2], [4, 2, 1, 1]], 'element_repr_type': 'PC', 'elementary': 2, 'eulerian_function': 3, 'exponent': 4, 'exponents_of_order': [3], 'factors_of_aut_order': [2], 'factors_of_order': [2], 'faithful_reps': [[2, 1, 1]], 'familial': True, 'frattini_label': '2.1', 'frattini_quotient': '4.2', 'hash': 3, 'hyperelementary': 2, 'inner_abelian': True, 'inner_cyclic': False, 'inner_exponent': 2, 'inner_gen_orders': [2, 2], 'inner_gens': [[1, 6], [5, 2]], 'inner_hash': 2, 'inner_nilpotent': True, 'inner_order': 4, 'inner_split': True, 'inner_tex': 'C_2^2', 'inner_used': [1, 2], 'irrC_degree': 2, 'irrQ_degree': 2, 'irrQ_dim': 2, 'irrR_degree': 2, 'irrep_stats': [[1, 4], [2, 1]], 'label': '8.3', 'linC_count': 1, 'linC_degree': 2, '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': 'D4', 'ngens': 2, 'nilpotency_class': 2, 'nilpotent': True, 'normal_counts': [0, 0, 0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 4, 'number_characteristic_subgroups': 4, 'number_conjugacy_classes': 5, 'number_divisions': 5, 'number_normal_subgroups': 6, 'number_subgroup_autclasses': 6, 'number_subgroup_classes': 8, 'number_subgroups': 10, 'old_label': None, 'order': 8, 'order_factorization_type': 3, 'order_stats': [[1, 1], [2, 5], [4, 2]], 'outer_abelian': True, 'outer_cyclic': True, 'outer_equivalence': False, 'outer_exponent': 2, 'outer_gen_orders': [2], 'outer_gen_pows': [2], 'outer_gens': [[3, 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': 2, 'perfect': False, 'permutation_degree': 4, 'pgroup': 2, 'primary_abelian_invariants': [2, 2], 'quasisimple': False, 'rank': 2, 'rational': True, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 1]], 'representations': {'PC': {'code': 294, 'gens': [1, 2], 'pres': [3, -2, 2, -2, 37, 16]}, 'GLZ': {'b': 3, 'd': 2, 'gens': [14, 46]}, 'Lie': [{'d': 2, 'q': 3, 'gens': [29, 56, 24], 'family': 'COPlus'}, {'d': 1, 'q': 4, 'gens': [7, 16, 1], 'family': 'ASigmaL'}], 'GLFp': {'d': 2, 'p': 3, 'gens': [12, 55, 56]}, 'Perm': {'d': 4, 'gens': [6, 16, 7]}}, 'schur_multiplier': [2], 'semidirect_product': True, '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': 'D_4', 'transitive_degree': 4, 'wreath_data': ['C_2', 'C_2', '2T1'], 'wreath_product': True}
