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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': True, 'ambient': '4752.h', 'ambient_counter': 8, 'ambient_order': 4752, 'ambient_tex': 'D_{132}:C_{18}', 'central': False, 'central_factor': False, 'centralizer_order': 72, 'characteristic': False, 'core_order': 18, 'counter': 175, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '4752.h.132.h1.b1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': False, 'old_label': '132.h1.b1', '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': 132, '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': '36.5', 'subgroup_hash': 5, 'subgroup_order': 36, 'subgroup_tex': 'C_2\\times C_{18}', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '4752.h', 'aut_centralizer_order': 80, 'aut_label': '132.h1', 'aut_quo_index': None, 'aut_stab_index': 66, 'aut_weyl_group': '12.5', 'aut_weyl_index': 5280, 'centralizer': '66.f1.b1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['12.j1.b1', '44.d1.b1', '66.c1.b1', '66.f1.b1', '66.g1.a1'], 'contains': ['264.b1.a1', '264.g1.b1', '396.h1.b1'], 'core': '264.b1.a1', 'coset_action_label': None, 'count': 33, 'diagramx': None, 'generators': [24778026504, 2127406316, 8560511, 19522081488], 'label': '4752.h.132.h1.b1', 'mobius_quo': None, 'mobius_sub': None, 'normal_closure': '4.b1.b1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '33.a1.a1', 'old_label': '132.h1.b1', 'projective_image': '264.38', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '132.h1.b1', 'subgroup_fusion': None, 'weyl_group': '2.1'}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': True, 'abelian_quotient': '36.5', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 6, 'aut_gen_orders': [6, 6], 'aut_gens': [[1, 2], [19, 14], [19, 17]], 'aut_group': '36.12', 'aut_hash': 12, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 36, 'aut_permdeg': 8, 'aut_perms': [751, 5761], 'aut_phi_ratio': 3.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 3, 1], [3, 1, 2, 1], [6, 1, 6, 1], [9, 1, 6, 1], [18, 1, 18, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_6\\times S_3', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 6, 'autcent_group': '36.12', 'autcent_hash': 12, 'autcent_nilpotent': False, 'autcent_order': 36, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_6\\times S_3', '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, 3], [3, 1, 2], [6, 1, 6], [9, 1, 6], [18, 1, 18]], 'center_label': '36.5', 'center_order': 36, 'central_product': True, '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', '3.1', '3.1'], 'composition_length': 4, 'conjugacy_classes_known': True, 'counter': 5, 'cyclic': False, 'derived_length': 1, 'dihedral': False, 'direct_factorization': [['2.1', 2], ['9.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [3, 1, 2, 1], [6, 1, 2, 3], [9, 1, 6, 1], [18, 1, 6, 3]], 'element_repr_type': 'PC', 'elementary': 2, 'eulerian_function': 12, 'exponent': 18, 'exponents_of_order': [2, 2], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [], 'familial': False, 'frattini_label': '3.1', 'frattini_quotient': '12.5', 'hash': 5, 'hyperelementary': 2, 'inner_abelian': True, 'inner_cyclic': True, 'inner_exponent': 1, 'inner_gen_orders': [1, 1], 'inner_gens': [[1, 2], [1, 2]], '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, 36]], 'label': '36.5', 'linC_count': 216, 'linC_degree': 2, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 7, 'linQ_degree_count': 6, 'linQ_dim': 7, 'linQ_dim_count': 6, 'linR_count': 18, 'linR_degree': 3, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C2*C18', 'ngens': 4, 'nilpotency_class': 1, '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': 6, 'number_characteristic_subgroups': 6, 'number_conjugacy_classes': 36, 'number_divisions': 12, 'number_normal_subgroups': 15, 'number_subgroup_autclasses': 9, 'number_subgroup_classes': 15, 'number_subgroups': 15, 'old_label': None, 'order': 36, 'order_factorization_type': 22, 'order_stats': [[1, 1], [2, 3], [3, 2], [6, 6], [9, 6], [18, 18]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 6, 'outer_gen_orders': [6, 6], 'outer_gen_pows': [0, 0], 'outer_gens': [[19, 14], [19, 17]], 'outer_group': '36.12', 'outer_hash': 12, 'outer_nilpotent': False, 'outer_order': 36, 'outer_permdeg': 8, 'outer_perms': [751, 5761], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_6\\times S_3', 'pc_rank': 2, 'perfect': False, 'permutation_degree': 13, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 9], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 4], [6, 4]], 'representations': {'PC': {'code': 319587, 'gens': [1, 2], 'pres': [4, -2, -2, -3, -3, 21, 46]}, 'GLFp': {'d': 2, 'p': 19, 'gens': [123463, 68592]}, 'Perm': {'d': 13, 'gens': [479001600, 3628800, 357120, 80884]}}, 'schur_multiplier': [2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 18], 'solvability_type': 1, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_2\\times C_{18}', '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': '72.18', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 660, 'aut_gen_orders': [30, 60, 30, 30, 30, 20, 60], 'aut_gens': [[20085218158, 3378821867, 158006], [5631369956, 1063703468, 43033609], [6569931495, 3441392746, 61408357], [11137597909, 21524346294, 49040219], [8071630041, 4442525157, 53148375], [688278810, 20585784533, 45940046], [8196771472, 20398072121, 52666417], [19709793673, 23902035630, 51939113]], 'aut_group': None, 'aut_hash': 8446468809167927362, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 63360, 'aut_permdeg': 140, 'aut_perms': [2796989794723060621287333003814897861881898028899043760989458456674387656048001923767892020704778643462930741518682557721325487772730808879832654963884084241042536524305987922717538466042249485193356295395118936320171642643885132043362343736, 11243351564314250459052369544593217242993430295366276471171037439311271126080610278101932006743339904155389571721520975877897358270161984140054868926373604010832612010219006342128802731089911631626319728714244316490405876757973027742194558448, 10341163202951751848766892108932230393231722891699212535532269285618710079836180509917766716045207700313350025173675159273706739581849898482495827200988603132622849214033772429789252312404199722958059698308236342764500527784961760244313652254, 13241632587907499504188964141646694333691239459878441133675300024602810313836693614687269296004502671047096252044049189924848800212937889669593981673281872097128522459282232070228370607391540541846062180259639663431167331184002769201649348006, 4785294253833984377805475080250001306113725209249609811413747752853797455027498125405366129719050462926862728099468748727734351155612563509510453372692584198639843917388764369461260982155797360877920957875512217198858180047286436175574909677, 2389408163975657385310478389261234214964510907671964248189496692934836824682140361051802934706024477991132493195439612107817770832014672674964454220944585186248941961285347572543073339434098706374310322440286633857089019866396281596534842209, 1385187938858170914316133128620910960918639686949933272607290087691777613768570971836159380652749667200404622996333991717300020663689108665644785218973554421142026852650702644082303046109054344721422570613085243303357756359285476944671109241], 'aut_phi_ratio': 44.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 2, 1, 1], [2, 66, 2, 1], [3, 1, 2, 1], [3, 2, 1, 1], [3, 2, 2, 1], [4, 1, 2, 1], [4, 2, 1, 1], [4, 66, 2, 1], [6, 1, 2, 1], [6, 2, 1, 1], [6, 2, 2, 3], [6, 2, 4, 1], [6, 66, 4, 1], [9, 1, 6, 1], [9, 2, 6, 1], [11, 2, 5, 1], [12, 1, 4, 1], [12, 2, 2, 3], [12, 2, 4, 2], [12, 66, 4, 1], [18, 1, 6, 1], [18, 2, 6, 2], [18, 2, 12, 1], [18, 66, 12, 1], [22, 2, 5, 1], [22, 2, 10, 1], [33, 2, 10, 2], [33, 2, 20, 1], [36, 1, 12, 1], [36, 2, 6, 1], [36, 2, 12, 2], [36, 66, 12, 1], [44, 2, 10, 2], [66, 2, 10, 2], [66, 2, 20, 3], [66, 2, 40, 1], [99, 2, 30, 1], [99, 2, 60, 1], [132, 2, 20, 4], [132, 2, 40, 2], [198, 2, 30, 1], [198, 2, 60, 2], [198, 2, 120, 1], [396, 2, 60, 2], [396, 2, 120, 2]], 'aut_supersolvable': True, 'aut_tex': 'C_{66}.C_{30}.C_2^5', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 6, 'autcent_group': '48.52', 'autcent_hash': 52, 'autcent_nilpotent': True, 'autcent_order': 48, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^3\\times C_6', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 330, 'autcentquo_group': '1320.142', 'autcentquo_hash': 142, 'autcentquo_nilpotent': False, 'autcentquo_order': 1320, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'D_6\\times F_{11}', 'cc_stats': [[1, 1, 1], [2, 1, 1], [2, 2, 1], [2, 66, 2], [3, 1, 2], [3, 2, 3], [4, 1, 2], [4, 2, 1], [4, 66, 2], [6, 1, 2], [6, 2, 11], [6, 66, 4], [9, 1, 6], [9, 2, 6], [11, 2, 5], [12, 1, 4], [12, 2, 14], [12, 66, 4], [18, 1, 6], [18, 2, 24], [18, 66, 12], [22, 2, 15], [33, 2, 40], [36, 1, 12], [36, 2, 30], [36, 66, 12], [44, 2, 20], [66, 2, 120], [99, 2, 90], [132, 2, 160], [198, 2, 270], [396, 2, 360]], 'center_label': '36.2', 'center_order': 36, 'central_product': True, 'central_quotient': '132.9', 'commutator_count': 1, 'commutator_label': '66.4', 'complements_known': True, 'complete': False, 'complex_characters_known': False, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '3.1', '3.1', '3.1', '11.1'], 'composition_length': 8, 'conjugacy_classes_known': True, 'counter': 8, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['528.149', 1], ['9.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 2, 1, 1], [2, 66, 1, 2], [3, 1, 2, 1], [3, 2, 1, 1], [3, 2, 2, 1], [4, 1, 2, 1], [4, 2, 1, 1], [4, 66, 1, 2], [6, 1, 2, 1], [6, 2, 1, 1], [6, 2, 2, 5], [6, 66, 2, 2], [9, 1, 6, 1], [9, 2, 6, 1], [11, 2, 5, 1], [12, 1, 4, 1], [12, 2, 2, 3], [12, 2, 4, 2], [12, 66, 2, 2], [18, 1, 6, 1], [18, 2, 6, 4], [18, 66, 6, 2], [22, 2, 5, 1], [22, 2, 10, 1], [33, 2, 10, 2], [33, 2, 20, 1], [36, 1, 12, 1], [36, 2, 6, 1], [36, 2, 12, 2], [36, 66, 6, 2], [44, 2, 10, 2], [66, 2, 10, 2], [66, 2, 20, 5], [99, 2, 30, 1], [99, 2, 60, 1], [132, 2, 20, 4], [132, 2, 40, 2], [198, 2, 30, 1], [198, 2, 60, 4], [396, 2, 60, 2], [396, 2, 120, 2]], 'element_repr_type': 'GLFp', 'elementary': 1, 'eulerian_function': 471744, 'exponent': 396, 'exponents_of_order': [4, 3, 1], 'factors_of_aut_order': [2, 3, 5, 11], 'factors_of_order': [2, 3, 11], 'faithful_reps': [[2, 0, 240]], 'familial': False, 'frattini_label': '6.2', 'frattini_quotient': '792.134', 'hash': 3521362650719687116, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 66, 'inner_gen_orders': [22, 66, 2], 'inner_gens': [[20085218158, 3378821867, 60098257], [20085218158, 3378821867, 22434867], [1564269646, 7821346679, 158006]], 'inner_hash': 9, 'inner_nilpotent': False, 'inner_order': 132, 'inner_split': True, 'inner_tex': 'D_{66}', 'inner_used': [1, 2, 3], 'irrC_degree': 2, 'irrQ_degree': 240, 'irrQ_dim': 240, 'irrR_degree': None, 'irrep_stats': [[1, 72], [2, 1170]], 'label': '4752.h', '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': 'D132:C18', '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], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 66, 'number_characteristic_subgroups': 81, 'number_conjugacy_classes': 1242, 'number_divisions': 76, 'number_normal_subgroups': 105, 'number_subgroup_autclasses': 208, 'number_subgroup_classes': 272, 'number_subgroups': 2320, 'old_label': None, 'order': 4752, 'order_factorization_type': 321, 'order_stats': [[1, 1], [2, 135], [3, 8], [4, 136], [6, 288], [9, 18], [11, 10], [12, 296], [18, 846], [22, 30], [33, 80], [36, 864], [44, 40], [66, 240], [99, 180], [132, 320], [198, 540], [396, 720]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 30, 'outer_gen_orders': [2, 2, 2, 2, 30], 'outer_gen_pows': [62570774, 62570774, 62570774, 62570774, 5819082193], 'outer_gens': [[23276327632, 7320780743, 158006], [23276327632, 17019250599, 158006], [23276327632, 17519816535, 158006], [7946488521, 3378821867, 158006], [23401469218, 4379954490, 52666417]], 'outer_group': '480.1213', 'outer_hash': 1213, 'outer_nilpotent': True, 'outer_order': 480, 'outer_permdeg': 18, 'outer_perms': [40279680, 355687428096000, 1307674368000, 362880, 6227026593], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^4\\times C_{30}', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 31, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 2, 9], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 8], [2, 12], [4, 5], [6, 8], [8, 4], [10, 4], [12, 4], [20, 8], [24, 3], [40, 5], [60, 4], [80, 4], [120, 4], [240, 3]], 'representations': {'PC': {'code': '95912109259417760139608623166727838487080807171684600119001', 'gens': [1, 2, 6], 'pres': [8, -2, -2, -3, -3, -11, -2, -2, -3, 41505, 41, 962, 90, 3843, 123, 14404, 209093, 141, 221766, 166, 202759]}, 'GLFp': {'d': 2, 'p': 397, 'gens': [20085218158, 3378821867, 158006]}, 'Perm': {'d': 31, 'gens': [265895334133363481073122580104280, 18598592312775672096228942937705, 558294921665220406931422093113387]}}, 'schur_multiplier': [2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 18], 'solvability_type': 7, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'D_{132}:C_{18}', 'transitive_degree': 792, 'wreath_data': None, 'wreath_product': False}
