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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': True, 'ambient': '25992.bj', 'ambient_counter': 36, 'ambient_order': 25992, 'ambient_tex': 'D_{19}^2:C_{18}', 'central': False, 'central_factor': False, 'centralizer_order': 36, 'characteristic': False, 'core_order': 1, 'counter': 87, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '25992.bj.2166.a1.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': False, 'old_label': '2166.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': 2166, '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': '12.5', 'subgroup_hash': 5, 'subgroup_order': 12, 'subgroup_tex': 'C_2\\times C_6', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '25992.bj', 'aut_centralizer_order': None, 'aut_label': '2166.a1', 'aut_quo_index': None, 'aut_stab_index': None, 'aut_weyl_group': None, 'aut_weyl_index': None, 'centralizer': '722.a1.a1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['114.a1.a1', '722.a1.a1', '1083.a1.a1'], 'contains': ['4332.a1.a1', '4332.b1.a1', '6498.a1.a1'], 'core': '25992.a1.a1', 'coset_action_label': None, 'count': 361, 'diagramx': [4358, -1, 6403, -1, 6736, -1, 8447, -1], 'generators': [14499, 10620, 6], 'label': '25992.bj.2166.a1.a1', 'mobius_quo': None, 'mobius_sub': -1, 'normal_closure': '6.a1.a1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '361.a1.a1', 'old_label': '2166.a1.a1', 'projective_image': '25992.bj', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '2166.a1.a1', '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': '12.5', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 6, 'aut_gen_orders': [2, 6], 'aut_gens': [[1, 2], [6, 9], [6, 11]], 'aut_group': '12.4', 'aut_hash': 4, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 12, 'aut_permdeg': 5, 'aut_perms': [6, 31], '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]], 'aut_supersolvable': True, 'aut_tex': 'D_6', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 6, 'autcent_group': '12.4', 'autcent_hash': 4, 'autcent_nilpotent': False, 'autcent_order': 12, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'D_6', '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]], 'center_label': '12.5', 'center_order': 12, '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'], 'composition_length': 3, 'conjugacy_classes_known': True, 'counter': 5, 'cyclic': False, 'derived_length': 1, 'dihedral': False, 'direct_factorization': [['2.1', 2], ['3.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [3, 1, 2, 1], [6, 1, 2, 3]], 'element_repr_type': 'PC', 'elementary': 2, 'eulerian_function': 4, 'exponent': 6, 'exponents_of_order': [2, 1], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [], 'familial': False, 'frattini_label': '1.1', '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, 12]], 'label': '12.5', 'linC_count': 24, 'linC_degree': 2, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 3, 'linQ_degree_count': 6, 'linQ_dim': 3, 'linQ_dim_count': 6, 'linR_count': 6, 'linR_degree': 3, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C2*C6', 'ngens': 3, 'nilpotency_class': 1, 'nilpotent': True, 'normal_counts': [0, 0, 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': 12, 'number_divisions': 8, 'number_normal_subgroups': 10, 'number_subgroup_autclasses': 6, 'number_subgroup_classes': 10, 'number_subgroups': 10, 'old_label': None, 'order': 12, 'order_factorization_type': 22, 'order_stats': [[1, 1], [2, 3], [3, 2], [6, 6]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 6, 'outer_gen_orders': [2, 6], 'outer_gen_pows': [0, 0], 'outer_gens': [[6, 9], [6, 11]], 'outer_group': '12.4', 'outer_hash': 4, 'outer_nilpotent': False, 'outer_order': 12, 'outer_permdeg': 5, 'outer_perms': [6, 31], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'D_6', 'pc_rank': 2, 'perfect': False, 'permutation_degree': 7, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 3], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 4]], 'representations': {'PC': {'code': 273, 'gens': [1, 2], 'pres': [3, -2, -2, -3, 16]}, 'GLZ': {'b': 3, 'd': 3, 'gens': [3362, 16507]}, 'GLFp': {'d': 2, 'p': 7, 'gens': [1030, 2064]}, 'Perm': {'d': 7, 'gens': [720, 24, 4]}}, 'schur_multiplier': [2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 6], '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_6', 'transitive_degree': 12, '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': '36.5', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 3, 'aut_exponent': 1368, 'aut_gen_orders': [72, 72, 18], 'aut_gens': [[1, 18, 72, 1368], [17443, 8874, 23256, 24984], [24355, 21762, 20520, 23976], [1981, 4446, 648, 14688]], 'aut_group': None, 'aut_hash': 6148826751490991875, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 51984, 'aut_permdeg': 437, 'aut_perms': [408544647600404212256818943708568826830909134353352796093806874356933065903780751693868076991804389662423853780276910719388098822194417166243278187358100799159135896664095187585010417198584917832668772162596209541865913693204488834586017797813294359179028353439453918008915011473480752692310661894181807939688715200668381591933299967624071839809981573058917910638425208010090878768314083487416921969046516144502766553641466977239505063933929561801544084170563725776884225437635904697027237758054983626664454641870854943387375868726504728796034522998432552617588466290003703105905237872444856102137661141558723689207655523847523433198958756903737671976814152683658767962390461819887436855248040568411679631539243693739899355807549577093141668122399965814011785867727162073467976862837651574173186819977130457059500087421593438365856640201266960748993876511944950454034556031417080451514844419537913172387751519689440127581327993024557237709595506910511333466976788075, 398550895071167894073004124136273910237884331604406285180367222657694067839411610434728974530203518455664436405456551387628111365552955733225632342631546833170768255419246905612081020025738928345606985235000438906144962945515356847910634621891713180030941230247536125192780192606904976264449419195117086299641627254071485846303808647123986445820505017702874163839415767054917151501870005855542014157442043445077775786197860245948839060102334968171022577598438425214047574657314198547176919584226086337682994607471250257201234908441416434582990979313282839322959169810030194915884139364998161720751564213501486111319362589664778903698489268954500546532476702115686826705642673183856433616840629968532053157058941963557984388466103836080097879884585171637219290157828859376866907670925426535598078944236061083658036970352369495776479930962561777350822418404439203698179532948917535382658097225978879455988900253497453290059816922574843217859289241373128954671298680136, 226172693307271001229789048032236725308208876794409195037133138309039087986523187484715945261778670004368795969427538580069545404399678834518350520309505949135726996649430102923384634026384118489441423024966986314507226231123585923372067482268039288466481824387860807396732633435936766036876370953260025236098631518110766305364739902314175399078007978678748956072081392734087641426724856465821482371178838795937806128866137449259298205425547900120351577178697844718994758233921298306590467854909929649804334039919858415591160062037015212213202254529187097241674732370680781736488184126467515336723733961815413807960661732758967365002883074565969576066559540037488748555903799879440941139305899293718284731096286264854525392800142659156076278295009197074066930595811484553182096943467941893926523896919806575983628623990363134480461435743277462453918292332996275587626838261743165379055301458501990332147447297784281401579951542201100957295395460332085110187026230900], 'aut_phi_ratio': 6.333333333333333, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 38, 2, 1], [2, 361, 1, 1], [3, 361, 1, 2], [4, 722, 1, 1], [6, 361, 1, 2], [6, 722, 2, 2], [9, 361, 1, 6], [12, 722, 1, 2], [18, 361, 1, 6], [18, 722, 2, 6], [19, 36, 2, 1], [19, 72, 2, 2], [36, 722, 1, 6], [38, 684, 2, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_{19}^2.C_{36}.C_2^2', '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': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 1368, 'autcentquo_group': None, 'autcentquo_hash': 6148826751490991875, 'autcentquo_nilpotent': False, 'autcentquo_order': 51984, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'C_{19}^2.C_{36}.C_2^2', 'cc_stats': [[1, 1, 1], [2, 38, 2], [2, 361, 1], [3, 361, 2], [4, 722, 1], [6, 361, 2], [6, 722, 4], [9, 361, 6], [12, 722, 2], [18, 361, 6], [18, 722, 12], [19, 36, 2], [19, 72, 4], [36, 722, 6], [38, 684, 2]], 'center_label': '1.1', 'center_order': 1, 'central_product': False, 'central_quotient': '25992.bj', 'commutator_count': 1, 'commutator_label': '722.4', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '3.1', '3.1', '19.1', '19.1'], 'composition_length': 7, 'conjugacy_classes_known': True, 'counter': 36, 'cyclic': False, 'derived_length': 3, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 38, 1, 2], [2, 361, 1, 1], [3, 361, 2, 1], [4, 722, 1, 1], [6, 361, 2, 1], [6, 722, 2, 2], [9, 361, 6, 1], [12, 722, 2, 1], [18, 361, 6, 1], [18, 722, 6, 2], [19, 36, 1, 2], [19, 72, 1, 4], [36, 722, 6, 1], [38, 684, 1, 2]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 4320, 'exponent': 684, 'exponents_of_order': [3, 2, 2], 'factors_of_aut_order': [2, 3, 19], 'factors_of_order': [2, 3, 19], 'faithful_reps': [[36, 1, 4], [72, 1, 4]], 'familial': False, 'frattini_label': '1.1', 'frattini_quotient': '25992.bj', 'hash': 3331103429084464448, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 684, 'inner_gen_orders': [18, 4, 19, 19], 'inner_gens': [[1, 24246, 19584, 17784], [24229, 18, 6768, 2016], [7849, 20682, 72, 1368], [9577, 738, 72, 1368]], 'inner_hash': 3331103429084464448, 'inner_nilpotent': False, 'inner_order': 25992, 'inner_split': False, 'inner_tex': 'D_{19}^2:C_{18}', 'inner_used': [1, 2], 'irrC_degree': 36, 'irrQ_degree': 36, 'irrQ_dim': 36, 'irrR_degree': 36, 'irrep_stats': [[1, 36], [2, 9], [36, 4], [72, 4]], 'label': '25992.bj', 'linC_count': 4, 'linC_degree': 36, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 36, 'linQ_degree_count': 4, 'linQ_dim': 36, 'linQ_dim_count': 4, 'linR_count': 4, 'linR_degree': 36, 'maximal_subgroups_known': True, 'metabelian': False, 'metacyclic': False, 'monomial': True, 'name': 'D19^2:C18', 'ngens': 7, '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], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 40, 'number_characteristic_subgroups': 13, 'number_conjugacy_classes': 53, 'number_divisions': 23, 'number_normal_subgroups': 19, 'number_subgroup_autclasses': 63, 'number_subgroup_classes': 102, 'number_subgroups': 11664, 'old_label': None, 'order': 25992, 'order_factorization_type': 321, 'order_stats': [[1, 1], [2, 437], [3, 722], [4, 722], [6, 3610], [9, 2166], [12, 1444], [18, 10830], [19, 360], [36, 4332], [38, 1368]], 'outer_abelian': True, 'outer_cyclic': True, 'outer_equivalence': False, 'outer_exponent': 2, 'outer_gen_orders': [2], 'outer_gen_pows': [22162], 'outer_gens': [[14491, 8442, 6840, 17568]], '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': 4, 'perfect': False, 'permutation_degree': 38, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 9], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 5], [4, 1], [6, 4], [12, 1], [36, 4], [72, 4]], 'representations': {'PC': {'code': '40982482990919089663600766416184331648319524383722826096904233817165304008350517658522744464148979', 'gens': [1, 4, 6, 7], 'pres': [7, -2, -3, -3, -2, -2, -19, 19, 14, 50, 678891, 225046, 90821, 80, 255784, 53561, 53148, 822533, 25716, 5563, 15818, 1545, 871422, 569785, 122912, 5515, 33550]}, 'Perm': {'d': 38, 'gens': [13933637039504757885110485832213800225661400, 285677560510209893698935868588048431477329440]}}, 'schur_multiplier': [2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 18], 'solvability_type': 11, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': 'D_{19}^2:C_{18}', 'transitive_degree': 38, 'wreath_data': None, 'wreath_product': False}
