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gps_subgroup_search • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'ambient': '5832.mu', 'ambient_counter': 333, 'ambient_order': 5832, 'ambient_tex': 'C_9:S_3\\wr C_3', 'central': False, 'central_factor': False, 'centralizer_order': 1, 'characteristic': False, 'core_order': 108, 'counter': 13, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '5832.mu.9.b1.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': False, 'metacyclic': False, 'minimal': False, 'minimal_normal': False, 'nilpotent': False, 'normal': False, 'old_label': '9.b1.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': 9, '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': '648.705', 'subgroup_hash': 705, 'subgroup_order': 648, 'subgroup_tex': 'S_3\\wr C_3', 'supersolvable': False, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '5832.mu', 'aut_centralizer_order': None, 'aut_label': '9.b1', 'aut_quo_index': None, 'aut_stab_index': None, 'aut_weyl_group': None, 'aut_weyl_index': None, 'centralizer': '5832.a1.a1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['3.b1.a1'], 'contains': ['18.m1.a1', '27.d1.a1', '36.o1.a1', '243.a1.a1'], 'core': '54.a1.a1', 'coset_action_label': None, 'count': 9, 'diagramx': [4162, -1, 3259, -1, 2803, -1, 2563, -1], 'generators': [3, 648, 216, 2, 1944, 54, 3564], 'label': '5832.mu.9.b1.a1', 'mobius_quo': None, 'mobius_sub': 0, 'normal_closure': '1.a1.a1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '9.b1.a1', 'old_label': '9.b1.a1', 'projective_image': '5832.mu', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '9.b1.a1', 'subgroup_fusion': None, 'weyl_group': '648.705'}
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gps_groups • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '6.2', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 4, 'aut_exponent': 36, 'aut_gen_orders': [2, 2, 3, 2, 2, 3, 3, 3], 'aut_gens': [[5065, 276480, 5041, 25, 4, 240, 80640], [5065, 276480, 5041, 25, 3, 144, 45360], [5065, 138690, 5064, 25, 240, 4, 80640], [5065, 276480, 25, 5064, 240, 80640, 4], [5065, 281544, 5041, 25, 3, 144, 80640], [5065, 276505, 5041, 25, 3, 240, 45360], [5069, 321844, 5045, 29, 4, 240, 80640], [85705, 357264, 85681, 25, 4, 240, 80640], [5305, 276723, 5041, 265, 4, 240, 80640]], 'aut_group': '1296.3490', 'aut_hash': 3490, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 1296, 'aut_permdeg': 81, 'aut_perms': [2780498122124859583833982519703092666664439975734538803591670151793166265726608596855869962860977993769659832365137567830, 268204293860183192026966690836636005894376738649176850484942156842932016720449244061209131162661935231702445368850064282, 4984672228134725557818045061466850293514936754643093371853862060631218735388203114042745374174981497506948944648249960962, 998047136611168967926428046490756120552854701437084364293650258556722303864406273479668862723236606667468797205232418012, 3283111348192806761113959627864059891005952454054851927058476543462864545884257780063754733891254616680348284607164220198, 2782392227437696684393194687838188984268925371198282382071157132095541070196022845963331267330463352038467001791095747249, 4520801118441418720905452156098611860022102492015043328561173572320132172770102856655684513599025440393454100435940511333, 1641468095932356816230661860554462302226103330876285686100074209152181023678845921785413510914115464239984937257152642623], 'aut_phi_ratio': 6.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 9, 1, 1], [2, 27, 1, 2], [3, 6, 1, 1], [3, 8, 1, 1], [3, 12, 1, 1], [3, 36, 2, 1], [6, 18, 2, 1], [6, 36, 1, 1], [6, 54, 1, 1], [6, 108, 2, 1], [9, 72, 2, 1]], 'aut_supersolvable': False, 'aut_tex': 'S_3\\wr S_3', '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': 36, 'autcentquo_group': '1296.3490', 'autcentquo_hash': 3490, 'autcentquo_nilpotent': False, 'autcentquo_order': 1296, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'S_3\\wr S_3', 'cc_stats': [[1, 1, 1], [2, 9, 1], [2, 27, 2], [3, 6, 1], [3, 8, 1], [3, 12, 1], [3, 36, 2], [6, 18, 2], [6, 36, 1], [6, 54, 1], [6, 108, 2], [9, 72, 2]], 'center_label': '1.1', 'center_order': 1, 'central_product': False, 'central_quotient': '648.705', 'commutator_count': 1, 'commutator_label': '108.40', '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'], 'composition_length': 7, 'conjugacy_classes_known': True, 'counter': 705, 'cyclic': False, 'derived_length': 3, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 9, 1, 1], [2, 27, 1, 2], [3, 6, 1, 1], [3, 8, 1, 1], [3, 12, 1, 1], [3, 36, 2, 1], [6, 18, 1, 2], [6, 36, 1, 1], [6, 54, 1, 1], [6, 108, 2, 1], [9, 72, 2, 1]], 'element_repr_type': 'Perm', 'elementary': 1, 'eulerian_function': 156, 'exponent': 18, 'exponents_of_order': [4, 3], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [[6, 1, 4], [8, 0, 2], [8, 1, 1], [12, 1, 2]], 'familial': False, 'frattini_label': '1.1', 'frattini_quotient': '648.705', 'hash': 705, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 18, 'inner_gen_orders': [2, 3, 2, 2, 3, 3, 3], 'inner_gens': [[5065, 276480, 5041, 25, 3, 144, 45360], [5065, 276480, 25, 5064, 240, 80640, 4], [5065, 276505, 5041, 25, 3, 240, 45360], [5065, 281544, 5041, 25, 3, 144, 80640], [5066, 357123, 5042, 26, 4, 240, 80640], [5161, 276628, 5041, 121, 4, 240, 80640], [40345, 322080, 40321, 25, 4, 240, 80640]], 'inner_hash': 705, 'inner_nilpotent': False, 'inner_order': 648, 'inner_split': True, 'inner_tex': 'S_3\\wr C_3', 'inner_used': [1, 2, 3, 5], 'irrC_degree': 6, 'irrQ_degree': 6, 'irrQ_dim': 6, 'irrR_degree': 6, 'irrep_stats': [[1, 6], [3, 2], [6, 4], [8, 3], [12, 2]], 'label': '648.705', 'linC_count': 4, 'linC_degree': 6, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 6, 'linQ_degree_count': 4, 'linQ_dim': 6, 'linQ_dim_count': 4, 'linR_count': 4, 'linR_degree': 6, 'maximal_subgroups_known': True, 'metabelian': False, 'metacyclic': False, 'monomial': True, 'name': 'S3wrC3', '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], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 13, 'number_characteristic_subgroups': 7, 'number_conjugacy_classes': 17, 'number_divisions': 14, 'number_normal_subgroups': 7, 'number_subgroup_autclasses': 68, 'number_subgroup_classes': 76, 'number_subgroups': 1208, 'old_label': None, 'order': 648, 'order_factorization_type': 33, 'order_stats': [[1, 1], [2, 63], [3, 98], [6, 342], [9, 144]], 'outer_abelian': True, 'outer_cyclic': True, 'outer_equivalence': False, 'outer_exponent': 2, 'outer_gen_orders': [2], 'outer_gen_pows': [0], 'outer_gens': [[5065, 138690, 5064, 25, 240, 4, 80640]], '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': 5, 'perfect': False, 'permutation_degree': 9, 'pgroup': 0, 'primary_abelian_invariants': [2, 3], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 2], [2, 2], [3, 2], [6, 4], [8, 1], [12, 2], [16, 1]], 'representations': {'PC': {'code': 16681187849555582041798990330483990139667211797550580903, 'gens': [1, 3, 4, 6, 7], 'pres': [7, -2, -3, -2, 2, -3, 3, 3, 14, 884, 387, 4203, 3622, 297, 80, 2524, 7571, 298, 18149, 1020, 530, 1182, 1777, 3548, 1791]}, 'GLZ': {'b': 3, 'd': 6, 'gens': [24970029429726835, 80608918917500176]}, 'GLFp': {'d': 4, 'p': 3, 'gens': [23554882, 14409982, 14520385, 10313546, 26213243, 26492364, 19221658]}, 'Perm': {'d': 9, 'gens': [5065, 276480, 5041, 25, 4, 240, 80640]}}, 'schur_multiplier': [2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [6], 'solvability_type': 11, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': 'S_3\\wr C_3', 'transitive_degree': 9, 'wreath_data': ['108.c1.a1', '216.d1.a1', '648.a1.a1', '3T1'], 'wreath_product': True}
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gps_groups • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '6.2', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 3, 'aut_exponent': 18, 'aut_gen_orders': [3, 6, 2], 'aut_gens': [[1, 6, 108, 648, 1944], [1909, 5820, 1026, 216, 1080], [3355, 4566, 3780, 648, 4536], [517, 2910, 1836, 1296, 3888]], 'aut_group': None, 'aut_hash': 6436928098408918977, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 34992, 'aut_permdeg': 189, 'aut_perms': [22088484261670653300655631658979494913321074930241449686988948433891424520353473269520421835011417939921228946750065763854725260162523954245067575942566188057521324924809391574336120199444446675146724416331885944039498578692249366602687314144706589405816958910690065434206093107435814259596329754792528113293069401274772177076918288316447200190879241, 9331246833004316731893566066725766996279183379622242894150068337241079075365516688523374533622561884867527485274216113804993568843259882105877931436520247689081836280819357237736909795010754423067194585207784407465537670582865940432541595658042510984114025992336766881117038691661162765629171784022917265947663478588009057947188248522140459438647533, 38394136633479505132556737948927476975704830814663669495292960841053561712597987059712111179667867396818304639871676138061895410579782362886225150065620735415611651558685565748119970706772301473535122618238195214910368307616165295575145028927643989260091717979284194684779756540761466499991611881939354663653509616057069626381002075172792512274596003], 'aut_phi_ratio': 18.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 27, 1, 1], [2, 81, 1, 1], [2, 243, 1, 1], [3, 2, 1, 1], [3, 6, 1, 1], [3, 8, 1, 1], [3, 8, 2, 1], [3, 12, 1, 2], [3, 24, 1, 1], [3, 108, 1, 2], [6, 54, 1, 2], [6, 108, 1, 1], [6, 162, 1, 2], [6, 324, 1, 1], [6, 972, 1, 2], [9, 6, 1, 1], [9, 12, 3, 1], [9, 24, 2, 1], [9, 24, 3, 1], [9, 216, 1, 4], [9, 216, 2, 2], [18, 54, 3, 1], [18, 108, 3, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_3^4.C_6^2.C_6.C_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': 18, 'autcentquo_group': None, 'autcentquo_hash': 6436928098408918977, 'autcentquo_nilpotent': False, 'autcentquo_order': 34992, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'C_3^4.C_6^2.C_6.C_2', 'cc_stats': [[1, 1, 1], [2, 27, 1], [2, 81, 1], [2, 243, 1], [3, 2, 1], [3, 6, 1], [3, 8, 3], [3, 12, 2], [3, 24, 1], [3, 108, 2], [6, 54, 2], [6, 108, 1], [6, 162, 2], [6, 324, 1], [6, 972, 2], [9, 6, 1], [9, 12, 3], [9, 24, 5], [9, 216, 8], [18, 54, 3], [18, 108, 3]], 'center_label': '1.1', 'center_order': 1, 'central_product': False, 'central_quotient': '5832.mu', 'commutator_count': 1, 'commutator_label': '972.797', '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', '3.1'], 'composition_length': 9, 'conjugacy_classes_known': True, 'counter': 333, 'cyclic': False, 'derived_length': 3, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 27, 1, 1], [2, 81, 1, 1], [2, 243, 1, 1], [3, 2, 1, 1], [3, 6, 1, 1], [3, 8, 1, 3], [3, 12, 1, 2], [3, 24, 1, 1], [3, 108, 2, 1], [6, 54, 1, 2], [6, 108, 1, 1], [6, 162, 1, 2], [6, 324, 1, 1], [6, 972, 2, 1], [9, 6, 1, 1], [9, 12, 3, 1], [9, 24, 1, 2], [9, 24, 3, 1], [9, 216, 2, 4], [18, 54, 3, 1], [18, 108, 3, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 312, 'exponent': 18, 'exponents_of_order': [6, 3], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [[12, 1, 6], [24, 1, 5]], 'familial': False, 'frattini_label': '3.1', 'frattini_quotient': '1944.3881', 'hash': 2690700921166908840, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 18, 'inner_gen_orders': [6, 18, 6, 3, 3], 'inner_gens': [[1, 5088, 4698, 216, 1080], [643, 6, 5076, 648, 4536], [919, 3462, 108, 1296, 3888], [1081, 6, 1404, 648, 1944], [3457, 5190, 3996, 648, 1944]], 'inner_hash': 2690700921166908840, 'inner_nilpotent': False, 'inner_order': 5832, 'inner_split': True, 'inner_tex': 'C_9:S_3\\wr C_3', 'inner_used': [1, 2], 'irrC_degree': 12, 'irrQ_degree': 24, 'irrQ_dim': 24, 'irrR_degree': 12, 'irrep_stats': [[1, 6], [2, 3], [3, 2], [6, 9], [8, 9], [12, 10], [24, 6]], 'label': '5832.mu', 'linC_count': 22, 'linC_degree': 12, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 12, 'linQ_degree_count': 4, 'linQ_dim': 12, 'linQ_dim_count': 4, 'linR_count': 22, 'linR_degree': 12, 'maximal_subgroups_known': True, 'metabelian': False, 'metacyclic': False, 'monomial': True, 'name': 'C9:S3wrC3', 'ngens': 9, '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], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 33, 'number_characteristic_subgroups': 14, 'number_conjugacy_classes': 45, 'number_divisions': 31, 'number_normal_subgroups': 14, 'number_subgroup_autclasses': 338, 'number_subgroup_classes': 386, 'number_subgroups': 23952, 'old_label': None, 'order': 5832, 'order_factorization_type': 33, 'order_stats': [[1, 1], [2, 351], [3, 296], [6, 2808], [9, 1890], [18, 486]], 'outer_abelian': True, 'outer_cyclic': True, 'outer_equivalence': False, 'outer_exponent': 6, 'outer_gen_orders': [6], 'outer_gen_pows': [0], 'outer_gens': [[1, 1362, 108, 648, 1944]], 'outer_group': '6.2', 'outer_hash': 2, 'outer_nilpotent': True, 'outer_order': 6, 'outer_permdeg': 5, 'outer_perms': [28], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_6', 'pc_rank': 5, 'perfect': False, 'permutation_degree': 18, 'pgroup': 0, 'primary_abelian_invariants': [2, 3], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 2], [2, 3], [3, 2], [4, 1], [6, 6], [8, 3], [12, 4], [16, 3], [18, 1], [24, 3], [36, 2], [72, 1]], 'representations': {'PC': {'code': '635036211090475380405253675097860495231038312999017381109815734417919551939421326739397604005403319', 'gens': [1, 3, 6, 8, 9], 'pres': [9, 2, 3, 2, 3, 3, 2, 3, 3, 3, 18, 137378, 20018, 74, 63075, 59196, 138, 3244, 253697, 112280, 45707, 3434, 158, 326598, 40839, 4560, 15559, 186640, 916, 87488, 113741, 61262, 2969]}, 'Perm': {'d': 18, 'gens': [355694134160527, 756010184924893]}}, 'schur_multiplier': [2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [6], 'solvability_type': 11, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': 'C_9:S_3\\wr C_3', 'transitive_degree': 36, 'wreath_data': None, 'wreath_product': False}
