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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': False, 'ambient': '1458.1361', 'ambient_counter': 1361, 'ambient_order': 1458, 'ambient_tex': '(C_3\\times C_9^2):C_6', 'central': False, 'central_factor': False, 'centralizer_order': 1, 'characteristic': False, 'core_order': 1, 'counter': 114, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '1458.1361.81.i1.b1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': False, 'minimal_normal': False, 'nilpotent': False, 'normal': False, 'old_label': '81.i1.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': 81, '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': '18.1', 'subgroup_hash': 1, 'subgroup_order': 18, 'subgroup_tex': 'D_9', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '1458.1361', 'aut_centralizer_order': 12, 'aut_label': '81.i1', 'aut_quo_index': None, 'aut_stab_index': 729, 'aut_weyl_group': '54.6', 'aut_weyl_index': 8748, 'centralizer': '1458.a1.a1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['27.d1.b1', '27.d1.e1', '27.d1.h1', '27.h1.a1'], 'contains': ['162.l1.b1', '243.e1.a1'], 'core': '1458.a1.a1', 'coset_action_label': None, 'count': 81, 'diagramx': [9757, -1, 6315, -1, 111, -1, 9551, -1], 'generators': [3, 702, 486], 'label': '1458.1361.81.i1.b1', 'mobius_quo': None, 'mobius_sub': 0, 'normal_closure': '3.a1.a1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '81.i1.b1', 'old_label': '81.i1.b1', 'projective_image': '1458.1361', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '81.i1.b1', 'subgroup_fusion': None, 'weyl_group': '18.1'}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': False, 'abelian_quotient': '2.1', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 18, 'aut_gen_orders': [6, 9], 'aut_gens': [[1, 2], [1, 4], [15, 2]], 'aut_group': '54.6', 'aut_hash': 6, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 54, 'aut_permdeg': 9, 'aut_perms': [100930, 59573], 'aut_phi_ratio': 9.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 9, 1, 1], [3, 2, 1, 1], [9, 2, 3, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_9:C_6', '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': '54.6', 'autcentquo_hash': 6, 'autcentquo_nilpotent': False, 'autcentquo_order': 54, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'C_9:C_6', 'cc_stats': [[1, 1, 1], [2, 9, 1], [3, 2, 1], [9, 2, 3]], 'center_label': '1.1', 'center_order': 1, 'central_product': False, 'central_quotient': '18.1', 'commutator_count': 1, 'commutator_label': '9.1', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '3.1', '3.1'], 'composition_length': 3, 'conjugacy_classes_known': True, 'counter': 1, 'cyclic': False, 'derived_length': 2, 'dihedral': True, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 9, 1, 1], [3, 2, 1, 1], [9, 2, 3, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 3, 'exponent': 18, 'exponents_of_order': [2, 1], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [[2, 1, 3]], 'familial': True, 'frattini_label': '3.1', 'frattini_quotient': '6.1', 'hash': 1, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 18, 'inner_gen_orders': [2, 9], 'inner_gens': [[1, 16], [5, 2]], 'inner_hash': 1, 'inner_nilpotent': False, 'inner_order': 18, 'inner_split': True, 'inner_tex': 'D_9', 'inner_used': [1, 2], 'irrC_degree': 2, 'irrQ_degree': 6, 'irrQ_dim': 6, 'irrR_degree': 2, 'irrep_stats': [[1, 2], [2, 4]], 'label': '18.1', 'linC_count': 3, 'linC_degree': 2, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 6, 'linQ_degree_count': 1, 'linQ_dim': 6, 'linQ_dim_count': 1, 'linR_count': 3, 'linR_degree': 2, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'D9', 'ngens': 3, 'nilpotency_class': -1, 'nilpotent': False, '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': 6, 'number_divisions': 4, 'number_normal_subgroups': 4, 'number_subgroup_autclasses': 6, 'number_subgroup_classes': 6, 'number_subgroups': 16, 'old_label': None, 'order': 18, 'order_factorization_type': 22, 'order_stats': [[1, 1], [2, 9], [3, 2], [9, 6]], 'outer_abelian': True, 'outer_cyclic': True, 'outer_equivalence': False, 'outer_exponent': 3, 'outer_gen_orders': [3], 'outer_gen_pows': [1], 'outer_gens': [[1, 4]], 'outer_group': '3.1', 'outer_hash': 1, 'outer_nilpotent': True, 'outer_order': 3, 'outer_permdeg': 3, 'outer_perms': [3], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_3', 'pc_rank': 2, 'perfect': False, 'permutation_degree': 9, 'pgroup': 0, 'primary_abelian_invariants': [2], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 2], [2, 1], [6, 1]], 'representations': {'PC': {'code': 106355, 'gens': [1, 2], 'pres': [3, -2, -3, -3, 97, 22, 110]}, 'GLZ': {'b': 3, 'd': 6, 'gens': [58301504795026264, 69483174513730069]}, 'Lie': [{'d': 2, 'q': 8, 'gens': [577, 613], 'family': 'CSOMinus'}], 'GLFp': {'d': 2, 'p': 17, 'gens': [38240, 78609]}, 'Perm': {'d': 9, 'gens': [5759, 357120, 80884]}}, 'schur_multiplier': [], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2], 'solvability_type': 6, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'D_9', 'transitive_degree': 9, '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': '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': [6, 18, 9, 18], 'aut_gens': [[1, 6, 18, 162], [713, 120, 126, 360], [1241, 12, 576, 690], [91, 60, 72, 660], [815, 114, 612, 438]], 'aut_group': None, 'aut_hash': 225813103164118894, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 472392, 'aut_permdeg': 297, 'aut_perms': [3395515274504520500622767564500478413646411682718085110490532881147257189626168293877210880804378258040171815258370197796726311393138762462633267834608499004059445224001227365315552971021910449428055420588838039476901050498932977247760715032374549244123612929426874367143625067289747117333824966654072580862659602129056682218990461600796303495668945841672576935482241877364971629544086893098867042213877929145449822631880822061535275895938670356162463969522811803485821670743854473059425230524851766658846647032735555659829216693713569727615912820998573068671992790108969561957356699565276749112972606669280, 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4387604475031261051561775798454550738963868096575459675586734009769451064795594023732655335737981747937555096245148569967684889554840400322275853593153881433752598758656401062764609395298031497625215396542753459125635506975960022559274265679390383433796237680077891799026116550510642357419250641264424251339392911479837650332698028830374724700417439685932395502776684841086853676972963820285818939347820701355641410236883636227098670304836228186421668619497997577133704923537310925049597450288142314624509353944334880167102497260949637256310122101419297203067102576897563290500896873483495131733128761655833, 10775604665427165591531473720857919387250790469595335539144943959156578426283714756203423223889842347351072182058766003438988680872133823705469051315690899965121343268469925267124427619628012164698158758925970713433630622460797391690679377021059589396950173443795608394899431858116618596928472615829329928453834204683673825881079591282353264596152802988360247241384482083872302563628876526778526250854412464412840004927004768777925121811310148307380363571062898320433861276544102609352430097947617864245550926167074326471466856272021783969178097382547205757532103119467992830702552736989268205824021878844306], 'aut_phi_ratio': 972.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 243, 1, 1], [3, 2, 1, 1], [3, 2, 3, 1], [3, 6, 1, 1], [3, 6, 2, 1], [3, 27, 2, 1], [3, 54, 2, 1], [3, 54, 6, 1], [6, 243, 2, 1], [9, 6, 3, 1], [9, 6, 6, 1], [9, 6, 27, 1]], 'aut_supersolvable': True, 'aut_tex': '(C_3^4\\times C_9).C_3^4.C_2^3', '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': 18, 'autcentquo_group': None, 'autcentquo_hash': 225813103164118894, 'autcentquo_nilpotent': False, 'autcentquo_order': 472392, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'C_3^4.C_3^4.C_3^2.C_2^3', 'cc_stats': [[1, 1, 1], [2, 243, 1], [3, 2, 4], [3, 6, 3], [3, 27, 2], [3, 54, 8], [6, 243, 2], [9, 6, 36]], 'center_label': '1.1', 'center_order': 1, 'central_product': False, 'central_quotient': '1458.1361', 'commutator_count': 1, 'commutator_label': '243.31', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '3.1', '3.1', '3.1', '3.1', '3.1', '3.1'], 'composition_length': 7, 'conjugacy_classes_known': True, 'counter': 1361, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 243, 1, 1], [3, 2, 1, 4], [3, 6, 1, 3], [3, 27, 2, 1], [3, 54, 2, 4], [6, 243, 2, 1], [9, 6, 3, 12]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 3276, 'exponent': 18, 'exponents_of_order': [6, 1], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [], 'familial': False, 'frattini_label': '27.2', 'frattini_quotient': '54.13', 'hash': 1361, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 18, 'inner_gen_orders': [6, 3, 9, 9], 'inner_gens': [[1, 12, 522, 1332], [13, 6, 18, 162], [1117, 6, 18, 162], [451, 6, 18, 162]], 'inner_hash': 1361, 'inner_nilpotent': False, 'inner_order': 1458, 'inner_split': True, 'inner_tex': '(C_3\\times C_9^2):C_6', 'inner_used': [1, 2, 3, 4], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 6], [2, 12], [6, 39]], 'label': '1458.1361', 'linC_count': 243, 'linC_degree': 8, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 20, 'linQ_degree_count': 27, 'linQ_dim': 20, 'linQ_dim_count': 27, 'linR_count': 81, 'linR_degree': 8, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': '(C3*C9^2):C6', 'ngens': 7, 'nilpotency_class': -1, 'nilpotent': False, 'normal_counts': [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': 13, 'number_conjugacy_classes': 57, 'number_divisions': 27, 'number_normal_subgroups': 26, 'number_subgroup_autclasses': 87, 'number_subgroup_classes': 173, 'number_subgroups': 4674, 'old_label': None, 'order': 1458, 'order_factorization_type': 51, 'order_stats': [[1, 1], [2, 243], [3, 512], [6, 486], [9, 216]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 6, 'outer_gen_orders': [6, 6, 3, 3], 'outer_gen_pows': [0, 0, 0, 0], 'outer_gens': [[5, 12, 1008, 1134], [5, 6, 1116, 174], [1, 114, 18, 162], [1, 6, 504, 1278]], 'outer_group': '324.117', 'outer_hash': 117, 'outer_nilpotent': False, 'outer_order': 324, 'outer_permdeg': 12, 'outer_perms': [8366667, 225198024, 1493304, 121204080], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_3^3:D_6', 'pc_rank': 4, 'perfect': False, 'permutation_degree': 30, 'pgroup': 0, 'primary_abelian_invariants': [2, 3], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 2], [2, 6], [4, 4], [6, 3], [18, 12]], 'representations': {'PC': {'code': 13452200397454095135002843271003256124855487, 'gens': [1, 3, 4, 6], 'pres': [7, 2, 3, 3, 3, 3, 3, 3, 14, 254, 14619, 7066, 108, 3784, 55949, 24582, 166, 52926, 14566]}, 'Perm': {'d': 30, 'gens': [68190099475615990881757550185, 9801602711421779188912864826160, 1230977384804728391721412975443, 19275866692321771034950234603107, 28434220498943166973290737299440, 18056337863174793418982720326560, 19275866692321771034950234603104]}}, 'schur_multiplier': [3, 9], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [6], 'solvability_type': 7, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': '(C_3\\times C_9^2):C_6', 'transitive_degree': 81, 'wreath_data': None, 'wreath_product': False}
