Formats: - HTML - YAML - JSON - 2026-07-18T12:50:28.456178
  • gps_subgroup_search • Show schema
    {'Agroup': True, 'Zgroup': False, 'abelian': False, 'ambient': '34992.lc', 'ambient_counter': 289, 'ambient_order': 34992, 'ambient_tex': 'C_3^5.S_3^2:C_2^2', 'central': False, 'central_factor': False, 'centralizer_order': 1, 'characteristic': False, 'core_order': 162, 'counter': 57, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '34992.lc.18.j1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': False, 'minimal': False, 'minimal_normal': False, 'nilpotent': False, 'normal': False, 'old_label': '18.j1', 'outer_equivalence': True, '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': 18, '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': '1944.3577', 'subgroup_hash': 3577, 'subgroup_order': 1944, 'subgroup_tex': 'C_9:S_3^3', 'supersolvable': True, 'sylow': 0}
  • gps_subgroup_data • Show schema
    {'ambient': '34992.lc', 'aut_centralizer_order': None, 'aut_label': '18.j1', 'aut_quo_index': None, 'aut_stab_index': None, 'aut_weyl_group': None, 'aut_weyl_index': None, 'centralizer': '34992.a1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['6.c1'], 'contains': ['36.o1', '36.p1', '36.r1', '36.ba1', '36.bk1', '54.f1', '54.j1', '54.k1'], 'core': '216.a1', 'coset_action_label': None, 'count': 18, 'diagramx': [9281, -1, 873, -1], 'generators': [48, 484528640370361896985, 168248014290637757567280, 82315575567582061023115, 561, 114720089607229997727302, 10252505913702136363440, 7], 'label': '34992.lc.18.j1', 'mobius_quo': None, 'mobius_sub': 0, 'normal_closure': '2.a1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '18.j1', 'old_label': '18.j1', 'projective_image': '34992.lc', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '18.j1', 'subgroup_fusion': None, 'weyl_group': '1944.3577'}
  • gps_groups • Show schema
    {'Agroup': True, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '8.5', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 4, 'aut_exponent': 36, 'aut_gen_orders': [12, 18, 6, 18], 'aut_gens': [[1, 2, 12, 72, 216], [150, 169, 1052, 72, 432], [3, 2, 1914, 24, 216], [102, 129, 980, 72, 1728], [27, 129, 549, 8, 432]], 'aut_group': None, 'aut_hash': 1698336522601683736, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 69984, 'aut_permdeg': 81, 'aut_perms': [2270117784466659645076119078179676462691584048747009597432362972816428654012067666591369724385144468614691046315963988778, 988611319025703334923216807888348252433386750425404533075680105517804362048312970182047919504584808378795327662947255811, 506356353642527312151871766280232879274496134343178487487109299711141517937722339477815436887361396415366179728984133079, 1326197743249510584328519166826848990383619424293389710820926555856486914258927810877818665559732352808281185931269770245], 'aut_phi_ratio': 108.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 9, 3, 1], [2, 27, 3, 1], [2, 243, 1, 1], [3, 2, 1, 1], [3, 2, 3, 1], [3, 4, 3, 2], [3, 8, 1, 1], [3, 8, 2, 1], [3, 8, 3, 1], [6, 18, 3, 2], [6, 36, 3, 1], [6, 54, 6, 1], [6, 108, 3, 1], [9, 2, 3, 1], [9, 4, 9, 1], [9, 8, 6, 1], [9, 8, 9, 1], [18, 18, 9, 1], [18, 36, 9, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_3^3:C_2^2.D_6\\times D_9:C_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': None, 'autcentquo_hash': 1698336522601683736, 'autcentquo_nilpotent': False, 'autcentquo_order': 69984, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'C_3^3:C_2^2.D_6\\times D_9:C_3', 'cc_stats': [[1, 1, 1], [2, 9, 3], [2, 27, 3], [2, 243, 1], [3, 2, 4], [3, 4, 6], [3, 8, 6], [6, 18, 6], [6, 36, 3], [6, 54, 6], [6, 108, 3], [9, 2, 3], [9, 4, 9], [9, 8, 15], [18, 18, 9], [18, 36, 9]], 'center_label': '1.1', 'center_order': 1, 'central_product': False, 'central_quotient': '1944.3577', 'commutator_count': 1, 'commutator_label': '243.61', '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'], 'composition_length': 8, 'conjugacy_classes_known': True, 'counter': 3577, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 9, 1, 3], [2, 27, 1, 3], [2, 243, 1, 1], [3, 2, 1, 4], [3, 4, 1, 6], [3, 8, 1, 6], [6, 18, 1, 6], [6, 36, 1, 3], [6, 54, 1, 6], [6, 108, 1, 3], [9, 2, 3, 1], [9, 4, 3, 3], [9, 8, 3, 5], [18, 18, 3, 3], [18, 36, 3, 3]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 21504, 'exponent': 18, 'exponents_of_order': [5, 3], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [[8, 1, 6]], 'familial': False, 'frattini_label': '3.1', 'frattini_quotient': '648.734', 'hash': 3577, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 18, 'inner_gen_orders': [2, 6, 6, 3, 9], 'inner_gens': [[1, 10, 12, 144, 216], [5, 2, 60, 144, 216], [1, 26, 12, 144, 1728], [145, 146, 156, 72, 216], [1, 2, 444, 72, 216]], 'inner_hash': 3577, 'inner_nilpotent': False, 'inner_order': 1944, 'inner_split': True, 'inner_tex': 'C_9:S_3^3', 'inner_used': [1, 2, 3, 4, 5], 'irrC_degree': 8, 'irrQ_degree': 24, 'irrQ_dim': 24, 'irrR_degree': 8, 'irrep_stats': [[1, 8], [2, 28], [4, 30], [8, 21]], 'label': '1944.3577', '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': 'C9:S3^3', 'ngens': 8, '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], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 22, 'number_characteristic_subgroups': 11, 'number_conjugacy_classes': 87, 'number_divisions': 57, 'number_normal_subgroups': 68, 'number_subgroup_autclasses': 238, 'number_subgroup_classes': 786, 'number_subgroups': 18968, 'old_label': None, 'order': 1944, 'order_factorization_type': 33, 'order_stats': [[1, 1], [2, 351], [3, 80], [6, 864], [9, 162], [18, 486]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': True, 'outer_exponent': 6, 'outer_gen_orders': [6, 6], 'outer_gen_pows': [0, 0], 'outer_gens': [[1, 151, 61, 4, 1512], [6, 79, 46, 48, 216]], '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': 5, 'perfect': False, 'permutation_degree': 18, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 2], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 8], [2, 16], [4, 12], [6, 4], [8, 6], [12, 6], [24, 5]], 'representations': {'PC': {'code': 13214942893176105084366695186150547778245180901935, 'gens': [1, 2, 4, 6, 7], 'pres': [8, -2, -2, -3, -2, -3, -3, -3, -3, 161, 41, 194, 971, 91, 972, 6917, 3469, 605, 8094, 222, 6943]}, 'Perm': {'d': 18, 'gens': [47089213925040, 5183, 7, 45360, 325, 435, 422391833740800, 800659321056000]}}, 'schur_multiplier': [2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 2], 'solvability_type': 7, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_9:S_3^3', 'transitive_degree': 36, 'wreath_data': None, 'wreath_product': False}
  • gps_groups • Show schema
    {'Agroup': False, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '8.5', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 4, 'aut_exponent': 36, 'aut_gen_orders': [4, 6, 6, 6, 6], 'aut_gens': [[82315575567582061023115, 29428993162140623573896, 51818418375351409197690], [324848675958606118731355, 60217904475962917690050, 273812375191113213603257], [336312377906980965638543, 445323416685879098591416, 341318279543935935190756], [336820931056493550742770, 56519768569085056050623, 293989736060895667564996], [315854102587144395859343, 300527706089594999716181, 34384238792558984816297], [233033165738178057988315, 447638284197888322654216, 290313532305741701877341]], 'aut_group': None, 'aut_hash': 2444291665725440200, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 419904, 'aut_permdeg': 648, 'aut_perms': 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'aut_phi_ratio': 36.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 9, 1, 1], [2, 81, 1, 1], [2, 162, 2, 2], [2, 729, 1, 1], [3, 4, 1, 4], [3, 8, 2, 2], [3, 16, 1, 2], [3, 18, 1, 1], [3, 36, 3, 1], [3, 72, 1, 2], [3, 72, 6, 1], [3, 144, 3, 1], [4, 1458, 2, 1], [6, 36, 1, 2], [6, 162, 1, 2], [6, 324, 1, 2], [6, 324, 2, 4], [6, 324, 3, 1], [6, 324, 6, 1], [6, 648, 1, 2], [6, 648, 2, 2], [6, 648, 6, 1], [6, 1458, 1, 1], [9, 36, 3, 1], [9, 72, 6, 1], [9, 144, 3, 1], [12, 1458, 2, 2], [18, 324, 3, 1], [18, 324, 6, 1], [18, 648, 6, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_3^3.C_3^4.(C_3\\times D_4^2)', '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': 36, 'autcentquo_group': None, 'autcentquo_hash': 2444291665725440200, 'autcentquo_nilpotent': False, 'autcentquo_order': 419904, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'C_3^5.C_3:S_3.C_6.C_2^4', 'cc_stats': [[1, 1, 1], [2, 9, 1], [2, 81, 1], [2, 162, 4], [2, 729, 1], [3, 4, 4], [3, 8, 4], [3, 16, 2], [3, 18, 1], [3, 36, 3], [3, 72, 8], [3, 144, 3], [4, 1458, 2], [6, 36, 2], [6, 162, 2], [6, 324, 19], [6, 648, 12], [6, 1458, 1], [9, 36, 3], [9, 72, 6], [9, 144, 3], [12, 1458, 4], [18, 324, 9], [18, 648, 6]], 'center_label': '1.1', 'center_order': 1, 'central_product': False, 'central_quotient': '34992.lc', 'commutator_count': 1, 'commutator_label': None, 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '3.1', '3.1', '3.1', '3.1', '3.1', '3.1', '3.1'], 'composition_length': 11, 'conjugacy_classes_known': True, 'counter': 289, 'cyclic': False, 'derived_length': 4, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 9, 1, 1], [2, 81, 1, 1], [2, 162, 1, 4], [2, 729, 1, 1], [3, 4, 1, 4], [3, 8, 1, 4], [3, 16, 1, 2], [3, 18, 1, 1], [3, 36, 1, 3], [3, 72, 1, 8], [3, 144, 1, 3], [4, 1458, 1, 2], [6, 36, 1, 2], [6, 162, 1, 2], [6, 324, 1, 19], [6, 648, 1, 12], [6, 1458, 1, 1], [9, 36, 3, 1], [9, 72, 3, 2], [9, 144, 3, 1], [12, 1458, 2, 2], [18, 324, 3, 3], [18, 648, 3, 2]], 'element_repr_type': 'Perm', 'elementary': 1, 'eulerian_function': 32659200, 'exponent': 36, 'exponents_of_order': [7, 4], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [[24, 1, 24], [48, 1, 6]], 'familial': False, 'frattini_label': '27.5', 'frattini_quotient': '1296.3531', 'hash': 3608991211474114175, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 36, 'inner_gen_orders': [18, 18, 4], 'inner_gens': [[82315575567582061023115, 233291257435443031413256, 353156427622273016493742], [233412800987445647501275, 29428993162140623573896, 343547796054111178930157], [300527706089594999716181, 30355467242711294033755, 51818418375351409197690]], 'inner_hash': 3608991211474114175, 'inner_nilpotent': False, 'inner_order': 34992, 'inner_split': True, 'inner_tex': 'C_3^5.S_3^2:C_2^2', 'inner_used': [1, 2, 3], 'irrC_degree': 24, 'irrQ_degree': 24, 'irrQ_dim': 24, 'irrR_degree': 24, 'irrep_stats': [[1, 8], [2, 2], [4, 16], [6, 8], [8, 8], [12, 24], [16, 2], [24, 28], [48, 6]], 'label': '34992.lc', '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': 'C3^5.S3^2: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], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 47, 'number_characteristic_subgroups': 20, 'number_conjugacy_classes': 102, 'number_divisions': 82, 'number_normal_subgroups': 30, 'number_subgroup_autclasses': 1516, 'number_subgroup_classes': 3623, 'number_subgroups': 800530, 'old_label': None, 'order': 34992, 'order_factorization_type': 33, 'order_stats': [[1, 1], [2, 1467], [3, 1214], [4, 2916], [6, 15786], [9, 972], [12, 5832], [18, 6804]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': True, 'outer_exponent': 6, 'outer_gen_orders': [2, 6], 'outer_gen_pows': [7, 0], 'outer_gens': [[82315575567582061022981, 29428993162140623573903, 51818418375351409197811], [92541190756176552907890, 267542597402267573075903, 83439573194829297887635]], 'outer_group': '12.5', 'outer_hash': 5, 'outer_nilpotent': True, 'outer_order': 12, 'outer_permdeg': 7, 'outer_perms': [24, 724], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2\\times C_6', 'pc_rank': 8, 'perfect': False, 'permutation_degree': 24, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 2], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 8], [2, 2], [4, 16], [6, 4], [8, 8], [12, 14], [16, 2], [24, 16], [36, 4], [48, 3], [72, 4], [144, 1]], 'representations': {'PC': {'code': '52088663688647032367479512349951695838858665310027437745809670167110789545224676103399918531349659679085339623889002747264607847268027205755426062337106800461567745939583', 'gens': [1, 2, 4, 6, 8, 9, 10, 11], 'pres': [11, -2, -2, -2, -2, -3, 3, -3, 3, 3, -3, 3, 14080, 186693, 56, 558758, 148195, 528894, 234897, 124, 22004, 73495, 44026, 224933, 68656, 63387, 27758, 7375, 258, 22182, 16649, 5572, 12690, 19037, 356408, 23207, 15199, 3627, 2138409, 1069220, 213871, 106962, 940917, 705704, 352879]}, 'Perm': {'d': 24, 'gens': [82315575567582061023115, 29428993162140623573896, 51818418375351409197690]}}, 'schur_multiplier': [2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 2], 'solvability_type': 12, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': 'C_3^5.S_3^2:C_2^2', 'transitive_degree': 36, 'wreath_data': None, 'wreath_product': False}