-
gps_subgroup_search • Show schema
Hide schema
{'Agroup': True, 'Zgroup': False, 'abelian': False, 'ambient': '3888.js', 'ambient_counter': 253, 'ambient_order': 3888, 'ambient_tex': 'C_3:S_3^4', 'central': False, 'central_factor': False, 'centralizer_order': 6, 'characteristic': False, 'core_order': 324, 'counter': 49, 'cyclic': False, 'direct': False, 'hall': 0, 'label': '3888.js.12.f1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': False, 'minimal': False, 'minimal_normal': False, 'nilpotent': False, 'normal': True, 'old_label': '12.f1', 'outer_equivalence': True, 'perfect': False, 'proper': True, 'quotient': '12.4', 'quotient_Agroup': True, 'quotient_abelian': False, 'quotient_cyclic': False, 'quotient_hash': 4, 'quotient_metabelian': True, 'quotient_nilpotent': False, 'quotient_order': 12, 'quotient_simple': False, 'quotient_solvable': True, 'quotient_supersolvable': True, 'quotient_tex': 'D_6', 'simple': False, 'solvable': True, 'special_labels': [], 'split': True, 'standard_generators': False, 'stem': False, 'subgroup': '324.170', 'subgroup_hash': 170, 'subgroup_order': 324, 'subgroup_tex': 'C_3^2:S_3^2', 'supersolvable': True, 'sylow': 0}
-
gps_subgroup_data • Show schema
Hide schema
{'ambient': '3888.js', 'aut_centralizer_order': 6, 'aut_label': '12.f1', 'aut_quo_index': 2, 'aut_stab_index': 6, 'aut_weyl_group': '2592.fs', 'aut_weyl_index': 36, 'centralizer': '648.a1', 'complements': ['324.h1', '324.j1', '324.k1', '324.bp1', '324.bq1', '324.w1', '324.x1', '324.o1', '324.z1', '324.ba1', '324.bb1', '324.bc1', '324.p1', '324.q1', '324.bn1', '324.s1', '324.t1'], 'conjugacy_class_count': 6, 'contained_in': ['4.e1', '6.a1', '6.c1', '6.n1'], 'contains': ['24.d1', '24.g1', '36.bx1', '36.ci1', '36.cj1', '36.ck1'], 'core': '12.f1', 'coset_action_label': None, 'count': 6, 'diagramx': None, 'generators': [93405312000, 3, 180583644210, 269, 148, 45360], 'label': '3888.js.12.f1', 'mobius_quo': None, 'mobius_sub': None, 'normal_closure': '12.f1', 'normal_contained_in': ['4.e1', '6.a1'], 'normal_contains': ['24.d1', '24.g1'], 'normalizer': '1.a1', 'old_label': '12.f1', 'projective_image': '3888.js', 'quotient_action_image': '4.2', 'quotient_action_kernel': '3.1', 'quotient_action_kernel_order': 3, 'quotient_fusion': None, 'short_label': '12.f1', 'subgroup_fusion': None, 'weyl_group': '648.733'}
-
gps_groups • Show schema
Hide schema
{'Agroup': True, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '4.2', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 5, 'aut_exponent': 24, 'aut_gen_orders': [3, 4, 6, 2, 4, 4, 3, 3, 6, 6, 2, 2], 'aut_gens': [[8507311, 44716501, 1216789, 1097596, 879733], [51764851, 44716501, 1216789, 1097596, 879733], [52003351, 1215961, 1219489, 1097596, 879733], [8510839, 1221361, 44717329, 879733, 1097596], [8510011, 44716501, 1222189, 1097596, 879733], [8512711, 1221361, 732529, 1097596, 879733], [52006051, 1221361, 881569, 33902551, 53803441], [51762151, 44716501, 735229, 1097596, 879733], [8512711, 44716501, 1216789, 1097596, 879733], [8512711, 1218661, 881569, 1097596, 879733], [19565596, 44713801, 33657244, 1097596, 879733], [51764851, 1218661, 44720029, 1097596, 879733], [8507311, 1215961, 44717329, 33902551, 879733]], 'aut_group': '31104.mm', 'aut_hash': 8544079931638136565, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 31104, 'aut_permdeg': 54, 'aut_perms': [4355599831554052545168591656912399278081296196380557505168866101107129, 222539037912302426343041585051562251804091885204731292628611042935976766, 48091163911320556070507609133057464795418456800108746203399983725732942, 17426758366318694238217689695974385599442978068073108213621592758583782, 230524290346982398731577600057975855803391719431890047550349102153102903, 12818447204980504091796830621555082777981247212421161022990260498550935, 21700180569248861625590670576366053847328793624302771206151802315800502, 227051273388535605846038827543822118445145808985358810105378940910110536, 226654033753220582968816089692341845594233024954298722900881240699871776, 126019896429659499331968782405410669513330819697191110710228485352320267, 4276374804914427197121225384416037200297136330859059052530497953941716, 82058417539207134045956152395595388832520555553997136853015524434374], 'aut_phi_ratio': 288.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 9, 1, 1], [2, 27, 2, 1], [3, 2, 2, 1], [3, 2, 4, 1], [3, 4, 1, 1], [3, 4, 8, 2], [6, 18, 4, 1], [6, 54, 2, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_3^4:(D_4\\times \\GL(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': 24, 'autcentquo_group': '31104.mm', 'autcentquo_hash': 8544079931638136565, 'autcentquo_nilpotent': False, 'autcentquo_order': 31104, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'C_3^4:(D_4\\times \\GL(2,3))', 'cc_stats': [[1, 1, 1], [2, 9, 1], [2, 27, 2], [3, 2, 6], [3, 4, 17], [6, 18, 4], [6, 54, 2]], 'center_label': '1.1', 'center_order': 1, 'central_product': False, 'central_quotient': '324.170', 'commutator_count': 1, 'commutator_label': '81.15', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '3.1', '3.1', '3.1', '3.1'], 'composition_length': 6, 'conjugacy_classes_known': True, 'counter': 170, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 9, 1, 1], [2, 27, 1, 2], [3, 2, 1, 6], [3, 4, 1, 9], [3, 4, 2, 4], [6, 18, 1, 4], [6, 54, 1, 2]], 'element_repr_type': 'GLZN', 'elementary': 1, 'eulerian_function': 336, 'exponent': 6, 'exponents_of_order': [4, 2], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [], 'familial': False, 'frattini_label': '1.1', 'frattini_quotient': '324.170', 'hash': 170, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 6, 'inner_gen_orders': [2, 3, 6, 3, 3], 'inner_gens': [[8507311, 1218661, 44720029, 33902551, 879733], [51762151, 44716501, 1216789, 1097596, 879733], [52003351, 44716501, 1216789, 33902551, 53803441], [40947766, 44716501, 33657244, 1097596, 879733], [8507311, 44716501, 53557297, 1097596, 879733]], 'inner_hash': 170, 'inner_nilpotent': False, 'inner_order': 324, 'inner_split': True, 'inner_tex': 'C_3^2:S_3^2', 'inner_used': [1, 2, 3, 4, 5], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 4], [2, 12], [4, 17]], 'label': '324.170', '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': 'C3^2:S3^2', 'ngens': 6, 'nilpotency_class': -1, 'nilpotent': False, 'normal_counts': [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': 10, 'number_characteristic_subgroups': 7, 'number_conjugacy_classes': 33, 'number_divisions': 29, 'number_normal_subgroups': 35, 'number_subgroup_autclasses': 58, 'number_subgroup_classes': 224, 'number_subgroups': 1432, 'old_label': None, 'order': 324, 'order_factorization_type': 32, 'order_stats': [[1, 1], [2, 63], [3, 80], [6, 180]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': True, 'outer_exponent': 24, 'outer_gen_orders': [3, 2, 2, 2, 4, 4], 'outer_gen_pows': [729001, 729001, 729829, 729001, 729001, 729001], 'outer_gens': [[8507311, 734401, 1216789, 1097596, 879733], [8507311, 734401, 44720029, 1097596, 879733], [8508139, 1218661, 44720029, 879733, 33902551], [8508139, 1218661, 44720029, 879733, 1097596], [8508139, 44713801, 735229, 879733, 1097596], [8507311, 1215961, 1219489, 1097596, 879733]], 'outer_group': '96.189', 'outer_hash': 189, 'outer_nilpotent': False, 'outer_order': 96, 'outer_permdeg': 10, 'outer_perms': [2592144, 177967, 1, 1270441, 2643169, 2419944], 'outer_solvable': True, 'outer_supersolvable': False, 'outer_tex': 'C_2\\times \\GL(2,3)', 'pc_rank': 5, 'perfect': False, 'permutation_degree': 12, 'pgroup': 0, 'primary_abelian_invariants': [2, 2], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 12], [4, 9], [8, 4]], 'representations': {'PC': {'code': 78354709127428954945815, 'gens': [1, 2, 3, 5, 6], 'pres': [6, -2, -3, -2, -3, -3, -3, 49, 542, 50, 579, 2164, 376, 1313]}, 'GLZN': {'d': 2, 'p': 90, 'gens': [729829, 879733, 1097596, 45079696, 731701, 19318996]}, 'Perm': {'d': 12, 'gens': [40723225, 41091145, 144, 147, 91450800, 130682160]}}, 'schur_multiplier': [6], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2], 'solvability_type': 7, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_3^2:S_3^2', 'transitive_degree': 36, 'wreath_data': None, 'wreath_product': False}
-
gps_groups • Show schema
Hide schema
{'Agroup': True, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '16.14', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 4, 'aut_exponent': 36, 'aut_gen_orders': [4, 18, 6, 4], 'aut_gens': [[3, 43545600, 45360, 39916800, 93405312000, 180667066290, 269, 40320, 144], [144, 45360, 79833600, 85680, 174356582400, 180583608694, 269, 39916800, 3], [174356582643, 45360, 43545600, 85680, 244, 87178296811, 6227021250, 39916800, 93405312243], [93405312148, 79833600, 45360, 3628800, 147, 87261754315, 180583603798, 5040, 174356582548], [174356582547, 79833600, 80640, 39916800, 148, 87181920694, 6227021275, 85680, 93405312147]], 'aut_group': '93312.is', 'aut_hash': 458068617261292163, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 93312, 'aut_permdeg': 33, 'aut_perms': [7488918491129848130499211209898814287, 7483182473869955279100711230071288207, 3692085122588095920907256051229287602, 715469638848146002050080652783461232], 'aut_phi_ratio': 72.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 3, 2, 1], [2, 9, 1, 1], [2, 9, 3, 1], [2, 27, 6, 1], [2, 81, 3, 1], [3, 2, 2, 1], [3, 2, 3, 1], [3, 4, 1, 1], [3, 4, 2, 1], [3, 4, 3, 1], [3, 4, 6, 1], [3, 8, 3, 1], [3, 8, 4, 1], [3, 8, 6, 1], [3, 16, 2, 1], [3, 16, 3, 1], [6, 6, 2, 1], [6, 6, 6, 1], [6, 12, 4, 1], [6, 12, 6, 2], [6, 18, 3, 2], [6, 18, 6, 1], [6, 24, 4, 1], [6, 24, 6, 1], [6, 36, 2, 1], [6, 36, 3, 2], [6, 36, 6, 1], [6, 54, 6, 2], [6, 72, 3, 1], [6, 108, 6, 1], [6, 162, 3, 1]], 'aut_supersolvable': False, 'aut_tex': 'S_3^5.D_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': 36, 'autcentquo_group': '93312.is', 'autcentquo_hash': 458068617261292163, 'autcentquo_nilpotent': False, 'autcentquo_order': 93312, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'S_3^5.D_6', 'cc_stats': [[1, 1, 1], [2, 3, 2], [2, 9, 4], [2, 27, 6], [2, 81, 3], [3, 2, 5], [3, 4, 12], [3, 8, 13], [3, 16, 5], [6, 6, 8], [6, 12, 16], [6, 18, 12], [6, 24, 10], [6, 36, 14], [6, 54, 12], [6, 72, 3], [6, 108, 6], [6, 162, 3]], 'center_label': '1.1', 'center_order': 1, 'central_product': True, 'central_quotient': '3888.js', 'commutator_count': 1, 'commutator_label': '243.67', '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'], 'composition_length': 9, 'conjugacy_classes_known': True, 'counter': 253, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['108.40', 1], ['6.1', 2]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 3, 1, 2], [2, 9, 1, 4], [2, 27, 1, 6], [2, 81, 1, 3], [3, 2, 1, 5], [3, 4, 1, 10], [3, 4, 2, 1], [3, 8, 1, 9], [3, 8, 2, 2], [3, 16, 1, 3], [3, 16, 2, 1], [6, 6, 1, 8], [6, 12, 1, 12], [6, 12, 2, 2], [6, 18, 1, 12], [6, 24, 1, 6], [6, 24, 2, 2], [6, 36, 1, 12], [6, 36, 2, 1], [6, 54, 1, 12], [6, 72, 1, 3], [6, 108, 1, 6], [6, 162, 1, 3]], 'element_repr_type': 'Perm', 'elementary': 1, 'eulerian_function': 623772240, 'exponent': 6, 'exponents_of_order': [5, 4], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [[16, 0, 2]], 'familial': False, 'frattini_label': '1.1', 'frattini_quotient': '3888.js', 'hash': 528584399898365654, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 6, 'inner_gen_orders': [3, 3, 3, 2, 3, 6, 2, 2, 3], 'inner_gens': [[3, 43545600, 45360, 39916800, 93405312000, 180667066054, 266, 40320, 144], [3, 43545600, 45360, 3628800, 93405312000, 180623520690, 269, 40320, 144], [3, 43545600, 45360, 39916800, 93405312000, 180667066290, 269, 5040, 144], [3, 79833600, 45360, 39916800, 93405312000, 180587232690, 269, 40320, 144], [3, 43545600, 45360, 39916800, 93405312000, 87261754290, 269, 40320, 144], [144, 79833600, 45360, 3628800, 174356582400, 180667066290, 265, 40320, 3], [4, 43545600, 45360, 39916800, 93405312000, 180667066194, 269, 40320, 240], [3, 43545600, 80640, 39916800, 93405312000, 180667066290, 269, 40320, 144], [3, 43545600, 45360, 39916800, 93405312000, 180667066197, 125, 40320, 144]], 'inner_hash': 528584399898365654, 'inner_nilpotent': False, 'inner_order': 3888, 'inner_split': True, 'inner_tex': 'C_3:S_3^4', 'inner_used': [1, 2, 3, 4, 5, 6, 7, 8], 'irrC_degree': 16, 'irrQ_degree': 32, 'irrQ_dim': 32, 'irrR_degree': 32, 'irrep_stats': [[1, 16], [2, 40], [4, 48], [8, 26], [16, 5]], 'label': '3888.js', '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': 'C3:S3^4', '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, 0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 36, 'number_characteristic_subgroups': 12, 'number_conjugacy_classes': 135, 'number_divisions': 126, 'number_normal_subgroups': 228, 'number_subgroup_autclasses': 766, 'number_subgroup_classes': 4266, 'number_subgroups': 87768, 'old_label': None, 'order': 3888, 'order_factorization_type': 33, 'order_stats': [[1, 1], [2, 447], [3, 242], [6, 3198]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': True, 'outer_exponent': 6, 'outer_gen_orders': [2, 2, 6], 'outer_gen_pows': [0, 269, 0], 'outer_gens': [[3, 43545600, 45360, 39916800, 174356582400, 180667066290, 269, 40320, 144], [240, 79833600, 45360, 39916800, 174356582400, 180587232715, 269, 40320, 3], [174356582548, 45360, 43545600, 40320, 147, 6227106956, 180583603650, 39916800, 93405312148]], 'outer_group': '24.14', 'outer_hash': 14, 'outer_nilpotent': False, 'outer_order': 24, 'outer_permdeg': 7, 'outer_perms': [7, 120, 856], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2\\times D_6', 'pc_rank': 6, 'perfect': False, 'permutation_degree': 15, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 2, 2], 'quasisimple': False, 'rank': 4, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 16], [2, 40], [4, 40], [8, 22], [16, 7], [32, 1]], 'representations': {'PC': {'code': '418182879202542983638411866285743712356080812124173732311110747', 'gens': [1, 2, 4, 6, 7, 9], 'pres': [9, -2, -2, -3, -2, -3, -3, -2, -3, -3, 1477, 46, 4106, 2388, 1425, 102, 1093, 680, 34035, 5703, 186, 31120, 5218, 1034]}, 'Perm': {'d': 15, 'gens': [3, 43545600, 45360, 39916800, 93405312000, 180667066290, 269, 40320, 144]}}, 'schur_multiplier': [2, 2, 2, 2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 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_3:S_3^4', 'transitive_degree': 48, 'wreath_data': None, 'wreath_product': False}
-
gps_groups • Show schema
Hide schema
{'Agroup': True, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '4.2', '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], [1, 10], [3, 2]], 'aut_group': '12.4', 'aut_hash': 4, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 12, 'aut_permdeg': 5, 'aut_perms': [6, 49], 'aut_phi_ratio': 3.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 3, 2, 1], [3, 2, 1, 1], [6, 2, 1, 1]], 'aut_supersolvable': True, 'aut_tex': 'D_6', 'autcent_abelian': True, 'autcent_cyclic': True, 'autcent_exponent': 2, 'autcent_group': '2.1', 'autcent_hash': 1, 'autcent_nilpotent': True, 'autcent_order': 2, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 6, 'autcentquo_group': '6.1', 'autcentquo_hash': 1, 'autcentquo_nilpotent': False, 'autcentquo_order': 6, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'S_3', 'cc_stats': [[1, 1, 1], [2, 1, 1], [2, 3, 2], [3, 2, 1], [6, 2, 1]], 'center_label': '2.1', 'center_order': 2, 'central_product': True, 'central_quotient': '6.1', 'commutator_count': 1, 'commutator_label': '3.1', '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': 4, 'cyclic': False, 'derived_length': 2, 'dihedral': True, 'direct_factorization': [['2.1', 1], ['6.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 3, 1, 2], [3, 2, 1, 1], [6, 2, 1, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 3, 'exponent': 6, 'exponents_of_order': [2, 1], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [[2, 1, 1]], 'familial': True, 'frattini_label': '1.1', 'frattini_quotient': '12.4', 'hash': 4, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 6, 'inner_gen_orders': [2, 3], 'inner_gens': [[1, 10], [5, 2]], 'inner_hash': 1, 'inner_nilpotent': False, 'inner_order': 6, 'inner_split': True, 'inner_tex': 'S_3', 'inner_used': [1, 2], 'irrC_degree': 2, 'irrQ_degree': 2, 'irrQ_dim': 2, 'irrR_degree': 2, 'irrep_stats': [[1, 4], [2, 2]], 'label': '12.4', 'linC_count': 1, 'linC_degree': 2, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 2, 'linQ_degree_count': 1, 'linQ_dim': 2, 'linQ_dim_count': 1, 'linR_count': 1, 'linR_degree': 2, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'D6', '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': 5, 'number_characteristic_subgroups': 5, 'number_conjugacy_classes': 6, 'number_divisions': 6, 'number_normal_subgroups': 7, 'number_subgroup_autclasses': 8, 'number_subgroup_classes': 10, 'number_subgroups': 16, 'old_label': None, 'order': 12, 'order_factorization_type': 22, 'order_stats': [[1, 1], [2, 7], [3, 2], [6, 2]], 'outer_abelian': True, 'outer_cyclic': True, 'outer_equivalence': False, 'outer_exponent': 2, 'outer_gen_orders': [2], 'outer_gen_pows': [0], 'outer_gens': [[7, 2]], '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': 2, 'perfect': False, 'permutation_degree': 5, 'pgroup': 0, 'primary_abelian_invariants': [2, 2], 'quasisimple': False, 'rank': 2, 'rational': True, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 2]], 'representations': {'PC': {'code': 43377, 'gens': [1, 2], 'pres': [3, -2, -2, -3, 61, 16, 74]}, 'GLZ': {'b': 3, 'd': 2, 'gens': [17, 35]}, 'GLFp': {'d': 2, 'p': 3, 'gens': [31, 55, 56]}, 'Perm': {'d': 5, 'gens': [6, 1, 30]}}, 'schur_multiplier': [2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2], 'solvability_type': 6, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'D_6', 'transitive_degree': 6, 'wreath_data': None, 'wreath_product': False}
