Formats: - HTML - YAML - JSON - 2026-07-17T21:50:54.151814
  • gps_groups • Show schema
    {'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, 6, 3, 9], 'aut_gens': [[1, 6, 18], [5, 66, 126], [1, 120, 156], [61, 6, 18], [25, 6, 18]], 'aut_group': '972.446', 'aut_hash': 446, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 972, 'aut_permdeg': 27, 'aut_perms': [3338739800221796969753122759, 3230676458013296413268959933, 1822918156030643663437189468, 6499141082497476730277681684], 'aut_phi_ratio': 18.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 27, 1, 1], [3, 2, 1, 1], [3, 6, 1, 1], [3, 9, 2, 1], [6, 27, 2, 1], [9, 6, 3, 1], [9, 18, 2, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_3^3.S_3^2', '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': '972.446', 'autcentquo_hash': 446, 'autcentquo_nilpotent': False, 'autcentquo_order': 972, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'C_3^3.S_3^2', 'cc_stats': [[1, 1, 1], [2, 27, 1], [3, 2, 1], [3, 6, 1], [3, 9, 2], [6, 27, 2], [9, 6, 3], [9, 18, 2]], 'center_label': '1.1', 'center_order': 1, 'central_product': False, 'central_quotient': '162.13', 'commutator_count': 1, 'commutator_label': '27.2', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '3.1', '3.1', '3.1', '3.1'], 'composition_length': 5, 'conjugacy_classes_known': True, 'counter': 13, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 27, 1, 1], [3, 2, 1, 1], [3, 6, 1, 1], [3, 9, 2, 1], [6, 27, 2, 1], [9, 6, 3, 1], [9, 18, 2, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 12, 'exponent': 18, 'exponents_of_order': [4, 1], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [[6, 1, 3]], 'familial': False, 'frattini_label': '9.2', 'frattini_quotient': '18.3', 'hash': 13, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 18, 'inner_gen_orders': [6, 3, 9], 'inner_gens': [[1, 120, 156], [67, 6, 18], [43, 6, 18]], 'inner_hash': 13, 'inner_nilpotent': False, 'inner_order': 162, 'inner_split': True, 'inner_tex': '\\He_3.S_3', 'inner_used': [1, 2, 3], 'irrC_degree': 6, 'irrQ_degree': 18, 'irrQ_dim': 18, 'irrR_degree': 6, 'irrep_stats': [[1, 6], [2, 3], [6, 4]], 'label': '162.13', 'linC_count': 3, 'linC_degree': 6, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 18, 'linQ_degree_count': 1, 'linQ_dim': 18, 'linQ_dim_count': 1, 'linR_count': 3, 'linR_degree': 6, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'He3.S3', 'ngens': 5, 'nilpotency_class': -1, 'nilpotent': False, 'normal_counts': [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': 8, 'number_characteristic_subgroups': 8, 'number_conjugacy_classes': 13, 'number_divisions': 8, 'number_normal_subgroups': 8, 'number_subgroup_autclasses': 22, 'number_subgroup_classes': 22, 'number_subgroups': 148, 'old_label': None, 'order': 162, 'order_factorization_type': 31, 'order_stats': [[1, 1], [2, 27], [3, 26], [6, 54], [9, 54]], 'outer_abelian': True, 'outer_cyclic': True, 'outer_equivalence': False, 'outer_exponent': 6, 'outer_gen_orders': [6], 'outer_gen_pows': [0], 'outer_gens': [[5, 114, 90]], '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': 3, 'perfect': False, 'permutation_degree': 27, 'pgroup': 0, 'primary_abelian_invariants': [2, 3], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 2], [2, 3], [4, 1], [6, 1], [18, 1]], 'representations': {'PC': {'code': 180175900447401868591, 'gens': [1, 3, 4], 'pres': [5, -2, -3, -3, 3, -3, 10, 1802, 862, 3123, 848, 78, 2704]}, 'Perm': {'d': 27, 'gens': [15593415495065233233131519, 26980884404807753019028, 8356737317491000254136320000, 2516538717432286673398548480, 806634631153204606767248884]}}, 'schur_multiplier': [3], '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': '\\He_3.S_3', 'transitive_degree': 27, 'wreath_data': None, 'wreath_product': False}
  • gps_conj_classes • Show schema
    • id: 108935343
      {'aut_label': '1A', 'centralizer': '1.a1.a1', 'counter': 1, 'group_counter': 13, 'group_order': 162, 'label': '1A', 'order': 1, 'powers': [1, 1], 'representative': 0, 'size': 1}
    • id: 108935348
      {'aut_label': '2A', 'centralizer': '27.a1.a1', 'counter': 2, 'group_counter': 13, 'group_order': 162, 'label': '2A', 'order': 2, 'powers': [1, 2], 'representative': 51, 'size': 27}
    • id: 108935349
      {'aut_label': '3A', 'centralizer': '2.a1.a1', 'counter': 3, 'group_counter': 13, 'group_order': 162, 'label': '3A', 'order': 3, 'powers': [3, 1], 'representative': 54, 'size': 2}
    • id: 108935350
      {'aut_label': '3B', 'centralizer': '6.b1.a1', 'counter': 4, 'group_counter': 13, 'group_order': 162, 'label': '3B', 'order': 3, 'powers': [4, 1], 'representative': 6, 'size': 6}
    • id: 108935351
      {'aut_label': '3C1', 'centralizer': '9.a1.a1', 'counter': 5, 'group_counter': 13, 'group_order': 162, 'label': '3C1', 'order': 3, 'powers': [6, 1], 'representative': 124, 'size': 9}
    • id: 108935352
      {'aut_label': '3C1', 'centralizer': '9.a1.a1', 'counter': 6, 'group_counter': 13, 'group_order': 162, 'label': '3C-1', 'order': 3, 'powers': [5, 1], 'representative': 8, 'size': 9}
    • id: 108935353
      {'aut_label': '6A1', 'centralizer': '27.a1.a1', 'counter': 7, 'group_counter': 13, 'group_order': 162, 'label': '6A1', 'order': 6, 'powers': [5, 2], 'representative': 47, 'size': 27}
    • id: 108935354
      {'aut_label': '6A1', 'centralizer': '27.a1.a1', 'counter': 8, 'group_counter': 13, 'group_order': 162, 'label': '6A-1', 'order': 6, 'powers': [6, 2], 'representative': 91, 'size': 27}
    • id: 108935355
      {'aut_label': '9A1', 'centralizer': '6.b1.a1', 'counter': 9, 'group_counter': 13, 'group_order': 162, 'label': '9A1', 'order': 9, 'powers': [10, 3], 'representative': 18, 'size': 6}
    • id: 108935344
      {'aut_label': '9A1', 'centralizer': '6.b1.a1', 'counter': 10, 'group_counter': 13, 'group_order': 162, 'label': '9A2', 'order': 9, 'powers': [11, 3], 'representative': 36, 'size': 6}
    • id: 108935345
      {'aut_label': '9A1', 'centralizer': '6.b1.a1', 'counter': 11, 'group_counter': 13, 'group_order': 162, 'label': '9A4', 'order': 9, 'powers': [9, 3], 'representative': 72, 'size': 6}
    • id: 108935346
      {'aut_label': '9B1', 'centralizer': '18.d1.a1', 'counter': 12, 'group_counter': 13, 'group_order': 162, 'label': '9B1', 'order': 9, 'powers': [13, 3], 'representative': 28, 'size': 18}
    • id: 108935347
      {'aut_label': '9B1', 'centralizer': '18.d1.a1', 'counter': 13, 'group_counter': 13, 'group_order': 162, 'label': '9B-1', 'order': 9, 'powers': [12, 3], 'representative': 146, 'size': 18}
  • gps_qchar • Show schema
    • id: 4075085
      {'cdim': 1, 'counter': 1, 'faithful': False, 'group': '162.13', 'group_order': 162, 'image': None, 'label': '162.13.1a', 'multiplicity': 1, 'nt': [1, 1], 'qdim': 1, 'qvalues': [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1], 'schur_index': 1}
    • id: 4075086
      {'cdim': 1, 'counter': 2, 'faithful': False, 'group': '162.13', 'group_order': 162, 'image': None, 'label': '162.13.1b', 'multiplicity': 1, 'nt': [2, 1], 'qdim': 1, 'qvalues': [1, -1, 1, 1, 1, 1, -1, -1, 1, 1, 1, 1, 1], 'schur_index': 1}
    • id: 4075087
      {'cdim': 1, 'counter': 3, 'faithful': False, 'group': '162.13', 'group_order': 162, 'image': None, 'label': '162.13.1c', 'multiplicity': 2, 'nt': [3, 1], 'qdim': 2, 'qvalues': [2, 2, 2, 2, -1, -1, -1, -1, 2, 2, 2, -1, -1], 'schur_index': 1}
    • id: 4075088
      {'cdim': 1, 'counter': 4, 'faithful': False, 'group': '162.13', 'group_order': 162, 'image': None, 'label': '162.13.1d', 'multiplicity': 2, 'nt': [6, 1], 'qdim': 2, 'qvalues': [2, -2, 2, 2, -1, -1, 1, 1, 2, 2, 2, -1, -1], 'schur_index': 1}
    • id: 4075089
      {'cdim': 2, 'counter': 5, 'faithful': False, 'group': '162.13', 'group_order': 162, 'image': None, 'label': '162.13.2a', 'multiplicity': 1, 'nt': [3, 2], 'qdim': 2, 'qvalues': [2, 0, 2, 2, 2, 2, 0, 0, -1, -1, -1, -1, -1], 'schur_index': 1}
    • id: 4075090
      {'cdim': 2, 'counter': 6, 'faithful': False, 'group': '162.13', 'group_order': 162, 'image': None, 'label': '162.13.2b', 'multiplicity': 2, 'nt': [6, 5], 'qdim': 4, 'qvalues': [4, 0, 4, 4, -2, -2, 0, 0, -2, -2, -2, 1, 1], 'schur_index': 1}
    • id: 4075091
      {'cdim': 6, 'counter': 7, 'faithful': False, 'group': '162.13', 'group_order': 162, 'image': None, 'label': '162.13.6a', 'multiplicity': 1, 'nt': [9, 11], 'qdim': 6, 'qvalues': [6, 0, 6, -3, 0, 0, 0, 0, 0, 0, 0, 0, 0], 'schur_index': 1}
    • id: 4075092
      {'cdim': 6, 'counter': 8, 'faithful': True, 'group': '162.13', 'group_order': 162, 'image': None, 'label': '162.13.6b', 'multiplicity': 3, 'nt': [27, 41], 'qdim': 18, 'qvalues': [18, 0, -9, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], 'schur_index': 1}
  • gps_char • Show schema
    • id: 5843040
      {'center': '1.a1.a1', 'center_index': 1, 'center_order': 162, 'counter': 1, 'cyclotomic_n': 1, 'dim': 1, 'faithful': False, 'field': [0, 1], 'group': '162.13', 'group_counter': 13, 'group_order': 162, 'image': None, 'image_isoclass': '1.1', 'image_order': 1, 'indicator': 1, 'kernel': '1.a1.a1', 'kernel_order': 162, 'label': '162.13.1a', 'nt': [1, 1], 'q_character': None, 'values': [[[1, 0]], [[1, 0]], [[1, 0]], [[1, 0]], [[1, 0]], [[1, 0]], [[1, 0]], [[1, 0]], [[1, 0]], [[1, 0]], [[1, 0]], [[1, 0]], [[1, 0]]]}
    • id: 5843045
      {'center': '1.a1.a1', 'center_index': 1, 'center_order': 162, 'counter': 2, 'cyclotomic_n': 1, 'dim': 1, 'faithful': False, 'field': [0, 1], 'group': '162.13', 'group_counter': 13, 'group_order': 162, 'image': None, 'image_isoclass': '2.1', 'image_order': 2, 'indicator': 1, 'kernel': '2.a1.a1', 'kernel_order': 81, 'label': '162.13.1b', 'nt': [2, 1], 'q_character': None, 'values': [[[1, 0]], [[-1, 0]], [[1, 0]], [[1, 0]], [[1, 0]], [[1, 0]], [[-1, 0]], [[-1, 0]], [[1, 0]], [[1, 0]], [[1, 0]], [[1, 0]], [[1, 0]]]}
    • id: 5843046
      {'center': '1.a1.a1', 'center_index': 1, 'center_order': 162, 'counter': 3, 'cyclotomic_n': 3, 'dim': 1, 'faithful': False, 'field': [1, -1, 1], 'group': '162.13', 'group_counter': 13, 'group_order': 162, 'image': None, 'image_isoclass': '3.1', 'image_order': 3, 'indicator': 0, 'kernel': '3.a1.a1', 'kernel_order': 54, 'label': '162.13.1c1', 'nt': [3, 1], 'q_character': None, 'values': [[[1, 0]], [[1, 0]], [[1, 0]], [[1, 0]], [[1, -1]], [[1, 1]], [[1, 1]], [[1, -1]], [[1, 0]], [[1, 0]], [[1, 0]], [[1, -1]], [[1, 1]]]}
    • id: 5843047
      {'center': '1.a1.a1', 'center_index': 1, 'center_order': 162, 'counter': 4, 'cyclotomic_n': 3, 'dim': 1, 'faithful': False, 'field': [1, -1, 1], 'group': '162.13', 'group_counter': 13, 'group_order': 162, 'image': None, 'image_isoclass': '3.1', 'image_order': 3, 'indicator': 0, 'kernel': '3.a1.a1', 'kernel_order': 54, 'label': '162.13.1c2', 'nt': [3, 2], 'q_character': None, 'values': [[[1, 0]], [[1, 0]], [[1, 0]], [[1, 0]], [[1, 1]], [[1, -1]], [[1, -1]], [[1, 1]], [[1, 0]], [[1, 0]], [[1, 0]], [[1, 1]], [[1, -1]]]}
    • id: 5843048
      {'center': '1.a1.a1', 'center_index': 1, 'center_order': 162, 'counter': 5, 'cyclotomic_n': 3, 'dim': 1, 'faithful': False, 'field': [1, -1, 1], 'group': '162.13', 'group_counter': 13, 'group_order': 162, 'image': None, 'image_isoclass': '6.2', 'image_order': 6, 'indicator': 0, 'kernel': '6.b1.a1', 'kernel_order': 27, 'label': '162.13.1d1', 'nt': [6, 1], 'q_character': None, 'values': [[[1, 0]], [[-1, 0]], [[1, 0]], [[1, 0]], [[1, -1]], [[1, 1]], [[-1, 1]], [[-1, -1]], [[1, 0]], [[1, 0]], [[1, 0]], [[1, -1]], [[1, 1]]]}
    • id: 5843049
      {'center': '1.a1.a1', 'center_index': 1, 'center_order': 162, 'counter': 6, 'cyclotomic_n': 3, 'dim': 1, 'faithful': False, 'field': [1, -1, 1], 'group': '162.13', 'group_counter': 13, 'group_order': 162, 'image': None, 'image_isoclass': '6.2', 'image_order': 6, 'indicator': 0, 'kernel': '6.b1.a1', 'kernel_order': 27, 'label': '162.13.1d2', 'nt': [6, 2], 'q_character': None, 'values': [[[1, 0]], [[-1, 0]], [[1, 0]], [[1, 0]], [[1, 1]], [[1, -1]], [[-1, -1]], [[-1, 1]], [[1, 0]], [[1, 0]], [[1, 0]], [[1, 1]], [[1, -1]]]}
    • id: 5843050
      {'center': '6.a1.a1', 'center_index': 6, 'center_order': 27, 'counter': 7, 'cyclotomic_n': 1, 'dim': 2, 'faithful': False, 'field': [0, 1], 'group': '162.13', 'group_counter': 13, 'group_order': 162, 'image': None, 'image_isoclass': '6.1', 'image_order': 6, 'indicator': 1, 'kernel': '6.a1.a1', 'kernel_order': 27, 'label': '162.13.2a', 'nt': [3, 2], 'q_character': None, 'values': [[[2, 0]], [[0, 0]], [[2, 0]], [[2, 0]], [[2, 0]], [[2, 0]], [[0, 0]], [[0, 0]], [[-1, 0]], [[-1, 0]], [[-1, 0]], [[-1, 0]], [[-1, 0]]]}
    • id: 5843051
      {'center': '6.a1.a1', 'center_index': 6, 'center_order': 27, 'counter': 8, 'cyclotomic_n': 3, 'dim': 2, 'faithful': False, 'field': [1, -1, 1], 'group': '162.13', 'group_counter': 13, 'group_order': 162, 'image': None, 'image_isoclass': '18.3', 'image_order': 18, 'indicator': 0, 'kernel': '18.a1.a1', 'kernel_order': 9, 'label': '162.13.2b1', 'nt': [6, 5], 'q_character': None, 'values': [[[2, 0]], [[0, 0]], [[2, 0]], [[2, 0]], [[2, -1]], [[2, 1]], [[0, 0]], [[0, 0]], [[-1, 0]], [[-1, 0]], [[-1, 0]], [[-1, -1]], [[-1, 1]]]}
    • id: 5843052
      {'center': '6.a1.a1', 'center_index': 6, 'center_order': 27, 'counter': 9, 'cyclotomic_n': 3, 'dim': 2, 'faithful': False, 'field': [1, -1, 1], 'group': '162.13', 'group_counter': 13, 'group_order': 162, 'image': None, 'image_isoclass': '18.3', 'image_order': 18, 'indicator': 0, 'kernel': '18.a1.a1', 'kernel_order': 9, 'label': '162.13.2b2', 'nt': [6, 2], 'q_character': None, 'values': [[[2, 0]], [[0, 0]], [[2, 0]], [[2, 0]], [[2, 1]], [[2, -1]], [[0, 0]], [[0, 0]], [[-1, 0]], [[-1, 0]], [[-1, 0]], [[-1, 1]], [[-1, -1]]]}
    • id: 5843041
      {'center': '54.a1.a1', 'center_index': 54, 'center_order': 3, 'counter': 10, 'cyclotomic_n': 1, 'dim': 6, 'faithful': False, 'field': [0, 1], 'group': '162.13', 'group_counter': 13, 'group_order': 162, 'image': None, 'image_isoclass': '54.5', 'image_order': 54, 'indicator': 1, 'kernel': '54.a1.a1', 'kernel_order': 3, 'label': '162.13.6a', 'nt': [9, 11], 'q_character': None, 'values': [[[6, 0]], [[0, 0]], [[6, 0]], [[-3, 0]], [[0, 0]], [[0, 0]], [[0, 0]], [[0, 0]], [[0, 0]], [[0, 0]], [[0, 0]], [[0, 0]], [[0, 0]]]}
    • id: 5843042
      {'center': '162.a1.a1', 'center_index': 162, 'center_order': 1, 'counter': 11, 'cyclotomic_n': 9, 'dim': 6, 'faithful': True, 'field': [-1, -3, 0, 1], 'group': '162.13', 'group_counter': 13, 'group_order': 162, 'image': None, 'image_isoclass': '162.13', 'image_order': 162, 'indicator': 1, 'kernel': '162.a1.a1', 'kernel_order': 1, 'label': '162.13.6b1', 'nt': [27, 41], 'q_character': None, 'values': [[[6, 0]], [[0, 0]], [[-3, 0]], [[0, 0]], [[0, 0]], [[0, 0]], [[0, 0]], [[0, 0]], [[-1, 1], [1, 2], [1, 4], [2, -4]], [[2, 1], [-2, 2], [1, 4], [-1, -4]], [[-1, 1], [1, 2], [-2, 4], [-1, -4]], [[0, 0]], [[0, 0]]]}
    • id: 5843043
      {'center': '162.a1.a1', 'center_index': 162, 'center_order': 1, 'counter': 12, 'cyclotomic_n': 9, 'dim': 6, 'faithful': True, 'field': [-1, -3, 0, 1], 'group': '162.13', 'group_counter': 13, 'group_order': 162, 'image': None, 'image_isoclass': '162.13', 'image_order': 162, 'indicator': 1, 'kernel': '162.a1.a1', 'kernel_order': 1, 'label': '162.13.6b2', 'nt': [27, 41], 'q_character': None, 'values': [[[6, 0]], [[0, 0]], [[-3, 0]], [[0, 0]], [[0, 0]], [[0, 0]], [[0, 0]], [[0, 0]], [[2, 1], [-2, 2], [1, 4], [-1, -4]], [[-1, 1], [1, 2], [-2, 4], [-1, -4]], [[-1, 1], [1, 2], [1, 4], [2, -4]], [[0, 0]], [[0, 0]]]}
    • id: 5843044
      {'center': '162.a1.a1', 'center_index': 162, 'center_order': 1, 'counter': 13, 'cyclotomic_n': 9, 'dim': 6, 'faithful': True, 'field': [-1, -3, 0, 1], 'group': '162.13', 'group_counter': 13, 'group_order': 162, 'image': None, 'image_isoclass': '162.13', 'image_order': 162, 'indicator': 1, 'kernel': '162.a1.a1', 'kernel_order': 1, 'label': '162.13.6b3', 'nt': [27, 41], 'q_character': None, 'values': [[[6, 0]], [[0, 0]], [[-3, 0]], [[0, 0]], [[0, 0]], [[0, 0]], [[0, 0]], [[0, 0]], [[-1, 1], [1, 2], [-2, 4], [-1, -4]], [[-1, 1], [1, 2], [1, 4], [2, -4]], [[2, 1], [-2, 2], [1, 4], [-1, -4]], [[0, 0]], [[0, 0]]]}
  • gps_subgroup_search • Show schema
    • id: 62891392
      {'Agroup': False, 'Zgroup': False, 'abelian': False, 'ambient': '162.13', 'ambient_counter': 13, 'ambient_order': 162, 'ambient_tex': '\\He_3.S_3', 'central': False, 'central_factor': False, 'centralizer_order': 1, 'characteristic': True, 'core_order': 162, 'counter': 1, 'cyclic': False, 'direct': True, 'hall': 6, 'label': '162.13.1.a1.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': False, 'minimal': False, 'minimal_normal': False, 'nilpotent': False, 'normal': True, 'old_label': '1.a1.a1', 'outer_equivalence': False, 'perfect': False, 'proper': False, 'quotient': '1.1', 'quotient_Agroup': True, 'quotient_abelian': True, 'quotient_cyclic': True, 'quotient_hash': 1, 'quotient_metabelian': True, 'quotient_nilpotent': True, 'quotient_order': 1, 'quotient_simple': False, 'quotient_solvable': True, 'quotient_supersolvable': True, 'quotient_tex': 'C_1', 'simple': False, 'solvable': True, 'special_labels': ['R', 'L0', 'D0', 'C0'], 'split': True, 'standard_generators': False, 'stem': False, 'subgroup': '162.13', 'subgroup_hash': 13, 'subgroup_order': 162, 'subgroup_tex': '\\He_3.S_3', 'supersolvable': True, 'sylow': 0}
    • id: 62891403
      {'Agroup': False, 'Zgroup': False, 'abelian': False, 'ambient': '162.13', 'ambient_counter': 13, 'ambient_order': 162, 'ambient_tex': '\\He_3.S_3', 'central': False, 'central_factor': False, 'centralizer_order': 3, 'characteristic': True, 'core_order': 81, 'counter': 2, 'cyclic': False, 'direct': False, 'hall': 3, 'label': '162.13.2.a1.a1', 'maximal': True, 'maximal_normal': True, 'metabelian': True, 'metacyclic': False, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': True, 'old_label': '2.a1.a1', 'outer_equivalence': False, 'perfect': False, 'proper': True, 'quotient': '2.1', 'quotient_Agroup': True, 'quotient_abelian': True, 'quotient_cyclic': True, 'quotient_hash': 1, 'quotient_metabelian': True, 'quotient_nilpotent': True, 'quotient_order': 2, 'quotient_simple': True, 'quotient_solvable': True, 'quotient_supersolvable': True, 'quotient_tex': 'C_2', 'simple': False, 'solvable': True, 'special_labels': ['F', 'C1'], 'split': True, 'standard_generators': False, 'stem': False, 'subgroup': '81.8', 'subgroup_hash': 8, 'subgroup_order': 81, 'subgroup_tex': 'C_3.\\He_3', 'supersolvable': True, 'sylow': 3}
    • id: 62891407
      {'Agroup': True, 'Zgroup': False, 'abelian': False, 'ambient': '162.13', 'ambient_counter': 13, 'ambient_order': 162, 'ambient_tex': '\\He_3.S_3', 'central': False, 'central_factor': False, 'centralizer_order': 1, 'characteristic': True, 'core_order': 54, 'counter': 3, 'cyclic': False, 'direct': False, 'hall': 0, 'label': '162.13.3.a1.a1', 'maximal': True, 'maximal_normal': True, 'metabelian': True, 'metacyclic': False, 'minimal': False, 'minimal_normal': False, 'nilpotent': False, 'normal': True, 'old_label': '3.a1.a1', 'outer_equivalence': False, 'perfect': False, 'proper': True, 'quotient': '3.1', 'quotient_Agroup': True, 'quotient_abelian': True, 'quotient_cyclic': True, 'quotient_hash': 1, 'quotient_metabelian': True, 'quotient_nilpotent': True, 'quotient_order': 3, 'quotient_simple': True, 'quotient_solvable': True, 'quotient_supersolvable': True, 'quotient_tex': 'C_3', 'simple': False, 'solvable': True, 'special_labels': [], 'split': True, 'standard_generators': False, 'stem': False, 'subgroup': '54.7', 'subgroup_hash': 7, 'subgroup_order': 54, 'subgroup_tex': 'C_3:D_9', 'supersolvable': True, 'sylow': 0}
    • id: 62891408
      {'Agroup': False, 'Zgroup': False, 'abelian': False, 'ambient': '162.13', 'ambient_counter': 13, 'ambient_order': 162, 'ambient_tex': '\\He_3.S_3', 'central': False, 'central_factor': False, 'centralizer_order': 1, 'characteristic': False, 'core_order': 27, 'counter': 4, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '162.13.3.b1.a1', 'maximal': True, 'maximal_normal': False, 'metabelian': True, 'metacyclic': False, 'minimal': False, 'minimal_normal': False, 'nilpotent': False, 'normal': False, 'old_label': '3.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': 3, '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': '54.5', 'subgroup_hash': 5, 'subgroup_order': 54, 'subgroup_tex': 'C_3^2:C_6', 'supersolvable': True, 'sylow': 0}
    • id: 62891409
      {'Agroup': False, 'Zgroup': False, 'abelian': False, 'ambient': '162.13', 'ambient_counter': 13, 'ambient_order': 162, 'ambient_tex': '\\He_3.S_3', 'central': False, 'central_factor': False, 'centralizer_order': 3, 'characteristic': True, 'core_order': 27, 'counter': 5, 'cyclic': False, 'direct': False, 'hall': 0, 'label': '162.13.6.a1.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': False, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': True, 'old_label': '6.a1.a1', 'outer_equivalence': False, 'perfect': False, 'proper': True, 'quotient': '6.1', 'quotient_Agroup': True, 'quotient_abelian': False, 'quotient_cyclic': False, 'quotient_hash': 1, 'quotient_metabelian': True, 'quotient_nilpotent': False, 'quotient_order': 6, 'quotient_simple': False, 'quotient_solvable': True, 'quotient_supersolvable': True, 'quotient_tex': 'S_3', 'simple': False, 'solvable': True, 'special_labels': [], 'split': False, 'standard_generators': False, 'stem': False, 'subgroup': '27.3', 'subgroup_hash': 3, 'subgroup_order': 27, 'subgroup_tex': '\\He_3', 'supersolvable': True, 'sylow': 0}
    • id: 62891410
      {'Agroup': True, 'Zgroup': False, 'abelian': True, 'ambient': '162.13', 'ambient_counter': 13, 'ambient_order': 162, 'ambient_tex': '\\He_3.S_3', 'central': False, 'central_factor': False, 'centralizer_order': 27, 'characteristic': True, 'core_order': 27, 'counter': 6, 'cyclic': False, 'direct': False, 'hall': 0, 'label': '162.13.6.b1.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': True, 'old_label': '6.b1.a1', 'outer_equivalence': False, 'perfect': False, 'proper': True, 'quotient': '6.2', 'quotient_Agroup': True, 'quotient_abelian': True, 'quotient_cyclic': True, 'quotient_hash': 2, 'quotient_metabelian': True, 'quotient_nilpotent': True, 'quotient_order': 6, 'quotient_simple': False, 'quotient_solvable': True, 'quotient_supersolvable': True, 'quotient_tex': 'C_6', 'simple': False, 'solvable': True, 'special_labels': ['D', 'L1', 'D1', 'C2'], 'split': True, 'standard_generators': False, 'stem': False, 'subgroup': '27.2', 'subgroup_hash': 2, 'subgroup_order': 27, 'subgroup_tex': 'C_3\\times C_9', 'supersolvable': True, 'sylow': 0}
    • id: 62891411
      {'Agroup': False, 'Zgroup': False, 'abelian': False, 'ambient': '162.13', 'ambient_counter': 13, 'ambient_order': 162, 'ambient_tex': '\\He_3.S_3', 'central': False, 'central_factor': False, 'centralizer_order': 3, 'characteristic': False, 'core_order': 9, 'counter': 7, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '162.13.6.c1.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': False, 'old_label': '6.c1.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': 6, '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': '27.4', 'subgroup_hash': 4, 'subgroup_order': 27, 'subgroup_tex': 'C_9:C_3', 'supersolvable': True, 'sylow': 0}
    • id: 62891412
      {'Agroup': True, 'Zgroup': False, 'abelian': False, 'ambient': '162.13', 'ambient_counter': 13, 'ambient_order': 162, 'ambient_tex': '\\He_3.S_3', 'central': False, 'central_factor': False, 'centralizer_order': 3, 'characteristic': False, 'core_order': 3, 'counter': 8, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '162.13.9.a1.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': False, 'minimal_normal': False, 'nilpotent': False, 'normal': False, 'old_label': '9.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': 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': '18.3', 'subgroup_hash': 3, 'subgroup_order': 18, 'subgroup_tex': 'C_3\\times S_3', 'supersolvable': True, 'sylow': 0}
    • id: 62891413
      {'Agroup': True, 'Zgroup': False, 'abelian': False, 'ambient': '162.13', 'ambient_counter': 13, 'ambient_order': 162, 'ambient_tex': '\\He_3.S_3', 'central': False, 'central_factor': False, 'centralizer_order': 1, 'characteristic': False, 'core_order': 9, 'counter': 9, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '162.13.9.b1.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, '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': '18.4', 'subgroup_hash': 4, 'subgroup_order': 18, 'subgroup_tex': 'C_3:S_3', 'supersolvable': True, 'sylow': 0}
    • id: 62891393
      {'Agroup': True, 'Zgroup': True, 'abelian': False, 'ambient': '162.13', 'ambient_counter': 13, 'ambient_order': 162, 'ambient_tex': '\\He_3.S_3', 'central': False, 'central_factor': False, 'centralizer_order': 1, 'characteristic': False, 'core_order': 3, 'counter': 10, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '162.13.9.c1.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': False, 'minimal_normal': False, 'nilpotent': False, 'normal': False, 'old_label': '9.c1.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': '18.1', 'subgroup_hash': 1, 'subgroup_order': 18, 'subgroup_tex': 'D_9', 'supersolvable': True, 'sylow': 0}
    • id: 62891394
      {'Agroup': True, 'Zgroup': False, 'abelian': True, 'ambient': '162.13', 'ambient_counter': 13, 'ambient_order': 162, 'ambient_tex': '\\He_3.S_3', 'central': False, 'central_factor': False, 'centralizer_order': 27, 'characteristic': True, 'core_order': 9, 'counter': 11, 'cyclic': False, 'direct': False, 'hall': 0, 'label': '162.13.18.a1.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': True, 'old_label': '18.a1.a1', 'outer_equivalence': False, 'perfect': False, 'proper': True, 'quotient': '18.3', 'quotient_Agroup': True, 'quotient_abelian': False, 'quotient_cyclic': False, 'quotient_hash': 3, 'quotient_metabelian': True, 'quotient_nilpotent': False, 'quotient_order': 18, 'quotient_simple': False, 'quotient_solvable': True, 'quotient_supersolvable': True, 'quotient_tex': 'C_3\\times S_3', 'simple': False, 'solvable': True, 'special_labels': ['Phi', 'C3'], 'split': False, 'standard_generators': False, 'stem': False, 'subgroup': '9.2', 'subgroup_hash': 2, 'subgroup_order': 9, 'subgroup_tex': 'C_3^2', 'supersolvable': True, 'sylow': 0}
    • id: 62891395
      {'Agroup': True, 'Zgroup': False, 'abelian': True, 'ambient': '162.13', 'ambient_counter': 13, 'ambient_order': 162, 'ambient_tex': '\\He_3.S_3', 'central': False, 'central_factor': False, 'centralizer_order': 9, 'characteristic': False, 'core_order': 3, 'counter': 12, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '162.13.18.b1.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': False, 'old_label': '18.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': 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': '9.2', 'subgroup_hash': 2, 'subgroup_order': 9, 'subgroup_tex': 'C_3^2', 'supersolvable': True, 'sylow': 0}
    • id: 62891396
      {'Agroup': True, 'Zgroup': True, 'abelian': True, 'ambient': '162.13', 'ambient_counter': 13, 'ambient_order': 162, 'ambient_tex': '\\He_3.S_3', 'central': False, 'central_factor': False, 'centralizer_order': 27, 'characteristic': False, 'core_order': 3, 'counter': 13, 'cyclic': True, 'direct': None, 'hall': 0, 'label': '162.13.18.c1.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': False, 'old_label': '18.c1.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': 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': '9.1', 'subgroup_hash': 1, 'subgroup_order': 9, 'subgroup_tex': 'C_9', 'supersolvable': True, 'sylow': 0}
    • id: 62891397
      {'Agroup': True, 'Zgroup': True, 'abelian': True, 'ambient': '162.13', 'ambient_counter': 13, 'ambient_order': 162, 'ambient_tex': '\\He_3.S_3', 'central': False, 'central_factor': False, 'centralizer_order': 9, 'characteristic': False, 'core_order': 3, 'counter': 14, 'cyclic': True, 'direct': None, 'hall': 0, 'label': '162.13.18.d1.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': False, 'old_label': '18.d1.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': 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': '9.1', 'subgroup_hash': 1, 'subgroup_order': 9, 'subgroup_tex': 'C_9', 'supersolvable': True, 'sylow': 0}
    • id: 62891398
      {'Agroup': True, 'Zgroup': True, 'abelian': True, 'ambient': '162.13', 'ambient_counter': 13, 'ambient_order': 162, 'ambient_tex': '\\He_3.S_3', 'central': False, 'central_factor': False, 'centralizer_order': 6, 'characteristic': False, 'core_order': 1, 'counter': 15, 'cyclic': True, 'direct': None, 'hall': 0, 'label': '162.13.27.a1.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': False, 'old_label': '27.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': 27, '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': '6.2', 'subgroup_hash': 2, 'subgroup_order': 6, 'subgroup_tex': 'C_6', 'supersolvable': True, 'sylow': 0}
    • id: 62891399
      {'Agroup': True, 'Zgroup': True, 'abelian': False, 'ambient': '162.13', 'ambient_counter': 13, 'ambient_order': 162, 'ambient_tex': '\\He_3.S_3', 'central': False, 'central_factor': False, 'centralizer_order': 3, 'characteristic': False, 'core_order': 3, 'counter': 16, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '162.13.27.b1.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': False, 'minimal_normal': False, 'nilpotent': False, 'normal': False, 'old_label': '27.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': 27, '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': '6.1', 'subgroup_hash': 1, 'subgroup_order': 6, 'subgroup_tex': 'S_3', 'supersolvable': True, 'sylow': 0}
    • id: 62891400
      {'Agroup': True, 'Zgroup': True, 'abelian': False, 'ambient': '162.13', 'ambient_counter': 13, 'ambient_order': 162, 'ambient_tex': '\\He_3.S_3', 'central': False, 'central_factor': False, 'centralizer_order': 1, 'characteristic': False, 'core_order': 1, 'counter': 17, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '162.13.27.c1.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': False, 'minimal_normal': False, 'nilpotent': False, 'normal': False, 'old_label': '27.c1.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': 27, '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': '6.1', 'subgroup_hash': 1, 'subgroup_order': 6, 'subgroup_tex': 'S_3', 'supersolvable': True, 'sylow': 0}
    • id: 62891401
      {'Agroup': True, 'Zgroup': True, 'abelian': True, 'ambient': '162.13', 'ambient_counter': 13, 'ambient_order': 162, 'ambient_tex': '\\He_3.S_3', 'central': False, 'central_factor': False, 'centralizer_order': 81, 'characteristic': True, 'core_order': 3, 'counter': 18, 'cyclic': True, 'direct': False, 'hall': 0, 'label': '162.13.54.a1.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': True, 'minimal_normal': True, 'nilpotent': True, 'normal': True, 'old_label': '54.a1.a1', 'outer_equivalence': False, 'perfect': False, 'proper': True, 'quotient': '54.5', 'quotient_Agroup': False, 'quotient_abelian': False, 'quotient_cyclic': False, 'quotient_hash': 5, 'quotient_metabelian': True, 'quotient_nilpotent': False, 'quotient_order': 54, 'quotient_simple': False, 'quotient_solvable': True, 'quotient_supersolvable': True, 'quotient_tex': 'C_3^2:C_6', 'simple': True, 'solvable': True, 'special_labels': ['S', 'C4'], 'split': False, 'standard_generators': False, 'stem': False, 'subgroup': '3.1', 'subgroup_hash': 1, 'subgroup_order': 3, 'subgroup_tex': 'C_3', 'supersolvable': True, 'sylow': 0}
    • id: 62891402
      {'Agroup': True, 'Zgroup': True, 'abelian': True, 'ambient': '162.13', 'ambient_counter': 13, 'ambient_order': 162, 'ambient_tex': '\\He_3.S_3', 'central': False, 'central_factor': False, 'centralizer_order': 27, 'characteristic': False, 'core_order': 1, 'counter': 19, 'cyclic': True, 'direct': None, 'hall': 0, 'label': '162.13.54.b1.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': True, 'minimal_normal': False, 'nilpotent': True, 'normal': False, 'old_label': '54.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': 54, 'quotient_simple': None, 'quotient_solvable': None, 'quotient_supersolvable': None, 'quotient_tex': None, 'simple': True, 'solvable': True, 'special_labels': [], 'split': None, 'standard_generators': False, 'stem': False, 'subgroup': '3.1', 'subgroup_hash': 1, 'subgroup_order': 3, 'subgroup_tex': 'C_3', 'supersolvable': True, 'sylow': 0}
    • id: 62891404
      {'Agroup': True, 'Zgroup': True, 'abelian': True, 'ambient': '162.13', 'ambient_counter': 13, 'ambient_order': 162, 'ambient_tex': '\\He_3.S_3', 'central': False, 'central_factor': False, 'centralizer_order': 18, 'characteristic': False, 'core_order': 1, 'counter': 20, 'cyclic': True, 'direct': None, 'hall': 0, 'label': '162.13.54.c1.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': True, 'minimal_normal': False, 'nilpotent': True, 'normal': False, 'old_label': '54.c1.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': 54, 'quotient_simple': None, 'quotient_solvable': None, 'quotient_supersolvable': None, 'quotient_tex': None, 'simple': True, 'solvable': True, 'special_labels': [], 'split': None, 'standard_generators': False, 'stem': False, 'subgroup': '3.1', 'subgroup_hash': 1, 'subgroup_order': 3, 'subgroup_tex': 'C_3', 'supersolvable': True, 'sylow': 0}
    • id: 62891405
      {'Agroup': True, 'Zgroup': True, 'abelian': True, 'ambient': '162.13', 'ambient_counter': 13, 'ambient_order': 162, 'ambient_tex': '\\He_3.S_3', 'central': False, 'central_factor': False, 'centralizer_order': 6, 'characteristic': False, 'core_order': 1, 'counter': 21, 'cyclic': True, 'direct': None, 'hall': 2, 'label': '162.13.81.a1.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': True, 'minimal_normal': False, 'nilpotent': True, 'normal': False, 'old_label': '81.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': 81, 'quotient_simple': None, 'quotient_solvable': None, 'quotient_supersolvable': None, 'quotient_tex': None, 'simple': True, 'solvable': True, 'special_labels': [], 'split': None, 'standard_generators': False, 'stem': False, 'subgroup': '2.1', 'subgroup_hash': 1, 'subgroup_order': 2, 'subgroup_tex': 'C_2', 'supersolvable': True, 'sylow': 2}
    • id: 62891406
      {'Agroup': True, 'Zgroup': True, 'abelian': True, 'ambient': '162.13', 'ambient_counter': 13, 'ambient_order': 162, 'ambient_tex': '\\He_3.S_3', 'central': True, 'central_factor': False, 'centralizer_order': 162, 'characteristic': True, 'core_order': 1, 'counter': 22, 'cyclic': True, 'direct': True, 'hall': 1, 'label': '162.13.162.a1.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': True, 'old_label': '162.a1.a1', 'outer_equivalence': False, 'perfect': True, 'proper': False, 'quotient': '162.13', 'quotient_Agroup': False, 'quotient_abelian': False, 'quotient_cyclic': False, 'quotient_hash': 13, 'quotient_metabelian': True, 'quotient_nilpotent': False, 'quotient_order': 162, 'quotient_simple': False, 'quotient_solvable': True, 'quotient_supersolvable': True, 'quotient_tex': '\\He_3.S_3', 'simple': False, 'solvable': True, 'special_labels': ['Z', 'PC', 'U0', 'D2', 'C5'], 'split': True, 'standard_generators': False, 'stem': True, 'subgroup': '1.1', 'subgroup_hash': 1, 'subgroup_order': 1, 'subgroup_tex': 'C_1', 'supersolvable': True, 'sylow': 1}
  • gps_subgroup_data • Show schema
    • id: 62891392
      {'ambient': '162.13', 'aut_centralizer_order': None, 'aut_label': '1.a1', 'aut_quo_index': None, 'aut_stab_index': None, 'aut_weyl_group': None, 'aut_weyl_index': None, 'centralizer': '162.a1.a1', 'complements': ['162.a1.a1'], 'conjugacy_class_count': 1, 'contained_in': [], 'contains': ['2.a1.a1', '3.a1.a1', '3.b1.a1'], 'core': '1.a1.a1', 'coset_action_label': None, 'count': 1, 'diagramx': [5310, 6076, 4249, 4234, 4393, 4532, 4393, 5418], 'generators': [3, 54, 2, 6, 18], 'label': '162.13.1.a1.a1', 'mobius_quo': 0, 'mobius_sub': 1, 'normal_closure': '1.a1.a1', 'normal_contained_in': [], 'normal_contains': ['2.a1.a1', '3.a1.a1'], 'normalizer': '1.a1.a1', 'old_label': '1.a1.a1', 'projective_image': '162.13', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '1.a1.a1', 'subgroup_fusion': None, 'weyl_group': '162.13'}
    • id: 62891393
      {'ambient': '162.13', 'aut_centralizer_order': 2, 'aut_label': '9.c1', 'aut_quo_index': None, 'aut_stab_index': 9, 'aut_weyl_group': '54.6', 'aut_weyl_index': 18, 'centralizer': '162.a1.a1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['3.a1.a1'], 'contains': ['18.c1.a1', '27.b1.a1'], 'core': '54.a1.a1', 'coset_action_label': None, 'count': 9, 'diagramx': [1637, -1, 507, -1, 4393, -1, 4393, -1], 'generators': [3, 18, 54], 'label': '162.13.9.c1.a1', 'mobius_quo': None, 'mobius_sub': 0, 'normal_closure': '3.a1.a1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '9.c1.a1', 'old_label': '9.c1.a1', 'projective_image': '162.13', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '9.c1.a1', 'subgroup_fusion': None, 'weyl_group': '18.1'}
    • id: 62891394
      {'ambient': '162.13', 'aut_centralizer_order': 81, 'aut_label': '18.a1', 'aut_quo_index': 1, 'aut_stab_index': 1, 'aut_weyl_group': '12.4', 'aut_weyl_index': 81, 'centralizer': '6.b1.a1', 'complements': [], 'conjugacy_class_count': 1, 'contained_in': ['6.a1.a1', '6.b1.a1', '6.c1.a1', '9.b1.a1'], 'contains': ['54.a1.a1', '54.b1.a1'], 'core': '18.a1.a1', 'coset_action_label': None, 'count': 1, 'diagramx': [4291, 5001, 4342, 4999, 4203, 4380, 4203, 5620], 'generators': [6, 54], 'label': '162.13.18.a1.a1', 'mobius_quo': 0, 'mobius_sub': -3, 'normal_closure': '18.a1.a1', 'normal_contained_in': ['6.a1.a1', '6.b1.a1'], 'normal_contains': ['54.a1.a1'], 'normalizer': '1.a1.a1', 'old_label': '18.a1.a1', 'projective_image': '162.13', 'quotient_action_image': '6.2', 'quotient_action_kernel': '3.1', 'quotient_action_kernel_order': 3, 'quotient_fusion': None, 'short_label': '18.a1.a1', 'subgroup_fusion': None, 'weyl_group': '6.2'}
    • id: 62891395
      {'ambient': '162.13', 'aut_centralizer_order': 27, 'aut_label': '18.b1', 'aut_quo_index': None, 'aut_stab_index': 3, 'aut_weyl_group': '12.4', 'aut_weyl_index': 81, 'centralizer': '18.b1.a1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['6.a1.a1', '9.a1.a1'], 'contains': ['54.a1.a1', '54.c1.a1'], 'core': '54.a1.a1', 'coset_action_label': None, 'count': 3, 'diagramx': [8263, -1, 8309, -1, 8096, -1, 8096, -1], 'generators': [2, 54], 'label': '162.13.18.b1.a1', 'mobius_quo': None, 'mobius_sub': 0, 'normal_closure': '6.a1.a1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '3.b1.a1', 'old_label': '18.b1.a1', 'projective_image': '162.13', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '18.b1.a1', 'subgroup_fusion': None, 'weyl_group': '6.1'}
    • id: 62891396
      {'ambient': '162.13', 'aut_centralizer_order': 54, 'aut_label': '18.c1', 'aut_quo_index': None, 'aut_stab_index': 3, 'aut_weyl_group': '6.2', 'aut_weyl_index': 162, 'centralizer': '6.b1.a1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['6.b1.a1', '9.c1.a1'], 'contains': ['54.a1.a1'], 'core': '54.a1.a1', 'coset_action_label': None, 'count': 3, 'diagramx': [2735, -1, 1542, -1, 2877, -1, 2877, -1], 'generators': [126], 'label': '162.13.18.c1.a1', 'mobius_quo': None, 'mobius_sub': 0, 'normal_closure': '6.b1.a1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '3.a1.a1', 'old_label': '18.c1.a1', 'projective_image': '162.13', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '18.c1.a1', 'subgroup_fusion': None, 'weyl_group': '2.1'}
    • id: 62891397
      {'ambient': '162.13', 'aut_centralizer_order': 27, 'aut_label': '18.d1', 'aut_quo_index': None, 'aut_stab_index': 6, 'aut_weyl_group': '6.2', 'aut_weyl_index': 162, 'centralizer': '18.d1.a1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['6.c1.a1'], 'contains': ['54.a1.a1'], 'core': '54.a1.a1', 'coset_action_label': None, 'count': 6, 'diagramx': [538, -1, 2895, -1, 1275, -1, 1275, -1], 'generators': [128], 'label': '162.13.18.d1.a1', 'mobius_quo': None, 'mobius_sub': 0, 'normal_closure': '2.a1.a1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '6.c1.a1', 'old_label': '18.d1.a1', 'projective_image': '162.13', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '18.d1.a1', 'subgroup_fusion': None, 'weyl_group': '3.1'}
    • id: 62891398
      {'ambient': '162.13', 'aut_centralizer_order': 18, 'aut_label': '27.a1', 'aut_quo_index': None, 'aut_stab_index': 27, 'aut_weyl_group': '2.1', 'aut_weyl_index': 486, 'centralizer': '27.a1.a1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['9.a1.a1'], 'contains': ['54.c1.a1', '81.a1.a1'], 'core': '162.a1.a1', 'coset_action_label': '27T72', 'count': 27, 'diagramx': [9462, -1, 9493, -1, 9423, -1, 9423, -1], 'generators': [3, 2], 'label': '162.13.27.a1.a1', 'mobius_quo': None, 'mobius_sub': 0, 'normal_closure': '1.a1.a1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '27.a1.a1', 'old_label': '27.a1.a1', 'projective_image': '162.13', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '27.a1.a1', 'subgroup_fusion': None, 'weyl_group': '1.1'}
    • id: 62891399
      {'ambient': '162.13', 'aut_centralizer_order': 18, 'aut_label': '27.b1', 'aut_quo_index': None, 'aut_stab_index': 9, 'aut_weyl_group': '6.1', 'aut_weyl_index': 162, 'centralizer': '54.c1.a1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['9.a1.a1', '9.b1.a1', '9.c1.a1'], 'contains': ['54.a1.a1', '81.a1.a1'], 'core': '54.a1.a1', 'coset_action_label': None, 'count': 9, 'diagramx': [4312, -1, 5358, -1, 5530, -1, 5530, -1], 'generators': [3, 54], 'label': '162.13.27.b1.a1', 'mobius_quo': None, 'mobius_sub': 0, 'normal_closure': '3.a1.a1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '9.a1.a1', 'old_label': '27.b1.a1', 'projective_image': '162.13', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '27.b1.a1', 'subgroup_fusion': None, 'weyl_group': '6.1'}
    • id: 62891400
      {'ambient': '162.13', 'aut_centralizer_order': 6, 'aut_label': '27.c1', 'aut_quo_index': None, 'aut_stab_index': 27, 'aut_weyl_group': '6.1', 'aut_weyl_index': 162, 'centralizer': '162.a1.a1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['9.b1.a1'], 'contains': ['54.b1.a1', '81.a1.a1'], 'core': '162.a1.a1', 'coset_action_label': '27T41', 'count': 27, 'diagramx': [7265, -1, 7425, -1, 6770, -1, 6770, -1], 'generators': [3, 6], 'label': '162.13.27.c1.a1', 'mobius_quo': None, 'mobius_sub': 0, 'normal_closure': '3.a1.a1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '27.c1.a1', 'old_label': '27.c1.a1', 'projective_image': '162.13', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '27.c1.a1', 'subgroup_fusion': None, 'weyl_group': '6.1'}
    • id: 62891401
      {'ambient': '162.13', 'aut_centralizer_order': 486, 'aut_label': '54.a1', 'aut_quo_index': 1, 'aut_stab_index': 1, 'aut_weyl_group': '2.1', 'aut_weyl_index': 486, 'centralizer': '2.a1.a1', 'complements': [], 'conjugacy_class_count': 1, 'contained_in': ['18.a1.a1', '18.b1.a1', '18.c1.a1', '18.d1.a1', '27.b1.a1'], 'contains': ['162.a1.a1'], 'core': '54.a1.a1', 'coset_action_label': None, 'count': 1, 'diagramx': [3592, 5001, 3966, 4999, 4255, 4380, 4255, 5620], 'generators': [54], 'label': '162.13.54.a1.a1', 'mobius_quo': -1, 'mobius_sub': 0, 'normal_closure': '54.a1.a1', 'normal_contained_in': ['18.a1.a1'], 'normal_contains': ['162.a1.a1'], 'normalizer': '1.a1.a1', 'old_label': '54.a1.a1', 'projective_image': '162.13', 'quotient_action_image': '2.1', 'quotient_action_kernel': '27.3', 'quotient_action_kernel_order': 27, 'quotient_fusion': None, 'short_label': '54.a1.a1', 'subgroup_fusion': None, 'weyl_group': '2.1'}
    • id: 62891402
      {'ambient': '162.13', 'aut_centralizer_order': 162, 'aut_label': '54.b1', 'aut_quo_index': None, 'aut_stab_index': 3, 'aut_weyl_group': '2.1', 'aut_weyl_index': 486, 'centralizer': '6.b1.a1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['18.a1.a1', '27.c1.a1'], 'contains': ['162.a1.a1'], 'core': '162.a1.a1', 'coset_action_label': None, 'count': 3, 'diagramx': [6167, -1, 5677, -1, 5530, -1, 5530, -1], 'generators': [6], 'label': '162.13.54.b1.a1', 'mobius_quo': None, 'mobius_sub': 0, 'normal_closure': '18.a1.a1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '3.a1.a1', 'old_label': '54.b1.a1', 'projective_image': '162.13', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '54.b1.a1', 'subgroup_fusion': None, 'weyl_group': '2.1'}
    • id: 62891403
      {'ambient': '162.13', 'aut_centralizer_order': 3, 'aut_label': '2.a1', 'aut_quo_index': 1, 'aut_stab_index': 1, 'aut_weyl_group': '324.117', 'aut_weyl_index': 3, 'centralizer': '54.a1.a1', 'complements': ['81.a1.a1'], 'conjugacy_class_count': 1, 'contained_in': ['1.a1.a1'], 'contains': ['6.a1.a1', '6.b1.a1', '6.c1.a1'], 'core': '2.a1.a1', 'coset_action_label': None, 'count': 1, 'diagramx': [5310, 3861, 4249, 5810, 2670, 2101, 2670, 7899], 'generators': [2, 126], 'label': '162.13.2.a1.a1', 'mobius_quo': 0, 'mobius_sub': -1, 'normal_closure': '2.a1.a1', 'normal_contained_in': ['1.a1.a1'], 'normal_contains': ['6.a1.a1', '6.b1.a1'], 'normalizer': '1.a1.a1', 'old_label': '2.a1.a1', 'projective_image': '162.13', 'quotient_action_image': '2.1', 'quotient_action_kernel': '1.1', 'quotient_action_kernel_order': 1, 'quotient_fusion': None, 'short_label': '2.a1.a1', 'subgroup_fusion': None, 'weyl_group': '54.5'}
    • id: 62891404
      {'ambient': '162.13', 'aut_centralizer_order': 54, 'aut_label': '54.c1', 'aut_quo_index': None, 'aut_stab_index': 9, 'aut_weyl_group': '2.1', 'aut_weyl_index': 486, 'centralizer': '9.a1.a1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['18.b1.a1', '27.a1.a1'], 'contains': ['162.a1.a1'], 'core': '162.a1.a1', 'coset_action_label': None, 'count': 9, 'diagramx': [8364, -1, 8459, -1, 8096, -1, 8096, -1], 'generators': [2], 'label': '162.13.54.c1.a1', 'mobius_quo': None, 'mobius_sub': 0, 'normal_closure': '6.a1.a1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '9.a1.a1', 'old_label': '54.c1.a1', 'projective_image': '162.13', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '54.c1.a1', 'subgroup_fusion': None, 'weyl_group': '1.1'}
    • id: 62891405
      {'ambient': '162.13', 'aut_centralizer_order': 36, 'aut_label': '81.a1', 'aut_quo_index': None, 'aut_stab_index': 27, 'aut_weyl_group': '1.1', 'aut_weyl_index': 972, 'centralizer': '27.a1.a1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['27.a1.a1', '27.b1.a1', '27.c1.a1'], 'contains': ['162.a1.a1'], 'core': '162.a1.a1', 'coset_action_label': None, 'count': 27, 'diagramx': [7265, -1, 6899, -1, 6770, -1, 6770, -1], 'generators': [3], 'label': '162.13.81.a1.a1', 'mobius_quo': None, 'mobius_sub': 0, 'normal_closure': '3.a1.a1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '27.a1.a1', 'old_label': '81.a1.a1', 'projective_image': '162.13', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '81.a1.a1', 'subgroup_fusion': None, 'weyl_group': '1.1'}
    • id: 62891406
      {'ambient': '162.13', 'aut_centralizer_order': None, 'aut_label': '162.a1', 'aut_quo_index': None, 'aut_stab_index': None, 'aut_weyl_group': None, 'aut_weyl_index': None, 'centralizer': '1.a1.a1', 'complements': ['1.a1.a1'], 'conjugacy_class_count': 1, 'contained_in': ['54.a1.a1', '54.b1.a1', '54.c1.a1', '81.a1.a1'], 'contains': [], 'core': '162.a1.a1', 'coset_action_label': None, 'count': 1, 'diagramx': [6726, 5001, 6034, 4999, 6150, 4380, 6150, 5620], 'generators': [], 'label': '162.13.162.a1.a1', 'mobius_quo': 1, 'mobius_sub': 0, 'normal_closure': '162.a1.a1', 'normal_contained_in': ['54.a1.a1'], 'normal_contains': [], 'normalizer': '1.a1.a1', 'old_label': '162.a1.a1', 'projective_image': '162.13', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '162.a1.a1', 'subgroup_fusion': None, 'weyl_group': '1.1'}
    • id: 62891407
      {'ambient': '162.13', 'aut_centralizer_order': 1, 'aut_label': '3.a1', 'aut_quo_index': 1, 'aut_stab_index': 1, 'aut_weyl_group': '972.446', 'aut_weyl_index': 1, 'centralizer': '162.a1.a1', 'complements': ['54.c1.a1'], 'conjugacy_class_count': 1, 'contained_in': ['1.a1.a1'], 'contains': ['6.b1.a1', '9.b1.a1', '9.c1.a1'], 'core': '3.a1.a1', 'coset_action_label': None, 'count': 1, 'diagramx': [3913, 8291, 2162, 2658, 4393, 7013, 4393, 2987], 'generators': [3, 6, 18, 54], 'label': '162.13.3.a1.a1', 'mobius_quo': 0, 'mobius_sub': -1, 'normal_closure': '3.a1.a1', 'normal_contained_in': ['1.a1.a1'], 'normal_contains': ['6.b1.a1'], 'normalizer': '1.a1.a1', 'old_label': '3.a1.a1', 'projective_image': '162.13', 'quotient_action_image': '3.1', 'quotient_action_kernel': '1.1', 'quotient_action_kernel_order': 1, 'quotient_fusion': None, 'short_label': '3.a1.a1', 'subgroup_fusion': None, 'weyl_group': '162.13'}
    • id: 62891408
      {'ambient': '162.13', 'aut_centralizer_order': 3, 'aut_label': '3.b1', 'aut_quo_index': None, 'aut_stab_index': 3, 'aut_weyl_group': '108.17', 'aut_weyl_index': 9, 'centralizer': '162.a1.a1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['1.a1.a1'], 'contains': ['6.a1.a1', '9.a1.a1', '9.b1.a1'], 'core': '6.a1.a1', 'coset_action_label': None, 'count': 3, 'diagramx': [7605, -1, 6804, -1, 6546, -1, 6546, -1], 'generators': [3, 2, 6, 54], 'label': '162.13.3.b1.a1', 'mobius_quo': None, 'mobius_sub': -1, 'normal_closure': '1.a1.a1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '3.b1.a1', 'old_label': '3.b1.a1', 'projective_image': '162.13', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '3.b1.a1', 'subgroup_fusion': None, 'weyl_group': '54.5'}
    • id: 62891409
      {'ambient': '162.13', 'aut_centralizer_order': 9, 'aut_label': '6.a1', 'aut_quo_index': 1, 'aut_stab_index': 1, 'aut_weyl_group': '108.17', 'aut_weyl_index': 9, 'centralizer': '54.a1.a1', 'complements': [], 'conjugacy_class_count': 1, 'contained_in': ['2.a1.a1', '3.b1.a1'], 'contains': ['18.a1.a1', '18.b1.a1'], 'core': '6.a1.a1', 'coset_action_label': None, 'count': 1, 'diagramx': [6746, 1709, 7219, 7342, 5633, 1747, 5633, 8253], 'generators': [2, 6], 'label': '162.13.6.a1.a1', 'mobius_quo': 0, 'mobius_sub': 3, 'normal_closure': '6.a1.a1', 'normal_contained_in': ['2.a1.a1'], 'normal_contains': ['18.a1.a1'], 'normalizer': '1.a1.a1', 'old_label': '6.a1.a1', 'projective_image': '162.13', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '6.a1.a1', 'subgroup_fusion': None, 'weyl_group': '54.5'}
    • id: 62891410
      {'ambient': '162.13', 'aut_centralizer_order': 27, 'aut_label': '6.b1', 'aut_quo_index': 1, 'aut_stab_index': 1, 'aut_weyl_group': '36.12', 'aut_weyl_index': 27, 'centralizer': '6.b1.a1', 'complements': ['27.a1.a1'], 'conjugacy_class_count': 1, 'contained_in': ['2.a1.a1', '3.a1.a1'], 'contains': ['18.a1.a1', '18.c1.a1'], 'core': '6.b1.a1', 'coset_action_label': None, 'count': 1, 'diagramx': [3913, 8291, 2105, 2658, 2601, 7013, 2601, 2987], 'generators': [6, 126], 'label': '162.13.6.b1.a1', 'mobius_quo': 0, 'mobius_sub': 1, 'normal_closure': '6.b1.a1', 'normal_contained_in': ['2.a1.a1', '3.a1.a1'], 'normal_contains': ['18.a1.a1'], 'normalizer': '1.a1.a1', 'old_label': '6.b1.a1', 'projective_image': '162.13', 'quotient_action_image': '6.2', 'quotient_action_kernel': '1.1', 'quotient_action_kernel_order': 1, 'quotient_fusion': None, 'short_label': '6.b1.a1', 'subgroup_fusion': None, 'weyl_group': '6.2'}
    • id: 62891411
      {'ambient': '162.13', 'aut_centralizer_order': 9, 'aut_label': '6.c1', 'aut_quo_index': None, 'aut_stab_index': 2, 'aut_weyl_group': '54.8', 'aut_weyl_index': 18, 'centralizer': '54.a1.a1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['2.a1.a1'], 'contains': ['18.a1.a1', '18.d1.a1'], 'core': '18.a1.a1', 'coset_action_label': None, 'count': 2, 'diagramx': [878, -1, 4249, -1, 637, -1, 637, -1], 'generators': [128, 6], 'label': '162.13.6.c1.a1', 'mobius_quo': None, 'mobius_sub': 0, 'normal_closure': '2.a1.a1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '2.a1.a1', 'old_label': '6.c1.a1', 'projective_image': '162.13', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '6.c1.a1', 'subgroup_fusion': None, 'weyl_group': '27.3'}
    • id: 62891412
      {'ambient': '162.13', 'aut_centralizer_order': 9, 'aut_label': '9.a1', 'aut_quo_index': None, 'aut_stab_index': 9, 'aut_weyl_group': '12.4', 'aut_weyl_index': 81, 'centralizer': '54.c1.a1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['3.b1.a1'], 'contains': ['18.b1.a1', '27.a1.a1', '27.b1.a1'], 'core': '54.a1.a1', 'coset_action_label': None, 'count': 9, 'diagramx': [9341, -1, 9042, -1, 8992, -1, 8992, -1], 'generators': [3, 2, 54], 'label': '162.13.9.a1.a1', 'mobius_quo': None, 'mobius_sub': 0, 'normal_closure': '1.a1.a1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '9.a1.a1', 'old_label': '9.a1.a1', 'projective_image': '162.13', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '9.a1.a1', 'subgroup_fusion': None, 'weyl_group': '6.1'}
    • id: 62891413
      {'ambient': '162.13', 'aut_centralizer_order': 3, 'aut_label': '9.b1', 'aut_quo_index': None, 'aut_stab_index': 3, 'aut_weyl_group': '108.17', 'aut_weyl_index': 9, 'centralizer': '162.a1.a1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['3.a1.a1', '3.b1.a1'], 'contains': ['18.a1.a1', '27.b1.a1', '27.c1.a1'], 'core': '18.a1.a1', 'coset_action_label': None, 'count': 3, 'diagramx': [6307, -1, 5470, -1, 7046, -1, 7046, -1], 'generators': [3, 6, 54], 'label': '162.13.9.b1.a1', 'mobius_quo': None, 'mobius_sub': 1, 'normal_closure': '3.a1.a1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '3.b1.a1', 'old_label': '9.b1.a1', 'projective_image': '162.13', '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': '54.5'}