Formats: - HTML - YAML - JSON - 2026-09-20T20:03:11.206045
  • gps_groups • Show schema
    {'Agroup': False, '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': 32, 'aut_gen_orders': [2, 8, 32], 'aut_gens': [[1, 2], [31, 62], [21, 6], [51, 34]], 'aut_group': '512.58302', 'aut_hash': 7772715459544409331, 'aut_nilpotency_class': 5, 'aut_nilpotent': True, 'aut_order': 512, 'aut_permdeg': 32, 'aut_perms': [209110625063666949217196394943844509, 166831302240608821283181734550591682, 192584446486825221723391051205525821], 'aut_phi_ratio': 16.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 16, 2, 1], [4, 2, 1, 1], [8, 2, 2, 1], [16, 2, 4, 1], [32, 2, 8, 1]], 'aut_supersolvable': True, 'aut_tex': 'D_{32}:C_8', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 2, 'autcent_group': '4.2', 'autcent_hash': 2, 'autcent_nilpotent': True, 'autcent_order': 4, 'autcent_solvable': True, 'autcent_split': False, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^2', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 16, 'autcentquo_group': '128.913', 'autcentquo_hash': 913, 'autcentquo_nilpotent': True, 'autcentquo_order': 128, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'D_{16}:C_4', 'cc_stats': [[1, 1, 1], [2, 1, 1], [2, 16, 2], [4, 2, 1], [8, 2, 2], [16, 2, 4], [32, 2, 8]], 'center_label': '2.1', 'center_order': 2, 'central_product': False, 'central_quotient': '32.18', 'commutator_count': 1, 'commutator_label': '16.1', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '2.1', '2.1'], 'composition_length': 6, 'conjugacy_classes_known': True, 'counter': 52, 'cyclic': False, 'derived_length': 2, 'dihedral': True, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 16, 1, 2], [4, 2, 1, 1], [8, 2, 2, 1], [16, 2, 4, 1], [32, 2, 8, 1]], 'element_repr_type': 'PC', 'elementary': 2, 'eulerian_function': 3, 'exponent': 32, 'exponents_of_order': [6], 'factors_of_aut_order': [2], 'factors_of_order': [2], 'faithful_reps': [[2, 1, 8]], 'familial': True, 'frattini_label': '16.1', 'frattini_quotient': '4.2', 'hash': 52, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 16, 'inner_gen_orders': [2, 16], 'inner_gens': [[1, 62], [5, 2]], 'inner_hash': 18, 'inner_nilpotent': True, 'inner_order': 32, 'inner_split': False, 'inner_tex': 'D_{16}', 'inner_used': [1, 2], 'irrC_degree': 2, 'irrQ_degree': 16, 'irrQ_dim': 16, 'irrR_degree': 2, 'irrep_stats': [[1, 4], [2, 15]], 'label': '64.52', 'linC_count': 8, 'linC_degree': 2, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 16, 'linQ_degree_count': 1, 'linQ_dim': 16, 'linQ_dim_count': 1, 'linR_count': 8, 'linR_degree': 2, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'D32', 'ngens': 2, 'nilpotency_class': 5, 'nilpotent': True, 'normal_counts': [0, 0, 0, 0, 0, 0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 7, 'number_characteristic_subgroups': 7, 'number_conjugacy_classes': 19, 'number_divisions': 8, 'number_normal_subgroups': 9, 'number_subgroup_autclasses': 12, 'number_subgroup_classes': 17, 'number_subgroups': 69, 'old_label': None, 'order': 64, 'order_factorization_type': 3, 'order_stats': [[1, 1], [2, 33], [4, 2], [8, 4], [16, 8], [32, 16]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 8, 'outer_gen_orders': [2, 8], 'outer_gen_pows': [2, 0], 'outer_gens': [[3, 2], [1, 10]], 'outer_group': '16.5', 'outer_hash': 5, 'outer_nilpotent': True, 'outer_order': 16, 'outer_permdeg': 10, 'outer_perms': [362880, 6057], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2\\times C_8', 'pc_rank': 2, 'perfect': False, 'permutation_degree': 32, 'pgroup': 2, 'primary_abelian_invariants': [2, 2], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 1], [4, 1], [8, 1], [16, 1]], 'representations': {'PC': {'code': 4836004266424402430, 'gens': [1, 2], 'pres': [6, -2, 2, -2, -2, -2, -2, 745, 31, 1082, 50, 1347, 69, 1444, 88]}, 'GLFp': {'d': 2, 'p': 31, 'gens': [893666, 893731]}, 'Perm': {'d': 32, 'gens': [135955991562369039230941840932864000, 8864346423732048428190622313017, 101966981911473416934578138255958456, 50973977447449949914510425377438038, 25199055155429228508132113875036937, 8231691320578226211046411712681647]}}, 'schur_multiplier': [2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2], 'solvability_type': 3, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'D_{32}', 'transitive_degree': 32, 'wreath_data': None, 'wreath_product': False}
  • gps_conj_classes • Show schema
    • id: 110757652
      {'aut_label': '1A', 'centralizer': '1.a1.a1', 'counter': 1, 'group_counter': 52, 'group_order': 64, 'label': '1A', 'order': 1, 'powers': [1], 'representative': 0, 'size': 1}
    • id: 110757663
      {'aut_label': '2A', 'centralizer': '1.a1.a1', 'counter': 2, 'group_counter': 52, 'group_order': 64, 'label': '2A', 'order': 2, 'powers': [1], 'representative': 32, 'size': 1}
    • id: 110757664
      {'aut_label': '2B', 'centralizer': '16.b1.a1', 'counter': 3, 'group_counter': 52, 'group_order': 64, 'label': '2B', 'order': 2, 'powers': [1], 'representative': 7, 'size': 16}
    • id: 110757665
      {'aut_label': '2B', 'centralizer': '16.b1.b1', 'counter': 4, 'group_counter': 52, 'group_order': 64, 'label': '2C', 'order': 2, 'powers': [1], 'representative': 17, 'size': 16}
    • id: 110757666
      {'aut_label': '4A', 'centralizer': '2.b1.a1', 'counter': 5, 'group_counter': 52, 'group_order': 64, 'label': '4A', 'order': 4, 'powers': [2], 'representative': 16, 'size': 2}
    • id: 110757667
      {'aut_label': '8A1', 'centralizer': '2.b1.a1', 'counter': 6, 'group_counter': 52, 'group_order': 64, 'label': '8A1', 'order': 8, 'powers': [5], 'representative': 8, 'size': 2}
    • id: 110757668
      {'aut_label': '8A1', 'centralizer': '2.b1.a1', 'counter': 7, 'group_counter': 52, 'group_order': 64, 'label': '8A3', 'order': 8, 'powers': [5], 'representative': 24, 'size': 2}
    • id: 110757669
      {'aut_label': '16A1', 'centralizer': '2.b1.a1', 'counter': 8, 'group_counter': 52, 'group_order': 64, 'label': '16A1', 'order': 16, 'powers': [6], 'representative': 4, 'size': 2}
    • id: 110757670
      {'aut_label': '16A1', 'centralizer': '2.b1.a1', 'counter': 9, 'group_counter': 52, 'group_order': 64, 'label': '16A3', 'order': 16, 'powers': [7], 'representative': 12, 'size': 2}
    • id: 110757653
      {'aut_label': '16A1', 'centralizer': '2.b1.a1', 'counter': 10, 'group_counter': 52, 'group_order': 64, 'label': '16A5', 'order': 16, 'powers': [7], 'representative': 20, 'size': 2}
    • id: 110757654
      {'aut_label': '16A1', 'centralizer': '2.b1.a1', 'counter': 11, 'group_counter': 52, 'group_order': 64, 'label': '16A7', 'order': 16, 'powers': [6], 'representative': 28, 'size': 2}
    • id: 110757655
      {'aut_label': '32A1', 'centralizer': '2.b1.a1', 'counter': 12, 'group_counter': 52, 'group_order': 64, 'label': '32A1', 'order': 32, 'powers': [8], 'representative': 2, 'size': 2}
    • id: 110757656
      {'aut_label': '32A1', 'centralizer': '2.b1.a1', 'counter': 13, 'group_counter': 52, 'group_order': 64, 'label': '32A3', 'order': 32, 'powers': [9], 'representative': 6, 'size': 2}
    • id: 110757657
      {'aut_label': '32A1', 'centralizer': '2.b1.a1', 'counter': 14, 'group_counter': 52, 'group_order': 64, 'label': '32A5', 'order': 32, 'powers': [10], 'representative': 10, 'size': 2}
    • id: 110757658
      {'aut_label': '32A1', 'centralizer': '2.b1.a1', 'counter': 15, 'group_counter': 52, 'group_order': 64, 'label': '32A7', 'order': 32, 'powers': [11], 'representative': 50, 'size': 2}
    • id: 110757659
      {'aut_label': '32A1', 'centralizer': '2.b1.a1', 'counter': 16, 'group_counter': 52, 'group_order': 64, 'label': '32A9', 'order': 32, 'powers': [11], 'representative': 18, 'size': 2}
    • id: 110757660
      {'aut_label': '32A1', 'centralizer': '2.b1.a1', 'counter': 17, 'group_counter': 52, 'group_order': 64, 'label': '32A11', 'order': 32, 'powers': [10], 'representative': 42, 'size': 2}
    • id: 110757661
      {'aut_label': '32A1', 'centralizer': '2.b1.a1', 'counter': 18, 'group_counter': 52, 'group_order': 64, 'label': '32A13', 'order': 32, 'powers': [9], 'representative': 26, 'size': 2}
    • id: 110757662
      {'aut_label': '32A1', 'centralizer': '2.b1.a1', 'counter': 19, 'group_counter': 52, 'group_order': 64, 'label': '32A15', 'order': 32, 'powers': [8], 'representative': 34, 'size': 2}
  • gps_qchar • Show schema
    • id: 4732953
      {'cdim': 1, 'counter': 1, 'faithful': False, 'group': '64.52', 'group_order': 64, 'image': None, 'label': '64.52.1a', 'multiplicity': 1, 'nt': [1, 1], 'qdim': 1, 'qvalues': [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1], 'schur_index': 1}
    • id: 4732954
      {'cdim': 1, 'counter': 2, 'faithful': False, 'group': '64.52', 'group_order': 64, 'image': None, 'label': '64.52.1b', 'multiplicity': 1, 'nt': [2, 1], 'qdim': 1, 'qvalues': [1, 1, -1, -1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1], 'schur_index': 1}
    • id: 4732955
      {'cdim': 1, 'counter': 3, 'faithful': False, 'group': '64.52', 'group_order': 64, 'image': None, 'label': '64.52.1c', 'multiplicity': 1, 'nt': [2, 1], 'qdim': 1, 'qvalues': [1, 1, -1, 1, 1, 1, 1, 1, 1, 1, 1, -1, -1, -1, -1, -1, -1, -1, -1], 'schur_index': 1}
    • id: 4732956
      {'cdim': 1, 'counter': 4, 'faithful': False, 'group': '64.52', 'group_order': 64, 'image': None, 'label': '64.52.1d', 'multiplicity': 1, 'nt': [2, 1], 'qdim': 1, 'qvalues': [1, 1, 1, -1, 1, 1, 1, 1, 1, 1, 1, -1, -1, -1, -1, -1, -1, -1, -1], 'schur_index': 1}
    • id: 4732957
      {'cdim': 2, 'counter': 5, 'faithful': False, 'group': '64.52', 'group_order': 64, 'image': None, 'label': '64.52.2a', 'multiplicity': 1, 'nt': [4, 3], 'qdim': 2, 'qvalues': [2, 2, 0, 0, 2, 2, 2, -2, -2, -2, -2, 0, 0, 0, 0, 0, 0, 0, 0], 'schur_index': 1}
    • id: 4732958
      {'cdim': 2, 'counter': 6, 'faithful': False, 'group': '64.52', 'group_order': 64, 'image': None, 'label': '64.52.2b', 'multiplicity': 2, 'nt': [8, 6], 'qdim': 4, 'qvalues': [4, 4, 0, 0, 4, -4, -4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], 'schur_index': 1}
    • id: 4732959
      {'cdim': 2, 'counter': 7, 'faithful': False, 'group': '64.52', 'group_order': 64, 'image': None, 'label': '64.52.2c', 'multiplicity': 4, 'nt': [16, 56], 'qdim': 8, 'qvalues': [8, 8, 0, 0, -8, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], 'schur_index': 1}
    • id: 4732960
      {'cdim': 2, 'counter': 8, 'faithful': True, 'group': '64.52', 'group_order': 64, 'image': None, 'label': '64.52.2d', 'multiplicity': 8, 'nt': [32, 374], 'qdim': 16, 'qvalues': [16, -16, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], 'schur_index': 1}
  • gps_char • Show schema
    • id: 7665111
      {'center': '1.a1.a1', 'center_index': 1, 'center_order': 64, 'counter': 1, 'cyclotomic_n': 1, 'dim': 1, 'faithful': False, 'field': [0, 1], 'group': '64.52', 'group_counter': 52, 'group_order': 64, 'image': None, 'image_isoclass': '1.1', 'image_order': 1, 'indicator': 1, 'kernel': '1.a1.a1', 'kernel_order': 64, 'label': '64.52.1a', '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]], [[1, 0]], [[1, 0]], [[1, 0]], [[1, 0]], [[1, 0]], [[1, 0]]]}
    • id: 7665122
      {'center': '1.a1.a1', 'center_index': 1, 'center_order': 64, 'counter': 2, 'cyclotomic_n': 1, 'dim': 1, 'faithful': False, 'field': [0, 1], 'group': '64.52', 'group_counter': 52, 'group_order': 64, 'image': None, 'image_isoclass': '2.1', 'image_order': 2, 'indicator': 1, 'kernel': '2.b1.a1', 'kernel_order': 32, 'label': '64.52.1b', '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]], [[1, 0]], [[1, 0]], [[1, 0]], [[1, 0]], [[1, 0]], [[1, 0]]]}
    • id: 7665123
      {'center': '1.a1.a1', 'center_index': 1, 'center_order': 64, 'counter': 3, 'cyclotomic_n': 1, 'dim': 1, 'faithful': False, 'field': [0, 1], 'group': '64.52', 'group_counter': 52, 'group_order': 64, 'image': None, 'image_isoclass': '2.1', 'image_order': 2, 'indicator': 1, 'kernel': '2.a1.b1', 'kernel_order': 32, 'label': '64.52.1c', '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]], [[-1, 0]], [[-1, 0]], [[-1, 0]], [[-1, 0]], [[-1, 0]], [[-1, 0]]]}
    • id: 7665124
      {'center': '1.a1.a1', 'center_index': 1, 'center_order': 64, 'counter': 4, 'cyclotomic_n': 1, 'dim': 1, 'faithful': False, 'field': [0, 1], 'group': '64.52', 'group_counter': 52, 'group_order': 64, 'image': None, 'image_isoclass': '2.1', 'image_order': 2, 'indicator': 1, 'kernel': '2.a1.a1', 'kernel_order': 32, 'label': '64.52.1d', '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]], [[-1, 0]], [[-1, 0]], [[-1, 0]], [[-1, 0]], [[-1, 0]], [[-1, 0]]]}
    • id: 7665125
      {'center': '4.a1.a1', 'center_index': 4, 'center_order': 16, 'counter': 5, 'cyclotomic_n': 1, 'dim': 2, 'faithful': False, 'field': [0, 1], 'group': '64.52', 'group_counter': 52, 'group_order': 64, 'image': None, 'image_isoclass': '8.3', 'image_order': 8, 'indicator': 1, 'kernel': '8.a1.a1', 'kernel_order': 8, 'label': '64.52.2a', 'nt': [4, 3], 'q_character': None, 'values': [[[2, 0]], [[2, 0]], [[0, 0]], [[0, 0]], [[2, 0]], [[2, 0]], [[2, 0]], [[-2, 0]], [[-2, 0]], [[-2, 0]], [[-2, 0]], [[0, 0]], [[0, 0]], [[0, 0]], [[0, 0]], [[0, 0]], [[0, 0]], [[0, 0]], [[0, 0]]]}
    • id: 7665126
      {'center': '8.a1.a1', 'center_index': 8, 'center_order': 8, 'counter': 6, 'cyclotomic_n': 8, 'dim': 2, 'faithful': False, 'field': [-2, 0, 1], 'group': '64.52', 'group_counter': 52, 'group_order': 64, 'image': None, 'image_isoclass': '16.7', 'image_order': 16, 'indicator': 1, 'kernel': '16.a1.a1', 'kernel_order': 4, 'label': '64.52.2b1', 'nt': [8, 6], 'q_character': None, 'values': [[[2, 0]], [[2, 0]], [[0, 0]], [[0, 0]], [[2, 0]], [[-2, 0]], [[-2, 0]], [[0, 0]], [[0, 0]], [[0, 0]], [[0, 0]], [[-1, 1], [-1, -1]], [[1, 1], [1, -1]], [[1, 1], [1, -1]], [[-1, 1], [-1, -1]], [[-1, 1], [-1, -1]], [[1, 1], [1, -1]], [[1, 1], [1, -1]], [[-1, 1], [-1, -1]]]}
    • id: 7665127
      {'center': '8.a1.a1', 'center_index': 8, 'center_order': 8, 'counter': 7, 'cyclotomic_n': 8, 'dim': 2, 'faithful': False, 'field': [-2, 0, 1], 'group': '64.52', 'group_counter': 52, 'group_order': 64, 'image': None, 'image_isoclass': '16.7', 'image_order': 16, 'indicator': 1, 'kernel': '16.a1.a1', 'kernel_order': 4, 'label': '64.52.2b2', 'nt': [8, 6], 'q_character': None, 'values': [[[2, 0]], [[2, 0]], [[0, 0]], [[0, 0]], [[2, 0]], [[-2, 0]], [[-2, 0]], [[0, 0]], [[0, 0]], [[0, 0]], [[0, 0]], [[1, 1], [1, -1]], [[-1, 1], [-1, -1]], [[-1, 1], [-1, -1]], [[1, 1], [1, -1]], [[1, 1], [1, -1]], [[-1, 1], [-1, -1]], [[-1, 1], [-1, -1]], [[1, 1], [1, -1]]]}
    • id: 7665128
      {'center': '16.a1.a1', 'center_index': 16, 'center_order': 4, 'counter': 8, 'cyclotomic_n': 16, 'dim': 2, 'faithful': False, 'field': [2, 0, -4, 0, 1], 'group': '64.52', 'group_counter': 52, 'group_order': 64, 'image': None, 'image_isoclass': '32.18', 'image_order': 32, 'indicator': 1, 'kernel': '32.a1.a1', 'kernel_order': 2, 'label': '64.52.2c1', 'nt': [16, 56], 'q_character': None, 'values': [[[2, 0]], [[2, 0]], [[0, 0]], [[0, 0]], [[-2, 0]], [[0, 0]], [[0, 0]], [[-1, 2], [-1, -2]], [[1, 2], [1, -2]], [[1, 2], [1, -2]], [[-1, 2], [-1, -2]], [[-1, 3], [-1, -3]], [[1, 1], [1, -1]], [[-1, 1], [-1, -1]], [[1, 3], [1, -3]], [[1, 3], [1, -3]], [[-1, 1], [-1, -1]], [[1, 1], [1, -1]], [[-1, 3], [-1, -3]]]}
    • id: 7665129
      {'center': '16.a1.a1', 'center_index': 16, 'center_order': 4, 'counter': 9, 'cyclotomic_n': 16, 'dim': 2, 'faithful': False, 'field': [2, 0, -4, 0, 1], 'group': '64.52', 'group_counter': 52, 'group_order': 64, 'image': None, 'image_isoclass': '32.18', 'image_order': 32, 'indicator': 1, 'kernel': '32.a1.a1', 'kernel_order': 2, 'label': '64.52.2c2', 'nt': [16, 56], 'q_character': None, 'values': [[[2, 0]], [[2, 0]], [[0, 0]], [[0, 0]], [[-2, 0]], [[0, 0]], [[0, 0]], [[-1, 2], [-1, -2]], [[1, 2], [1, -2]], [[1, 2], [1, -2]], [[-1, 2], [-1, -2]], [[1, 3], [1, -3]], [[-1, 1], [-1, -1]], [[1, 1], [1, -1]], [[-1, 3], [-1, -3]], [[-1, 3], [-1, -3]], [[1, 1], [1, -1]], [[-1, 1], [-1, -1]], [[1, 3], [1, -3]]]}
    • id: 7665112
      {'center': '16.a1.a1', 'center_index': 16, 'center_order': 4, 'counter': 10, 'cyclotomic_n': 16, 'dim': 2, 'faithful': False, 'field': [2, 0, -4, 0, 1], 'group': '64.52', 'group_counter': 52, 'group_order': 64, 'image': None, 'image_isoclass': '32.18', 'image_order': 32, 'indicator': 1, 'kernel': '32.a1.a1', 'kernel_order': 2, 'label': '64.52.2c3', 'nt': [16, 56], 'q_character': None, 'values': [[[2, 0]], [[2, 0]], [[0, 0]], [[0, 0]], [[-2, 0]], [[0, 0]], [[0, 0]], [[1, 2], [1, -2]], [[-1, 2], [-1, -2]], [[-1, 2], [-1, -2]], [[1, 2], [1, -2]], [[-1, 1], [-1, -1]], [[-1, 3], [-1, -3]], [[1, 3], [1, -3]], [[1, 1], [1, -1]], [[1, 1], [1, -1]], [[1, 3], [1, -3]], [[-1, 3], [-1, -3]], [[-1, 1], [-1, -1]]]}
    • id: 7665113
      {'center': '16.a1.a1', 'center_index': 16, 'center_order': 4, 'counter': 11, 'cyclotomic_n': 16, 'dim': 2, 'faithful': False, 'field': [2, 0, -4, 0, 1], 'group': '64.52', 'group_counter': 52, 'group_order': 64, 'image': None, 'image_isoclass': '32.18', 'image_order': 32, 'indicator': 1, 'kernel': '32.a1.a1', 'kernel_order': 2, 'label': '64.52.2c4', 'nt': [16, 56], 'q_character': None, 'values': [[[2, 0]], [[2, 0]], [[0, 0]], [[0, 0]], [[-2, 0]], [[0, 0]], [[0, 0]], [[1, 2], [1, -2]], [[-1, 2], [-1, -2]], [[-1, 2], [-1, -2]], [[1, 2], [1, -2]], [[1, 1], [1, -1]], [[1, 3], [1, -3]], [[-1, 3], [-1, -3]], [[-1, 1], [-1, -1]], [[-1, 1], [-1, -1]], [[-1, 3], [-1, -3]], [[1, 3], [1, -3]], [[1, 1], [1, -1]]]}
    • id: 7665114
      {'center': '32.a1.a1', 'center_index': 32, 'center_order': 2, 'counter': 12, 'cyclotomic_n': 32, 'dim': 2, 'faithful': True, 'field': [2, 0, -16, 0, 20, 0, -8, 0, 1], 'group': '64.52', 'group_counter': 52, 'group_order': 64, 'image': None, 'image_isoclass': '64.52', 'image_order': 64, 'indicator': 1, 'kernel': '64.a1.a1', 'kernel_order': 1, 'label': '64.52.2d1', 'nt': [32, 374], 'q_character': None, 'values': [[[2, 0]], [[-2, 0]], [[0, 0]], [[0, 0]], [[0, 0]], [[-1, 4], [-1, -4]], [[1, 4], [1, -4]], [[-1, 6], [-1, -6]], [[1, 2], [1, -2]], [[-1, 2], [-1, -2]], [[1, 6], [1, -6]], [[-1, 5], [-1, -5]], [[1, 1], [1, -1]], [[1, 7], [1, -7]], [[1, 3], [1, -3]], [[-1, 3], [-1, -3]], [[-1, 7], [-1, -7]], [[-1, 1], [-1, -1]], [[1, 5], [1, -5]]]}
    • id: 7665115
      {'center': '32.a1.a1', 'center_index': 32, 'center_order': 2, 'counter': 13, 'cyclotomic_n': 32, 'dim': 2, 'faithful': True, 'field': [2, 0, -16, 0, 20, 0, -8, 0, 1], 'group': '64.52', 'group_counter': 52, 'group_order': 64, 'image': None, 'image_isoclass': '64.52', 'image_order': 64, 'indicator': 1, 'kernel': '64.a1.a1', 'kernel_order': 1, 'label': '64.52.2d2', 'nt': [32, 374], 'q_character': None, 'values': [[[2, 0]], [[-2, 0]], [[0, 0]], [[0, 0]], [[0, 0]], [[-1, 4], [-1, -4]], [[1, 4], [1, -4]], [[-1, 6], [-1, -6]], [[1, 2], [1, -2]], [[-1, 2], [-1, -2]], [[1, 6], [1, -6]], [[1, 5], [1, -5]], [[-1, 1], [-1, -1]], [[-1, 7], [-1, -7]], [[-1, 3], [-1, -3]], [[1, 3], [1, -3]], [[1, 7], [1, -7]], [[1, 1], [1, -1]], [[-1, 5], [-1, -5]]]}
    • id: 7665116
      {'center': '32.a1.a1', 'center_index': 32, 'center_order': 2, 'counter': 14, 'cyclotomic_n': 32, 'dim': 2, 'faithful': True, 'field': [2, 0, -16, 0, 20, 0, -8, 0, 1], 'group': '64.52', 'group_counter': 52, 'group_order': 64, 'image': None, 'image_isoclass': '64.52', 'image_order': 64, 'indicator': 1, 'kernel': '64.a1.a1', 'kernel_order': 1, 'label': '64.52.2d3', 'nt': [32, 374], 'q_character': None, 'values': [[[2, 0]], [[-2, 0]], [[0, 0]], [[0, 0]], [[0, 0]], [[-1, 4], [-1, -4]], [[1, 4], [1, -4]], [[1, 6], [1, -6]], [[-1, 2], [-1, -2]], [[1, 2], [1, -2]], [[-1, 6], [-1, -6]], [[-1, 3], [-1, -3]], [[1, 7], [1, -7]], [[-1, 1], [-1, -1]], [[-1, 5], [-1, -5]], [[1, 5], [1, -5]], [[1, 1], [1, -1]], [[-1, 7], [-1, -7]], [[1, 3], [1, -3]]]}
    • id: 7665117
      {'center': '32.a1.a1', 'center_index': 32, 'center_order': 2, 'counter': 15, 'cyclotomic_n': 32, 'dim': 2, 'faithful': True, 'field': [2, 0, -16, 0, 20, 0, -8, 0, 1], 'group': '64.52', 'group_counter': 52, 'group_order': 64, 'image': None, 'image_isoclass': '64.52', 'image_order': 64, 'indicator': 1, 'kernel': '64.a1.a1', 'kernel_order': 1, 'label': '64.52.2d4', 'nt': [32, 374], 'q_character': None, 'values': [[[2, 0]], [[-2, 0]], [[0, 0]], [[0, 0]], [[0, 0]], [[-1, 4], [-1, -4]], [[1, 4], [1, -4]], [[1, 6], [1, -6]], [[-1, 2], [-1, -2]], [[1, 2], [1, -2]], [[-1, 6], [-1, -6]], [[1, 3], [1, -3]], [[-1, 7], [-1, -7]], [[1, 1], [1, -1]], [[1, 5], [1, -5]], [[-1, 5], [-1, -5]], [[-1, 1], [-1, -1]], [[1, 7], [1, -7]], [[-1, 3], [-1, -3]]]}
    • id: 7665118
      {'center': '32.a1.a1', 'center_index': 32, 'center_order': 2, 'counter': 16, 'cyclotomic_n': 32, 'dim': 2, 'faithful': True, 'field': [2, 0, -16, 0, 20, 0, -8, 0, 1], 'group': '64.52', 'group_counter': 52, 'group_order': 64, 'image': None, 'image_isoclass': '64.52', 'image_order': 64, 'indicator': 1, 'kernel': '64.a1.a1', 'kernel_order': 1, 'label': '64.52.2d5', 'nt': [32, 374], 'q_character': None, 'values': [[[2, 0]], [[-2, 0]], [[0, 0]], [[0, 0]], [[0, 0]], [[1, 4], [1, -4]], [[-1, 4], [-1, -4]], [[-1, 2], [-1, -2]], [[-1, 6], [-1, -6]], [[1, 6], [1, -6]], [[1, 2], [1, -2]], [[-1, 7], [-1, -7]], [[1, 5], [1, -5]], [[1, 3], [1, -3]], [[-1, 1], [-1, -1]], [[1, 1], [1, -1]], [[-1, 3], [-1, -3]], [[-1, 5], [-1, -5]], [[1, 7], [1, -7]]]}
    • id: 7665119
      {'center': '32.a1.a1', 'center_index': 32, 'center_order': 2, 'counter': 17, 'cyclotomic_n': 32, 'dim': 2, 'faithful': True, 'field': [2, 0, -16, 0, 20, 0, -8, 0, 1], 'group': '64.52', 'group_counter': 52, 'group_order': 64, 'image': None, 'image_isoclass': '64.52', 'image_order': 64, 'indicator': 1, 'kernel': '64.a1.a1', 'kernel_order': 1, 'label': '64.52.2d6', 'nt': [32, 374], 'q_character': None, 'values': [[[2, 0]], [[-2, 0]], [[0, 0]], [[0, 0]], [[0, 0]], [[1, 4], [1, -4]], [[-1, 4], [-1, -4]], [[-1, 2], [-1, -2]], [[-1, 6], [-1, -6]], [[1, 6], [1, -6]], [[1, 2], [1, -2]], [[1, 7], [1, -7]], [[-1, 5], [-1, -5]], [[-1, 3], [-1, -3]], [[1, 1], [1, -1]], [[-1, 1], [-1, -1]], [[1, 3], [1, -3]], [[1, 5], [1, -5]], [[-1, 7], [-1, -7]]]}
    • id: 7665120
      {'center': '32.a1.a1', 'center_index': 32, 'center_order': 2, 'counter': 18, 'cyclotomic_n': 32, 'dim': 2, 'faithful': True, 'field': [2, 0, -16, 0, 20, 0, -8, 0, 1], 'group': '64.52', 'group_counter': 52, 'group_order': 64, 'image': None, 'image_isoclass': '64.52', 'image_order': 64, 'indicator': 1, 'kernel': '64.a1.a1', 'kernel_order': 1, 'label': '64.52.2d7', 'nt': [32, 374], 'q_character': None, 'values': [[[2, 0]], [[-2, 0]], [[0, 0]], [[0, 0]], [[0, 0]], [[1, 4], [1, -4]], [[-1, 4], [-1, -4]], [[1, 2], [1, -2]], [[1, 6], [1, -6]], [[-1, 6], [-1, -6]], [[-1, 2], [-1, -2]], [[-1, 1], [-1, -1]], [[-1, 3], [-1, -3]], [[1, 5], [1, -5]], [[1, 7], [1, -7]], [[-1, 7], [-1, -7]], [[-1, 5], [-1, -5]], [[1, 3], [1, -3]], [[1, 1], [1, -1]]]}
    • id: 7665121
      {'center': '32.a1.a1', 'center_index': 32, 'center_order': 2, 'counter': 19, 'cyclotomic_n': 32, 'dim': 2, 'faithful': True, 'field': [2, 0, -16, 0, 20, 0, -8, 0, 1], 'group': '64.52', 'group_counter': 52, 'group_order': 64, 'image': None, 'image_isoclass': '64.52', 'image_order': 64, 'indicator': 1, 'kernel': '64.a1.a1', 'kernel_order': 1, 'label': '64.52.2d8', 'nt': [32, 374], 'q_character': None, 'values': [[[2, 0]], [[-2, 0]], [[0, 0]], [[0, 0]], [[0, 0]], [[1, 4], [1, -4]], [[-1, 4], [-1, -4]], [[1, 2], [1, -2]], [[1, 6], [1, -6]], [[-1, 6], [-1, -6]], [[-1, 2], [-1, -2]], [[1, 1], [1, -1]], [[1, 3], [1, -3]], [[-1, 5], [-1, -5]], [[-1, 7], [-1, -7]], [[1, 7], [1, -7]], [[1, 5], [1, -5]], [[-1, 3], [-1, -3]], [[-1, 1], [-1, -1]]]}
  • gps_subgroup_search • Show schema
    • id: 67936646
      {'Agroup': False, 'Zgroup': False, 'abelian': False, 'ambient': '64.52', 'ambient_counter': 52, 'ambient_order': 64, 'ambient_tex': 'D_{32}', 'central': False, 'central_factor': False, 'centralizer_order': 2, 'characteristic': True, 'core_order': 64, 'counter': 1, 'cyclic': False, 'direct': True, 'hall': 2, 'label': '64.52.1.a1.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, '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': ['F', 'R', 'U0', 'L0', 'D0', 'C0'], 'split': True, 'standard_generators': False, 'stem': False, 'subgroup': '64.52', 'subgroup_hash': 52, 'subgroup_order': 64, 'subgroup_tex': 'D_{32}', 'supersolvable': True, 'sylow': 2}
    • id: 67936655
      {'Agroup': False, 'Zgroup': False, 'abelian': False, 'ambient': '64.52', 'ambient_counter': 52, 'ambient_order': 64, 'ambient_tex': 'D_{32}', 'central': False, 'central_factor': False, 'centralizer_order': 2, 'characteristic': False, 'core_order': 32, 'counter': 2, 'cyclic': False, 'direct': False, 'hall': 0, 'label': '64.52.2.a1.a1', 'maximal': True, 'maximal_normal': True, 'metabelian': True, 'metacyclic': True, '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': [], 'split': True, 'standard_generators': False, 'stem': False, 'subgroup': '32.18', 'subgroup_hash': 18, 'subgroup_order': 32, 'subgroup_tex': 'D_{16}', 'supersolvable': True, 'sylow': 0}
    • id: 67936656
      {'Agroup': False, 'Zgroup': False, 'abelian': False, 'ambient': '64.52', 'ambient_counter': 52, 'ambient_order': 64, 'ambient_tex': 'D_{32}', 'central': False, 'central_factor': False, 'centralizer_order': 2, 'characteristic': False, 'core_order': 32, 'counter': 3, 'cyclic': False, 'direct': False, 'hall': 0, 'label': '64.52.2.a1.b1', 'maximal': True, 'maximal_normal': True, 'metabelian': True, 'metacyclic': True, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': True, 'old_label': '2.a1.b1', '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': [], 'split': True, 'standard_generators': False, 'stem': False, 'subgroup': '32.18', 'subgroup_hash': 18, 'subgroup_order': 32, 'subgroup_tex': 'D_{16}', 'supersolvable': True, 'sylow': 0}
    • id: 67936657
      {'Agroup': True, 'Zgroup': True, 'abelian': True, 'ambient': '64.52', 'ambient_counter': 52, 'ambient_order': 64, 'ambient_tex': 'D_{32}', 'central': False, 'central_factor': False, 'centralizer_order': 32, 'characteristic': True, 'core_order': 32, 'counter': 4, 'cyclic': True, 'direct': False, 'hall': 0, 'label': '64.52.2.b1.a1', 'maximal': True, 'maximal_normal': True, 'metabelian': True, 'metacyclic': True, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': True, 'old_label': '2.b1.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': ['C1'], 'split': True, 'standard_generators': False, 'stem': False, 'subgroup': '32.1', 'subgroup_hash': 1, 'subgroup_order': 32, 'subgroup_tex': 'C_{32}', 'supersolvable': True, 'sylow': 0}
    • id: 67936658
      {'Agroup': True, 'Zgroup': True, 'abelian': True, 'ambient': '64.52', 'ambient_counter': 52, 'ambient_order': 64, 'ambient_tex': 'D_{32}', 'central': False, 'central_factor': False, 'centralizer_order': 32, 'characteristic': True, 'core_order': 16, 'counter': 5, 'cyclic': True, 'direct': False, 'hall': 0, 'label': '64.52.4.a1.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': True, 'old_label': '4.a1.a1', 'outer_equivalence': False, 'perfect': False, 'proper': True, 'quotient': '4.2', 'quotient_Agroup': True, 'quotient_abelian': True, 'quotient_cyclic': False, 'quotient_hash': 2, 'quotient_metabelian': True, 'quotient_nilpotent': True, 'quotient_order': 4, 'quotient_simple': False, 'quotient_solvable': True, 'quotient_supersolvable': True, 'quotient_tex': 'C_2^2', 'simple': False, 'solvable': True, 'special_labels': ['D', 'Phi', 'U1', 'L1', 'D1', 'C2'], 'split': False, 'standard_generators': False, 'stem': False, 'subgroup': '16.1', 'subgroup_hash': 1, 'subgroup_order': 16, 'subgroup_tex': 'C_{16}', 'supersolvable': True, 'sylow': 0}
    • id: 67936659
      {'Agroup': False, 'Zgroup': False, 'abelian': False, 'ambient': '64.52', 'ambient_counter': 52, 'ambient_order': 64, 'ambient_tex': 'D_{32}', 'central': False, 'central_factor': False, 'centralizer_order': 2, 'characteristic': False, 'core_order': 8, 'counter': 6, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '64.52.4.b1.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': False, 'old_label': '4.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': 4, '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': '16.7', 'subgroup_hash': 7, 'subgroup_order': 16, 'subgroup_tex': 'D_8', 'supersolvable': True, 'sylow': 0}
    • id: 67936660
      {'Agroup': False, 'Zgroup': False, 'abelian': False, 'ambient': '64.52', 'ambient_counter': 52, 'ambient_order': 64, 'ambient_tex': 'D_{32}', 'central': False, 'central_factor': False, 'centralizer_order': 2, 'characteristic': False, 'core_order': 8, 'counter': 7, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '64.52.4.b1.b1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': False, 'old_label': '4.b1.b1', 'outer_equivalence': False, 'perfect': False, 'proper': True, 'quotient': None, 'quotient_Agroup': None, 'quotient_abelian': None, 'quotient_cyclic': None, 'quotient_hash': None, 'quotient_metabelian': None, 'quotient_nilpotent': None, 'quotient_order': 4, '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': '16.7', 'subgroup_hash': 7, 'subgroup_order': 16, 'subgroup_tex': 'D_8', 'supersolvable': True, 'sylow': 0}
    • id: 67936661
      {'Agroup': True, 'Zgroup': True, 'abelian': True, 'ambient': '64.52', 'ambient_counter': 52, 'ambient_order': 64, 'ambient_tex': 'D_{32}', 'central': False, 'central_factor': False, 'centralizer_order': 32, 'characteristic': True, 'core_order': 8, 'counter': 8, 'cyclic': True, 'direct': False, 'hall': 0, 'label': '64.52.8.a1.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': True, 'old_label': '8.a1.a1', 'outer_equivalence': False, 'perfect': False, 'proper': True, 'quotient': '8.3', 'quotient_Agroup': False, 'quotient_abelian': False, 'quotient_cyclic': False, 'quotient_hash': 3, 'quotient_metabelian': True, 'quotient_nilpotent': True, 'quotient_order': 8, 'quotient_simple': False, 'quotient_solvable': True, 'quotient_supersolvable': True, 'quotient_tex': 'D_4', 'simple': False, 'solvable': True, 'special_labels': ['U2', 'L2', 'C3'], 'split': False, 'standard_generators': False, 'stem': False, 'subgroup': '8.1', 'subgroup_hash': 1, 'subgroup_order': 8, 'subgroup_tex': 'C_8', 'supersolvable': True, 'sylow': 0}
    • id: 67936662
      {'Agroup': False, 'Zgroup': False, 'abelian': False, 'ambient': '64.52', 'ambient_counter': 52, 'ambient_order': 64, 'ambient_tex': 'D_{32}', 'central': False, 'central_factor': False, 'centralizer_order': 2, 'characteristic': False, 'core_order': 4, 'counter': 9, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '64.52.8.b1.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': False, 'old_label': '8.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': 8, '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': '8.3', 'subgroup_hash': 3, 'subgroup_order': 8, 'subgroup_tex': 'D_4', 'supersolvable': True, 'sylow': 0}
    • id: 67936647
      {'Agroup': False, 'Zgroup': False, 'abelian': False, 'ambient': '64.52', 'ambient_counter': 52, 'ambient_order': 64, 'ambient_tex': 'D_{32}', 'central': False, 'central_factor': False, 'centralizer_order': 2, 'characteristic': False, 'core_order': 4, 'counter': 10, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '64.52.8.b1.b1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': False, 'old_label': '8.b1.b1', 'outer_equivalence': False, 'perfect': False, 'proper': True, 'quotient': None, 'quotient_Agroup': None, 'quotient_abelian': None, 'quotient_cyclic': None, 'quotient_hash': None, 'quotient_metabelian': None, 'quotient_nilpotent': None, 'quotient_order': 8, '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': '8.3', 'subgroup_hash': 3, 'subgroup_order': 8, 'subgroup_tex': 'D_4', 'supersolvable': True, 'sylow': 0}
    • id: 67936648
      {'Agroup': True, 'Zgroup': True, 'abelian': True, 'ambient': '64.52', 'ambient_counter': 52, 'ambient_order': 64, 'ambient_tex': 'D_{32}', 'central': False, 'central_factor': False, 'centralizer_order': 32, 'characteristic': True, 'core_order': 4, 'counter': 11, 'cyclic': True, 'direct': False, 'hall': 0, 'label': '64.52.16.a1.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': True, 'old_label': '16.a1.a1', 'outer_equivalence': False, 'perfect': False, 'proper': True, 'quotient': '16.7', 'quotient_Agroup': False, 'quotient_abelian': False, 'quotient_cyclic': False, 'quotient_hash': 7, 'quotient_metabelian': True, 'quotient_nilpotent': True, 'quotient_order': 16, 'quotient_simple': False, 'quotient_solvable': True, 'quotient_supersolvable': True, 'quotient_tex': 'D_8', 'simple': False, 'solvable': True, 'special_labels': ['U3', 'L3', 'C4'], 'split': False, 'standard_generators': False, 'stem': False, 'subgroup': '4.1', 'subgroup_hash': 1, 'subgroup_order': 4, 'subgroup_tex': 'C_4', 'supersolvable': True, 'sylow': 0}
    • id: 67936649
      {'Agroup': True, 'Zgroup': False, 'abelian': True, 'ambient': '64.52', 'ambient_counter': 52, 'ambient_order': 64, 'ambient_tex': 'D_{32}', 'central': False, 'central_factor': False, 'centralizer_order': 4, 'characteristic': False, 'core_order': 2, 'counter': 12, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '64.52.16.b1.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': False, 'old_label': '16.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': 16, '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': '4.2', 'subgroup_hash': 2, 'subgroup_order': 4, 'subgroup_tex': 'C_2^2', 'supersolvable': True, 'sylow': 0}
    • id: 67936650
      {'Agroup': True, 'Zgroup': False, 'abelian': True, 'ambient': '64.52', 'ambient_counter': 52, 'ambient_order': 64, 'ambient_tex': 'D_{32}', 'central': False, 'central_factor': False, 'centralizer_order': 4, 'characteristic': False, 'core_order': 2, 'counter': 13, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '64.52.16.b1.b1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': False, 'old_label': '16.b1.b1', 'outer_equivalence': False, 'perfect': False, 'proper': True, 'quotient': None, 'quotient_Agroup': None, 'quotient_abelian': None, 'quotient_cyclic': None, 'quotient_hash': None, 'quotient_metabelian': None, 'quotient_nilpotent': None, 'quotient_order': 16, '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': '4.2', 'subgroup_hash': 2, 'subgroup_order': 4, 'subgroup_tex': 'C_2^2', 'supersolvable': True, 'sylow': 0}
    • id: 67936651
      {'Agroup': True, 'Zgroup': True, 'abelian': True, 'ambient': '64.52', 'ambient_counter': 52, 'ambient_order': 64, 'ambient_tex': 'D_{32}', 'central': True, 'central_factor': False, 'centralizer_order': 64, 'characteristic': True, 'core_order': 2, 'counter': 14, 'cyclic': True, 'direct': False, 'hall': 0, 'label': '64.52.32.a1.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': True, 'minimal_normal': True, 'nilpotent': True, 'normal': True, 'old_label': '32.a1.a1', 'outer_equivalence': False, 'perfect': False, 'proper': True, 'quotient': '32.18', 'quotient_Agroup': False, 'quotient_abelian': False, 'quotient_cyclic': False, 'quotient_hash': 18, 'quotient_metabelian': True, 'quotient_nilpotent': True, 'quotient_order': 32, 'quotient_simple': False, 'quotient_solvable': True, 'quotient_supersolvable': True, 'quotient_tex': 'D_{16}', 'simple': True, 'solvable': True, 'special_labels': ['Z', 'S', 'U4', 'L4', 'C5'], 'split': False, 'standard_generators': False, 'stem': True, 'subgroup': '2.1', 'subgroup_hash': 1, 'subgroup_order': 2, 'subgroup_tex': 'C_2', 'supersolvable': True, 'sylow': 0}
    • id: 67936652
      {'Agroup': True, 'Zgroup': True, 'abelian': True, 'ambient': '64.52', 'ambient_counter': 52, 'ambient_order': 64, 'ambient_tex': 'D_{32}', 'central': False, 'central_factor': False, 'centralizer_order': 4, 'characteristic': False, 'core_order': 1, 'counter': 15, 'cyclic': True, 'direct': None, 'hall': 0, 'label': '64.52.32.b1.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': True, 'minimal_normal': False, 'nilpotent': True, 'normal': False, 'old_label': '32.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': 32, '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': 0}
    • id: 67936653
      {'Agroup': True, 'Zgroup': True, 'abelian': True, 'ambient': '64.52', 'ambient_counter': 52, 'ambient_order': 64, 'ambient_tex': 'D_{32}', 'central': False, 'central_factor': False, 'centralizer_order': 4, 'characteristic': False, 'core_order': 1, 'counter': 16, 'cyclic': True, 'direct': None, 'hall': 0, 'label': '64.52.32.b1.b1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': True, 'minimal_normal': False, 'nilpotent': True, 'normal': False, 'old_label': '32.b1.b1', 'outer_equivalence': False, 'perfect': False, 'proper': True, 'quotient': None, 'quotient_Agroup': None, 'quotient_abelian': None, 'quotient_cyclic': None, 'quotient_hash': None, 'quotient_metabelian': None, 'quotient_nilpotent': None, 'quotient_order': 32, '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': 0}
    • id: 67936654
      {'Agroup': True, 'Zgroup': True, 'abelian': True, 'ambient': '64.52', 'ambient_counter': 52, 'ambient_order': 64, 'ambient_tex': 'D_{32}', 'central': True, 'central_factor': False, 'centralizer_order': 64, 'characteristic': True, 'core_order': 1, 'counter': 17, 'cyclic': True, 'direct': True, 'hall': 1, 'label': '64.52.64.a1.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': True, 'old_label': '64.a1.a1', 'outer_equivalence': False, 'perfect': True, 'proper': False, 'quotient': '64.52', 'quotient_Agroup': False, 'quotient_abelian': False, 'quotient_cyclic': False, 'quotient_hash': 52, 'quotient_metabelian': True, 'quotient_nilpotent': True, 'quotient_order': 64, 'quotient_simple': False, 'quotient_solvable': True, 'quotient_supersolvable': True, 'quotient_tex': 'D_{32}', 'simple': False, 'solvable': True, 'special_labels': ['PC', 'U5', 'L5', 'D2', 'C6'], '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: 67936646
      {'ambient': '64.52', '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': '32.a1.a1', 'complements': ['64.a1.a1'], 'conjugacy_class_count': 1, 'contained_in': [], 'contains': ['2.a1.a1', '2.a1.b1', '2.b1.a1'], 'core': '1.a1.a1', 'coset_action_label': None, 'count': 1, 'diagramx': [4969, 5032, 5000, 5000, 4953, 4953, 5000, 5000], 'generators': [1, 2], 'label': '64.52.1.a1.a1', 'mobius_quo': 0, 'mobius_sub': 1, 'normal_closure': '1.a1.a1', 'normal_contained_in': [], 'normal_contains': ['2.a1.a1', '2.a1.b1', '2.b1.a1'], 'normalizer': '1.a1.a1', 'old_label': '1.a1.a1', 'projective_image': '32.18', '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': '32.18'}
    • id: 67936647
      {'ambient': '64.52', 'aut_centralizer_order': 8, 'aut_label': '8.b1', 'aut_quo_index': None, 'aut_stab_index': 8, 'aut_weyl_group': '8.3', 'aut_weyl_index': 64, 'centralizer': '32.a1.a1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['4.b1.b1'], 'contains': ['16.a1.a1', '16.b1.b1'], 'core': '16.a1.a1', 'coset_action_label': None, 'count': 4, 'diagramx': [7702, -1, 1907, -1, 7774, -1, 1908, -1], 'generators': [1, 48], 'label': '64.52.8.b1.b1', 'mobius_quo': None, 'mobius_sub': 0, 'normal_closure': '2.a1.b1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '4.b1.b1', 'old_label': '8.b1.b1', 'projective_image': '32.18', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '8.b1.b1', 'subgroup_fusion': None, 'weyl_group': '8.3'}
    • id: 67936648
      {'ambient': '64.52', 'aut_centralizer_order': 256, 'aut_label': '16.a1', 'aut_quo_index': 1, 'aut_stab_index': 1, 'aut_weyl_group': '2.1', 'aut_weyl_index': 256, 'centralizer': '2.b1.a1', 'complements': [], 'conjugacy_class_count': 1, 'contained_in': ['8.a1.a1', '8.b1.a1', '8.b1.b1'], 'contains': ['32.a1.a1'], 'core': '16.a1.a1', 'coset_action_label': None, 'count': 1, 'diagramx': [3044, 5032, 5000, 5000, 3135, 4953, 5000, 5000], 'generators': [48], 'label': '64.52.16.a1.a1', 'mobius_quo': 0, 'mobius_sub': 0, 'normal_closure': '16.a1.a1', 'normal_contained_in': ['8.a1.a1'], 'normal_contains': ['32.a1.a1'], 'normalizer': '1.a1.a1', 'old_label': '16.a1.a1', 'projective_image': '32.18', 'quotient_action_image': '2.1', 'quotient_action_kernel': '8.1', 'quotient_action_kernel_order': 8, 'quotient_fusion': None, 'short_label': '16.a1.a1', 'subgroup_fusion': None, 'weyl_group': '2.1'}
    • id: 67936649
      {'ambient': '64.52', 'aut_centralizer_order': 16, 'aut_label': '16.b1', 'aut_quo_index': None, 'aut_stab_index': 16, 'aut_weyl_group': '2.1', 'aut_weyl_index': 256, 'centralizer': '16.b1.a1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['8.b1.a1'], 'contains': ['32.a1.a1', '32.b1.a1'], 'core': '32.a1.a1', 'coset_action_label': None, 'count': 8, 'diagramx': [7826, -1, 8093, -1, 7962, -1, 8092, -1], 'generators': [3, 32], 'label': '64.52.16.b1.a1', 'mobius_quo': None, 'mobius_sub': 0, 'normal_closure': '2.a1.a1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '8.b1.a1', 'old_label': '16.b1.a1', 'projective_image': '32.18', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '16.b1.a1', 'subgroup_fusion': None, 'weyl_group': '2.1'}
    • id: 67936650
      {'ambient': '64.52', 'aut_centralizer_order': 16, 'aut_label': '16.b1', 'aut_quo_index': None, 'aut_stab_index': 16, 'aut_weyl_group': '2.1', 'aut_weyl_index': 256, 'centralizer': '16.b1.b1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['8.b1.b1'], 'contains': ['32.a1.a1', '32.b1.b1'], 'core': '32.a1.a1', 'coset_action_label': None, 'count': 8, 'diagramx': [7826, -1, 1907, -1, 7962, -1, 1908, -1], 'generators': [1, 32], 'label': '64.52.16.b1.b1', 'mobius_quo': None, 'mobius_sub': 0, 'normal_closure': '2.a1.b1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '8.b1.b1', 'old_label': '16.b1.b1', 'projective_image': '32.18', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '16.b1.b1', 'subgroup_fusion': None, 'weyl_group': '2.1'}
    • id: 67936651
      {'ambient': '64.52', 'aut_centralizer_order': 512, 'aut_label': '32.a1', 'aut_quo_index': 1, 'aut_stab_index': 1, 'aut_weyl_group': '1.1', 'aut_weyl_index': 512, 'centralizer': '1.a1.a1', 'complements': [], 'conjugacy_class_count': 1, 'contained_in': ['16.a1.a1', '16.b1.a1', '16.b1.b1'], 'contains': ['64.a1.a1'], 'core': '32.a1.a1', 'coset_action_label': None, 'count': 1, 'diagramx': [3354, 5032, 5000, 5000, 3448, 4953, 5000, 5000], 'generators': [32], 'label': '64.52.32.a1.a1', 'mobius_quo': -1, 'mobius_sub': 0, 'normal_closure': '32.a1.a1', 'normal_contained_in': ['16.a1.a1'], 'normal_contains': ['64.a1.a1'], 'normalizer': '1.a1.a1', 'old_label': '32.a1.a1', 'projective_image': '32.18', 'quotient_action_image': '1.1', 'quotient_action_kernel': '32.18', 'quotient_action_kernel_order': 32, 'quotient_fusion': None, 'short_label': '32.a1.a1', 'subgroup_fusion': None, 'weyl_group': '1.1'}
    • id: 67936652
      {'ambient': '64.52', 'aut_centralizer_order': 16, 'aut_label': '32.b1', 'aut_quo_index': None, 'aut_stab_index': 32, 'aut_weyl_group': '1.1', 'aut_weyl_index': 512, 'centralizer': '16.b1.a1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['16.b1.a1'], 'contains': ['64.a1.a1'], 'core': '64.a1.a1', 'coset_action_label': '32T374', 'count': 16, 'diagramx': [7826, -1, 8013, -1, 7962, -1, 7972, -1], 'generators': [3], 'label': '64.52.32.b1.a1', 'mobius_quo': None, 'mobius_sub': 0, 'normal_closure': '2.a1.a1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '16.b1.a1', 'old_label': '32.b1.a1', 'projective_image': '64.52', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '32.b1.a1', 'subgroup_fusion': None, 'weyl_group': '1.1'}
    • id: 67936653
      {'ambient': '64.52', 'aut_centralizer_order': 16, 'aut_label': '32.b1', 'aut_quo_index': None, 'aut_stab_index': 32, 'aut_weyl_group': '1.1', 'aut_weyl_index': 512, 'centralizer': '16.b1.b1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['16.b1.b1'], 'contains': ['64.a1.a1'], 'core': '64.a1.a1', 'coset_action_label': '32T374', 'count': 16, 'diagramx': [7826, -1, 2028, -1, 7962, -1, 2028, -1], 'generators': [1], 'label': '64.52.32.b1.b1', 'mobius_quo': None, 'mobius_sub': 0, 'normal_closure': '2.a1.b1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '16.b1.b1', 'old_label': '32.b1.b1', 'projective_image': '64.52', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '32.b1.b1', 'subgroup_fusion': None, 'weyl_group': '1.1'}
    • id: 67936654
      {'ambient': '64.52', 'aut_centralizer_order': None, 'aut_label': '64.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': ['32.a1.a1', '32.b1.a1', '32.b1.b1'], 'contains': [], 'core': '64.a1.a1', 'coset_action_label': None, 'count': 1, 'diagramx': [5591, 5032, 5000, 5000, 5705, 4953, 5000, 5000], 'generators': [], 'label': '64.52.64.a1.a1', 'mobius_quo': 1, 'mobius_sub': 0, 'normal_closure': '64.a1.a1', 'normal_contained_in': ['32.a1.a1'], 'normal_contains': [], 'normalizer': '1.a1.a1', 'old_label': '64.a1.a1', 'projective_image': '64.52', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '64.a1.a1', 'subgroup_fusion': None, 'weyl_group': '1.1'}
    • id: 67936655
      {'ambient': '64.52', 'aut_centralizer_order': 2, 'aut_label': '2.a1', 'aut_quo_index': 1, 'aut_stab_index': 2, 'aut_weyl_group': '128.913', 'aut_weyl_index': 4, 'centralizer': '32.a1.a1', 'complements': ['32.b1.b1'], 'conjugacy_class_count': 1, 'contained_in': ['1.a1.a1'], 'contains': ['4.a1.a1', '4.b1.a1'], 'core': '2.a1.a1', 'coset_action_label': None, 'count': 1, 'diagramx': [7702, 7799, 8575, 8575, 7774, 7774, 8574, 8574], 'generators': [3, 60], 'label': '64.52.2.a1.a1', 'mobius_quo': 0, 'mobius_sub': -1, 'normal_closure': '2.a1.a1', 'normal_contained_in': ['1.a1.a1'], 'normal_contains': ['4.a1.a1'], 'normalizer': '1.a1.a1', 'old_label': '2.a1.a1', 'projective_image': '32.18', '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': '32.18'}
    • id: 67936656
      {'ambient': '64.52', 'aut_centralizer_order': 2, 'aut_label': '2.a1', 'aut_quo_index': 1, 'aut_stab_index': 2, 'aut_weyl_group': '128.913', 'aut_weyl_index': 4, 'centralizer': '32.a1.a1', 'complements': ['32.b1.a1'], 'conjugacy_class_count': 1, 'contained_in': ['1.a1.a1'], 'contains': ['4.a1.a1', '4.b1.b1'], 'core': '2.a1.b1', 'coset_action_label': None, 'count': 1, 'diagramx': [7702, 7799, 1425, 1425, 7774, 7774, 1426, 1426], 'generators': [1, 60], 'label': '64.52.2.a1.b1', 'mobius_quo': 0, 'mobius_sub': -1, 'normal_closure': '2.a1.b1', 'normal_contained_in': ['1.a1.a1'], 'normal_contains': ['4.a1.a1'], 'normalizer': '1.a1.a1', 'old_label': '2.a1.b1', 'projective_image': '32.18', 'quotient_action_image': '2.1', 'quotient_action_kernel': '1.1', 'quotient_action_kernel_order': 1, 'quotient_fusion': None, 'short_label': '2.a1.b1', 'subgroup_fusion': None, 'weyl_group': '32.18'}
    • id: 67936657
      {'ambient': '64.52', 'aut_centralizer_order': 32, 'aut_label': '2.b1', 'aut_quo_index': 1, 'aut_stab_index': 1, 'aut_weyl_group': '16.5', 'aut_weyl_index': 32, 'centralizer': '2.b1.a1', 'complements': ['32.b1.b1', '32.b1.a1'], 'conjugacy_class_count': 1, 'contained_in': ['1.a1.a1'], 'contains': ['4.a1.a1'], 'core': '2.b1.a1', 'coset_action_label': None, 'count': 1, 'diagramx': [2174, 2201, 5000, 5000, 2194, 2194, 5000, 5000], 'generators': [2], 'label': '64.52.2.b1.a1', 'mobius_quo': 0, 'mobius_sub': -1, 'normal_closure': '2.b1.a1', 'normal_contained_in': ['1.a1.a1'], 'normal_contains': ['4.a1.a1'], 'normalizer': '1.a1.a1', 'old_label': '2.b1.a1', 'projective_image': '32.18', 'quotient_action_image': '2.1', 'quotient_action_kernel': '1.1', 'quotient_action_kernel_order': 1, 'quotient_fusion': None, 'short_label': '2.b1.a1', 'subgroup_fusion': None, 'weyl_group': '2.1'}
    • id: 67936658
      {'ambient': '64.52', 'aut_centralizer_order': 64, 'aut_label': '4.a1', 'aut_quo_index': 3, 'aut_stab_index': 1, 'aut_weyl_group': '8.2', 'aut_weyl_index': 64, 'centralizer': '2.b1.a1', 'complements': [], 'conjugacy_class_count': 1, 'contained_in': ['2.a1.a1', '2.a1.b1', '2.b1.a1'], 'contains': ['8.a1.a1'], 'core': '4.a1.a1', 'coset_action_label': None, 'count': 1, 'diagramx': [2732, 5032, 5000, 5000, 2759, 4953, 5000, 5000], 'generators': [60], 'label': '64.52.4.a1.a1', 'mobius_quo': 0, 'mobius_sub': 2, 'normal_closure': '4.a1.a1', 'normal_contained_in': ['2.a1.a1', '2.a1.b1', '2.b1.a1'], 'normal_contains': ['8.a1.a1'], 'normalizer': '1.a1.a1', 'old_label': '4.a1.a1', 'projective_image': '32.18', 'quotient_action_image': '2.1', 'quotient_action_kernel': '2.1', 'quotient_action_kernel_order': 2, 'quotient_fusion': None, 'short_label': '4.a1.a1', 'subgroup_fusion': None, 'weyl_group': '2.1'}
    • id: 67936659
      {'ambient': '64.52', 'aut_centralizer_order': 4, 'aut_label': '4.b1', 'aut_quo_index': None, 'aut_stab_index': 4, 'aut_weyl_group': '32.43', 'aut_weyl_index': 16, 'centralizer': '32.a1.a1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['2.a1.a1'], 'contains': ['8.a1.a1', '8.b1.a1'], 'core': '8.a1.a1', 'coset_action_label': None, 'count': 2, 'diagramx': [7702, -1, 8213, -1, 7774, -1, 8213, -1], 'generators': [3, 8], 'label': '64.52.4.b1.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': '4.b1.a1', 'projective_image': '32.18', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '4.b1.a1', 'subgroup_fusion': None, 'weyl_group': '16.7'}
    • id: 67936660
      {'ambient': '64.52', 'aut_centralizer_order': 4, 'aut_label': '4.b1', 'aut_quo_index': None, 'aut_stab_index': 4, 'aut_weyl_group': '32.43', 'aut_weyl_index': 16, 'centralizer': '32.a1.a1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['2.a1.b1'], 'contains': ['8.a1.a1', '8.b1.b1'], 'core': '8.a1.a1', 'coset_action_label': None, 'count': 2, 'diagramx': [7702, -1, 1787, -1, 7774, -1, 1787, -1], 'generators': [1, 8], 'label': '64.52.4.b1.b1', 'mobius_quo': None, 'mobius_sub': 0, 'normal_closure': '2.a1.b1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '2.a1.b1', 'old_label': '4.b1.b1', 'projective_image': '32.18', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '4.b1.b1', 'subgroup_fusion': None, 'weyl_group': '16.7'}
    • id: 67936661
      {'ambient': '64.52', 'aut_centralizer_order': 128, 'aut_label': '8.a1', 'aut_quo_index': 1, 'aut_stab_index': 1, 'aut_weyl_group': '4.2', 'aut_weyl_index': 128, 'centralizer': '2.b1.a1', 'complements': [], 'conjugacy_class_count': 1, 'contained_in': ['4.a1.a1', '4.b1.a1', '4.b1.b1'], 'contains': ['16.a1.a1'], 'core': '8.a1.a1', 'coset_action_label': None, 'count': 1, 'diagramx': [3044, 5032, 5000, 5000, 3135, 4953, 5000, 5000], 'generators': [8], 'label': '64.52.8.a1.a1', 'mobius_quo': 0, 'mobius_sub': 0, 'normal_closure': '8.a1.a1', 'normal_contained_in': ['4.a1.a1'], 'normal_contains': ['16.a1.a1'], 'normalizer': '1.a1.a1', 'old_label': '8.a1.a1', 'projective_image': '32.18', 'quotient_action_image': '2.1', 'quotient_action_kernel': '4.1', 'quotient_action_kernel_order': 4, 'quotient_fusion': None, 'short_label': '8.a1.a1', 'subgroup_fusion': None, 'weyl_group': '2.1'}
    • id: 67936662
      {'ambient': '64.52', 'aut_centralizer_order': 8, 'aut_label': '8.b1', 'aut_quo_index': None, 'aut_stab_index': 8, 'aut_weyl_group': '8.3', 'aut_weyl_index': 64, 'centralizer': '32.a1.a1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['4.b1.a1'], 'contains': ['16.a1.a1', '16.b1.a1'], 'core': '16.a1.a1', 'coset_action_label': None, 'count': 4, 'diagramx': [7702, -1, 8093, -1, 7774, -1, 8092, -1], 'generators': [3, 48], 'label': '64.52.8.b1.a1', 'mobius_quo': None, 'mobius_sub': 0, 'normal_closure': '2.a1.a1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '4.b1.a1', 'old_label': '8.b1.a1', 'projective_image': '32.18', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '8.b1.a1', 'subgroup_fusion': None, 'weyl_group': '8.3'}