Formats: - HTML - YAML - JSON - 2026-09-21T01:57:25.127253
  • 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': 8, 'aut_gen_orders': [4, 2, 8], 'aut_gens': [[1, 2], [1, 10], [1, 14], [21, 2]], 'aut_group': '64.123', 'aut_hash': 123, 'aut_nilpotency_class': 3, 'aut_nilpotent': True, 'aut_order': 64, 'aut_permdeg': 12, 'aut_perms': [3714489, 8356336, 52266376], 'aut_phi_ratio': 4.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 8, 1, 1], [4, 2, 1, 1], [4, 8, 1, 1], [8, 2, 2, 1], [16, 2, 4, 1]], 'aut_supersolvable': True, 'aut_tex': 'D_8:C_4', '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': 4, 'autcentquo_group': '16.11', 'autcentquo_hash': 11, 'autcentquo_nilpotent': True, 'autcentquo_order': 16, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'C_2\\times D_4', 'cc_stats': [[1, 1, 1], [2, 1, 1], [2, 8, 1], [4, 2, 1], [4, 8, 1], [8, 2, 2], [16, 2, 4]], 'center_label': '2.1', 'center_order': 2, 'central_product': False, 'central_quotient': '16.7', 'commutator_count': 1, 'commutator_label': '8.1', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '2.1'], 'composition_length': 5, 'conjugacy_classes_known': True, 'counter': 19, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 8, 1, 1], [4, 2, 1, 1], [4, 8, 1, 1], [8, 2, 2, 1], [16, 2, 4, 1]], 'element_repr_type': 'PC', 'elementary': 2, 'eulerian_function': 6, 'exponent': 16, 'exponents_of_order': [5], 'factors_of_aut_order': [2], 'factors_of_order': [2], 'faithful_reps': [[2, 0, 4]], 'familial': True, 'frattini_label': '8.1', 'frattini_quotient': '4.2', 'hash': 19, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 8, 'inner_gen_orders': [2, 8], 'inner_gens': [[1, 14], [21, 2]], 'inner_hash': 7, 'inner_nilpotent': True, 'inner_order': 16, 'inner_split': True, 'inner_tex': 'D_8', 'inner_used': [1, 2], 'irrC_degree': 2, 'irrQ_degree': 8, 'irrQ_dim': 8, 'irrR_degree': 4, 'irrep_stats': [[1, 4], [2, 7]], 'label': '32.19', 'linC_count': 4, 'linC_degree': 2, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 8, 'linQ_degree_count': 1, 'linQ_dim': 8, 'linQ_dim_count': 1, 'linR_count': 2, 'linR_degree': 4, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'SD32', 'ngens': 2, 'nilpotency_class': 4, 'nilpotent': True, 'normal_counts': [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': 8, 'number_conjugacy_classes': 11, 'number_divisions': 7, 'number_normal_subgroups': 8, 'number_subgroup_autclasses': 13, 'number_subgroup_classes': 13, 'number_subgroups': 28, 'old_label': None, 'order': 32, 'order_factorization_type': 3, 'order_stats': [[1, 1], [2, 9], [4, 10], [8, 4], [16, 8]], 'outer_abelian': True, 'outer_cyclic': True, 'outer_equivalence': False, 'outer_exponent': 4, 'outer_gen_orders': [4], 'outer_gen_pows': [0], 'outer_gens': [[1, 10]], 'outer_group': '4.1', 'outer_hash': 1, 'outer_nilpotent': True, 'outer_order': 4, 'outer_permdeg': 4, 'outer_perms': [9], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_4', 'pc_rank': 2, 'perfect': False, 'permutation_degree': 16, '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]], 'representations': {'PC': {'code': 604534538366, 'gens': [1, 2], 'pres': [5, -2, 2, -2, -2, -2, 141, 26, 422, 42, 483, 58]}, 'GLFp': {'d': 2, 'p': 7, 'gens': [2059, 1102]}, 'Perm': {'d': 16, 'gens': [12526762711680, 6314147152, 8402167089238, 4097506706057, 1313941673647]}}, 'schur_multiplier': [], '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': '\\SD_{32}', 'transitive_degree': 16, 'wreath_data': None, 'wreath_product': False}
  • gps_conj_classes • Show schema
    • id: 109738036
      {'aut_label': '1A', 'centralizer': '1.a1.a1', 'counter': 1, 'group_counter': 19, 'group_order': 32, 'label': '1A', 'order': 1, 'powers': [1], 'representative': 0, 'size': 1}
    • id: 109738039
      {'aut_label': '2A', 'centralizer': '1.a1.a1', 'counter': 2, 'group_counter': 19, 'group_order': 32, 'label': '2A', 'order': 2, 'powers': [1], 'representative': 16, 'size': 1}
    • id: 109738040
      {'aut_label': '2B', 'centralizer': '8.b1.a1', 'counter': 3, 'group_counter': 19, 'group_order': 32, 'label': '2B', 'order': 2, 'powers': [1], 'representative': 1, 'size': 8}
    • id: 109738041
      {'aut_label': '4A', 'centralizer': '2.c1.a1', 'counter': 4, 'group_counter': 19, 'group_order': 32, 'label': '4A', 'order': 4, 'powers': [2], 'representative': 8, 'size': 2}
    • id: 109738042
      {'aut_label': '4B', 'centralizer': '8.c1.a1', 'counter': 5, 'group_counter': 19, 'group_order': 32, 'label': '4B', 'order': 4, 'powers': [2], 'representative': 7, 'size': 8}
    • id: 109738043
      {'aut_label': '8A1', 'centralizer': '2.c1.a1', 'counter': 6, 'group_counter': 19, 'group_order': 32, 'label': '8A1', 'order': 8, 'powers': [4], 'representative': 4, 'size': 2}
    • id: 109738044
      {'aut_label': '8A1', 'centralizer': '2.c1.a1', 'counter': 7, 'group_counter': 19, 'group_order': 32, 'label': '8A3', 'order': 8, 'powers': [4], 'representative': 12, 'size': 2}
    • id: 109738045
      {'aut_label': '16A1', 'centralizer': '2.c1.a1', 'counter': 8, 'group_counter': 19, 'group_order': 32, 'label': '16A1', 'order': 16, 'powers': [6], 'representative': 2, 'size': 2}
    • id: 109738046
      {'aut_label': '16A1', 'centralizer': '2.c1.a1', 'counter': 9, 'group_counter': 19, 'group_order': 32, 'label': '16A-1', 'order': 16, 'powers': [6], 'representative': 18, 'size': 2}
    • id: 109738037
      {'aut_label': '16A1', 'centralizer': '2.c1.a1', 'counter': 10, 'group_counter': 19, 'group_order': 32, 'label': '16A3', 'order': 16, 'powers': [7], 'representative': 10, 'size': 2}
    • id: 109738038
      {'aut_label': '16A1', 'centralizer': '2.c1.a1', 'counter': 11, 'group_counter': 19, 'group_order': 32, 'label': '16A-3', 'order': 16, 'powers': [7], 'representative': 26, 'size': 2}
  • gps_qchar • Show schema
    • id: 4381524
      {'cdim': 1, 'counter': 1, 'faithful': False, 'group': '32.19', 'group_order': 32, 'image': None, 'label': '32.19.1a', 'multiplicity': 1, 'nt': [1, 1], 'qdim': 1, 'qvalues': [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1], 'schur_index': 1}
    • id: 4381525
      {'cdim': 1, 'counter': 2, 'faithful': False, 'group': '32.19', 'group_order': 32, 'image': None, 'label': '32.19.1b', 'multiplicity': 1, 'nt': [2, 1], 'qdim': 1, 'qvalues': [1, 1, -1, 1, -1, 1, 1, 1, 1, 1, 1], 'schur_index': 1}
    • id: 4381526
      {'cdim': 1, 'counter': 3, 'faithful': False, 'group': '32.19', 'group_order': 32, 'image': None, 'label': '32.19.1c', 'multiplicity': 1, 'nt': [2, 1], 'qdim': 1, 'qvalues': [1, 1, -1, 1, 1, 1, 1, -1, -1, -1, -1], 'schur_index': 1}
    • id: 4381527
      {'cdim': 1, 'counter': 4, 'faithful': False, 'group': '32.19', 'group_order': 32, 'image': None, 'label': '32.19.1d', 'multiplicity': 1, 'nt': [2, 1], 'qdim': 1, 'qvalues': [1, 1, 1, 1, -1, 1, 1, -1, -1, -1, -1], 'schur_index': 1}
    • id: 4381528
      {'cdim': 2, 'counter': 5, 'faithful': False, 'group': '32.19', 'group_order': 32, 'image': None, 'label': '32.19.2a', 'multiplicity': 1, 'nt': [4, 3], 'qdim': 2, 'qvalues': [2, 2, 0, 2, 0, -2, -2, 0, 0, 0, 0], 'schur_index': 1}
    • id: 4381529
      {'cdim': 2, 'counter': 6, 'faithful': False, 'group': '32.19', 'group_order': 32, 'image': None, 'label': '32.19.2b', 'multiplicity': 2, 'nt': [8, 6], 'qdim': 4, 'qvalues': [4, 4, 0, -4, 0, 0, 0, 0, 0, 0, 0], 'schur_index': 1}
    • id: 4381530
      {'cdim': 2, 'counter': 7, 'faithful': True, 'group': '32.19', 'group_order': 32, 'image': None, 'label': '32.19.2c', 'multiplicity': 4, 'nt': [16, 55], 'qdim': 8, 'qvalues': [8, -8, 0, 0, 0, 0, 0, 0, 0, 0, 0], 'schur_index': 1}
  • gps_char • Show schema
    • id: 6645674
      {'center': '1.a1.a1', 'center_index': 1, 'center_order': 32, 'counter': 1, 'cyclotomic_n': 1, 'dim': 1, 'faithful': False, 'field': [0, 1], 'group': '32.19', 'group_counter': 19, 'group_order': 32, 'image': None, 'image_isoclass': '1.1', 'image_order': 1, 'indicator': 1, 'kernel': '1.a1.a1', 'kernel_order': 32, 'label': '32.19.1a', '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]]]}
    • id: 6645677
      {'center': '1.a1.a1', 'center_index': 1, 'center_order': 32, 'counter': 2, 'cyclotomic_n': 1, 'dim': 1, 'faithful': False, 'field': [0, 1], 'group': '32.19', 'group_counter': 19, 'group_order': 32, 'image': None, 'image_isoclass': '2.1', 'image_order': 2, 'indicator': 1, 'kernel': '2.c1.a1', 'kernel_order': 16, 'label': '32.19.1b', '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]]]}
    • id: 6645678
      {'center': '1.a1.a1', 'center_index': 1, 'center_order': 32, 'counter': 3, 'cyclotomic_n': 1, 'dim': 1, 'faithful': False, 'field': [0, 1], 'group': '32.19', 'group_counter': 19, 'group_order': 32, 'image': None, 'image_isoclass': '2.1', 'image_order': 2, 'indicator': 1, 'kernel': '2.b1.a1', 'kernel_order': 16, 'label': '32.19.1c', '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]]]}
    • id: 6645679
      {'center': '1.a1.a1', 'center_index': 1, 'center_order': 32, 'counter': 4, 'cyclotomic_n': 1, 'dim': 1, 'faithful': False, 'field': [0, 1], 'group': '32.19', 'group_counter': 19, 'group_order': 32, 'image': None, 'image_isoclass': '2.1', 'image_order': 2, 'indicator': 1, 'kernel': '2.a1.a1', 'kernel_order': 16, 'label': '32.19.1d', '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]]]}
    • id: 6645680
      {'center': '4.a1.a1', 'center_index': 4, 'center_order': 8, 'counter': 5, 'cyclotomic_n': 1, 'dim': 2, 'faithful': False, 'field': [0, 1], 'group': '32.19', 'group_counter': 19, 'group_order': 32, 'image': None, 'image_isoclass': '8.3', 'image_order': 8, 'indicator': 1, 'kernel': '8.a1.a1', 'kernel_order': 4, 'label': '32.19.2a', 'nt': [4, 3], 'q_character': None, 'values': [[[2, 0]], [[2, 0]], [[0, 0]], [[2, 0]], [[0, 0]], [[-2, 0]], [[-2, 0]], [[0, 0]], [[0, 0]], [[0, 0]], [[0, 0]]]}
    • id: 6645681
      {'center': '8.a1.a1', 'center_index': 8, 'center_order': 4, 'counter': 6, 'cyclotomic_n': 8, 'dim': 2, 'faithful': False, 'field': [-2, 0, 1], 'group': '32.19', 'group_counter': 19, 'group_order': 32, 'image': None, 'image_isoclass': '16.7', 'image_order': 16, 'indicator': 1, 'kernel': '16.a1.a1', 'kernel_order': 2, 'label': '32.19.2b1', 'nt': [8, 6], 'q_character': None, 'values': [[[2, 0]], [[2, 0]], [[0, 0]], [[-2, 0]], [[0, 0]], [[0, 0]], [[0, 0]], [[-1, 1], [-1, -1]], [[-1, 1], [-1, -1]], [[1, 1], [1, -1]], [[1, 1], [1, -1]]]}
    • id: 6645682
      {'center': '8.a1.a1', 'center_index': 8, 'center_order': 4, 'counter': 7, 'cyclotomic_n': 8, 'dim': 2, 'faithful': False, 'field': [-2, 0, 1], 'group': '32.19', 'group_counter': 19, 'group_order': 32, 'image': None, 'image_isoclass': '16.7', 'image_order': 16, 'indicator': 1, 'kernel': '16.a1.a1', 'kernel_order': 2, 'label': '32.19.2b2', 'nt': [8, 6], 'q_character': None, 'values': [[[2, 0]], [[2, 0]], [[0, 0]], [[-2, 0]], [[0, 0]], [[0, 0]], [[0, 0]], [[1, 1], [1, -1]], [[1, 1], [1, -1]], [[-1, 1], [-1, -1]], [[-1, 1], [-1, -1]]]}
    • id: 6645683
      {'center': '16.a1.a1', 'center_index': 16, 'center_order': 2, 'counter': 8, 'cyclotomic_n': 16, 'dim': 2, 'faithful': True, 'field': [2, 0, 4, 0, 1], 'group': '32.19', 'group_counter': 19, 'group_order': 32, 'image': None, 'image_isoclass': '32.19', 'image_order': 32, 'indicator': 0, 'kernel': '32.a1.a1', 'kernel_order': 1, 'label': '32.19.2c1', 'nt': [16, 55], 'q_character': None, 'values': [[[2, 0]], [[-2, 0]], [[0, 0]], [[0, 0]], [[0, 0]], [[-1, 2], [-1, -2]], [[1, 2], [1, -2]], [[-1, 3], [-1, 5]], [[1, 3], [1, 5]], [[1, 1], [1, 7]], [[-1, 1], [-1, 7]]]}
    • id: 6645684
      {'center': '16.a1.a1', 'center_index': 16, 'center_order': 2, 'counter': 9, 'cyclotomic_n': 16, 'dim': 2, 'faithful': True, 'field': [2, 0, 4, 0, 1], 'group': '32.19', 'group_counter': 19, 'group_order': 32, 'image': None, 'image_isoclass': '32.19', 'image_order': 32, 'indicator': 0, 'kernel': '32.a1.a1', 'kernel_order': 1, 'label': '32.19.2c2', 'nt': [16, 4], 'q_character': None, 'values': [[[2, 0]], [[-2, 0]], [[0, 0]], [[0, 0]], [[0, 0]], [[-1, 2], [-1, -2]], [[1, 2], [1, -2]], [[1, 3], [1, 5]], [[-1, 3], [-1, 5]], [[-1, 1], [-1, 7]], [[1, 1], [1, 7]]]}
    • id: 6645675
      {'center': '16.a1.a1', 'center_index': 16, 'center_order': 2, 'counter': 10, 'cyclotomic_n': 16, 'dim': 2, 'faithful': True, 'field': [2, 0, 4, 0, 1], 'group': '32.19', 'group_counter': 19, 'group_order': 32, 'image': None, 'image_isoclass': '32.19', 'image_order': 32, 'indicator': 0, 'kernel': '32.a1.a1', 'kernel_order': 1, 'label': '32.19.2c3', 'nt': [16, 55], 'q_character': None, 'values': [[[2, 0]], [[-2, 0]], [[0, 0]], [[0, 0]], [[0, 0]], [[1, 2], [1, -2]], [[-1, 2], [-1, -2]], [[-1, 1], [-1, 7]], [[1, 1], [1, 7]], [[-1, 3], [-1, 5]], [[1, 3], [1, 5]]]}
    • id: 6645676
      {'center': '16.a1.a1', 'center_index': 16, 'center_order': 2, 'counter': 11, 'cyclotomic_n': 16, 'dim': 2, 'faithful': True, 'field': [2, 0, 4, 0, 1], 'group': '32.19', 'group_counter': 19, 'group_order': 32, 'image': None, 'image_isoclass': '32.19', 'image_order': 32, 'indicator': 0, 'kernel': '32.a1.a1', 'kernel_order': 1, 'label': '32.19.2c4', 'nt': [16, 4], 'q_character': None, 'values': [[[2, 0]], [[-2, 0]], [[0, 0]], [[0, 0]], [[0, 0]], [[1, 2], [1, -2]], [[-1, 2], [-1, -2]], [[1, 1], [1, 7]], [[-1, 1], [-1, 7]], [[1, 3], [1, 5]], [[-1, 3], [-1, 5]]]}
  • gps_subgroup_search • Show schema
    • id: 65316283
      {'Agroup': False, 'Zgroup': False, 'abelian': False, 'ambient': '32.19', 'ambient_counter': 19, 'ambient_order': 32, 'ambient_tex': '\\SD_{32}', 'central': False, 'central_factor': False, 'centralizer_order': 2, 'characteristic': True, 'core_order': 32, 'counter': 1, 'cyclic': False, 'direct': True, 'hall': 2, 'label': '32.19.1.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': '32.19', 'subgroup_hash': 19, 'subgroup_order': 32, 'subgroup_tex': '\\SD_{32}', 'supersolvable': True, 'sylow': 2}
    • id: 65316288
      {'Agroup': False, 'Zgroup': False, 'abelian': False, 'ambient': '32.19', 'ambient_counter': 19, 'ambient_order': 32, 'ambient_tex': '\\SD_{32}', 'central': False, 'central_factor': False, 'centralizer_order': 2, 'characteristic': True, 'core_order': 16, 'counter': 2, 'cyclic': False, 'direct': False, 'hall': 0, 'label': '32.19.2.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': False, 'standard_generators': False, 'stem': False, 'subgroup': '16.7', 'subgroup_hash': 7, 'subgroup_order': 16, 'subgroup_tex': 'D_8', 'supersolvable': True, 'sylow': 0}
    • id: 65316289
      {'Agroup': False, 'Zgroup': False, 'abelian': False, 'ambient': '32.19', 'ambient_counter': 19, 'ambient_order': 32, 'ambient_tex': '\\SD_{32}', 'central': False, 'central_factor': False, 'centralizer_order': 2, 'characteristic': True, 'core_order': 16, 'counter': 3, 'cyclic': False, 'direct': False, 'hall': 0, 'label': '32.19.2.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': [], 'split': True, 'standard_generators': False, 'stem': False, 'subgroup': '16.9', 'subgroup_hash': 9, 'subgroup_order': 16, 'subgroup_tex': 'Q_{16}', 'supersolvable': True, 'sylow': 0}
    • id: 65316290
      {'Agroup': True, 'Zgroup': True, 'abelian': True, 'ambient': '32.19', 'ambient_counter': 19, 'ambient_order': 32, 'ambient_tex': '\\SD_{32}', 'central': False, 'central_factor': False, 'centralizer_order': 16, 'characteristic': True, 'core_order': 16, 'counter': 4, 'cyclic': True, 'direct': False, 'hall': 0, 'label': '32.19.2.c1.a1', 'maximal': True, 'maximal_normal': True, 'metabelian': True, 'metacyclic': True, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': True, 'old_label': '2.c1.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': '16.1', 'subgroup_hash': 1, 'subgroup_order': 16, 'subgroup_tex': 'C_{16}', 'supersolvable': True, 'sylow': 0}
    • id: 65316291
      {'Agroup': True, 'Zgroup': True, 'abelian': True, 'ambient': '32.19', 'ambient_counter': 19, 'ambient_order': 32, 'ambient_tex': '\\SD_{32}', 'central': False, 'central_factor': False, 'centralizer_order': 16, 'characteristic': True, 'core_order': 8, 'counter': 5, 'cyclic': True, 'direct': False, 'hall': 0, 'label': '32.19.4.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': '8.1', 'subgroup_hash': 1, 'subgroup_order': 8, 'subgroup_tex': 'C_8', 'supersolvable': True, 'sylow': 0}
    • id: 65316292
      {'Agroup': False, 'Zgroup': False, 'abelian': False, 'ambient': '32.19', 'ambient_counter': 19, 'ambient_order': 32, 'ambient_tex': '\\SD_{32}', 'central': False, 'central_factor': False, 'centralizer_order': 2, 'characteristic': False, 'core_order': 4, 'counter': 6, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '32.19.4.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': '8.3', 'subgroup_hash': 3, 'subgroup_order': 8, 'subgroup_tex': 'D_4', 'supersolvable': True, 'sylow': 0}
    • id: 65316293
      {'Agroup': False, 'Zgroup': False, 'abelian': False, 'ambient': '32.19', 'ambient_counter': 19, 'ambient_order': 32, 'ambient_tex': '\\SD_{32}', 'central': False, 'central_factor': False, 'centralizer_order': 2, 'characteristic': False, 'core_order': 4, 'counter': 7, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '32.19.4.c1.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': False, 'old_label': '4.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': 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': '8.4', 'subgroup_hash': 4, 'subgroup_order': 8, 'subgroup_tex': 'Q_8', 'supersolvable': True, 'sylow': 0}
    • id: 65316294
      {'Agroup': True, 'Zgroup': True, 'abelian': True, 'ambient': '32.19', 'ambient_counter': 19, 'ambient_order': 32, 'ambient_tex': '\\SD_{32}', 'central': False, 'central_factor': False, 'centralizer_order': 16, 'characteristic': True, 'core_order': 4, 'counter': 8, 'cyclic': True, 'direct': False, 'hall': 0, 'label': '32.19.8.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': '4.1', 'subgroup_hash': 1, 'subgroup_order': 4, 'subgroup_tex': 'C_4', 'supersolvable': True, 'sylow': 0}
    • id: 65316295
      {'Agroup': True, 'Zgroup': False, 'abelian': True, 'ambient': '32.19', 'ambient_counter': 19, 'ambient_order': 32, 'ambient_tex': '\\SD_{32}', 'central': False, 'central_factor': False, 'centralizer_order': 4, 'characteristic': False, 'core_order': 2, 'counter': 9, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '32.19.8.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': '4.2', 'subgroup_hash': 2, 'subgroup_order': 4, 'subgroup_tex': 'C_2^2', 'supersolvable': True, 'sylow': 0}
    • id: 65316284
      {'Agroup': True, 'Zgroup': True, 'abelian': True, 'ambient': '32.19', 'ambient_counter': 19, 'ambient_order': 32, 'ambient_tex': '\\SD_{32}', 'central': False, 'central_factor': False, 'centralizer_order': 4, 'characteristic': False, 'core_order': 2, 'counter': 10, 'cyclic': True, 'direct': None, 'hall': 0, 'label': '32.19.8.c1.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': False, 'old_label': '8.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': 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': '4.1', 'subgroup_hash': 1, 'subgroup_order': 4, 'subgroup_tex': 'C_4', 'supersolvable': True, 'sylow': 0}
    • id: 65316285
      {'Agroup': True, 'Zgroup': True, 'abelian': True, 'ambient': '32.19', 'ambient_counter': 19, 'ambient_order': 32, 'ambient_tex': '\\SD_{32}', 'central': True, 'central_factor': False, 'centralizer_order': 32, 'characteristic': True, 'core_order': 2, 'counter': 11, 'cyclic': True, 'direct': False, 'hall': 0, 'label': '32.19.16.a1.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': True, 'minimal_normal': True, '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': True, 'solvable': True, 'special_labels': ['Z', 'S', 'U3', 'L3', 'C4'], '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: 65316286
      {'Agroup': True, 'Zgroup': True, 'abelian': True, 'ambient': '32.19', 'ambient_counter': 19, 'ambient_order': 32, 'ambient_tex': '\\SD_{32}', 'central': False, 'central_factor': False, 'centralizer_order': 4, 'characteristic': False, 'core_order': 1, 'counter': 12, 'cyclic': True, 'direct': None, 'hall': 0, 'label': '32.19.16.b1.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': True, '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': 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: 65316287
      {'Agroup': True, 'Zgroup': True, 'abelian': True, 'ambient': '32.19', 'ambient_counter': 19, 'ambient_order': 32, 'ambient_tex': '\\SD_{32}', 'central': True, 'central_factor': False, 'centralizer_order': 32, 'characteristic': True, 'core_order': 1, 'counter': 13, 'cyclic': True, 'direct': True, 'hall': 1, 'label': '32.19.32.a1.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': True, 'old_label': '32.a1.a1', 'outer_equivalence': False, 'perfect': True, 'proper': False, 'quotient': '32.19', 'quotient_Agroup': False, 'quotient_abelian': False, 'quotient_cyclic': False, 'quotient_hash': 19, 'quotient_metabelian': True, 'quotient_nilpotent': True, 'quotient_order': 32, 'quotient_simple': False, 'quotient_solvable': True, 'quotient_supersolvable': True, 'quotient_tex': '\\SD_{32}', 'simple': False, 'solvable': True, 'special_labels': ['PC', 'U4', 'L4', '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: 65316283
      {'ambient': '32.19', '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': '16.a1.a1', 'complements': ['32.a1.a1'], 'conjugacy_class_count': 1, 'contained_in': [], 'contains': ['2.a1.a1', '2.b1.a1', '2.c1.a1'], 'core': '1.a1.a1', 'coset_action_label': None, 'count': 1, 'diagramx': [4801, 4822, 4801, 4822, 4668, 4655, 4668, 4655], 'generators': [1, 2], 'label': '32.19.1.a1.a1', 'mobius_quo': 0, 'mobius_sub': 1, 'normal_closure': '1.a1.a1', 'normal_contained_in': [], 'normal_contains': ['2.a1.a1', '2.b1.a1', '2.c1.a1'], 'normalizer': '1.a1.a1', 'old_label': '1.a1.a1', 'projective_image': '16.7', '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': '16.7'}
    • id: 65316284
      {'ambient': '32.19', 'aut_centralizer_order': 8, 'aut_label': '8.c1', 'aut_quo_index': None, 'aut_stab_index': 4, 'aut_weyl_group': '2.1', 'aut_weyl_index': 32, 'centralizer': '8.c1.a1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['4.c1.a1'], 'contains': ['16.a1.a1'], 'core': '16.a1.a1', 'coset_action_label': None, 'count': 4, 'diagramx': [7974, -1, 7974, -1, 7752, -1, 7752, -1], 'generators': [3], 'label': '32.19.8.c1.a1', 'mobius_quo': None, 'mobius_sub': 0, 'normal_closure': '2.b1.a1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '4.c1.a1', 'old_label': '8.c1.a1', 'projective_image': '16.7', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '8.c1.a1', 'subgroup_fusion': None, 'weyl_group': '2.1'}
    • id: 65316285
      {'ambient': '32.19', 'aut_centralizer_order': 64, 'aut_label': '16.a1', 'aut_quo_index': 2, 'aut_stab_index': 1, 'aut_weyl_group': '1.1', 'aut_weyl_index': 64, 'centralizer': '1.a1.a1', 'complements': [], 'conjugacy_class_count': 1, 'contained_in': ['8.a1.a1', '8.b1.a1', '8.c1.a1'], 'contains': ['32.a1.a1'], 'core': '16.a1.a1', 'coset_action_label': None, 'count': 1, 'diagramx': [4714, 4822, 4714, 4822, 4582, 4655, 4582, 4655], 'generators': [16], 'label': '32.19.16.a1.a1', 'mobius_quo': -1, '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': '16.7', 'quotient_action_image': '1.1', 'quotient_action_kernel': '16.7', 'quotient_action_kernel_order': 16, 'quotient_fusion': None, 'short_label': '16.a1.a1', 'subgroup_fusion': None, 'weyl_group': '1.1'}
    • id: 65316286
      {'ambient': '32.19', 'aut_centralizer_order': 8, 'aut_label': '16.b1', 'aut_quo_index': None, 'aut_stab_index': 8, 'aut_weyl_group': '1.1', 'aut_weyl_index': 64, 'centralizer': '8.b1.a1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['8.b1.a1'], 'contains': ['32.a1.a1'], 'core': '32.a1.a1', 'coset_action_label': '16T55', 'count': 8, 'diagramx': [1498, -1, 1498, -1, 1456, -1, 1456, -1], 'generators': [1], 'label': '32.19.16.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': '32.19', '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': '1.1'}
    • id: 65316287
      {'ambient': '32.19', 'aut_centralizer_order': None, 'aut_label': '32.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': ['16.a1.a1', '16.b1.a1'], 'contains': [], 'core': '32.a1.a1', 'coset_action_label': '32T30', 'count': 1, 'diagramx': [3128, 4822, 3128, 4822, 2998, 4655, 2998, 4655], 'generators': [], 'label': '32.19.32.a1.a1', 'mobius_quo': 1, 'mobius_sub': 0, 'normal_closure': '32.a1.a1', 'normal_contained_in': ['16.a1.a1'], 'normal_contains': [], 'normalizer': '1.a1.a1', 'old_label': '32.a1.a1', 'projective_image': '32.19', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '32.a1.a1', 'subgroup_fusion': None, 'weyl_group': '1.1'}
    • id: 65316288
      {'ambient': '32.19', 'aut_centralizer_order': 2, 'aut_label': '2.a1', 'aut_quo_index': 1, 'aut_stab_index': 1, 'aut_weyl_group': '32.43', 'aut_weyl_index': 2, 'centralizer': '16.a1.a1', 'complements': [], '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': [1277, 1205, 1277, 1205, 1242, 1164, 1242, 1164], 'generators': [1, 12], 'label': '32.19.2.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': '16.7', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '2.a1.a1', 'subgroup_fusion': None, 'weyl_group': '16.7'}
    • id: 65316289
      {'ambient': '32.19', 'aut_centralizer_order': 2, 'aut_label': '2.b1', 'aut_quo_index': 1, 'aut_stab_index': 1, 'aut_weyl_group': '32.43', 'aut_weyl_index': 2, 'centralizer': '16.a1.a1', 'complements': ['16.b1.a1'], 'conjugacy_class_count': 1, 'contained_in': ['1.a1.a1'], 'contains': ['4.a1.a1', '4.c1.a1'], 'core': '2.b1.a1', 'coset_action_label': None, 'count': 1, 'diagramx': [8723, 4822, 8723, 4822, 8480, 4655, 8480, 4655], 'generators': [3, 12], 'label': '32.19.2.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': '16.7', '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': '16.7'}
    • id: 65316290
      {'ambient': '32.19', 'aut_centralizer_order': 8, 'aut_label': '2.c1', 'aut_quo_index': 1, 'aut_stab_index': 1, 'aut_weyl_group': '8.2', 'aut_weyl_index': 8, 'centralizer': '2.c1.a1', 'complements': ['16.b1.a1'], 'conjugacy_class_count': 1, 'contained_in': ['1.a1.a1'], 'contains': ['4.a1.a1'], 'core': '2.c1.a1', 'coset_action_label': None, 'count': 1, 'diagramx': [4801, 8795, 4801, 8795, 4668, 8491, 4668, 8491], 'generators': [2], 'label': '32.19.2.c1.a1', 'mobius_quo': 0, 'mobius_sub': -1, 'normal_closure': '2.c1.a1', 'normal_contained_in': ['1.a1.a1'], 'normal_contains': ['4.a1.a1'], 'normalizer': '1.a1.a1', 'old_label': '2.c1.a1', 'projective_image': '16.7', 'quotient_action_image': '2.1', 'quotient_action_kernel': '1.1', 'quotient_action_kernel_order': 1, 'quotient_fusion': None, 'short_label': '2.c1.a1', 'subgroup_fusion': None, 'weyl_group': '2.1'}
    • id: 65316291
      {'ambient': '32.19', 'aut_centralizer_order': 16, 'aut_label': '4.a1', 'aut_quo_index': 6, 'aut_stab_index': 1, 'aut_weyl_group': '4.2', 'aut_weyl_index': 16, 'centralizer': '2.c1.a1', 'complements': [], 'conjugacy_class_count': 1, 'contained_in': ['2.a1.a1', '2.b1.a1', '2.c1.a1'], 'contains': ['8.a1.a1'], 'core': '4.a1.a1', 'coset_action_label': None, 'count': 1, 'diagramx': [4801, 4822, 4801, 4822, 4668, 4655, 4668, 4655], 'generators': [12], 'label': '32.19.4.a1.a1', 'mobius_quo': 0, 'mobius_sub': 2, 'normal_closure': '4.a1.a1', 'normal_contained_in': ['2.a1.a1', '2.b1.a1', '2.c1.a1'], 'normal_contains': ['8.a1.a1'], 'normalizer': '1.a1.a1', 'old_label': '4.a1.a1', 'projective_image': '16.7', '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: 65316292
      {'ambient': '32.19', 'aut_centralizer_order': 4, 'aut_label': '4.b1', 'aut_quo_index': None, 'aut_stab_index': 2, 'aut_weyl_group': '8.3', 'aut_weyl_index': 8, 'centralizer': '16.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': [1409, -1, 1409, -1, 1370, -1, 1370, -1], 'generators': [1, 8], 'label': '32.19.4.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': '16.7', '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': '8.3'}
    • id: 65316293
      {'ambient': '32.19', 'aut_centralizer_order': 4, 'aut_label': '4.c1', 'aut_quo_index': None, 'aut_stab_index': 2, 'aut_weyl_group': '8.3', 'aut_weyl_index': 8, 'centralizer': '16.a1.a1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['2.b1.a1'], 'contains': ['8.a1.a1', '8.c1.a1'], 'core': '8.a1.a1', 'coset_action_label': None, 'count': 2, 'diagramx': [7974, -1, 7974, -1, 7752, -1, 7752, -1], 'generators': [3, 8], 'label': '32.19.4.c1.a1', 'mobius_quo': None, 'mobius_sub': 0, 'normal_closure': '2.b1.a1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '2.b1.a1', 'old_label': '4.c1.a1', 'projective_image': '16.7', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '4.c1.a1', 'subgroup_fusion': None, 'weyl_group': '8.3'}
    • id: 65316294
      {'ambient': '32.19', 'aut_centralizer_order': 32, 'aut_label': '8.a1', 'aut_quo_index': 2, 'aut_stab_index': 1, 'aut_weyl_group': '2.1', 'aut_weyl_index': 32, 'centralizer': '2.c1.a1', 'complements': [], 'conjugacy_class_count': 1, 'contained_in': ['4.a1.a1', '4.b1.a1', '4.c1.a1'], 'contains': ['16.a1.a1'], 'core': '8.a1.a1', 'coset_action_label': None, 'count': 1, 'diagramx': [4801, 4822, 4801, 4822, 4668, 4655, 4668, 4655], 'generators': [8], 'label': '32.19.8.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': '16.7', '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: 65316295
      {'ambient': '32.19', 'aut_centralizer_order': 8, 'aut_label': '8.b1', 'aut_quo_index': None, 'aut_stab_index': 4, 'aut_weyl_group': '2.1', 'aut_weyl_index': 32, 'centralizer': '8.b1.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': [1409, -1, 1409, -1, 1370, -1, 1370, -1], 'generators': [1, 16], 'label': '32.19.8.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': '16.7', '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': '2.1'}