Formats: - HTML - YAML - JSON - 2026-09-21T01:57:24.916552
  • gps_groups • Show schema
    {'Agroup': True, 'Zgroup': True, 'abelian': False, 'abelian_quotient': '2.1', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 110, 'aut_gen_orders': [10, 11], 'aut_gens': [[1, 2], [21, 14], [7, 2]], 'aut_group': '110.1', 'aut_hash': 1, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 110, 'aut_permdeg': 11, 'aut_perms': [1289144, 17617417], 'aut_phi_ratio': 11.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 11, 1, 1], [11, 2, 5, 1]], 'aut_supersolvable': True, 'aut_tex': 'F_{11}', 'autcent_abelian': True, 'autcent_cyclic': True, 'autcent_exponent': 1, 'autcent_group': '1.1', 'autcent_hash': 1, 'autcent_nilpotent': True, 'autcent_order': 1, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_1', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 110, 'autcentquo_group': '110.1', 'autcentquo_hash': 1, 'autcentquo_nilpotent': False, 'autcentquo_order': 110, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'F_{11}', 'cc_stats': [[1, 1, 1], [2, 11, 1], [11, 2, 5]], 'center_label': '1.1', 'center_order': 1, 'central_product': False, 'central_quotient': '22.1', 'commutator_count': 1, 'commutator_label': '11.1', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '11.1'], 'composition_length': 2, 'conjugacy_classes_known': True, 'counter': 1, 'cyclic': False, 'derived_length': 2, 'dihedral': True, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 11, 1, 1], [11, 2, 5, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 3, 'exponent': 22, 'exponents_of_order': [1, 1], 'factors_of_aut_order': [2, 5, 11], 'factors_of_order': [2, 11], 'faithful_reps': [[2, 1, 5]], 'familial': True, 'frattini_label': '1.1', 'frattini_quotient': '22.1', 'hash': 1, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 22, 'inner_gen_orders': [2, 11], 'inner_gens': [[1, 20], [5, 2]], 'inner_hash': 1, 'inner_nilpotent': False, 'inner_order': 22, 'inner_split': True, 'inner_tex': 'D_{11}', 'inner_used': [1, 2], 'irrC_degree': 2, 'irrQ_degree': 10, 'irrQ_dim': 10, 'irrR_degree': 2, 'irrep_stats': [[1, 2], [2, 5]], 'label': '22.1', 'linC_count': 5, 'linC_degree': 2, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 10, 'linQ_degree_count': 1, 'linQ_dim': 10, 'linQ_dim_count': 1, 'linR_count': 5, 'linR_degree': 2, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'D11', 'ngens': 2, 'nilpotency_class': -1, 'nilpotent': False, 'normal_counts': [0, 0, 0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 3, 'number_characteristic_subgroups': 3, 'number_conjugacy_classes': 7, 'number_divisions': 3, 'number_normal_subgroups': 3, 'number_subgroup_autclasses': 4, 'number_subgroup_classes': 4, 'number_subgroups': 14, 'old_label': None, 'order': 22, 'order_factorization_type': 11, 'order_stats': [[1, 1], [2, 11], [11, 10]], 'outer_abelian': True, 'outer_cyclic': True, 'outer_equivalence': False, 'outer_exponent': 5, 'outer_gen_orders': [5], 'outer_gen_pows': [1], 'outer_gens': [[1, 4]], 'outer_group': '5.1', 'outer_hash': 1, 'outer_nilpotent': True, 'outer_order': 5, 'outer_permdeg': 5, 'outer_perms': [33], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_5', 'pc_rank': 2, 'perfect': False, 'permutation_degree': 11, 'pgroup': 0, 'primary_abelian_invariants': [2], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 2], [10, 1]], 'representations': {'PC': {'code': 3809, 'gens': [1, 2], 'pres': [2, -2, -11, 81]}, 'GLFp': {'d': 2, 'p': 11, 'gens': [1343, 13311]}, 'Perm': {'d': 11, 'gens': [3628799, 36288000]}}, 'schur_multiplier': [], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2], 'solvability_type': 6, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'D_{11}', 'transitive_degree': 11, 'wreath_data': None, 'wreath_product': False}
  • gps_conj_classes • Show schema
    • id: 109281298
      {'aut_label': '1A', 'centralizer': '1.a1.a1', 'counter': 1, 'group_counter': 1, 'group_order': 22, 'label': '1A', 'order': 1, 'powers': [1, 1, 1], 'representative': 0, 'size': 1}
    • id: 109281299
      {'aut_label': '2A', 'centralizer': '11.a1.a1', 'counter': 2, 'group_counter': 1, 'group_order': 22, 'label': '2A', 'order': 2, 'powers': [1, 2, 2], 'representative': 7, 'size': 11}
    • id: 109281300
      {'aut_label': '11A1', 'centralizer': '2.a1.a1', 'counter': 3, 'group_counter': 1, 'group_order': 22, 'label': '11A1', 'order': 11, 'powers': [4, 7, 1], 'representative': 2, 'size': 2}
    • id: 109281301
      {'aut_label': '11A1', 'centralizer': '2.a1.a1', 'counter': 4, 'group_counter': 1, 'group_order': 22, 'label': '11A2', 'order': 11, 'powers': [6, 3, 1], 'representative': 4, 'size': 2}
    • id: 109281302
      {'aut_label': '11A1', 'centralizer': '2.a1.a1', 'counter': 5, 'group_counter': 1, 'group_order': 22, 'label': '11A3', 'order': 11, 'powers': [7, 6, 1], 'representative': 6, 'size': 2}
    • id: 109281303
      {'aut_label': '11A1', 'centralizer': '2.a1.a1', 'counter': 6, 'group_counter': 1, 'group_order': 22, 'label': '11A4', 'order': 11, 'powers': [5, 4, 1], 'representative': 8, 'size': 2}
    • id: 109281304
      {'aut_label': '11A1', 'centralizer': '2.a1.a1', 'counter': 7, 'group_counter': 1, 'group_order': 22, 'label': '11A5', 'order': 11, 'powers': [3, 5, 1], 'representative': 10, 'size': 2}
  • gps_qchar • Show schema
    • id: 4208906
      {'cdim': 1, 'counter': 1, 'faithful': False, 'group': '22.1', 'group_order': 22, 'image': None, 'label': '22.1.1a', 'multiplicity': 1, 'nt': [1, 1], 'qdim': 1, 'qvalues': [1, 1, 1, 1, 1, 1, 1], 'schur_index': 1}
    • id: 4208907
      {'cdim': 1, 'counter': 2, 'faithful': False, 'group': '22.1', 'group_order': 22, 'image': None, 'label': '22.1.1b', 'multiplicity': 1, 'nt': [2, 1], 'qdim': 1, 'qvalues': [1, -1, 1, 1, 1, 1, 1], 'schur_index': 1}
    • id: 4208908
      {'cdim': 2, 'counter': 3, 'faithful': True, 'group': '22.1', 'group_order': 22, 'image': None, 'label': '22.1.2a', 'multiplicity': 5, 'nt': [11, 2], 'qdim': 10, 'qvalues': [10, 0, -1, -1, -1, -1, -1], 'schur_index': 1}
  • gps_char • Show schema
    • id: 6188995
      {'center': '1.a1.a1', 'center_index': 1, 'center_order': 22, 'counter': 1, 'cyclotomic_n': 1, 'dim': 1, 'faithful': False, 'field': [0, 1], 'group': '22.1', 'group_counter': 1, 'group_order': 22, 'image': None, 'image_isoclass': '1.1', 'image_order': 1, 'indicator': 1, 'kernel': '1.a1.a1', 'kernel_order': 22, 'label': '22.1.1a', 'nt': [1, 1], 'q_character': None, 'values': [[[1, 0]], [[1, 0]], [[1, 0]], [[1, 0]], [[1, 0]], [[1, 0]], [[1, 0]]]}
    • id: 6188996
      {'center': '1.a1.a1', 'center_index': 1, 'center_order': 22, 'counter': 2, 'cyclotomic_n': 1, 'dim': 1, 'faithful': False, 'field': [0, 1], 'group': '22.1', 'group_counter': 1, 'group_order': 22, 'image': None, 'image_isoclass': '2.1', 'image_order': 2, 'indicator': 1, 'kernel': '2.a1.a1', 'kernel_order': 11, 'label': '22.1.1b', 'nt': [2, 1], 'q_character': None, 'values': [[[1, 0]], [[-1, 0]], [[1, 0]], [[1, 0]], [[1, 0]], [[1, 0]], [[1, 0]]]}
    • id: 6188997
      {'center': '22.a1.a1', 'center_index': 22, 'center_order': 1, 'counter': 3, 'cyclotomic_n': 11, 'dim': 2, 'faithful': True, 'field': [-1, 3, 3, -4, -1, 1], 'group': '22.1', 'group_counter': 1, 'group_order': 22, 'image': None, 'image_isoclass': '22.1', 'image_order': 22, 'indicator': 1, 'kernel': '22.a1.a1', 'kernel_order': 1, 'label': '22.1.2a1', 'nt': [11, 2], 'q_character': None, 'values': [[[2, 0]], [[0, 0]], [[1, 5], [1, -5]], [[1, 1], [1, -1]], [[1, 4], [1, -4]], [[1, 2], [1, -2]], [[1, 3], [1, -3]]]}
    • id: 6188998
      {'center': '22.a1.a1', 'center_index': 22, 'center_order': 1, 'counter': 4, 'cyclotomic_n': 11, 'dim': 2, 'faithful': True, 'field': [-1, 3, 3, -4, -1, 1], 'group': '22.1', 'group_counter': 1, 'group_order': 22, 'image': None, 'image_isoclass': '22.1', 'image_order': 22, 'indicator': 1, 'kernel': '22.a1.a1', 'kernel_order': 1, 'label': '22.1.2a2', 'nt': [11, 2], 'q_character': None, 'values': [[[2, 0]], [[0, 0]], [[1, 4], [1, -4]], [[1, 3], [1, -3]], [[1, 1], [1, -1]], [[1, 5], [1, -5]], [[1, 2], [1, -2]]]}
    • id: 6188999
      {'center': '22.a1.a1', 'center_index': 22, 'center_order': 1, 'counter': 5, 'cyclotomic_n': 11, 'dim': 2, 'faithful': True, 'field': [-1, 3, 3, -4, -1, 1], 'group': '22.1', 'group_counter': 1, 'group_order': 22, 'image': None, 'image_isoclass': '22.1', 'image_order': 22, 'indicator': 1, 'kernel': '22.a1.a1', 'kernel_order': 1, 'label': '22.1.2a3', 'nt': [11, 2], 'q_character': None, 'values': [[[2, 0]], [[0, 0]], [[1, 3], [1, -3]], [[1, 5], [1, -5]], [[1, 2], [1, -2]], [[1, 1], [1, -1]], [[1, 4], [1, -4]]]}
    • id: 6189000
      {'center': '22.a1.a1', 'center_index': 22, 'center_order': 1, 'counter': 6, 'cyclotomic_n': 11, 'dim': 2, 'faithful': True, 'field': [-1, 3, 3, -4, -1, 1], 'group': '22.1', 'group_counter': 1, 'group_order': 22, 'image': None, 'image_isoclass': '22.1', 'image_order': 22, 'indicator': 1, 'kernel': '22.a1.a1', 'kernel_order': 1, 'label': '22.1.2a4', 'nt': [11, 2], 'q_character': None, 'values': [[[2, 0]], [[0, 0]], [[1, 2], [1, -2]], [[1, 4], [1, -4]], [[1, 5], [1, -5]], [[1, 3], [1, -3]], [[1, 1], [1, -1]]]}
    • id: 6189001
      {'center': '22.a1.a1', 'center_index': 22, 'center_order': 1, 'counter': 7, 'cyclotomic_n': 11, 'dim': 2, 'faithful': True, 'field': [-1, 3, 3, -4, -1, 1], 'group': '22.1', 'group_counter': 1, 'group_order': 22, 'image': None, 'image_isoclass': '22.1', 'image_order': 22, 'indicator': 1, 'kernel': '22.a1.a1', 'kernel_order': 1, 'label': '22.1.2a5', 'nt': [11, 2], 'q_character': None, 'values': [[[2, 0]], [[0, 0]], [[1, 1], [1, -1]], [[1, 2], [1, -2]], [[1, 3], [1, -3]], [[1, 4], [1, -4]], [[1, 5], [1, -5]]]}
  • gps_subgroup_search • Show schema
    • id: 63987075
      {'Agroup': True, 'Zgroup': True, 'abelian': False, 'ambient': '22.1', 'ambient_counter': 1, 'ambient_order': 22, 'ambient_tex': 'D_{11}', 'central': False, 'central_factor': False, 'centralizer_order': 1, 'characteristic': True, 'core_order': 22, 'counter': 1, 'cyclic': False, 'direct': True, 'hall': 22, 'label': '22.1.1.a1.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': False, 'minimal_normal': False, 'nilpotent': False, 'normal': True, 'old_label': '1.a1.a1', 'outer_equivalence': False, 'perfect': False, 'proper': False, 'quotient': '1.1', 'quotient_Agroup': True, 'quotient_abelian': True, 'quotient_cyclic': True, 'quotient_hash': 1, 'quotient_metabelian': True, 'quotient_nilpotent': True, 'quotient_order': 1, 'quotient_simple': False, 'quotient_solvable': True, 'quotient_supersolvable': True, 'quotient_tex': 'C_1', 'simple': False, 'solvable': True, 'special_labels': ['R', 'L0', 'D0', 'C0'], 'split': True, 'standard_generators': False, 'stem': False, 'subgroup': '22.1', 'subgroup_hash': 1, 'subgroup_order': 22, 'subgroup_tex': 'D_{11}', 'supersolvable': True, 'sylow': 0}
    • id: 63987076
      {'Agroup': True, 'Zgroup': True, 'abelian': True, 'ambient': '22.1', 'ambient_counter': 1, 'ambient_order': 22, 'ambient_tex': 'D_{11}', 'central': False, 'central_factor': False, 'centralizer_order': 11, 'characteristic': True, 'core_order': 11, 'counter': 2, 'cyclic': True, 'direct': False, 'hall': 11, 'label': '22.1.2.a1.a1', 'maximal': True, 'maximal_normal': True, 'metabelian': True, 'metacyclic': True, 'minimal': True, 'minimal_normal': True, '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': True, 'solvable': True, 'special_labels': ['D', 'F', 'S', 'L1', 'D1', 'C1'], 'split': True, 'standard_generators': False, 'stem': False, 'subgroup': '11.1', 'subgroup_hash': 1, 'subgroup_order': 11, 'subgroup_tex': 'C_{11}', 'supersolvable': True, 'sylow': 11}
    • id: 63987077
      {'Agroup': True, 'Zgroup': True, 'abelian': True, 'ambient': '22.1', 'ambient_counter': 1, 'ambient_order': 22, 'ambient_tex': 'D_{11}', 'central': False, 'central_factor': False, 'centralizer_order': 2, 'characteristic': False, 'core_order': 1, 'counter': 3, 'cyclic': True, 'direct': None, 'hall': 2, 'label': '22.1.11.a1.a1', 'maximal': True, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': True, 'minimal_normal': False, 'nilpotent': True, 'normal': False, 'old_label': '11.a1.a1', 'outer_equivalence': False, 'perfect': False, 'proper': True, 'quotient': None, 'quotient_Agroup': None, 'quotient_abelian': None, 'quotient_cyclic': None, 'quotient_hash': None, 'quotient_metabelian': None, 'quotient_nilpotent': None, 'quotient_order': 11, 'quotient_simple': None, 'quotient_solvable': None, 'quotient_supersolvable': None, 'quotient_tex': None, 'simple': True, 'solvable': True, 'special_labels': [], 'split': None, 'standard_generators': False, 'stem': False, 'subgroup': '2.1', 'subgroup_hash': 1, 'subgroup_order': 2, 'subgroup_tex': 'C_2', 'supersolvable': True, 'sylow': 2}
    • id: 63987078
      {'Agroup': True, 'Zgroup': True, 'abelian': True, 'ambient': '22.1', 'ambient_counter': 1, 'ambient_order': 22, 'ambient_tex': 'D_{11}', 'central': True, 'central_factor': False, 'centralizer_order': 22, 'characteristic': True, 'core_order': 1, 'counter': 4, 'cyclic': True, 'direct': True, 'hall': 1, 'label': '22.1.22.a1.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': True, 'old_label': '22.a1.a1', 'outer_equivalence': False, 'perfect': True, 'proper': False, 'quotient': '22.1', 'quotient_Agroup': True, 'quotient_abelian': False, 'quotient_cyclic': False, 'quotient_hash': 1, 'quotient_metabelian': True, 'quotient_nilpotent': False, 'quotient_order': 22, 'quotient_simple': False, 'quotient_solvable': True, 'quotient_supersolvable': True, 'quotient_tex': 'D_{11}', 'simple': False, 'solvable': True, 'special_labels': ['Z', 'Phi', 'PC', 'U0', 'D2', 'C2'], '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: 63987075
      {'ambient': '22.1', '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': '22.a1.a1', 'complements': ['22.a1.a1'], 'conjugacy_class_count': 1, 'contained_in': [], 'contains': ['2.a1.a1', '11.a1.a1'], 'core': '1.a1.a1', 'coset_action_label': None, 'count': 1, 'diagramx': [4503, 5000, 6261, 5000, 5282, 5000, 4718, 5000], 'generators': [1, 2], 'label': '22.1.1.a1.a1', 'mobius_quo': 0, 'mobius_sub': 1, 'normal_closure': '1.a1.a1', 'normal_contained_in': [], 'normal_contains': ['2.a1.a1'], 'normalizer': '1.a1.a1', 'old_label': '1.a1.a1', 'projective_image': '22.1', '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': '22.1'}
    • id: 63987076
      {'ambient': '22.1', 'aut_centralizer_order': 11, 'aut_label': '2.a1', 'aut_quo_index': 1, 'aut_stab_index': 1, 'aut_weyl_group': '10.2', 'aut_weyl_index': 11, 'centralizer': '2.a1.a1', 'complements': ['11.a1.a1'], 'conjugacy_class_count': 1, 'contained_in': ['1.a1.a1'], 'contains': ['22.a1.a1'], 'core': '2.a1.a1', 'coset_action_label': None, 'count': 1, 'diagramx': [2672, 5000, 7652, 5000, 2465, 5000, 7535, 5000], 'generators': [2], 'label': '22.1.2.a1.a1', 'mobius_quo': -1, 'mobius_sub': -1, 'normal_closure': '2.a1.a1', 'normal_contained_in': ['1.a1.a1'], 'normal_contains': ['22.a1.a1'], 'normalizer': '1.a1.a1', 'old_label': '2.a1.a1', 'projective_image': '22.1', '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': '2.1'}
    • id: 63987077
      {'ambient': '22.1', 'aut_centralizer_order': 10, 'aut_label': '11.a1', 'aut_quo_index': None, 'aut_stab_index': 11, 'aut_weyl_group': '1.1', 'aut_weyl_index': 110, 'centralizer': '11.a1.a1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['1.a1.a1'], 'contains': ['22.a1.a1'], 'core': '22.a1.a1', 'coset_action_label': '11T2', 'count': 11, 'diagramx': [7328, -1, 2348, -1, 8099, -1, 1901, -1], 'generators': [1], 'label': '22.1.11.a1.a1', 'mobius_quo': None, 'mobius_sub': -1, 'normal_closure': '1.a1.a1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '11.a1.a1', 'old_label': '11.a1.a1', 'projective_image': '22.1', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '11.a1.a1', 'subgroup_fusion': None, 'weyl_group': '1.1'}
    • id: 63987078
      {'ambient': '22.1', 'aut_centralizer_order': None, 'aut_label': '22.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': ['2.a1.a1', '11.a1.a1'], 'contains': [], 'core': '22.a1.a1', 'coset_action_label': '22T2', 'count': 1, 'diagramx': [5190, 5000, 4870, 5000, 5282, 5000, 4718, 5000], 'generators': [], 'label': '22.1.22.a1.a1', 'mobius_quo': 1, 'mobius_sub': 11, 'normal_closure': '22.a1.a1', 'normal_contained_in': ['2.a1.a1'], 'normal_contains': [], 'normalizer': '1.a1.a1', 'old_label': '22.a1.a1', 'projective_image': '22.1', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '22.a1.a1', 'subgroup_fusion': None, 'weyl_group': '1.1'}