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gps_groups • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '8.2', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 4, 'aut_gen_orders': [2, 2, 2, 4], 'aut_gens': [[1, 2, 4], [11, 2, 4], [9, 2, 4], [3, 2, 12], [9, 2, 13]], 'aut_group': '32.27', 'aut_hash': 27, 'aut_nilpotency_class': 2, 'aut_nilpotent': True, 'aut_order': 32, 'aut_permdeg': 8, 'aut_perms': [36273, 10905, 38579, 6322], 'aut_phi_ratio': 4.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 1, 2, 1], [2, 2, 2, 1], [4, 2, 4, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_2^2\\wr C_2', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 2, 'autcent_group': '16.14', 'autcent_hash': 14, 'autcent_nilpotent': True, 'autcent_order': 16, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^4', 'autcentquo_abelian': True, 'autcentquo_cyclic': True, 'autcentquo_exponent': 2, 'autcentquo_group': '2.1', 'autcentquo_hash': 1, 'autcentquo_nilpotent': True, 'autcentquo_order': 2, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'C_2', 'cc_stats': [[1, 1, 1], [2, 1, 3], [2, 2, 2], [4, 2, 4]], 'center_label': '4.2', 'center_order': 4, 'central_product': False, 'central_quotient': '4.2', 'commutator_count': 1, 'commutator_label': '2.1', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1'], 'composition_length': 4, 'conjugacy_classes_known': True, 'counter': 3, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [2, 2, 1, 2], [4, 2, 2, 2]], 'element_repr_type': 'PC', 'elementary': 2, 'eulerian_function': 3, 'exponent': 4, 'exponents_of_order': [4], 'factors_of_aut_order': [2], 'factors_of_order': [2], 'faithful_reps': [], 'familial': False, 'frattini_label': '4.2', 'frattini_quotient': '4.2', 'hash': 3, 'hyperelementary': 2, 'inner_abelian': True, 'inner_cyclic': False, 'inner_exponent': 2, 'inner_gen_orders': [2, 1, 2], 'inner_gens': [[1, 2, 6], [1, 2, 4], [3, 2, 4]], 'inner_hash': 2, 'inner_nilpotent': True, 'inner_order': 4, 'inner_split': True, 'inner_tex': 'C_2^2', 'inner_used': [1, 3], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 8], [2, 2]], 'label': '16.3', 'linC_count': 8, 'linC_degree': 3, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 4, 'linQ_degree_count': 5, 'linQ_dim': 4, 'linQ_dim_count': 5, 'linR_count': 5, 'linR_degree': 4, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'C2^2:C4', 'ngens': 2, 'nilpotency_class': 2, 'nilpotent': True, 'normal_counts': [0, 0, 0, 0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 5, 'number_characteristic_subgroups': 5, 'number_conjugacy_classes': 10, 'number_divisions': 8, 'number_normal_subgroups': 11, 'number_subgroup_autclasses': 11, 'number_subgroup_classes': 17, 'number_subgroups': 23, 'old_label': None, 'order': 16, 'order_factorization_type': 3, 'order_stats': [[1, 1], [2, 7], [4, 8]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 4, 'outer_gen_orders': [2, 4], 'outer_gen_pows': [0, 0], 'outer_gens': [[1, 2, 5], [9, 2, 13]], 'outer_group': '8.3', 'outer_hash': 3, 'outer_nilpotent': True, 'outer_order': 8, 'outer_permdeg': 4, 'outer_perms': [7, 9], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'D_4', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 8, 'pgroup': 2, 'primary_abelian_invariants': [2, 4], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 4]], 'representations': {'PC': {'code': 33300, 'gens': [1, 2, 3], 'pres': [4, 2, 2, 2, 2, 74, 34]}, 'GLZ': {'b': 3, 'd': 4, 'gens': [16917454, 35813460]}, 'GLFp': {'d': 3, 'p': 5, 'gens': [1152128, 611879, 438254, 1565004]}, 'Perm': {'d': 8, 'gens': [28776, 5183, 11536, 5167]}}, 'schur_multiplier': [2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 4], 'solvability_type': 4, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_2^2:C_4', 'transitive_degree': 8, 'wreath_data': None, 'wreath_product': False}
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gps_conj_classes • Show schema
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id: 52281894
{'aut_label': '1A', 'centralizer': '1.a1.a1', 'counter': 1, 'group_counter': 3, 'group_order': 16, 'label': '1A', 'order': 1, 'powers': [1], 'representative': 0, 'size': 1}
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id: 52281896
{'aut_label': '2A', 'centralizer': '1.a1.a1', 'counter': 2, 'group_counter': 3, 'group_order': 16, 'label': '2A', 'order': 2, 'powers': [1], 'representative': 2, 'size': 1}
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id: 52281897
{'aut_label': '2B', 'centralizer': '1.a1.a1', 'counter': 3, 'group_counter': 3, 'group_order': 16, 'label': '2B', 'order': 2, 'powers': [1], 'representative': 8, 'size': 1}
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id: 52281898
{'aut_label': '2B', 'centralizer': '1.a1.a1', 'counter': 4, 'group_counter': 3, 'group_order': 16, 'label': '2C', 'order': 2, 'powers': [1], 'representative': 10, 'size': 1}
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id: 52281899
{'aut_label': '2D', 'centralizer': '2.a1.a1', 'counter': 5, 'group_counter': 3, 'group_order': 16, 'label': '2D', 'order': 2, 'powers': [1], 'representative': 1, 'size': 2}
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id: 52281900
{'aut_label': '2D', 'centralizer': '2.a1.a1', 'counter': 6, 'group_counter': 3, 'group_order': 16, 'label': '2E', 'order': 2, 'powers': [1], 'representative': 9, 'size': 2}
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id: 52281901
{'aut_label': '4A1', 'centralizer': '2.b1.a1', 'counter': 7, 'group_counter': 3, 'group_order': 16, 'label': '4A1', 'order': 4, 'powers': [3], 'representative': 4, 'size': 2}
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id: 52281902
{'aut_label': '4A1', 'centralizer': '2.b1.a1', 'counter': 8, 'group_counter': 3, 'group_order': 16, 'label': '4A-1', 'order': 4, 'powers': [3], 'representative': 12, 'size': 2}
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id: 52281903
{'aut_label': '4A1', 'centralizer': '2.b1.b1', 'counter': 9, 'group_counter': 3, 'group_order': 16, 'label': '4B1', 'order': 4, 'powers': [4], 'representative': 5, 'size': 2}
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id: 52281895
{'aut_label': '4A1', 'centralizer': '2.b1.b1', 'counter': 10, 'group_counter': 3, 'group_order': 16, 'label': '4B-1', 'order': 4, 'powers': [4], 'representative': 13, 'size': 2}
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gps_qchar • Show schema
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id: 171100
{'cdim': 1, 'counter': 1, 'faithful': False, 'group': '16.3', 'group_order': 16, 'image': None, 'label': '16.3.1a', 'multiplicity': 1, 'nt': [1, 1], 'qdim': 1, 'qvalues': [1, 1, 1, 1, 1, 1, 1, 1, 1, 1], 'schur_index': 1}
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id: 171101
{'cdim': 1, 'counter': 2, 'faithful': False, 'group': '16.3', 'group_order': 16, 'image': None, 'label': '16.3.1b', 'multiplicity': 1, 'nt': [2, 1], 'qdim': 1, 'qvalues': [1, 1, 1, 1, -1, -1, -1, -1, 1, 1], 'schur_index': 1}
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id: 171102
{'cdim': 1, 'counter': 3, 'faithful': False, 'group': '16.3', 'group_order': 16, 'image': None, 'label': '16.3.1c', 'multiplicity': 1, 'nt': [2, 1], 'qdim': 1, 'qvalues': [1, 1, 1, 1, -1, -1, 1, 1, -1, -1], 'schur_index': 1}
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id: 171103
{'cdim': 1, 'counter': 4, 'faithful': False, 'group': '16.3', 'group_order': 16, 'image': None, 'label': '16.3.1d', 'multiplicity': 1, 'nt': [2, 1], 'qdim': 1, 'qvalues': [1, 1, 1, 1, 1, 1, -1, -1, -1, -1], 'schur_index': 1}
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id: 171104
{'cdim': 1, 'counter': 5, 'faithful': False, 'group': '16.3', 'group_order': 16, 'image': None, 'label': '16.3.1e', 'multiplicity': 2, 'nt': [4, 1], 'qdim': 2, 'qvalues': [2, 2, -2, -2, -2, 2, 0, 0, 0, 0], 'schur_index': 1}
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id: 171105
{'cdim': 1, 'counter': 6, 'faithful': False, 'group': '16.3', 'group_order': 16, 'image': None, 'label': '16.3.1f', 'multiplicity': 2, 'nt': [4, 1], 'qdim': 2, 'qvalues': [2, 2, -2, -2, 2, -2, 0, 0, 0, 0], 'schur_index': 1}
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id: 171106
{'cdim': 2, 'counter': 7, 'faithful': False, 'group': '16.3', 'group_order': 16, 'image': None, 'label': '16.3.2a', 'multiplicity': 1, 'nt': [4, 3], 'qdim': 2, 'qvalues': [2, -2, -2, 2, 0, 0, 0, 0, 0, 0], 'schur_index': 1}
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id: 171107
{'cdim': 2, 'counter': 8, 'faithful': False, 'group': '16.3', 'group_order': 16, 'image': None, 'label': '16.3.2b', 'multiplicity': 1, 'nt': [4, 3], 'qdim': 2, 'qvalues': [2, -2, 2, -2, 0, 0, 0, 0, 0, 0], 'schur_index': 1}
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gps_char • Show schema
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id: 203973
{'center': '1.a1.a1', 'center_index': 1, 'center_order': 16, 'counter': 1, 'cyclotomic_n': 1, 'dim': 1, 'faithful': False, 'field': [0, 1], 'group': '16.3', 'group_counter': 3, 'group_order': 16, 'image': None, 'image_isoclass': '1.1', 'image_order': 1, 'indicator': 1, 'kernel': '1.a1.a1', 'kernel_order': 16, 'label': '16.3.1a', '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]]]}
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id: 203975
{'center': '1.a1.a1', 'center_index': 1, 'center_order': 16, 'counter': 2, 'cyclotomic_n': 1, 'dim': 1, 'faithful': False, 'field': [0, 1], 'group': '16.3', 'group_counter': 3, 'group_order': 16, 'image': None, 'image_isoclass': '2.1', 'image_order': 2, 'indicator': 1, 'kernel': '2.b1.b1', 'kernel_order': 8, 'label': '16.3.1b', '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]]]}
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id: 203976
{'center': '1.a1.a1', 'center_index': 1, 'center_order': 16, 'counter': 3, 'cyclotomic_n': 1, 'dim': 1, 'faithful': False, 'field': [0, 1], 'group': '16.3', 'group_counter': 3, 'group_order': 16, 'image': None, 'image_isoclass': '2.1', 'image_order': 2, 'indicator': 1, 'kernel': '2.b1.a1', 'kernel_order': 8, 'label': '16.3.1c', '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]]]}
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id: 203977
{'center': '1.a1.a1', 'center_index': 1, 'center_order': 16, 'counter': 4, 'cyclotomic_n': 1, 'dim': 1, 'faithful': False, 'field': [0, 1], 'group': '16.3', 'group_counter': 3, 'group_order': 16, 'image': None, 'image_isoclass': '2.1', 'image_order': 2, 'indicator': 1, 'kernel': '2.a1.a1', 'kernel_order': 8, 'label': '16.3.1d', '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]]]}
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id: 203978
{'center': '1.a1.a1', 'center_index': 1, 'center_order': 16, 'counter': 5, 'cyclotomic_n': 4, 'dim': 1, 'faithful': False, 'field': [1, 0, 1], 'group': '16.3', 'group_counter': 3, 'group_order': 16, 'image': None, 'image_isoclass': '4.1', 'image_order': 4, 'indicator': 0, 'kernel': '4.b1.b1', 'kernel_order': 4, 'label': '16.3.1e1', 'nt': [4, 1], 'q_character': None, 'values': [[[1, 0]], [[1, 0]], [[-1, 0]], [[-1, 0]], [[-1, 0]], [[1, 0]], [[-1, 1]], [[1, 1]], [[1, 1]], [[-1, 1]]]}
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id: 203979
{'center': '1.a1.a1', 'center_index': 1, 'center_order': 16, 'counter': 6, 'cyclotomic_n': 4, 'dim': 1, 'faithful': False, 'field': [1, 0, 1], 'group': '16.3', 'group_counter': 3, 'group_order': 16, 'image': None, 'image_isoclass': '4.1', 'image_order': 4, 'indicator': 0, 'kernel': '4.b1.b1', 'kernel_order': 4, 'label': '16.3.1e2', 'nt': [4, 2], 'q_character': None, 'values': [[[1, 0]], [[1, 0]], [[-1, 0]], [[-1, 0]], [[-1, 0]], [[1, 0]], [[1, 1]], [[-1, 1]], [[-1, 1]], [[1, 1]]]}
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id: 203980
{'center': '1.a1.a1', 'center_index': 1, 'center_order': 16, 'counter': 7, 'cyclotomic_n': 4, 'dim': 1, 'faithful': False, 'field': [1, 0, 1], 'group': '16.3', 'group_counter': 3, 'group_order': 16, 'image': None, 'image_isoclass': '4.1', 'image_order': 4, 'indicator': 0, 'kernel': '4.b1.a1', 'kernel_order': 4, 'label': '16.3.1f1', 'nt': [4, 1], 'q_character': None, 'values': [[[1, 0]], [[1, 0]], [[-1, 0]], [[-1, 0]], [[1, 0]], [[-1, 0]], [[-1, 1]], [[1, 1]], [[-1, 1]], [[1, 1]]]}
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id: 203981
{'center': '1.a1.a1', 'center_index': 1, 'center_order': 16, 'counter': 8, 'cyclotomic_n': 4, 'dim': 1, 'faithful': False, 'field': [1, 0, 1], 'group': '16.3', 'group_counter': 3, 'group_order': 16, 'image': None, 'image_isoclass': '4.1', 'image_order': 4, 'indicator': 0, 'kernel': '4.b1.a1', 'kernel_order': 4, 'label': '16.3.1f2', 'nt': [4, 2], 'q_character': None, 'values': [[[1, 0]], [[1, 0]], [[-1, 0]], [[-1, 0]], [[1, 0]], [[-1, 0]], [[1, 1]], [[-1, 1]], [[1, 1]], [[-1, 1]]]}
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id: 203982
{'center': '4.a1.a1', 'center_index': 4, 'center_order': 4, 'counter': 9, 'cyclotomic_n': 1, 'dim': 2, 'faithful': False, 'field': [0, 1], 'group': '16.3', 'group_counter': 3, 'group_order': 16, 'image': None, 'image_isoclass': '8.3', 'image_order': 8, 'indicator': 1, 'kernel': '8.b1.b1', 'kernel_order': 2, 'label': '16.3.2a', 'nt': [4, 3], 'q_character': None, 'values': [[[2, 0]], [[-2, 0]], [[-2, 0]], [[2, 0]], [[0, 0]], [[0, 0]], [[0, 0]], [[0, 0]], [[0, 0]], [[0, 0]]]}
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id: 203974
{'center': '4.a1.a1', 'center_index': 4, 'center_order': 4, 'counter': 10, 'cyclotomic_n': 1, 'dim': 2, 'faithful': False, 'field': [0, 1], 'group': '16.3', 'group_counter': 3, 'group_order': 16, 'image': None, 'image_isoclass': '8.3', 'image_order': 8, 'indicator': 1, 'kernel': '8.b1.a1', 'kernel_order': 2, 'label': '16.3.2b', 'nt': [4, 3], 'q_character': None, 'values': [[[2, 0]], [[-2, 0]], [[2, 0]], [[-2, 0]], [[0, 0]], [[0, 0]], [[0, 0]], [[0, 0]], [[0, 0]], [[0, 0]]]}
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gps_subgroup_search • Show schema
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id: 13000837
{'Agroup': False, 'Zgroup': False, 'abelian': False, 'ambient': '16.3', 'ambient_counter': 3, 'ambient_order': 16, 'ambient_tex': 'C_2^2:C_4', 'central': False, 'central_factor': False, 'centralizer_order': 4, 'characteristic': True, 'core_order': 16, 'counter': 1, 'cyclic': False, 'direct': True, 'hall': 2, 'label': '16.3.1.a1.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': False, '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': '16.3', 'subgroup_hash': 3, 'subgroup_order': 16, 'subgroup_tex': 'C_2^2:C_4', 'supersolvable': True, 'sylow': 2}
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id: 13000846
{'Agroup': True, 'Zgroup': False, 'abelian': True, 'ambient': '16.3', 'ambient_counter': 3, 'ambient_order': 16, 'ambient_tex': 'C_2^2:C_4', 'central': False, 'central_factor': False, 'centralizer_order': 8, 'characteristic': True, 'core_order': 8, 'counter': 2, 'cyclic': False, 'direct': False, 'hall': 0, 'label': '16.3.2.a1.a1', 'maximal': True, 'maximal_normal': True, 'metabelian': True, 'metacyclic': False, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': True, 'old_label': '2.a1.a1', 'outer_equivalence': False, 'perfect': False, 'proper': True, 'quotient': '2.1', 'quotient_Agroup': True, 'quotient_abelian': True, 'quotient_cyclic': True, 'quotient_hash': 1, 'quotient_metabelian': True, 'quotient_nilpotent': True, 'quotient_order': 2, 'quotient_simple': True, 'quotient_solvable': True, 'quotient_supersolvable': True, 'quotient_tex': 'C_2', 'simple': False, 'solvable': True, 'special_labels': [], 'split': False, 'standard_generators': False, 'stem': False, 'subgroup': '8.5', 'subgroup_hash': 5, 'subgroup_order': 8, 'subgroup_tex': 'C_2^3', 'supersolvable': True, 'sylow': 0}
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id: 13000847
{'Agroup': True, 'Zgroup': False, 'abelian': True, 'ambient': '16.3', 'ambient_counter': 3, 'ambient_order': 16, 'ambient_tex': 'C_2^2:C_4', 'central': False, 'central_factor': False, 'centralizer_order': 8, 'characteristic': False, 'core_order': 8, 'counter': 3, 'cyclic': False, 'direct': False, 'hall': 0, 'label': '16.3.2.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': '8.2', 'subgroup_hash': 2, 'subgroup_order': 8, 'subgroup_tex': 'C_2\\times C_4', 'supersolvable': True, 'sylow': 0}
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id: 13000848
{'Agroup': True, 'Zgroup': False, 'abelian': True, 'ambient': '16.3', 'ambient_counter': 3, 'ambient_order': 16, 'ambient_tex': 'C_2^2:C_4', 'central': False, 'central_factor': False, 'centralizer_order': 8, 'characteristic': False, 'core_order': 8, 'counter': 4, 'cyclic': False, 'direct': False, 'hall': 0, 'label': '16.3.2.b1.b1', 'maximal': True, 'maximal_normal': True, 'metabelian': True, 'metacyclic': True, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': True, 'old_label': '2.b1.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': '8.2', 'subgroup_hash': 2, 'subgroup_order': 8, 'subgroup_tex': 'C_2\\times C_4', 'supersolvable': True, 'sylow': 0}
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id: 13000849
{'Agroup': True, 'Zgroup': False, 'abelian': True, 'ambient': '16.3', 'ambient_counter': 3, 'ambient_order': 16, 'ambient_tex': 'C_2^2:C_4', 'central': True, 'central_factor': False, 'centralizer_order': 16, 'characteristic': True, 'core_order': 4, 'counter': 5, 'cyclic': False, 'direct': False, 'hall': 0, 'label': '16.3.4.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': ['Z', 'Phi', 'S', 'U1', 'C2'], 'split': False, 'standard_generators': False, 'stem': False, 'subgroup': '4.2', 'subgroup_hash': 2, 'subgroup_order': 4, 'subgroup_tex': 'C_2^2', 'supersolvable': True, 'sylow': 0}
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id: 13000850
{'Agroup': True, 'Zgroup': False, 'abelian': True, 'ambient': '16.3', 'ambient_counter': 3, 'ambient_order': 16, 'ambient_tex': 'C_2^2:C_4', 'central': False, 'central_factor': False, 'centralizer_order': 8, 'characteristic': False, 'core_order': 4, 'counter': 6, 'cyclic': False, 'direct': False, 'hall': 0, 'label': '16.3.4.b1.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': True, 'old_label': '4.b1.a1', 'outer_equivalence': False, 'perfect': False, 'proper': True, 'quotient': '4.1', 'quotient_Agroup': True, 'quotient_abelian': True, 'quotient_cyclic': True, 'quotient_hash': 1, 'quotient_metabelian': True, 'quotient_nilpotent': True, 'quotient_order': 4, 'quotient_simple': False, 'quotient_solvable': True, 'quotient_supersolvable': True, 'quotient_tex': 'C_4', 'simple': False, 'solvable': True, 'special_labels': [], 'split': True, 'standard_generators': False, 'stem': False, 'subgroup': '4.2', 'subgroup_hash': 2, 'subgroup_order': 4, 'subgroup_tex': 'C_2^2', 'supersolvable': True, 'sylow': 0}
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id: 13000851
{'Agroup': True, 'Zgroup': False, 'abelian': True, 'ambient': '16.3', 'ambient_counter': 3, 'ambient_order': 16, 'ambient_tex': 'C_2^2:C_4', 'central': False, 'central_factor': False, 'centralizer_order': 8, 'characteristic': False, 'core_order': 4, 'counter': 7, 'cyclic': False, 'direct': False, 'hall': 0, 'label': '16.3.4.b1.b1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': True, 'old_label': '4.b1.b1', 'outer_equivalence': False, 'perfect': False, 'proper': True, 'quotient': '4.1', 'quotient_Agroup': True, 'quotient_abelian': True, 'quotient_cyclic': True, 'quotient_hash': 1, 'quotient_metabelian': True, 'quotient_nilpotent': True, 'quotient_order': 4, 'quotient_simple': False, 'quotient_solvable': True, 'quotient_supersolvable': True, 'quotient_tex': 'C_4', 'simple': False, 'solvable': True, 'special_labels': [], 'split': True, 'standard_generators': False, 'stem': False, 'subgroup': '4.2', 'subgroup_hash': 2, 'subgroup_order': 4, 'subgroup_tex': 'C_2^2', 'supersolvable': True, 'sylow': 0}
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id: 13000852
{'Agroup': True, 'Zgroup': False, 'abelian': True, 'ambient': '16.3', 'ambient_counter': 3, 'ambient_order': 16, 'ambient_tex': 'C_2^2:C_4', 'central': False, 'central_factor': False, 'centralizer_order': 8, 'characteristic': False, 'core_order': 2, 'counter': 8, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '16.3.4.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': '4.2', 'subgroup_hash': 2, 'subgroup_order': 4, 'subgroup_tex': 'C_2^2', 'supersolvable': True, 'sylow': 0}
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id: 13000853
{'Agroup': True, 'Zgroup': False, 'abelian': True, 'ambient': '16.3', 'ambient_counter': 3, 'ambient_order': 16, 'ambient_tex': 'C_2^2:C_4', 'central': False, 'central_factor': False, 'centralizer_order': 8, 'characteristic': False, 'core_order': 2, 'counter': 9, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '16.3.4.c1.b1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': False, 'old_label': '4.c1.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': '4.2', 'subgroup_hash': 2, 'subgroup_order': 4, 'subgroup_tex': 'C_2^2', 'supersolvable': True, 'sylow': 0}
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id: 13000838
{'Agroup': True, 'Zgroup': True, 'abelian': True, 'ambient': '16.3', 'ambient_counter': 3, 'ambient_order': 16, 'ambient_tex': 'C_2^2:C_4', 'central': False, 'central_factor': False, 'centralizer_order': 8, 'characteristic': False, 'core_order': 2, 'counter': 10, 'cyclic': True, 'direct': None, 'hall': 0, 'label': '16.3.4.d1.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': False, 'old_label': '4.d1.a1', 'outer_equivalence': False, 'perfect': False, 'proper': True, 'quotient': None, 'quotient_Agroup': None, 'quotient_abelian': None, 'quotient_cyclic': None, 'quotient_hash': None, 'quotient_metabelian': None, 'quotient_nilpotent': None, 'quotient_order': 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': '4.1', 'subgroup_hash': 1, 'subgroup_order': 4, 'subgroup_tex': 'C_4', 'supersolvable': True, 'sylow': 0}
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id: 13000839
{'Agroup': True, 'Zgroup': True, 'abelian': True, 'ambient': '16.3', 'ambient_counter': 3, 'ambient_order': 16, 'ambient_tex': 'C_2^2:C_4', 'central': False, 'central_factor': False, 'centralizer_order': 8, 'characteristic': False, 'core_order': 2, 'counter': 11, 'cyclic': True, 'direct': None, 'hall': 0, 'label': '16.3.4.d1.b1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': False, 'old_label': '4.d1.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': '4.1', 'subgroup_hash': 1, 'subgroup_order': 4, 'subgroup_tex': 'C_4', 'supersolvable': True, 'sylow': 0}
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id: 13000840
{'Agroup': True, 'Zgroup': True, 'abelian': True, 'ambient': '16.3', 'ambient_counter': 3, 'ambient_order': 16, 'ambient_tex': 'C_2^2:C_4', 'central': True, 'central_factor': False, 'centralizer_order': 16, 'characteristic': True, 'core_order': 2, 'counter': 12, 'cyclic': True, 'direct': False, 'hall': 0, 'label': '16.3.8.a1.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': True, 'minimal_normal': True, 'nilpotent': True, 'normal': True, 'old_label': '8.a1.a1', 'outer_equivalence': False, 'perfect': False, 'proper': True, 'quotient': '8.2', 'quotient_Agroup': True, 'quotient_abelian': True, 'quotient_cyclic': False, 'quotient_hash': 2, 'quotient_metabelian': True, 'quotient_nilpotent': True, 'quotient_order': 8, 'quotient_simple': False, 'quotient_solvable': True, 'quotient_supersolvable': True, 'quotient_tex': 'C_2\\times C_4', 'simple': True, 'solvable': True, 'special_labels': ['D', 'L1', 'D1'], 'split': False, 'standard_generators': False, 'stem': True, 'subgroup': '2.1', 'subgroup_hash': 1, 'subgroup_order': 2, 'subgroup_tex': 'C_2', 'supersolvable': True, 'sylow': 0}
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id: 13000841
{'Agroup': True, 'Zgroup': True, 'abelian': True, 'ambient': '16.3', 'ambient_counter': 3, 'ambient_order': 16, 'ambient_tex': 'C_2^2:C_4', 'central': True, 'central_factor': False, 'centralizer_order': 16, 'characteristic': False, 'core_order': 2, 'counter': 13, 'cyclic': True, 'direct': False, 'hall': 0, 'label': '16.3.8.b1.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': True, 'minimal_normal': True, 'nilpotent': True, 'normal': True, 'old_label': '8.b1.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': True, 'solvable': True, 'special_labels': ['C3'], 'split': False, 'standard_generators': False, 'stem': False, 'subgroup': '2.1', 'subgroup_hash': 1, 'subgroup_order': 2, 'subgroup_tex': 'C_2', 'supersolvable': True, 'sylow': 0}
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id: 13000842
{'Agroup': True, 'Zgroup': True, 'abelian': True, 'ambient': '16.3', 'ambient_counter': 3, 'ambient_order': 16, 'ambient_tex': 'C_2^2:C_4', 'central': True, 'central_factor': False, 'centralizer_order': 16, 'characteristic': False, 'core_order': 2, 'counter': 14, 'cyclic': True, 'direct': False, 'hall': 0, 'label': '16.3.8.b1.b1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': True, 'minimal_normal': True, 'nilpotent': True, 'normal': True, 'old_label': '8.b1.b1', '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': True, 'solvable': True, 'special_labels': [], 'split': False, 'standard_generators': False, 'stem': False, 'subgroup': '2.1', 'subgroup_hash': 1, 'subgroup_order': 2, 'subgroup_tex': 'C_2', 'supersolvable': True, 'sylow': 0}
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id: 13000843
{'Agroup': True, 'Zgroup': True, 'abelian': True, 'ambient': '16.3', 'ambient_counter': 3, 'ambient_order': 16, 'ambient_tex': 'C_2^2:C_4', 'central': False, 'central_factor': False, 'centralizer_order': 8, 'characteristic': False, 'core_order': 1, 'counter': 15, 'cyclic': True, 'direct': None, 'hall': 0, 'label': '16.3.8.c1.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': True, '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': 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}
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id: 13000844
{'Agroup': True, 'Zgroup': True, 'abelian': True, 'ambient': '16.3', 'ambient_counter': 3, 'ambient_order': 16, 'ambient_tex': 'C_2^2:C_4', 'central': False, 'central_factor': False, 'centralizer_order': 8, 'characteristic': False, 'core_order': 1, 'counter': 16, 'cyclic': True, 'direct': None, 'hall': 0, 'label': '16.3.8.c1.b1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': True, 'minimal_normal': False, 'nilpotent': True, 'normal': False, 'old_label': '8.c1.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': 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}
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id: 13000845
{'Agroup': True, 'Zgroup': True, 'abelian': True, 'ambient': '16.3', 'ambient_counter': 3, 'ambient_order': 16, 'ambient_tex': 'C_2^2:C_4', 'central': True, 'central_factor': False, 'centralizer_order': 16, 'characteristic': True, 'core_order': 1, 'counter': 17, 'cyclic': True, 'direct': True, 'hall': 1, 'label': '16.3.16.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': True, 'proper': False, 'quotient': '16.3', 'quotient_Agroup': False, 'quotient_abelian': False, 'quotient_cyclic': False, 'quotient_hash': 3, 'quotient_metabelian': True, 'quotient_nilpotent': True, 'quotient_order': 16, 'quotient_simple': False, 'quotient_solvable': True, 'quotient_supersolvable': True, 'quotient_tex': 'C_2^2:C_4', 'simple': False, 'solvable': True, 'special_labels': ['PC', 'U2', 'L2', 'D2', 'C4'], 'split': True, 'standard_generators': False, 'stem': True, 'subgroup': '1.1', 'subgroup_hash': 1, 'subgroup_order': 1, 'subgroup_tex': 'C_1', 'supersolvable': True, 'sylow': 1}
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gps_subgroup_data • Show schema
Hide schema
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id: 13000844
{'ambient': '16.3', 'aut_centralizer_order': 8, 'aut_label': '8.c1', 'aut_quo_index': None, 'aut_stab_index': 4, 'aut_weyl_group': '1.1', 'aut_weyl_index': 32, 'centralizer': '2.a1.a1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['4.b1.b1', '4.c1.a1', '4.c1.b1'], 'contains': ['16.a1.a1'], 'core': '16.a1.a1', 'coset_action_label': '8T10', 'count': 2, 'diagramx': [8406, -1, 6243, -1, 8266, -1, 6345, -1], 'generators': [9], 'label': '16.3.8.c1.b1', 'mobius_quo': None, 'mobius_sub': 0, 'normal_closure': '4.b1.b1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '2.a1.a1', 'old_label': '8.c1.b1', 'projective_image': '16.3', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '8.c1.b1', 'subgroup_fusion': None, 'weyl_group': '1.1'}
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id: 13000845
{'ambient': '16.3', 'aut_centralizer_order': None, 'aut_label': '16.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': ['8.a1.a1', '8.b1.a1', '8.b1.b1', '8.c1.a1', '8.c1.b1'], 'contains': [], 'core': '16.a1.a1', 'coset_action_label': '16T10', 'count': 1, 'diagramx': [5966, 5374, 4866, 4904, 5867, 6100, 4945, 5243], 'generators': [], 'label': '16.3.16.a1.a1', 'mobius_quo': 1, 'mobius_sub': 0, 'normal_closure': '16.a1.a1', 'normal_contained_in': ['8.a1.a1', '8.b1.a1', '8.b1.b1'], 'normal_contains': [], 'normalizer': '1.a1.a1', 'old_label': '16.a1.a1', 'projective_image': '16.3', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '16.a1.a1', 'subgroup_fusion': None, 'weyl_group': '1.1'}
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id: 13000837
{'ambient': '16.3', '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': '4.a1.a1', 'complements': ['16.a1.a1'], 'conjugacy_class_count': 1, 'contained_in': [], 'contains': ['2.a1.a1', '2.b1.a1', '2.b1.b1'], 'core': '1.a1.a1', 'coset_action_label': None, 'count': 1, 'diagramx': [4474, 5946, 3488, 4665, 4367, 5650, 3545, 4987], 'generators': [1, 4], 'label': '16.3.1.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.b1.b1'], 'normalizer': '1.a1.a1', 'old_label': '1.a1.a1', 'projective_image': '4.2', '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': '4.2'}
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id: 13000838
{'ambient': '16.3', 'aut_centralizer_order': 4, 'aut_label': '4.d1', 'aut_quo_index': None, 'aut_stab_index': 4, 'aut_weyl_group': '2.1', 'aut_weyl_index': 16, 'centralizer': '2.b1.a1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['2.b1.a1'], 'contains': ['8.b1.a1'], 'core': '8.b1.a1', 'coset_action_label': None, 'count': 2, 'diagramx': [916, -1, 3488, -1, 900, -1, 3545, -1], 'generators': [4], 'label': '16.3.4.d1.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.d1.a1', 'projective_image': '8.3', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '4.d1.a1', 'subgroup_fusion': None, 'weyl_group': '1.1'}
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id: 13000839
{'ambient': '16.3', 'aut_centralizer_order': 4, 'aut_label': '4.d1', 'aut_quo_index': None, 'aut_stab_index': 4, 'aut_weyl_group': '2.1', 'aut_weyl_index': 16, 'centralizer': '2.b1.b1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['2.b1.b1'], 'contains': ['8.b1.b1'], 'core': '8.b1.b1', 'coset_action_label': None, 'count': 2, 'diagramx': [916, -1, 733, -1, 900, -1, 745, -1], 'generators': [5], 'label': '16.3.4.d1.b1', 'mobius_quo': None, 'mobius_sub': 0, 'normal_closure': '2.b1.b1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '2.b1.b1', 'old_label': '4.d1.b1', 'projective_image': '8.3', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '4.d1.b1', 'subgroup_fusion': None, 'weyl_group': '1.1'}
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id: 13000840
{'ambient': '16.3', 'aut_centralizer_order': 32, 'aut_label': '8.a1', 'aut_quo_index': 1, 'aut_stab_index': 1, 'aut_weyl_group': '1.1', 'aut_weyl_index': 32, 'centralizer': '1.a1.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': [5966, 6960, 4866, 6627, 5867, 7900, 4945, 7084], 'generators': [2], 'label': '16.3.8.a1.a1', 'mobius_quo': -1, 'mobius_sub': 0, 'normal_closure': '8.a1.a1', 'normal_contained_in': ['4.a1.a1', '4.b1.b1', '4.b1.a1'], 'normal_contains': ['16.a1.a1'], 'normalizer': '1.a1.a1', 'old_label': '8.a1.a1', 'projective_image': '8.2', 'quotient_action_image': '1.1', 'quotient_action_kernel': '8.2', 'quotient_action_kernel_order': 8, 'quotient_fusion': None, 'short_label': '8.a1.a1', 'subgroup_fusion': None, 'weyl_group': '1.1'}
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id: 13000841
{'ambient': '16.3', 'aut_centralizer_order': 16, 'aut_label': '8.b1', 'aut_quo_index': 2, 'aut_stab_index': 2, 'aut_weyl_group': '1.1', 'aut_weyl_index': 32, 'centralizer': '1.a1.a1', 'complements': [], 'conjugacy_class_count': 1, 'contained_in': ['4.a1.a1', '4.c1.a1', '4.d1.a1'], 'contains': ['16.a1.a1'], 'core': '8.b1.a1', 'coset_action_label': None, 'count': 1, 'diagramx': [3525, 3789, 3524, 3182, 3467, 4300, 3600, 3401], 'generators': [8], 'label': '16.3.8.b1.a1', 'mobius_quo': -1, 'mobius_sub': 0, 'normal_closure': '8.b1.a1', 'normal_contained_in': ['4.a1.a1'], 'normal_contains': ['16.a1.a1'], 'normalizer': '1.a1.a1', 'old_label': '8.b1.a1', 'projective_image': '8.3', 'quotient_action_image': '1.1', 'quotient_action_kernel': '8.3', 'quotient_action_kernel_order': 8, 'quotient_fusion': None, 'short_label': '8.b1.a1', 'subgroup_fusion': None, 'weyl_group': '1.1'}
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id: 13000842
{'ambient': '16.3', 'aut_centralizer_order': 16, 'aut_label': '8.b1', 'aut_quo_index': 2, 'aut_stab_index': 2, 'aut_weyl_group': '1.1', 'aut_weyl_index': 32, 'centralizer': '1.a1.a1', 'complements': [], 'conjugacy_class_count': 1, 'contained_in': ['4.a1.a1', '4.c1.b1', '4.d1.b1'], 'contains': ['16.a1.a1'], 'core': '8.b1.b1', 'coset_action_label': None, 'count': 1, 'diagramx': [3525, 3789, 2200, 4904, 3467, 4300, 2200, 5243], 'generators': [10], 'label': '16.3.8.b1.b1', 'mobius_quo': -1, 'mobius_sub': 0, 'normal_closure': '8.b1.b1', 'normal_contained_in': ['4.a1.a1'], 'normal_contains': ['16.a1.a1'], 'normalizer': '1.a1.a1', 'old_label': '8.b1.b1', 'projective_image': '8.3', 'quotient_action_image': '1.1', 'quotient_action_kernel': '8.3', 'quotient_action_kernel_order': 8, 'quotient_fusion': None, 'short_label': '8.b1.b1', 'subgroup_fusion': None, 'weyl_group': '1.1'}
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id: 13000843
{'ambient': '16.3', 'aut_centralizer_order': 8, 'aut_label': '8.c1', 'aut_quo_index': None, 'aut_stab_index': 4, 'aut_weyl_group': '1.1', 'aut_weyl_index': 32, 'centralizer': '2.a1.a1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['4.b1.a1', '4.c1.a1', '4.c1.b1'], 'contains': ['16.a1.a1'], 'core': '16.a1.a1', 'coset_action_label': '8T10', 'count': 2, 'diagramx': [8406, -1, 7675, -1, 8266, -1, 7782, -1], 'generators': [1], 'label': '16.3.8.c1.a1', 'mobius_quo': None, 'mobius_sub': 0, 'normal_closure': '4.b1.a1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '2.a1.a1', 'old_label': '8.c1.a1', 'projective_image': '16.3', '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': '1.1'}
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id: 13000846
{'ambient': '16.3', 'aut_centralizer_order': 4, 'aut_label': '2.a1', 'aut_quo_index': 1, 'aut_stab_index': 1, 'aut_weyl_group': '8.3', 'aut_weyl_index': 4, 'centralizer': '2.a1.a1', 'complements': [], 'conjugacy_class_count': 1, 'contained_in': ['1.a1.a1'], 'contains': ['4.a1.a1', '4.b1.a1', '4.b1.b1', '4.c1.a1', '4.c1.b1'], 'core': '2.a1.a1', 'coset_action_label': None, 'count': 1, 'diagramx': [6305, 7401, 5885, 7273, 6200, 8400, 5982, 7775], 'generators': [1, 2, 8], 'label': '16.3.2.a1.a1', 'mobius_quo': 0, 'mobius_sub': -1, 'normal_closure': '2.a1.a1', 'normal_contained_in': ['1.a1.a1'], 'normal_contains': ['4.a1.a1', '4.b1.b1', '4.b1.a1'], 'normalizer': '1.a1.a1', 'old_label': '2.a1.a1', 'projective_image': '4.2', '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'}
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id: 13000847
{'ambient': '16.3', 'aut_centralizer_order': 4, 'aut_label': '2.b1', 'aut_quo_index': 1, 'aut_stab_index': 2, 'aut_weyl_group': '4.2', 'aut_weyl_index': 8, 'centralizer': '2.b1.a1', 'complements': ['8.c1.a1', '8.c1.b1'], 'conjugacy_class_count': 1, 'contained_in': ['1.a1.a1'], 'contains': ['4.a1.a1', '4.d1.a1'], 'core': '2.b1.a1', 'coset_action_label': None, 'count': 1, 'diagramx': [2610, 2599, 3488, 1411, 2567, 2950, 3545, 1509], 'generators': [4, 2], 'label': '16.3.2.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': '4.2', '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'}
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id: 13000848
{'ambient': '16.3', 'aut_centralizer_order': 4, 'aut_label': '2.b1', 'aut_quo_index': 1, 'aut_stab_index': 2, 'aut_weyl_group': '4.2', 'aut_weyl_index': 8, 'centralizer': '2.b1.b1', 'complements': ['8.c1.a1', '8.c1.b1'], 'conjugacy_class_count': 1, 'contained_in': ['1.a1.a1'], 'contains': ['4.a1.a1', '4.d1.b1'], 'core': '2.b1.b1', 'coset_action_label': None, 'count': 1, 'diagramx': [2610, 2599, 1056, 4665, 2567, 2950, 1073, 4987], 'generators': [5, 8], 'label': '16.3.2.b1.b1', 'mobius_quo': 0, 'mobius_sub': -1, 'normal_closure': '2.b1.b1', 'normal_contained_in': ['1.a1.a1'], 'normal_contains': ['4.a1.a1'], 'normalizer': '1.a1.a1', 'old_label': '2.b1.b1', 'projective_image': '4.2', 'quotient_action_image': '2.1', 'quotient_action_kernel': '1.1', 'quotient_action_kernel_order': 1, 'quotient_fusion': None, 'short_label': '2.b1.b1', 'subgroup_fusion': None, 'weyl_group': '2.1'}
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id: 13000849
{'ambient': '16.3', 'aut_centralizer_order': 16, 'aut_label': '4.a1', 'aut_quo_index': 3, 'aut_stab_index': 1, 'aut_weyl_group': '2.1', 'aut_weyl_index': 16, 'centralizer': '1.a1.a1', 'complements': [], 'conjugacy_class_count': 1, 'contained_in': ['2.a1.a1', '2.b1.a1', '2.b1.b1'], 'contains': ['8.a1.a1', '8.b1.a1', '8.b1.b1'], 'core': '4.a1.a1', 'coset_action_label': None, 'count': 1, 'diagramx': [3525, 3789, 2111, 4665, 3467, 4300, 2145, 4987], 'generators': [2, 8], 'label': '16.3.4.a1.a1', 'mobius_quo': 2, 'mobius_sub': 2, 'normal_closure': '4.a1.a1', 'normal_contained_in': ['2.a1.a1', '2.b1.a1', '2.b1.b1'], 'normal_contains': ['8.a1.a1', '8.b1.a1', '8.b1.b1'], 'normalizer': '1.a1.a1', 'old_label': '4.a1.a1', 'projective_image': '4.2', 'quotient_action_image': '1.1', 'quotient_action_kernel': '4.2', 'quotient_action_kernel_order': 4, 'quotient_fusion': None, 'short_label': '4.a1.a1', 'subgroup_fusion': None, 'weyl_group': '1.1'}
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id: 13000850
{'ambient': '16.3', 'aut_centralizer_order': 8, 'aut_label': '4.b1', 'aut_quo_index': 1, 'aut_stab_index': 2, 'aut_weyl_group': '2.1', 'aut_weyl_index': 16, 'centralizer': '2.a1.a1', 'complements': ['4.d1.a1', '4.d1.b1'], 'conjugacy_class_count': 1, 'contained_in': ['2.a1.a1'], 'contains': ['8.a1.a1', '8.c1.a1'], 'core': '4.b1.a1', 'coset_action_label': None, 'count': 1, 'diagramx': [9084, 7401, 9267, 6627, 8933, 8400, 9418, 7084], 'generators': [1, 2], 'label': '16.3.4.b1.a1', 'mobius_quo': 0, 'mobius_sub': 0, 'normal_closure': '4.b1.a1', 'normal_contained_in': ['2.a1.a1'], 'normal_contains': ['8.a1.a1'], 'normalizer': '1.a1.a1', 'old_label': '4.b1.a1', 'projective_image': '8.2', 'quotient_action_image': '2.1', 'quotient_action_kernel': '2.1', 'quotient_action_kernel_order': 2, 'quotient_fusion': None, 'short_label': '4.b1.a1', 'subgroup_fusion': None, 'weyl_group': '2.1'}
-
id: 13000851
{'ambient': '16.3', 'aut_centralizer_order': 8, 'aut_label': '4.b1', 'aut_quo_index': 1, 'aut_stab_index': 2, 'aut_weyl_group': '2.1', 'aut_weyl_index': 16, 'centralizer': '2.a1.a1', 'complements': ['4.d1.a1', '4.d1.b1'], 'conjugacy_class_count': 1, 'contained_in': ['2.a1.a1'], 'contains': ['8.a1.a1', '8.c1.b1'], 'core': '4.b1.b1', 'coset_action_label': None, 'count': 1, 'diagramx': [9084, 7401, 4866, 8589, 8933, 8400, 4945, 9181], 'generators': [9, 2], 'label': '16.3.4.b1.b1', 'mobius_quo': 0, 'mobius_sub': 0, 'normal_closure': '4.b1.b1', 'normal_contained_in': ['2.a1.a1'], 'normal_contains': ['8.a1.a1'], 'normalizer': '1.a1.a1', 'old_label': '4.b1.b1', 'projective_image': '8.2', 'quotient_action_image': '2.1', 'quotient_action_kernel': '2.1', 'quotient_action_kernel_order': 2, 'quotient_fusion': None, 'short_label': '4.b1.b1', 'subgroup_fusion': None, 'weyl_group': '2.1'}
-
id: 13000852
{'ambient': '16.3', 'aut_centralizer_order': 4, 'aut_label': '4.c1', 'aut_quo_index': None, 'aut_stab_index': 4, 'aut_weyl_group': '2.1', 'aut_weyl_index': 16, 'centralizer': '2.a1.a1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['2.a1.a1'], 'contains': ['8.b1.a1', '8.c1.a1', '8.c1.b1'], 'core': '8.b1.a1', 'coset_action_label': None, 'count': 2, 'diagramx': [6305, -1, 7800, -1, 6200, -1, 7927, -1], 'generators': [1, 8], 'label': '16.3.4.c1.a1', 'mobius_quo': None, 'mobius_sub': 0, 'normal_closure': '2.a1.a1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '2.a1.a1', 'old_label': '4.c1.a1', 'projective_image': '8.3', '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': '1.1'}
-
id: 13000853
{'ambient': '16.3', 'aut_centralizer_order': 4, 'aut_label': '4.c1', 'aut_quo_index': None, 'aut_stab_index': 4, 'aut_weyl_group': '2.1', 'aut_weyl_index': 16, 'centralizer': '2.a1.a1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['2.a1.a1'], 'contains': ['8.b1.b1', '8.c1.a1', '8.c1.b1'], 'core': '8.b1.b1', 'coset_action_label': None, 'count': 2, 'diagramx': [6305, -1, 6333, -1, 6200, -1, 6436, -1], 'generators': [1, 10], 'label': '16.3.4.c1.b1', '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.c1.b1', 'projective_image': '8.3', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '4.c1.b1', 'subgroup_fusion': None, 'weyl_group': '1.1'}
