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gps_groups • Show schema
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{'Agroup': True, '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': 42, 'aut_gen_orders': [2, 6, 14], 'aut_gens': [[1, 2], [29, 2], [1, 34], [41, 30]], 'aut_group': '168.47', 'aut_hash': 47, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 168, 'aut_permdeg': 11, 'aut_perms': [7, 1829640, 8871256], 'aut_phi_ratio': 7.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 7, 2, 1], [4, 1, 2, 1], [4, 7, 2, 1], [7, 2, 3, 1], [14, 2, 3, 1], [28, 2, 6, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_2^2\\times F_7', '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': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^2', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 42, 'autcentquo_group': '42.1', 'autcentquo_hash': 1, 'autcentquo_nilpotent': False, 'autcentquo_order': 42, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'F_7', 'cc_stats': [[1, 1, 1], [2, 1, 1], [2, 7, 2], [4, 1, 2], [4, 7, 2], [7, 2, 3], [14, 2, 3], [28, 2, 6]], 'center_label': '4.1', 'center_order': 4, 'central_product': True, 'central_quotient': '14.1', 'commutator_count': 1, 'commutator_label': '7.1', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '7.1'], 'composition_length': 4, 'conjugacy_classes_known': True, 'counter': 4, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['14.1', 1], ['4.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 7, 1, 2], [4, 1, 2, 1], [4, 7, 2, 1], [7, 2, 3, 1], [14, 2, 3, 1], [28, 2, 6, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 6, 'exponent': 28, 'exponents_of_order': [3, 1], 'factors_of_aut_order': [2, 3, 7], 'factors_of_order': [2, 7], 'faithful_reps': [[2, 0, 6]], 'familial': False, 'frattini_label': '2.1', 'frattini_quotient': '28.3', 'hash': 4, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 14, 'inner_gen_orders': [2, 7], 'inner_gens': [[1, 26], [33, 2]], 'inner_hash': 1, 'inner_nilpotent': False, 'inner_order': 14, 'inner_split': True, 'inner_tex': 'D_7', 'inner_used': [1, 2], 'irrC_degree': 2, 'irrQ_degree': 12, 'irrQ_dim': 12, 'irrR_degree': 4, 'irrep_stats': [[1, 8], [2, 12]], 'label': '56.4', 'linC_count': 6, 'linC_degree': 2, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 8, 'linQ_degree_count': 4, 'linQ_dim': 8, 'linQ_dim_count': 4, 'linR_count': 15, 'linR_degree': 4, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C4*D7', 'ngens': 4, 'nilpotency_class': -1, 'nilpotent': False, 'normal_counts': [0, 0, 0, 0, 0, 0, 0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 8, 'number_characteristic_subgroups': 9, 'number_conjugacy_classes': 20, 'number_divisions': 9, 'number_normal_subgroups': 11, 'number_subgroup_autclasses': 14, 'number_subgroup_classes': 16, 'number_subgroups': 46, 'old_label': None, 'order': 56, 'order_factorization_type': 31, 'order_stats': [[1, 1], [2, 15], [4, 16], [7, 6], [14, 6], [28, 12]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 6, 'outer_gen_orders': [2, 6], 'outer_gen_pows': [0, 0], 'outer_gens': [[1, 30], [29, 34]], 'outer_group': '12.5', 'outer_hash': 5, 'outer_nilpotent': True, 'outer_order': 12, 'outer_permdeg': 7, 'outer_perms': [720, 28], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2\\times C_6', 'pc_rank': 2, 'perfect': False, 'permutation_degree': 11, 'pgroup': 0, 'primary_abelian_invariants': [2, 4], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 2], [6, 2], [12, 1]], 'representations': {'PC': {'code': 33333964118501, 'gens': [1, 2], 'pres': [4, -2, -2, -2, -7, 209, 21, 626, 34, 771]}, 'GLFp': {'d': 2, 'p': 13, 'gens': [26365, 24763]}, 'Perm': {'d': 11, 'gens': [368047, 17, 7, 4364760]}}, 'schur_multiplier': [2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 4], 'solvability_type': 6, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_4\\times D_7', 'transitive_degree': 28, 'wreath_data': None, 'wreath_product': False}
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gps_conj_classes • Show schema
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id: 110625514
{'aut_label': '1A', 'centralizer': '1.a1.a1', 'counter': 1, 'group_counter': 4, 'group_order': 56, 'label': '1A', 'order': 1, 'powers': [1, 1, 1], 'representative': 0, 'size': 1}
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id: 110625525
{'aut_label': '2A', 'centralizer': '1.a1.a1', 'counter': 2, 'group_counter': 4, 'group_order': 56, 'label': '2A', 'order': 2, 'powers': [1, 2, 2], 'representative': 28, 'size': 1}
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id: 110625527
{'aut_label': '2B', 'centralizer': '7.a1.a1', 'counter': 3, 'group_counter': 4, 'group_order': 56, 'label': '2B', 'order': 2, 'powers': [1, 3, 3], 'representative': 1, 'size': 7}
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id: 110625528
{'aut_label': '2B', 'centralizer': '7.a1.a1', 'counter': 4, 'group_counter': 4, 'group_order': 56, 'label': '2C', 'order': 2, 'powers': [1, 4, 4], 'representative': 5, 'size': 7}
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id: 110625529
{'aut_label': '4A1', 'centralizer': '1.a1.a1', 'counter': 5, 'group_counter': 4, 'group_order': 56, 'label': '4A1', 'order': 4, 'powers': [2, 6, 6], 'representative': 14, 'size': 1}
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id: 110625530
{'aut_label': '4A1', 'centralizer': '1.a1.a1', 'counter': 6, 'group_counter': 4, 'group_order': 56, 'label': '4A-1', 'order': 4, 'powers': [2, 5, 5], 'representative': 42, 'size': 1}
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id: 110625531
{'aut_label': '4B1', 'centralizer': '7.a1.a1', 'counter': 7, 'group_counter': 4, 'group_order': 56, 'label': '4B1', 'order': 4, 'powers': [2, 8, 8], 'representative': 3, 'size': 7}
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id: 110625532
{'aut_label': '4B1', 'centralizer': '7.a1.a1', 'counter': 8, 'group_counter': 4, 'group_order': 56, 'label': '4B-1', 'order': 4, 'powers': [2, 7, 7], 'representative': 7, 'size': 7}
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id: 110625533
{'aut_label': '7A1', 'centralizer': '2.b1.a1', 'counter': 9, 'group_counter': 4, 'group_order': 56, 'label': '7A1', 'order': 7, 'powers': [10, 11, 1], 'representative': 8, 'size': 2}
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id: 110625515
{'aut_label': '7A1', 'centralizer': '2.b1.a1', 'counter': 10, 'group_counter': 4, 'group_order': 56, 'label': '7A2', 'order': 7, 'powers': [11, 9, 1], 'representative': 16, 'size': 2}
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id: 110625516
{'aut_label': '7A1', 'centralizer': '2.b1.a1', 'counter': 11, 'group_counter': 4, 'group_order': 56, 'label': '7A3', 'order': 7, 'powers': [9, 10, 1], 'representative': 24, 'size': 2}
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id: 110625517
{'aut_label': '14A1', 'centralizer': '2.b1.a1', 'counter': 12, 'group_counter': 4, 'group_order': 56, 'label': '14A1', 'order': 14, 'powers': [9, 13, 2], 'representative': 4, 'size': 2}
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id: 110625518
{'aut_label': '14A1', 'centralizer': '2.b1.a1', 'counter': 13, 'group_counter': 4, 'group_order': 56, 'label': '14A3', 'order': 14, 'powers': [11, 14, 2], 'representative': 12, 'size': 2}
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id: 110625519
{'aut_label': '14A1', 'centralizer': '2.b1.a1', 'counter': 14, 'group_counter': 4, 'group_order': 56, 'label': '14A5', 'order': 14, 'powers': [10, 12, 2], 'representative': 20, 'size': 2}
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id: 110625520
{'aut_label': '28A1', 'centralizer': '2.b1.a1', 'counter': 15, 'group_counter': 4, 'group_order': 56, 'label': '28A1', 'order': 28, 'powers': [12, 17, 5], 'representative': 2, 'size': 2}
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id: 110625521
{'aut_label': '28A1', 'centralizer': '2.b1.a1', 'counter': 16, 'group_counter': 4, 'group_order': 56, 'label': '28A-1', 'order': 28, 'powers': [12, 18, 6], 'representative': 30, 'size': 2}
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id: 110625522
{'aut_label': '28A1', 'centralizer': '2.b1.a1', 'counter': 17, 'group_counter': 4, 'group_order': 56, 'label': '28A3', 'order': 28, 'powers': [13, 19, 6], 'representative': 6, 'size': 2}
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id: 110625523
{'aut_label': '28A1', 'centralizer': '2.b1.a1', 'counter': 18, 'group_counter': 4, 'group_order': 56, 'label': '28A-3', 'order': 28, 'powers': [13, 20, 5], 'representative': 34, 'size': 2}
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id: 110625524
{'aut_label': '28A1', 'centralizer': '2.b1.a1', 'counter': 19, 'group_counter': 4, 'group_order': 56, 'label': '28A5', 'order': 28, 'powers': [14, 16, 5], 'representative': 10, 'size': 2}
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id: 110625526
{'aut_label': '28A1', 'centralizer': '2.b1.a1', 'counter': 20, 'group_counter': 4, 'group_order': 56, 'label': '28A-5', 'order': 28, 'powers': [14, 15, 6], 'representative': 38, 'size': 2}
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gps_qchar • Show schema
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id: 4686385
{'cdim': 1, 'counter': 1, 'faithful': False, 'group': '56.4', 'group_order': 56, 'image': None, 'label': '56.4.1a', 'multiplicity': 1, 'nt': [1, 1], 'qdim': 1, 'qvalues': [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1], 'schur_index': 1}
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id: 4686386
{'cdim': 1, 'counter': 2, 'faithful': False, 'group': '56.4', 'group_order': 56, 'image': None, 'label': '56.4.1b', 'multiplicity': 1, 'nt': [2, 1], 'qdim': 1, 'qvalues': [1, 1, -1, -1, -1, -1, 1, 1, 1, 1, 1, 1, 1, 1, -1, -1, -1, -1, -1, -1], 'schur_index': 1}
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id: 4686387
{'cdim': 1, 'counter': 3, 'faithful': False, 'group': '56.4', 'group_order': 56, 'image': None, 'label': '56.4.1c', 'multiplicity': 1, 'nt': [2, 1], 'qdim': 1, 'qvalues': [1, 1, -1, -1, 1, 1, -1, -1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1], 'schur_index': 1}
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id: 4686388
{'cdim': 1, 'counter': 4, 'faithful': False, 'group': '56.4', 'group_order': 56, 'image': None, 'label': '56.4.1d', 'multiplicity': 1, 'nt': [2, 1], 'qdim': 1, 'qvalues': [1, 1, 1, 1, -1, -1, -1, -1, 1, 1, 1, 1, 1, 1, -1, -1, -1, -1, -1, -1], 'schur_index': 1}
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id: 4686389
{'cdim': 1, 'counter': 5, 'faithful': False, 'group': '56.4', 'group_order': 56, 'image': None, 'label': '56.4.1e', 'multiplicity': 2, 'nt': [4, 1], 'qdim': 2, 'qvalues': [2, -2, -2, 2, 0, 0, 0, 0, 2, 2, 2, -2, -2, -2, 0, 0, 0, 0, 0, 0], 'schur_index': 1}
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id: 4686390
{'cdim': 1, 'counter': 6, 'faithful': False, 'group': '56.4', 'group_order': 56, 'image': None, 'label': '56.4.1f', 'multiplicity': 2, 'nt': [4, 1], 'qdim': 2, 'qvalues': [2, -2, 2, -2, 0, 0, 0, 0, 2, 2, 2, -2, -2, -2, 0, 0, 0, 0, 0, 0], 'schur_index': 1}
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id: 4686391
{'cdim': 2, 'counter': 7, 'faithful': False, 'group': '56.4', 'group_order': 56, 'image': None, 'label': '56.4.2a', 'multiplicity': 3, 'nt': [7, 2], 'qdim': 6, 'qvalues': [6, 6, 0, 0, 6, 6, 0, 0, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1], 'schur_index': 1}
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id: 4686392
{'cdim': 2, 'counter': 8, 'faithful': False, 'group': '56.4', 'group_order': 56, 'image': None, 'label': '56.4.2b', 'multiplicity': 3, 'nt': [14, 3], 'qdim': 6, 'qvalues': [6, 6, 0, 0, -6, -6, 0, 0, -1, -1, -1, -1, -1, -1, 1, 1, 1, 1, 1, 1], 'schur_index': 1}
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id: 4686393
{'cdim': 2, 'counter': 9, 'faithful': True, 'group': '56.4', 'group_order': 56, 'image': None, 'label': '56.4.2c', 'multiplicity': 6, 'nt': [28, 8], 'qdim': 12, 'qvalues': [12, -12, 0, 0, 0, 0, 0, 0, -2, -2, -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: 7532973
{'center': '1.a1.a1', 'center_index': 1, 'center_order': 56, 'counter': 1, 'cyclotomic_n': 1, 'dim': 1, 'faithful': False, 'field': [0, 1], 'group': '56.4', 'group_counter': 4, 'group_order': 56, 'image': None, 'image_isoclass': '1.1', 'image_order': 1, 'indicator': 1, 'kernel': '1.a1.a1', 'kernel_order': 56, 'label': '56.4.1a', 'nt': [1, 1], 'q_character': None, 'values': [[[1, 0]], [[1, 0]], [[1, 0]], [[1, 0]], [[1, 0]], [[1, 0]], [[1, 0]], [[1, 0]], [[1, 0]], [[1, 0]], [[1, 0]], [[1, 0]], [[1, 0]], [[1, 0]], [[1, 0]], [[1, 0]], [[1, 0]], [[1, 0]], [[1, 0]], [[1, 0]]]}
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id: 7532984
{'center': '1.a1.a1', 'center_index': 1, 'center_order': 56, 'counter': 2, 'cyclotomic_n': 1, 'dim': 1, 'faithful': False, 'field': [0, 1], 'group': '56.4', 'group_counter': 4, 'group_order': 56, 'image': None, 'image_isoclass': '2.1', 'image_order': 2, 'indicator': 1, 'kernel': '2.c1.a1', 'kernel_order': 28, 'label': '56.4.1b', 'nt': [2, 1], 'q_character': None, 'values': [[[1, 0]], [[1, 0]], [[-1, 0]], [[-1, 0]], [[-1, 0]], [[-1, 0]], [[1, 0]], [[1, 0]], [[1, 0]], [[1, 0]], [[1, 0]], [[1, 0]], [[1, 0]], [[1, 0]], [[-1, 0]], [[-1, 0]], [[-1, 0]], [[-1, 0]], [[-1, 0]], [[-1, 0]]]}
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id: 7532986
{'center': '1.a1.a1', 'center_index': 1, 'center_order': 56, 'counter': 3, 'cyclotomic_n': 1, 'dim': 1, 'faithful': False, 'field': [0, 1], 'group': '56.4', 'group_counter': 4, 'group_order': 56, 'image': None, 'image_isoclass': '2.1', 'image_order': 2, 'indicator': 1, 'kernel': '2.b1.a1', 'kernel_order': 28, 'label': '56.4.1c', 'nt': [2, 1], 'q_character': None, 'values': [[[1, 0]], [[1, 0]], [[-1, 0]], [[-1, 0]], [[1, 0]], [[1, 0]], [[-1, 0]], [[-1, 0]], [[1, 0]], [[1, 0]], [[1, 0]], [[1, 0]], [[1, 0]], [[1, 0]], [[1, 0]], [[1, 0]], [[1, 0]], [[1, 0]], [[1, 0]], [[1, 0]]]}
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id: 7532987
{'center': '1.a1.a1', 'center_index': 1, 'center_order': 56, 'counter': 4, 'cyclotomic_n': 1, 'dim': 1, 'faithful': False, 'field': [0, 1], 'group': '56.4', 'group_counter': 4, 'group_order': 56, 'image': None, 'image_isoclass': '2.1', 'image_order': 2, 'indicator': 1, 'kernel': '2.a1.a1', 'kernel_order': 28, 'label': '56.4.1d', 'nt': [2, 1], 'q_character': None, 'values': [[[1, 0]], [[1, 0]], [[1, 0]], [[1, 0]], [[-1, 0]], [[-1, 0]], [[-1, 0]], [[-1, 0]], [[1, 0]], [[1, 0]], [[1, 0]], [[1, 0]], [[1, 0]], [[1, 0]], [[-1, 0]], [[-1, 0]], [[-1, 0]], [[-1, 0]], [[-1, 0]], [[-1, 0]]]}
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id: 7532988
{'center': '1.a1.a1', 'center_index': 1, 'center_order': 56, 'counter': 5, 'cyclotomic_n': 4, 'dim': 1, 'faithful': False, 'field': [1, 0, 1], 'group': '56.4', 'group_counter': 4, 'group_order': 56, 'image': None, 'image_isoclass': '4.1', 'image_order': 4, 'indicator': 0, 'kernel': '4.b1.b1', 'kernel_order': 14, 'label': '56.4.1e1', 'nt': [4, 1], 'q_character': None, 'values': [[[1, 0]], [[-1, 0]], [[-1, 0]], [[1, 0]], [[-1, 1]], [[1, 1]], [[-1, 1]], [[1, 1]], [[1, 0]], [[1, 0]], [[1, 0]], [[-1, 0]], [[-1, 0]], [[-1, 0]], [[-1, 1]], [[1, 1]], [[1, 1]], [[1, 1]], [[-1, 1]], [[-1, 1]]]}
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id: 7532989
{'center': '1.a1.a1', 'center_index': 1, 'center_order': 56, 'counter': 6, 'cyclotomic_n': 4, 'dim': 1, 'faithful': False, 'field': [1, 0, 1], 'group': '56.4', 'group_counter': 4, 'group_order': 56, 'image': None, 'image_isoclass': '4.1', 'image_order': 4, 'indicator': 0, 'kernel': '4.b1.b1', 'kernel_order': 14, 'label': '56.4.1e2', 'nt': [4, 2], 'q_character': None, 'values': [[[1, 0]], [[-1, 0]], [[-1, 0]], [[1, 0]], [[1, 1]], [[-1, 1]], [[1, 1]], [[-1, 1]], [[1, 0]], [[1, 0]], [[1, 0]], [[-1, 0]], [[-1, 0]], [[-1, 0]], [[1, 1]], [[-1, 1]], [[-1, 1]], [[-1, 1]], [[1, 1]], [[1, 1]]]}
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id: 7532990
{'center': '1.a1.a1', 'center_index': 1, 'center_order': 56, 'counter': 7, 'cyclotomic_n': 4, 'dim': 1, 'faithful': False, 'field': [1, 0, 1], 'group': '56.4', 'group_counter': 4, 'group_order': 56, 'image': None, 'image_isoclass': '4.1', 'image_order': 4, 'indicator': 0, 'kernel': '4.b1.a1', 'kernel_order': 14, 'label': '56.4.1f1', 'nt': [4, 1], 'q_character': None, 'values': [[[1, 0]], [[-1, 0]], [[1, 0]], [[-1, 0]], [[-1, 1]], [[1, 1]], [[1, 1]], [[-1, 1]], [[1, 0]], [[1, 0]], [[1, 0]], [[-1, 0]], [[-1, 0]], [[-1, 0]], [[-1, 1]], [[1, 1]], [[1, 1]], [[1, 1]], [[-1, 1]], [[-1, 1]]]}
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id: 7532991
{'center': '1.a1.a1', 'center_index': 1, 'center_order': 56, 'counter': 8, 'cyclotomic_n': 4, 'dim': 1, 'faithful': False, 'field': [1, 0, 1], 'group': '56.4', 'group_counter': 4, 'group_order': 56, 'image': None, 'image_isoclass': '4.1', 'image_order': 4, 'indicator': 0, 'kernel': '4.b1.a1', 'kernel_order': 14, 'label': '56.4.1f2', 'nt': [4, 2], 'q_character': None, 'values': [[[1, 0]], [[-1, 0]], [[1, 0]], [[-1, 0]], [[1, 1]], [[-1, 1]], [[-1, 1]], [[1, 1]], [[1, 0]], [[1, 0]], [[1, 0]], [[-1, 0]], [[-1, 0]], [[-1, 0]], [[1, 1]], [[-1, 1]], [[-1, 1]], [[-1, 1]], [[1, 1]], [[1, 1]]]}
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id: 7532992
{'center': '14.a1.a1', 'center_index': 14, 'center_order': 4, 'counter': 9, 'cyclotomic_n': 7, 'dim': 2, 'faithful': False, 'field': [1, -2, -1, 1], 'group': '56.4', 'group_counter': 4, 'group_order': 56, 'image': None, 'image_isoclass': '14.1', 'image_order': 14, 'indicator': 1, 'kernel': '14.a1.a1', 'kernel_order': 4, 'label': '56.4.2a1', 'nt': [7, 2], 'q_character': None, 'values': [[[2, 0]], [[2, 0]], [[0, 0]], [[0, 0]], [[2, 0]], [[2, 0]], [[0, 0]], [[0, 0]], [[1, 3], [1, -3]], [[1, 1], [1, -1]], [[1, 2], [1, -2]], [[1, 2], [1, -2]], [[1, 1], [1, -1]], [[1, 3], [1, -3]], [[1, 2], [1, -2]], [[1, 1], [1, -1]], [[1, 3], [1, -3]], [[1, 2], [1, -2]], [[1, 3], [1, -3]], [[1, 1], [1, -1]]]}
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id: 7532974
{'center': '14.a1.a1', 'center_index': 14, 'center_order': 4, 'counter': 10, 'cyclotomic_n': 7, 'dim': 2, 'faithful': False, 'field': [1, -2, -1, 1], 'group': '56.4', 'group_counter': 4, 'group_order': 56, 'image': None, 'image_isoclass': '14.1', 'image_order': 14, 'indicator': 1, 'kernel': '14.a1.a1', 'kernel_order': 4, 'label': '56.4.2a2', 'nt': [7, 2], 'q_character': None, 'values': [[[2, 0]], [[2, 0]], [[0, 0]], [[0, 0]], [[2, 0]], [[2, 0]], [[0, 0]], [[0, 0]], [[1, 2], [1, -2]], [[1, 3], [1, -3]], [[1, 1], [1, -1]], [[1, 1], [1, -1]], [[1, 3], [1, -3]], [[1, 2], [1, -2]], [[1, 1], [1, -1]], [[1, 3], [1, -3]], [[1, 2], [1, -2]], [[1, 1], [1, -1]], [[1, 2], [1, -2]], [[1, 3], [1, -3]]]}
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id: 7532975
{'center': '14.a1.a1', 'center_index': 14, 'center_order': 4, 'counter': 11, 'cyclotomic_n': 7, 'dim': 2, 'faithful': False, 'field': [1, -2, -1, 1], 'group': '56.4', 'group_counter': 4, 'group_order': 56, 'image': None, 'image_isoclass': '14.1', 'image_order': 14, 'indicator': 1, 'kernel': '14.a1.a1', 'kernel_order': 4, 'label': '56.4.2a3', 'nt': [7, 2], 'q_character': None, 'values': [[[2, 0]], [[2, 0]], [[0, 0]], [[0, 0]], [[2, 0]], [[2, 0]], [[0, 0]], [[0, 0]], [[1, 1], [1, -1]], [[1, 2], [1, -2]], [[1, 3], [1, -3]], [[1, 3], [1, -3]], [[1, 2], [1, -2]], [[1, 1], [1, -1]], [[1, 3], [1, -3]], [[1, 2], [1, -2]], [[1, 1], [1, -1]], [[1, 3], [1, -3]], [[1, 1], [1, -1]], [[1, 2], [1, -2]]]}
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id: 7532976
{'center': '14.a1.a1', 'center_index': 14, 'center_order': 4, 'counter': 12, 'cyclotomic_n': 7, 'dim': 2, 'faithful': False, 'field': [1, -2, -1, 1], 'group': '56.4', 'group_counter': 4, 'group_order': 56, 'image': None, 'image_isoclass': '28.3', 'image_order': 28, 'indicator': 1, 'kernel': '28.a1.a1', 'kernel_order': 2, 'label': '56.4.2b1', 'nt': [14, 3], 'q_character': None, 'values': [[[2, 0]], [[2, 0]], [[0, 0]], [[0, 0]], [[-2, 0]], [[-2, 0]], [[0, 0]], [[0, 0]], [[1, 3], [1, -3]], [[1, 1], [1, -1]], [[1, 2], [1, -2]], [[1, 2], [1, -2]], [[1, 1], [1, -1]], [[1, 3], [1, -3]], [[-1, 2], [-1, -2]], [[-1, 1], [-1, -1]], [[-1, 3], [-1, -3]], [[-1, 2], [-1, -2]], [[-1, 3], [-1, -3]], [[-1, 1], [-1, -1]]]}
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id: 7532977
{'center': '14.a1.a1', 'center_index': 14, 'center_order': 4, 'counter': 13, 'cyclotomic_n': 7, 'dim': 2, 'faithful': False, 'field': [1, -2, -1, 1], 'group': '56.4', 'group_counter': 4, 'group_order': 56, 'image': None, 'image_isoclass': '28.3', 'image_order': 28, 'indicator': 1, 'kernel': '28.a1.a1', 'kernel_order': 2, 'label': '56.4.2b2', 'nt': [14, 3], 'q_character': None, 'values': [[[2, 0]], [[2, 0]], [[0, 0]], [[0, 0]], [[-2, 0]], [[-2, 0]], [[0, 0]], [[0, 0]], [[1, 2], [1, -2]], [[1, 3], [1, -3]], [[1, 1], [1, -1]], [[1, 1], [1, -1]], [[1, 3], [1, -3]], [[1, 2], [1, -2]], [[-1, 1], [-1, -1]], [[-1, 3], [-1, -3]], [[-1, 2], [-1, -2]], [[-1, 1], [-1, -1]], [[-1, 2], [-1, -2]], [[-1, 3], [-1, -3]]]}
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id: 7532978
{'center': '14.a1.a1', 'center_index': 14, 'center_order': 4, 'counter': 14, 'cyclotomic_n': 7, 'dim': 2, 'faithful': False, 'field': [1, -2, -1, 1], 'group': '56.4', 'group_counter': 4, 'group_order': 56, 'image': None, 'image_isoclass': '28.3', 'image_order': 28, 'indicator': 1, 'kernel': '28.a1.a1', 'kernel_order': 2, 'label': '56.4.2b3', 'nt': [14, 3], 'q_character': None, 'values': [[[2, 0]], [[2, 0]], [[0, 0]], [[0, 0]], [[-2, 0]], [[-2, 0]], [[0, 0]], [[0, 0]], [[1, 1], [1, -1]], [[1, 2], [1, -2]], [[1, 3], [1, -3]], [[1, 3], [1, -3]], [[1, 2], [1, -2]], [[1, 1], [1, -1]], [[-1, 3], [-1, -3]], [[-1, 2], [-1, -2]], [[-1, 1], [-1, -1]], [[-1, 3], [-1, -3]], [[-1, 1], [-1, -1]], [[-1, 2], [-1, -2]]]}
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id: 7532979
{'center': '14.a1.a1', 'center_index': 14, 'center_order': 4, 'counter': 15, 'cyclotomic_n': 28, 'dim': 2, 'faithful': True, 'field': [1, 0, 6, 0, 5, 0, 1], 'group': '56.4', 'group_counter': 4, 'group_order': 56, 'image': None, 'image_isoclass': '56.4', 'image_order': 56, 'indicator': 0, 'kernel': '56.a1.a1', 'kernel_order': 1, 'label': '56.4.2c1', 'nt': [28, 8], 'q_character': None, 'values': [[[2, 0]], [[-2, 0]], [[0, 0]], [[0, 0]], [[-2, 7]], [[2, 7]], [[0, 0]], [[0, 0]], [[-1, 2], [-1, -2]], [[1, 4], [1, -4]], [[-1, 6], [-1, -6]], [[1, 6], [1, -6]], [[-1, 4], [-1, -4]], [[1, 2], [1, -2]], [[1, 3], [-1, 5], [1, 7], [-1, 9], [1, 11]], [[1, 3], [1, 11]], [[-1, 5], [-1, 9]], [[-1, 3], [1, 5], [-1, 7], [1, 9], [-1, 11]], [[1, 5], [1, 9]], [[-1, 3], [-1, 11]]]}
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id: 7532980
{'center': '14.a1.a1', 'center_index': 14, 'center_order': 4, 'counter': 16, 'cyclotomic_n': 28, 'dim': 2, 'faithful': True, 'field': [1, 0, 6, 0, 5, 0, 1], 'group': '56.4', 'group_counter': 4, 'group_order': 56, 'image': None, 'image_isoclass': '56.4', 'image_order': 56, 'indicator': 0, 'kernel': '56.a1.a1', 'kernel_order': 1, 'label': '56.4.2c2', 'nt': [28, 6], 'q_character': None, 'values': [[[2, 0]], [[-2, 0]], [[0, 0]], [[0, 0]], [[2, 7]], [[-2, 7]], [[0, 0]], [[0, 0]], [[-1, 2], [-1, -2]], [[1, 4], [1, -4]], [[-1, 6], [-1, -6]], [[1, 6], [1, -6]], [[-1, 4], [-1, -4]], [[1, 2], [1, -2]], [[-1, 3], [1, 5], [-1, 7], [1, 9], [-1, 11]], [[-1, 3], [-1, 11]], [[1, 5], [1, 9]], [[1, 3], [-1, 5], [1, 7], [-1, 9], [1, 11]], [[-1, 5], [-1, 9]], [[1, 3], [1, 11]]]}
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id: 7532981
{'center': '14.a1.a1', 'center_index': 14, 'center_order': 4, 'counter': 17, 'cyclotomic_n': 28, 'dim': 2, 'faithful': True, 'field': [1, 0, 6, 0, 5, 0, 1], 'group': '56.4', 'group_counter': 4, 'group_order': 56, 'image': None, 'image_isoclass': '56.4', 'image_order': 56, 'indicator': 0, 'kernel': '56.a1.a1', 'kernel_order': 1, 'label': '56.4.2c3', 'nt': [28, 8], 'q_character': None, 'values': [[[2, 0]], [[-2, 0]], [[0, 0]], [[0, 0]], [[-2, 7]], [[2, 7]], [[0, 0]], [[0, 0]], [[-1, 6], [-1, -6]], [[-1, 2], [-1, -2]], [[1, 4], [1, -4]], [[-1, 4], [-1, -4]], [[1, 2], [1, -2]], [[1, 6], [1, -6]], [[-1, 3], [-1, 11]], [[-1, 5], [-1, 9]], [[-1, 3], [1, 5], [-1, 7], [1, 9], [-1, 11]], [[1, 3], [1, 11]], [[1, 3], [-1, 5], [1, 7], [-1, 9], [1, 11]], [[1, 5], [1, 9]]]}
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id: 7532982
{'center': '14.a1.a1', 'center_index': 14, 'center_order': 4, 'counter': 18, 'cyclotomic_n': 28, 'dim': 2, 'faithful': True, 'field': [1, 0, 6, 0, 5, 0, 1], 'group': '56.4', 'group_counter': 4, 'group_order': 56, 'image': None, 'image_isoclass': '56.4', 'image_order': 56, 'indicator': 0, 'kernel': '56.a1.a1', 'kernel_order': 1, 'label': '56.4.2c4', 'nt': [28, 6], 'q_character': None, 'values': [[[2, 0]], [[-2, 0]], [[0, 0]], [[0, 0]], [[2, 7]], [[-2, 7]], [[0, 0]], [[0, 0]], [[-1, 6], [-1, -6]], [[-1, 2], [-1, -2]], [[1, 4], [1, -4]], [[-1, 4], [-1, -4]], [[1, 2], [1, -2]], [[1, 6], [1, -6]], [[1, 3], [1, 11]], [[1, 5], [1, 9]], [[1, 3], [-1, 5], [1, 7], [-1, 9], [1, 11]], [[-1, 3], [-1, 11]], [[-1, 3], [1, 5], [-1, 7], [1, 9], [-1, 11]], [[-1, 5], [-1, 9]]]}
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id: 7532983
{'center': '14.a1.a1', 'center_index': 14, 'center_order': 4, 'counter': 19, 'cyclotomic_n': 28, 'dim': 2, 'faithful': True, 'field': [1, 0, 6, 0, 5, 0, 1], 'group': '56.4', 'group_counter': 4, 'group_order': 56, 'image': None, 'image_isoclass': '56.4', 'image_order': 56, 'indicator': 0, 'kernel': '56.a1.a1', 'kernel_order': 1, 'label': '56.4.2c5', 'nt': [28, 8], 'q_character': None, 'values': [[[2, 0]], [[-2, 0]], [[0, 0]], [[0, 0]], [[-2, 7]], [[2, 7]], [[0, 0]], [[0, 0]], [[1, 4], [1, -4]], [[-1, 6], [-1, -6]], [[-1, 2], [-1, -2]], [[1, 2], [1, -2]], [[1, 6], [1, -6]], [[-1, 4], [-1, -4]], [[1, 5], [1, 9]], [[-1, 3], [1, 5], [-1, 7], [1, 9], [-1, 11]], [[1, 3], [1, 11]], [[-1, 5], [-1, 9]], [[-1, 3], [-1, 11]], [[1, 3], [-1, 5], [1, 7], [-1, 9], [1, 11]]]}
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id: 7532985
{'center': '14.a1.a1', 'center_index': 14, 'center_order': 4, 'counter': 20, 'cyclotomic_n': 28, 'dim': 2, 'faithful': True, 'field': [1, 0, 6, 0, 5, 0, 1], 'group': '56.4', 'group_counter': 4, 'group_order': 56, 'image': None, 'image_isoclass': '56.4', 'image_order': 56, 'indicator': 0, 'kernel': '56.a1.a1', 'kernel_order': 1, 'label': '56.4.2c6', 'nt': [28, 6], 'q_character': None, 'values': [[[2, 0]], [[-2, 0]], [[0, 0]], [[0, 0]], [[2, 7]], [[-2, 7]], [[0, 0]], [[0, 0]], [[1, 4], [1, -4]], [[-1, 6], [-1, -6]], [[-1, 2], [-1, -2]], [[1, 2], [1, -2]], [[1, 6], [1, -6]], [[-1, 4], [-1, -4]], [[-1, 5], [-1, 9]], [[1, 3], [-1, 5], [1, 7], [-1, 9], [1, 11]], [[-1, 3], [-1, 11]], [[1, 5], [1, 9]], [[1, 3], [1, 11]], [[-1, 3], [1, 5], [-1, 7], [1, 9], [-1, 11]]]}
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gps_subgroup_search • Show schema
Hide schema
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id: 67612378
{'Agroup': True, 'Zgroup': False, 'abelian': False, 'ambient': '56.4', 'ambient_counter': 4, 'ambient_order': 56, 'ambient_tex': 'C_4\\times D_7', 'central': False, 'central_factor': False, 'centralizer_order': 4, 'characteristic': True, 'core_order': 56, 'counter': 1, 'cyclic': False, 'direct': True, 'hall': 14, 'label': '56.4.1.a1.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': None, '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': '56.4', 'subgroup_hash': 4, 'subgroup_order': 56, 'subgroup_tex': 'C_4\\times D_7', 'supersolvable': True, 'sylow': 0}
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id: 67612386
{'Agroup': True, 'Zgroup': False, 'abelian': False, 'ambient': '56.4', 'ambient_counter': 4, 'ambient_order': 56, 'ambient_tex': 'C_4\\times D_7', 'central': False, 'central_factor': True, 'centralizer_order': 4, 'characteristic': True, 'core_order': 28, 'counter': 2, 'cyclic': False, 'direct': False, 'hall': 0, 'label': '56.4.2.a1.a1', 'maximal': True, 'maximal_normal': True, 'metabelian': True, 'metacyclic': None, 'minimal': False, 'minimal_normal': False, 'nilpotent': False, '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': ['C1'], 'split': False, 'standard_generators': False, 'stem': False, 'subgroup': '28.3', 'subgroup_hash': 3, 'subgroup_order': 28, 'subgroup_tex': 'D_{14}', 'supersolvable': True, 'sylow': 0}
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id: 67612387
{'Agroup': True, 'Zgroup': True, 'abelian': True, 'ambient': '56.4', 'ambient_counter': 4, 'ambient_order': 56, 'ambient_tex': 'C_4\\times D_7', 'central': False, 'central_factor': False, 'centralizer_order': 28, 'characteristic': True, 'core_order': 28, 'counter': 3, 'cyclic': True, 'direct': False, 'hall': 0, 'label': '56.4.2.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': ['F'], 'split': True, 'standard_generators': False, 'stem': False, 'subgroup': '28.2', 'subgroup_hash': 2, 'subgroup_order': 28, 'subgroup_tex': 'C_{28}', 'supersolvable': True, 'sylow': 0}
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id: 67612388
{'Agroup': True, 'Zgroup': True, 'abelian': False, 'ambient': '56.4', 'ambient_counter': 4, 'ambient_order': 56, 'ambient_tex': 'C_4\\times D_7', 'central': False, 'central_factor': True, 'centralizer_order': 4, 'characteristic': True, 'core_order': 28, 'counter': 4, 'cyclic': False, 'direct': False, 'hall': 0, 'label': '56.4.2.c1.a1', 'maximal': True, 'maximal_normal': True, 'metabelian': True, 'metacyclic': True, 'minimal': False, 'minimal_normal': False, 'nilpotent': False, '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': [], 'split': True, 'standard_generators': False, 'stem': False, 'subgroup': '28.1', 'subgroup_hash': 1, 'subgroup_order': 28, 'subgroup_tex': 'C_7:C_4', 'supersolvable': True, 'sylow': 0}
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id: 67612389
{'Agroup': True, 'Zgroup': True, 'abelian': True, 'ambient': '56.4', 'ambient_counter': 4, 'ambient_order': 56, 'ambient_tex': 'C_4\\times D_7', 'central': False, 'central_factor': False, 'centralizer_order': 28, 'characteristic': True, 'core_order': 14, 'counter': 5, 'cyclic': True, 'direct': False, 'hall': 0, 'label': '56.4.4.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': ['S', 'C2'], 'split': False, 'standard_generators': False, 'stem': False, 'subgroup': '14.2', 'subgroup_hash': 2, 'subgroup_order': 14, 'subgroup_tex': 'C_{14}', 'supersolvable': True, 'sylow': 0}
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id: 67612390
{'Agroup': True, 'Zgroup': True, 'abelian': False, 'ambient': '56.4', 'ambient_counter': 4, 'ambient_order': 56, 'ambient_tex': 'C_4\\times D_7', 'central': False, 'central_factor': True, 'centralizer_order': 4, 'characteristic': False, 'core_order': 14, 'counter': 6, 'cyclic': False, 'direct': True, 'hall': 0, 'label': '56.4.4.b1.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': False, 'minimal_normal': False, 'nilpotent': False, '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': '14.1', 'subgroup_hash': 1, 'subgroup_order': 14, 'subgroup_tex': 'D_7', 'supersolvable': True, 'sylow': 0}
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id: 67612391
{'Agroup': True, 'Zgroup': True, 'abelian': False, 'ambient': '56.4', 'ambient_counter': 4, 'ambient_order': 56, 'ambient_tex': 'C_4\\times D_7', 'central': False, 'central_factor': True, 'centralizer_order': 4, 'characteristic': False, 'core_order': 14, 'counter': 7, 'cyclic': False, 'direct': True, 'hall': 0, 'label': '56.4.4.b1.b1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': False, 'minimal_normal': False, 'nilpotent': False, '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': '14.1', 'subgroup_hash': 1, 'subgroup_order': 14, 'subgroup_tex': 'D_7', 'supersolvable': True, 'sylow': 0}
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id: 67612392
{'Agroup': True, 'Zgroup': False, 'abelian': True, 'ambient': '56.4', 'ambient_counter': 4, 'ambient_order': 56, 'ambient_tex': 'C_4\\times D_7', 'central': False, 'central_factor': False, 'centralizer_order': 8, 'characteristic': False, 'core_order': 4, 'counter': 8, 'cyclic': False, 'direct': None, 'hall': 2, 'label': '56.4.7.a1.a1', 'maximal': True, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': False, 'old_label': '7.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': 7, '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': '8.2', 'subgroup_hash': 2, 'subgroup_order': 8, 'subgroup_tex': 'C_2\\times C_4', 'supersolvable': True, 'sylow': 2}
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id: 67612393
{'Agroup': True, 'Zgroup': True, 'abelian': True, 'ambient': '56.4', 'ambient_counter': 4, 'ambient_order': 56, 'ambient_tex': 'C_4\\times D_7', 'central': False, 'central_factor': False, 'centralizer_order': 28, 'characteristic': True, 'core_order': 7, 'counter': 9, 'cyclic': True, 'direct': False, 'hall': 7, 'label': '56.4.8.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', 'C3'], 'split': True, 'standard_generators': False, 'stem': False, 'subgroup': '7.1', 'subgroup_hash': 1, 'subgroup_order': 7, 'subgroup_tex': 'C_7', 'supersolvable': True, 'sylow': 7}
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id: 67612379
{'Agroup': True, 'Zgroup': True, 'abelian': True, 'ambient': '56.4', 'ambient_counter': 4, 'ambient_order': 56, 'ambient_tex': 'C_4\\times D_7', 'central': True, 'central_factor': False, 'centralizer_order': 56, 'characteristic': True, 'core_order': 4, 'counter': 10, 'cyclic': True, 'direct': True, 'hall': 0, 'label': '56.4.14.a1.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': True, 'old_label': '14.a1.a1', 'outer_equivalence': False, 'perfect': False, 'proper': True, 'quotient': '14.1', 'quotient_Agroup': True, 'quotient_abelian': False, 'quotient_cyclic': False, 'quotient_hash': 1, 'quotient_metabelian': True, 'quotient_nilpotent': False, 'quotient_order': 14, 'quotient_simple': False, 'quotient_solvable': True, 'quotient_supersolvable': True, 'quotient_tex': 'D_7', 'simple': False, 'solvable': True, 'special_labels': ['Z', 'U0'], 'split': True, '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: 67612380
{'Agroup': True, 'Zgroup': False, 'abelian': True, 'ambient': '56.4', 'ambient_counter': 4, 'ambient_order': 56, 'ambient_tex': 'C_4\\times D_7', 'central': False, 'central_factor': False, 'centralizer_order': 8, 'characteristic': False, 'core_order': 2, 'counter': 11, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '56.4.14.b1.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': False, 'old_label': '14.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': 14, '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: 67612381
{'Agroup': True, 'Zgroup': True, 'abelian': True, 'ambient': '56.4', 'ambient_counter': 4, 'ambient_order': 56, 'ambient_tex': 'C_4\\times D_7', 'central': False, 'central_factor': False, 'centralizer_order': 8, 'characteristic': False, 'core_order': 2, 'counter': 12, 'cyclic': True, 'direct': None, 'hall': 0, 'label': '56.4.14.c1.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': False, 'old_label': '14.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': 14, '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: 67612382
{'Agroup': True, 'Zgroup': True, 'abelian': True, 'ambient': '56.4', 'ambient_counter': 4, 'ambient_order': 56, 'ambient_tex': 'C_4\\times D_7', 'central': True, 'central_factor': False, 'centralizer_order': 56, 'characteristic': True, 'core_order': 2, 'counter': 13, 'cyclic': True, 'direct': False, 'hall': 0, 'label': '56.4.28.a1.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': True, 'minimal_normal': True, 'nilpotent': True, 'normal': True, 'old_label': '28.a1.a1', 'outer_equivalence': False, 'perfect': False, 'proper': True, 'quotient': '28.3', 'quotient_Agroup': True, 'quotient_abelian': False, 'quotient_cyclic': False, 'quotient_hash': 3, 'quotient_metabelian': True, 'quotient_nilpotent': False, 'quotient_order': 28, 'quotient_simple': False, 'quotient_solvable': True, 'quotient_supersolvable': True, 'quotient_tex': 'D_{14}', 'simple': True, 'solvable': True, 'special_labels': ['Phi'], '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: 67612383
{'Agroup': True, 'Zgroup': True, 'abelian': True, 'ambient': '56.4', 'ambient_counter': 4, 'ambient_order': 56, 'ambient_tex': 'C_4\\times D_7', 'central': False, 'central_factor': False, 'centralizer_order': 8, 'characteristic': False, 'core_order': 1, 'counter': 14, 'cyclic': True, 'direct': None, 'hall': 0, 'label': '56.4.28.b1.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': True, 'minimal_normal': False, 'nilpotent': True, 'normal': False, 'old_label': '28.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': 28, '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: 67612384
{'Agroup': True, 'Zgroup': True, 'abelian': True, 'ambient': '56.4', 'ambient_counter': 4, 'ambient_order': 56, 'ambient_tex': 'C_4\\times D_7', 'central': False, 'central_factor': False, 'centralizer_order': 8, 'characteristic': False, 'core_order': 1, 'counter': 15, 'cyclic': True, 'direct': None, 'hall': 0, 'label': '56.4.28.b1.b1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': True, 'minimal_normal': False, 'nilpotent': True, 'normal': False, 'old_label': '28.b1.b1', 'outer_equivalence': False, 'perfect': False, 'proper': True, 'quotient': None, 'quotient_Agroup': None, 'quotient_abelian': None, 'quotient_cyclic': None, 'quotient_hash': None, 'quotient_metabelian': None, 'quotient_nilpotent': None, 'quotient_order': 28, '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: 67612385
{'Agroup': True, 'Zgroup': True, 'abelian': True, 'ambient': '56.4', 'ambient_counter': 4, 'ambient_order': 56, 'ambient_tex': 'C_4\\times D_7', 'central': True, 'central_factor': False, 'centralizer_order': 56, 'characteristic': True, 'core_order': 1, 'counter': 16, 'cyclic': True, 'direct': True, 'hall': 1, 'label': '56.4.56.a1.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': True, 'old_label': '56.a1.a1', 'outer_equivalence': False, 'perfect': True, 'proper': False, 'quotient': '56.4', 'quotient_Agroup': True, 'quotient_abelian': False, 'quotient_cyclic': False, 'quotient_hash': 4, 'quotient_metabelian': True, 'quotient_nilpotent': False, 'quotient_order': 56, 'quotient_simple': False, 'quotient_solvable': True, 'quotient_supersolvable': True, 'quotient_tex': 'C_4\\times D_7', 'simple': False, 'solvable': True, 'special_labels': ['PC', 'U1', '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: 67612378
{'ambient': '56.4', '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': '14.a1.a1', 'complements': ['56.a1.a1'], 'conjugacy_class_count': 1, 'contained_in': [], 'contains': ['2.a1.a1', '2.b1.a1', '2.c1.a1', '7.a1.a1'], 'core': '1.a1.a1', 'coset_action_label': None, 'count': 1, 'diagramx': [7313, 5020, 8014, 5952, 5538, 5030, 4207, 5847], 'generators': [1, 28, 8, 14], 'label': '56.4.1.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': '14.1', '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': '14.1'}
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id: 67612379
{'ambient': '56.4', 'aut_centralizer_order': 84, 'aut_label': '14.a1', 'aut_quo_index': 1, 'aut_stab_index': 1, 'aut_weyl_group': '2.1', 'aut_weyl_index': 84, 'centralizer': '1.a1.a1', 'complements': ['4.b1.a1', '4.b1.b1'], 'conjugacy_class_count': 1, 'contained_in': ['2.b1.a1', '7.a1.a1'], 'contains': ['28.a1.a1'], 'core': '14.a1.a1', 'coset_action_label': None, 'count': 1, 'diagramx': [2331, 7560, 7397, 8686, 1256, 8409, 3069, 8771], 'generators': [14], 'label': '56.4.14.a1.a1', 'mobius_quo': 0, 'mobius_sub': 7, 'normal_closure': '14.a1.a1', 'normal_contained_in': ['2.b1.a1'], 'normal_contains': ['28.a1.a1'], 'normalizer': '1.a1.a1', 'old_label': '14.a1.a1', 'projective_image': '14.1', 'quotient_action_image': '1.1', 'quotient_action_kernel': '14.1', 'quotient_action_kernel_order': 14, 'quotient_fusion': None, 'short_label': '14.a1.a1', 'subgroup_fusion': None, 'weyl_group': '1.1'}
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id: 67612380
{'ambient': '56.4', 'aut_centralizer_order': 12, 'aut_label': '14.b1', 'aut_quo_index': None, 'aut_stab_index': 7, 'aut_weyl_group': '2.1', 'aut_weyl_index': 84, 'centralizer': '7.a1.a1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['2.a1.a1', '7.a1.a1'], 'contains': ['28.a1.a1', '28.b1.a1', '28.b1.b1'], 'core': '28.a1.a1', 'coset_action_label': None, 'count': 7, 'diagramx': [9033, -1, 4292, -1, 5077, -1, 6423, -1], 'generators': [1, 28], 'label': '56.4.14.b1.a1', 'mobius_quo': None, 'mobius_sub': 1, 'normal_closure': '2.a1.a1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '7.a1.a1', 'old_label': '14.b1.a1', 'projective_image': '28.3', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '14.b1.a1', 'subgroup_fusion': None, 'weyl_group': '1.1'}
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id: 67612381
{'ambient': '56.4', 'aut_centralizer_order': 12, 'aut_label': '14.c1', 'aut_quo_index': None, 'aut_stab_index': 7, 'aut_weyl_group': '2.1', 'aut_weyl_index': 84, 'centralizer': '7.a1.a1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['2.c1.a1', '7.a1.a1'], 'contains': ['28.a1.a1'], 'core': '28.a1.a1', 'coset_action_label': None, 'count': 7, 'diagramx': [6273, -1, 9384, -1, 3103, -1, 1606, -1], 'generators': [3], 'label': '56.4.14.c1.a1', 'mobius_quo': None, 'mobius_sub': 1, 'normal_closure': '2.c1.a1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '7.a1.a1', 'old_label': '14.c1.a1', 'projective_image': '28.3', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '14.c1.a1', 'subgroup_fusion': None, 'weyl_group': '1.1'}
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id: 67612382
{'ambient': '56.4', 'aut_centralizer_order': 168, 'aut_label': '28.a1', 'aut_quo_index': 2, 'aut_stab_index': 1, 'aut_weyl_group': '1.1', 'aut_weyl_index': 168, 'centralizer': '1.a1.a1', 'complements': [], 'conjugacy_class_count': 1, 'contained_in': ['4.a1.a1', '14.a1.a1', '14.b1.a1', '14.c1.a1'], 'contains': ['56.a1.a1'], 'core': '28.a1.a1', 'coset_action_label': None, 'count': 1, 'diagramx': [5305, 5179, 5662, 6193, 4308, 6461, 4695, 5947], 'generators': [28], 'label': '56.4.28.a1.a1', 'mobius_quo': -1, 'mobius_sub': -14, 'normal_closure': '28.a1.a1', 'normal_contained_in': ['4.a1.a1', '14.a1.a1'], 'normal_contains': ['56.a1.a1'], 'normalizer': '1.a1.a1', 'old_label': '28.a1.a1', 'projective_image': '28.3', 'quotient_action_image': '1.1', 'quotient_action_kernel': '28.3', 'quotient_action_kernel_order': 28, 'quotient_fusion': None, 'short_label': '28.a1.a1', 'subgroup_fusion': None, 'weyl_group': '1.1'}
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id: 67612383
{'ambient': '56.4', 'aut_centralizer_order': 12, 'aut_label': '28.b1', 'aut_quo_index': None, 'aut_stab_index': 14, 'aut_weyl_group': '1.1', 'aut_weyl_index': 168, 'centralizer': '7.a1.a1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['4.b1.a1', '14.b1.a1'], 'contains': ['56.a1.a1'], 'core': '56.a1.a1', 'coset_action_label': '28T8', 'count': 7, 'diagramx': [9033, -1, 616, -1, 6154, -1, 6463, -1], 'generators': [1], 'label': '56.4.28.b1.a1', 'mobius_quo': None, 'mobius_sub': 0, 'normal_closure': '4.b1.a1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '7.a1.a1', 'old_label': '28.b1.a1', 'projective_image': '56.4', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '28.b1.a1', 'subgroup_fusion': None, 'weyl_group': '1.1'}
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id: 67612384
{'ambient': '56.4', 'aut_centralizer_order': 12, 'aut_label': '28.b1', 'aut_quo_index': None, 'aut_stab_index': 14, 'aut_weyl_group': '1.1', 'aut_weyl_index': 168, 'centralizer': '7.a1.a1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['4.b1.b1', '14.b1.a1'], 'contains': ['56.a1.a1'], 'core': '56.a1.a1', 'coset_action_label': '28T8', 'count': 7, 'diagramx': [9033, -1, 3127, -1, 6154, -1, 9390, -1], 'generators': [5], 'label': '56.4.28.b1.b1', 'mobius_quo': None, 'mobius_sub': 0, 'normal_closure': '4.b1.b1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '7.a1.a1', 'old_label': '28.b1.b1', 'projective_image': '56.4', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '28.b1.b1', 'subgroup_fusion': None, 'weyl_group': '1.1'}
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id: 67612385
{'ambient': '56.4', 'aut_centralizer_order': None, 'aut_label': '56.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', '28.a1.a1', '28.b1.a1', '28.b1.b1'], 'contains': [], 'core': '56.a1.a1', 'coset_action_label': None, 'count': 1, 'diagramx': [5305, 4107, 2511, 5259, 6154, 5030, 7195, 4618], 'generators': [], 'label': '56.4.56.a1.a1', 'mobius_quo': 1, 'mobius_sub': 0, 'normal_closure': '56.a1.a1', 'normal_contained_in': ['8.a1.a1', '28.a1.a1'], 'normal_contains': [], 'normalizer': '1.a1.a1', 'old_label': '56.a1.a1', 'projective_image': '56.4', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '56.a1.a1', 'subgroup_fusion': None, 'weyl_group': '1.1'}
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id: 67612386
{'ambient': '56.4', 'aut_centralizer_order': 2, 'aut_label': '2.a1', 'aut_quo_index': 1, 'aut_stab_index': 1, 'aut_weyl_group': '84.7', 'aut_weyl_index': 2, 'centralizer': '14.a1.a1', 'complements': [], 'conjugacy_class_count': 1, 'contained_in': ['1.a1.a1'], 'contains': ['4.a1.a1', '4.b1.a1', '4.b1.b1', '14.b1.a1'], 'core': '2.a1.a1', 'coset_action_label': None, 'count': 1, 'diagramx': [8925, 1409, 4521, 2802, 9026, 1412, 7155, 2824], 'generators': [1, 8, 28], 'label': '56.4.2.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': '28.3', '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': '14.1'}
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id: 67612387
{'ambient': '56.4', 'aut_centralizer_order': 14, 'aut_label': '2.b1', 'aut_quo_index': 1, 'aut_stab_index': 1, 'aut_weyl_group': '12.5', 'aut_weyl_index': 14, 'centralizer': '2.b1.a1', 'complements': ['28.b1.a1', '28.b1.b1'], 'conjugacy_class_count': 1, 'contained_in': ['1.a1.a1'], 'contains': ['4.a1.a1', '14.a1.a1'], 'core': '2.b1.a1', 'coset_action_label': None, 'count': 1, 'diagramx': [2437, 8591, 6987, 9065, 4385, 8608, 5346, 8837], 'generators': [14, 28, 8], 'label': '56.4.2.b1.a1', 'mobius_quo': 0, 'mobius_sub': -1, 'normal_closure': '2.b1.a1', 'normal_contained_in': ['1.a1.a1'], 'normal_contains': ['4.a1.a1', '14.a1.a1'], 'normalizer': '1.a1.a1', 'old_label': '2.b1.a1', 'projective_image': '14.1', '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: 67612388
{'ambient': '56.4', 'aut_centralizer_order': 2, 'aut_label': '2.c1', 'aut_quo_index': 1, 'aut_stab_index': 1, 'aut_weyl_group': '84.7', 'aut_weyl_index': 2, 'centralizer': '14.a1.a1', 'complements': ['28.b1.a1', '28.b1.b1'], 'conjugacy_class_count': 1, 'contained_in': ['1.a1.a1'], 'contains': ['4.a1.a1', '14.c1.a1'], 'core': '2.c1.a1', 'coset_action_label': None, 'count': 1, 'diagramx': [5664, 5020, 9041, 5952, 6692, 5030, 752, 5847], 'generators': [3, 8, 28], 'label': '56.4.2.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': '28.3', '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': '14.1'}
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id: 67612389
{'ambient': '56.4', 'aut_centralizer_order': 28, 'aut_label': '4.a1', 'aut_quo_index': 6, 'aut_stab_index': 1, 'aut_weyl_group': '6.2', 'aut_weyl_index': 28, 'centralizer': '2.b1.a1', 'complements': [], 'conjugacy_class_count': 1, 'contained_in': ['2.a1.a1', '2.b1.a1', '2.c1.a1'], 'contains': ['8.a1.a1', '28.a1.a1'], 'core': '4.a1.a1', 'coset_action_label': None, 'count': 1, 'diagramx': [3406, 5020, 5959, 6193, 7256, 5030, 4695, 5947], 'generators': [28, 8], 'label': '56.4.4.a1.a1', 'mobius_quo': 1, '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', '28.a1.a1'], 'normalizer': '1.a1.a1', 'old_label': '4.a1.a1', 'projective_image': '28.3', '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'}
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id: 67612390
{'ambient': '56.4', 'aut_centralizer_order': 2, 'aut_label': '4.b1', 'aut_quo_index': 1, 'aut_stab_index': 2, 'aut_weyl_group': '42.1', 'aut_weyl_index': 4, 'centralizer': '14.a1.a1', 'complements': ['14.a1.a1', '14.c1.a1'], 'conjugacy_class_count': 1, 'contained_in': ['2.a1.a1'], 'contains': ['8.a1.a1', '28.b1.a1'], 'core': '4.b1.a1', 'coset_action_label': None, 'count': 1, 'diagramx': [8925, 1647, 1188, 935, 9308, 1650, 7988, 897], 'generators': [1, 8], 'label': '56.4.4.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': '56.4', 'quotient_action_image': '1.1', 'quotient_action_kernel': '4.1', 'quotient_action_kernel_order': 4, 'quotient_fusion': None, 'short_label': '4.b1.a1', 'subgroup_fusion': None, 'weyl_group': '14.1'}
-
id: 67612391
{'ambient': '56.4', 'aut_centralizer_order': 2, 'aut_label': '4.b1', 'aut_quo_index': 1, 'aut_stab_index': 2, 'aut_weyl_group': '42.1', 'aut_weyl_index': 4, 'centralizer': '14.a1.a1', 'complements': ['14.a1.a1', '14.c1.a1'], 'conjugacy_class_count': 1, 'contained_in': ['2.a1.a1'], 'contains': ['8.a1.a1', '28.b1.b1'], 'core': '4.b1.b1', 'coset_action_label': None, 'count': 1, 'diagramx': [8925, 1647, 2922, 3425, 9308, 1650, 9451, 3289], 'generators': [5, 8], 'label': '56.4.4.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': '56.4', 'quotient_action_image': '1.1', 'quotient_action_kernel': '4.1', 'quotient_action_kernel_order': 4, 'quotient_fusion': None, 'short_label': '4.b1.b1', 'subgroup_fusion': None, 'weyl_group': '14.1'}
-
id: 67612392
{'ambient': '56.4', 'aut_centralizer_order': 6, 'aut_label': '7.a1', 'aut_quo_index': None, 'aut_stab_index': 7, 'aut_weyl_group': '4.2', 'aut_weyl_index': 42, 'centralizer': '7.a1.a1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['1.a1.a1'], 'contains': ['14.a1.a1', '14.b1.a1', '14.c1.a1'], 'core': '14.a1.a1', 'coset_action_label': None, 'count': 7, 'diagramx': [8783, -1, 9384, -1, 1513, -1, 3069, -1], 'generators': [1, 14], 'label': '56.4.7.a1.a1', 'mobius_quo': None, 'mobius_sub': -1, 'normal_closure': '1.a1.a1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '7.a1.a1', 'old_label': '7.a1.a1', 'projective_image': '14.1', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '7.a1.a1', 'subgroup_fusion': None, 'weyl_group': '1.1'}
-
id: 67612393
{'ambient': '56.4', 'aut_centralizer_order': 28, 'aut_label': '8.a1', 'aut_quo_index': 2, 'aut_stab_index': 1, 'aut_weyl_group': '6.2', 'aut_weyl_index': 28, 'centralizer': '2.b1.a1', 'complements': ['7.a1.a1'], 'conjugacy_class_count': 1, 'contained_in': ['4.a1.a1', '4.b1.a1', '4.b1.b1'], 'contains': ['56.a1.a1'], 'core': '8.a1.a1', 'coset_action_label': None, 'count': 1, 'diagramx': [967, 2996, 1667, 3875, 8000, 3598, 7927, 3422], 'generators': [8], 'label': '56.4.8.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': ['56.a1.a1'], 'normalizer': '1.a1.a1', 'old_label': '8.a1.a1', 'projective_image': '56.4', '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'}
