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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': 20, 'aut_gen_orders': [2, 4, 20], 'aut_gens': [[1, 4], [61, 44], [1, 52], [45, 4]], 'aut_group': '160.207', 'aut_hash': 207, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 160, 'aut_permdeg': 9, 'aut_perms': [1, 6487, 162742], 'aut_phi_ratio': 5.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 5, 1, 2], [4, 2, 1, 1], [4, 10, 1, 1], [4, 10, 2, 2], [5, 4, 1, 1], [10, 4, 1, 1], [20, 4, 2, 1]], 'aut_supersolvable': True, 'aut_tex': 'D_4\\times F_5', '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': 20, 'autcentquo_group': '40.12', 'autcentquo_hash': 12, 'autcentquo_nilpotent': False, 'autcentquo_order': 40, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'C_2\\times F_5', 'cc_stats': [[1, 1, 1], [2, 1, 1], [2, 5, 2], [4, 2, 1], [4, 10, 5], [5, 4, 1], [10, 4, 1], [20, 4, 2]], 'center_label': '2.1', 'center_order': 2, 'central_product': False, 'central_quotient': '40.12', 'commutator_count': 1, 'commutator_label': '10.2', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '5.1'], 'composition_length': 5, 'conjugacy_classes_known': True, 'counter': 31, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 5, 1, 2], [4, 2, 1, 1], [4, 10, 1, 1], [4, 10, 2, 2], [5, 4, 1, 1], [10, 4, 1, 1], [20, 4, 2, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 12, 'exponent': 20, 'exponents_of_order': [4, 1], 'factors_of_aut_order': [2, 5], 'factors_of_order': [2, 5], 'faithful_reps': [[4, 0, 2]], 'familial': False, 'frattini_label': '2.1', 'frattini_quotient': '40.12', 'hash': 31, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 20, 'inner_gen_orders': [4, 10], 'inner_gens': [[1, 12], [73, 4]], 'inner_hash': 12, 'inner_nilpotent': False, 'inner_order': 40, 'inner_split': False, 'inner_tex': 'C_2\\times F_5', 'inner_used': [1, 2], 'irrC_degree': 4, 'irrQ_degree': 8, 'irrQ_dim': 8, 'irrR_degree': 8, 'irrep_stats': [[1, 8], [2, 2], [4, 4]], 'label': '80.31', 'linC_count': 2, 'linC_degree': 4, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 6, 'linQ_degree_count': 4, 'linQ_dim': 6, 'linQ_dim_count': 2, 'linR_count': 2, 'linR_degree': 6, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C20:C4', 'ngens': 5, 'nilpotency_class': -1, 'nilpotent': False, 'normal_counts': [0, 0, 0, 0, 0, 0, 0, 0, 0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 11, 'number_characteristic_subgroups': 12, 'number_conjugacy_classes': 14, 'number_divisions': 11, 'number_normal_subgroups': 14, 'number_subgroup_autclasses': 22, 'number_subgroup_classes': 26, 'number_subgroups': 78, 'old_label': None, 'order': 80, 'order_factorization_type': 31, 'order_stats': [[1, 1], [2, 11], [4, 52], [5, 4], [10, 4], [20, 8]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 2, 'outer_gen_orders': [2, 2], 'outer_gen_pows': [20, 0], 'outer_gens': [[21, 4], [1, 44]], 'outer_group': '4.2', 'outer_hash': 2, 'outer_nilpotent': True, 'outer_order': 4, 'outer_permdeg': 4, 'outer_perms': [1, 6], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^2', 'pc_rank': 2, 'perfect': False, 'permutation_degree': 9, 'pgroup': 0, 'primary_abelian_invariants': [2, 4], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 4], [4, 2], [8, 1]], 'representations': {'PC': {'code': 85858774671193928610819, 'gens': [1, 3], 'pres': [5, -2, -2, -2, -2, -5, 10, 182, 277, 42, 483, 728, 58, 1204, 809]}, 'GLZ': {'b': 3, 'd': 6, 'gens': [25038649734177915, 91655869359786907]}, 'GLFp': {'d': 4, 'p': 3, 'gens': [33433269, 32577956, 27187973, 28816400, 22871412]}, 'Perm': {'d': 9, 'gens': [5887, 13, 11520, 23, 46800]}}, '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_{20}:C_4', 'transitive_degree': 20, 'wreath_data': None, 'wreath_product': False}
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gps_conj_classes • Show schema
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id: 110998536
{'aut_label': '1A', 'centralizer': '1.a1.a1', 'counter': 1, 'group_counter': 31, 'group_order': 80, 'label': '1A', 'order': 1, 'powers': [1, 1], 'representative': 0, 'size': 1}
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id: 110998542
{'aut_label': '2A', 'centralizer': '1.a1.a1', 'counter': 2, 'group_counter': 31, 'group_order': 80, 'label': '2A', 'order': 2, 'powers': [1, 2], 'representative': 40, 'size': 1}
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id: 110998543
{'aut_label': '2B', 'centralizer': '5.a1.a1', 'counter': 3, 'group_counter': 31, 'group_order': 80, 'label': '2B', 'order': 2, 'powers': [1, 3], 'representative': 10, 'size': 5}
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id: 110998544
{'aut_label': '2C', 'centralizer': '5.a1.a1', 'counter': 4, 'group_counter': 31, 'group_order': 80, 'label': '2C', 'order': 2, 'powers': [1, 4], 'representative': 34, 'size': 5}
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id: 110998545
{'aut_label': '4A', 'centralizer': '2.a1.a1', 'counter': 5, 'group_counter': 31, 'group_order': 80, 'label': '4A', 'order': 4, 'powers': [2, 5], 'representative': 20, 'size': 2}
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id: 110998546
{'aut_label': '4B', 'centralizer': '10.a1.a1', 'counter': 6, 'group_counter': 31, 'group_order': 80, 'label': '4B', 'order': 4, 'powers': [2, 6], 'representative': 14, 'size': 10}
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id: 110998547
{'aut_label': '4C1', 'centralizer': '10.b1.a1', 'counter': 7, 'group_counter': 31, 'group_order': 80, 'label': '4C1', 'order': 4, 'powers': [4, 7], 'representative': 9, 'size': 10}
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id: 110998548
{'aut_label': '4C-1', 'centralizer': '10.b1.a1', 'counter': 8, 'group_counter': 31, 'group_order': 80, 'label': '4C-1', 'order': 4, 'powers': [4, 8], 'representative': 27, 'size': 10}
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id: 110998549
{'aut_label': '4C-1', 'centralizer': '10.b1.b1', 'counter': 9, 'group_counter': 31, 'group_order': 80, 'label': '4D1', 'order': 4, 'powers': [4, 9], 'representative': 31, 'size': 10}
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id: 110998537
{'aut_label': '4C1', 'centralizer': '10.b1.b1', 'counter': 10, 'group_counter': 31, 'group_order': 80, 'label': '4D-1', 'order': 4, 'powers': [4, 10], 'representative': 77, 'size': 10}
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id: 110998538
{'aut_label': '5A', 'centralizer': '4.b1.a1', 'counter': 11, 'group_counter': 31, 'group_order': 80, 'label': '5A', 'order': 5, 'powers': [11, 1], 'representative': 16, 'size': 4}
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id: 110998539
{'aut_label': '10A', 'centralizer': '4.b1.a1', 'counter': 12, 'group_counter': 31, 'group_order': 80, 'label': '10A', 'order': 10, 'powers': [11, 2], 'representative': 8, 'size': 4}
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id: 110998540
{'aut_label': '20A1', 'centralizer': '4.b1.a1', 'counter': 13, 'group_counter': 31, 'group_order': 80, 'label': '20A1', 'order': 20, 'powers': [12, 5], 'representative': 4, 'size': 4}
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id: 110998541
{'aut_label': '20A1', 'centralizer': '4.b1.a1', 'counter': 14, 'group_counter': 31, 'group_order': 80, 'label': '20A-1', 'order': 20, 'powers': [12, 5], 'representative': 52, 'size': 4}
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gps_qchar • Show schema
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id: 4816126
{'cdim': 1, 'counter': 1, 'faithful': False, 'group': '80.31', 'group_order': 80, 'image': None, 'label': '80.31.1a', 'multiplicity': 1, 'nt': [1, 1], 'qdim': 1, 'qvalues': [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1], 'schur_index': 1}
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id: 4816129
{'cdim': 1, 'counter': 2, 'faithful': False, 'group': '80.31', 'group_order': 80, 'image': None, 'label': '80.31.1b', 'multiplicity': 1, 'nt': [2, 1], 'qdim': 1, 'qvalues': [1, 1, 1, 1, -1, -1, -1, -1, 1, 1, 1, 1, -1, -1], 'schur_index': 1}
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id: 4816130
{'cdim': 1, 'counter': 3, 'faithful': False, 'group': '80.31', 'group_order': 80, 'image': None, 'label': '80.31.1c', 'multiplicity': 1, 'nt': [2, 1], 'qdim': 1, 'qvalues': [1, 1, 1, 1, -1, 1, 1, -1, -1, -1, 1, 1, -1, -1], 'schur_index': 1}
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id: 4816131
{'cdim': 1, 'counter': 4, 'faithful': False, 'group': '80.31', 'group_order': 80, 'image': None, 'label': '80.31.1d', 'multiplicity': 1, 'nt': [2, 1], 'qdim': 1, 'qvalues': [1, 1, 1, 1, 1, -1, -1, 1, -1, -1, 1, 1, 1, 1], 'schur_index': 1}
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id: 4816132
{'cdim': 1, 'counter': 5, 'faithful': False, 'group': '80.31', 'group_order': 80, 'image': None, 'label': '80.31.1e', 'multiplicity': 2, 'nt': [4, 1], 'qdim': 2, 'qvalues': [2, 2, -2, -2, -2, 0, 0, 2, 0, 0, 2, 2, -2, -2], 'schur_index': 1}
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id: 4816133
{'cdim': 1, 'counter': 6, 'faithful': False, 'group': '80.31', 'group_order': 80, 'image': None, 'label': '80.31.1f', 'multiplicity': 2, 'nt': [4, 1], 'qdim': 2, 'qvalues': [2, 2, -2, -2, 2, 0, 0, -2, 0, 0, 2, 2, 2, 2], 'schur_index': 1}
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id: 4816134
{'cdim': 2, 'counter': 7, 'faithful': False, 'group': '80.31', 'group_order': 80, 'image': None, 'label': '80.31.2a', 'multiplicity': 1, 'nt': [4, 3], 'qdim': 2, 'qvalues': [2, -2, -2, 2, 0, 0, 0, 0, 0, 0, 2, -2, 0, 0], 'schur_index': 1}
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id: 4816135
{'cdim': 2, 'counter': 8, 'faithful': False, 'group': '80.31', 'group_order': 80, 'image': None, 'label': '80.31.2b', 'multiplicity': 1, 'nt': [8, 5], 'qdim': 2, 'qvalues': [2, -2, 2, -2, 0, 0, 0, 0, 0, 0, 2, -2, 0, 0], 'schur_index': 2}
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id: 4816136
{'cdim': 4, 'counter': 9, 'faithful': False, 'group': '80.31', 'group_order': 80, 'image': None, 'label': '80.31.4a', 'multiplicity': 1, 'nt': [5, 3], 'qdim': 4, 'qvalues': [4, 4, 0, 0, 4, 0, 0, 0, 0, 0, -1, -1, -1, -1], 'schur_index': 1}
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id: 4816127
{'cdim': 4, 'counter': 10, 'faithful': False, 'group': '80.31', 'group_order': 80, 'image': None, 'label': '80.31.4b', 'multiplicity': 1, 'nt': [10, 5], 'qdim': 4, 'qvalues': [4, 4, 0, 0, -4, 0, 0, 0, 0, 0, -1, -1, 1, 1], 'schur_index': 1}
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id: 4816128
{'cdim': 4, 'counter': 11, 'faithful': True, 'group': '80.31', 'group_order': 80, 'image': None, 'label': '80.31.4c', 'multiplicity': 2, 'nt': [20, 18], 'qdim': 8, 'qvalues': [8, -8, 0, 0, 0, 0, 0, 0, 0, 0, -2, 2, 0, 0], 'schur_index': 1}
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gps_char • Show schema
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id: 7900374
{'center': '1.a1.a1', 'center_index': 1, 'center_order': 80, 'counter': 1, 'cyclotomic_n': 1, 'dim': 1, 'faithful': False, 'field': [0, 1], 'group': '80.31', 'group_counter': 31, 'group_order': 80, 'image': None, 'image_isoclass': '1.1', 'image_order': 1, 'indicator': 1, 'kernel': '1.a1.a1', 'kernel_order': 80, 'label': '80.31.1a', '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]]]}
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id: 7900380
{'center': '1.a1.a1', 'center_index': 1, 'center_order': 80, 'counter': 2, 'cyclotomic_n': 1, 'dim': 1, 'faithful': False, 'field': [0, 1], 'group': '80.31', 'group_counter': 31, 'group_order': 80, 'image': None, 'image_isoclass': '2.1', 'image_order': 2, 'indicator': 1, 'kernel': '2.b1.a1', 'kernel_order': 40, 'label': '80.31.1b', '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]]]}
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id: 7900381
{'center': '1.a1.a1', 'center_index': 1, 'center_order': 80, 'counter': 3, 'cyclotomic_n': 1, 'dim': 1, 'faithful': False, 'field': [0, 1], 'group': '80.31', 'group_counter': 31, 'group_order': 80, 'image': None, 'image_isoclass': '2.1', 'image_order': 2, 'indicator': 1, 'kernel': '2.b1.b1', 'kernel_order': 40, 'label': '80.31.1c', '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]]]}
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id: 7900382
{'center': '1.a1.a1', 'center_index': 1, 'center_order': 80, 'counter': 4, 'cyclotomic_n': 1, 'dim': 1, 'faithful': False, 'field': [0, 1], 'group': '80.31', 'group_counter': 31, 'group_order': 80, 'image': None, 'image_isoclass': '2.1', 'image_order': 2, 'indicator': 1, 'kernel': '2.a1.a1', 'kernel_order': 40, 'label': '80.31.1d', '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]]]}
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id: 7900383
{'center': '1.a1.a1', 'center_index': 1, 'center_order': 80, 'counter': 5, 'cyclotomic_n': 4, 'dim': 1, 'faithful': False, 'field': [1, 0, 1], 'group': '80.31', 'group_counter': 31, 'group_order': 80, 'image': None, 'image_isoclass': '4.1', 'image_order': 4, 'indicator': 0, 'kernel': '4.c1.a1', 'kernel_order': 20, 'label': '80.31.1e1', 'nt': [4, 1], 'q_character': None, 'values': [[[1, 0]], [[1, 0]], [[-1, 0]], [[-1, 0]], [[-1, 0]], [[-1, 1]], [[1, 1]], [[1, 0]], [[-1, 1]], [[1, 1]], [[1, 0]], [[1, 0]], [[-1, 0]], [[-1, 0]]]}
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id: 7900384
{'center': '1.a1.a1', 'center_index': 1, 'center_order': 80, 'counter': 6, 'cyclotomic_n': 4, 'dim': 1, 'faithful': False, 'field': [1, 0, 1], 'group': '80.31', 'group_counter': 31, 'group_order': 80, 'image': None, 'image_isoclass': '4.1', 'image_order': 4, 'indicator': 0, 'kernel': '4.c1.a1', 'kernel_order': 20, 'label': '80.31.1e2', 'nt': [4, 2], 'q_character': None, 'values': [[[1, 0]], [[1, 0]], [[-1, 0]], [[-1, 0]], [[-1, 0]], [[1, 1]], [[-1, 1]], [[1, 0]], [[1, 1]], [[-1, 1]], [[1, 0]], [[1, 0]], [[-1, 0]], [[-1, 0]]]}
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id: 7900385
{'center': '1.a1.a1', 'center_index': 1, 'center_order': 80, 'counter': 7, 'cyclotomic_n': 4, 'dim': 1, 'faithful': False, 'field': [1, 0, 1], 'group': '80.31', 'group_counter': 31, 'group_order': 80, 'image': None, 'image_isoclass': '4.1', 'image_order': 4, 'indicator': 0, 'kernel': '4.b1.a1', 'kernel_order': 20, 'label': '80.31.1f1', 'nt': [4, 1], 'q_character': None, 'values': [[[1, 0]], [[1, 0]], [[-1, 0]], [[-1, 0]], [[1, 0]], [[-1, 1]], [[1, 1]], [[-1, 0]], [[1, 1]], [[-1, 1]], [[1, 0]], [[1, 0]], [[1, 0]], [[1, 0]]]}
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id: 7900386
{'center': '1.a1.a1', 'center_index': 1, 'center_order': 80, 'counter': 8, 'cyclotomic_n': 4, 'dim': 1, 'faithful': False, 'field': [1, 0, 1], 'group': '80.31', 'group_counter': 31, 'group_order': 80, 'image': None, 'image_isoclass': '4.1', 'image_order': 4, 'indicator': 0, 'kernel': '4.b1.a1', 'kernel_order': 20, 'label': '80.31.1f2', 'nt': [4, 2], 'q_character': None, 'values': [[[1, 0]], [[1, 0]], [[-1, 0]], [[-1, 0]], [[1, 0]], [[1, 1]], [[-1, 1]], [[-1, 0]], [[-1, 1]], [[1, 1]], [[1, 0]], [[1, 0]], [[1, 0]], [[1, 0]]]}
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id: 7900387
{'center': '4.a1.a1', 'center_index': 4, 'center_order': 20, 'counter': 9, 'cyclotomic_n': 1, 'dim': 2, 'faithful': False, 'field': [0, 1], 'group': '80.31', 'group_counter': 31, 'group_order': 80, 'image': None, 'image_isoclass': '8.3', 'image_order': 8, 'indicator': 1, 'kernel': '8.c1.a1', 'kernel_order': 10, 'label': '80.31.2a', '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]], [[2, 0]], [[-2, 0]], [[0, 0]], [[0, 0]]]}
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id: 7900375
{'center': '4.a1.a1', 'center_index': 4, 'center_order': 20, 'counter': 10, 'cyclotomic_n': 1, 'dim': 2, 'faithful': False, 'field': [0, 1], 'group': '80.31', 'group_counter': 31, 'group_order': 80, 'image': None, 'image_isoclass': '8.4', 'image_order': 8, 'indicator': -1, 'kernel': '8.b1.a1', 'kernel_order': 10, 'label': '80.31.2b', 'nt': [8, 5], 'q_character': None, 'values': [[[2, 0]], [[-2, 0]], [[2, 0]], [[-2, 0]], [[0, 0]], [[0, 0]], [[0, 0]], [[0, 0]], [[0, 0]], [[0, 0]], [[2, 0]], [[-2, 0]], [[0, 0]], [[0, 0]]]}
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id: 7900376
{'center': '20.a1.a1', 'center_index': 20, 'center_order': 4, 'counter': 11, 'cyclotomic_n': 1, 'dim': 4, 'faithful': False, 'field': [0, 1], 'group': '80.31', 'group_counter': 31, 'group_order': 80, 'image': None, 'image_isoclass': '20.3', 'image_order': 20, 'indicator': 1, 'kernel': '20.a1.a1', 'kernel_order': 4, 'label': '80.31.4a', 'nt': [5, 3], 'q_character': None, 'values': [[[4, 0]], [[4, 0]], [[0, 0]], [[0, 0]], [[4, 0]], [[0, 0]], [[0, 0]], [[0, 0]], [[0, 0]], [[0, 0]], [[-1, 0]], [[-1, 0]], [[-1, 0]], [[-1, 0]]]}
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id: 7900377
{'center': '20.a1.a1', 'center_index': 20, 'center_order': 4, 'counter': 12, 'cyclotomic_n': 1, 'dim': 4, 'faithful': False, 'field': [0, 1], 'group': '80.31', 'group_counter': 31, 'group_order': 80, 'image': None, 'image_isoclass': '40.12', 'image_order': 40, 'indicator': 1, 'kernel': '40.a1.a1', 'kernel_order': 2, 'label': '80.31.4b', 'nt': [10, 5], 'q_character': None, 'values': [[[4, 0]], [[4, 0]], [[0, 0]], [[0, 0]], [[-4, 0]], [[0, 0]], [[0, 0]], [[0, 0]], [[0, 0]], [[0, 0]], [[-1, 0]], [[-1, 0]], [[1, 0]], [[1, 0]]]}
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id: 7900378
{'center': '40.a1.a1', 'center_index': 40, 'center_order': 2, 'counter': 13, 'cyclotomic_n': 20, 'dim': 4, 'faithful': True, 'field': [5, 0, 1], 'group': '80.31', 'group_counter': 31, 'group_order': 80, 'image': None, 'image_isoclass': '80.31', 'image_order': 80, 'indicator': 0, 'kernel': '80.a1.a1', 'kernel_order': 1, 'label': '80.31.4c1', 'nt': [20, 18], 'q_character': None, 'values': [[[4, 0]], [[-4, 0]], [[0, 0]], [[0, 0]], [[0, 0]], [[0, 0]], [[0, 0]], [[0, 0]], [[0, 0]], [[0, 0]], [[-1, 0]], [[1, 0]], [[-2, 3], [1, 5], [-2, 7]], [[2, 3], [-1, 5], [2, 7]]]}
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id: 7900379
{'center': '40.a1.a1', 'center_index': 40, 'center_order': 2, 'counter': 14, 'cyclotomic_n': 20, 'dim': 4, 'faithful': True, 'field': [5, 0, 1], 'group': '80.31', 'group_counter': 31, 'group_order': 80, 'image': None, 'image_isoclass': '80.31', 'image_order': 80, 'indicator': 0, 'kernel': '80.a1.a1', 'kernel_order': 1, 'label': '80.31.4c2', 'nt': [20, 2], 'q_character': None, 'values': [[[4, 0]], [[-4, 0]], [[0, 0]], [[0, 0]], [[0, 0]], [[0, 0]], [[0, 0]], [[0, 0]], [[0, 0]], [[0, 0]], [[-1, 0]], [[1, 0]], [[2, 3], [-1, 5], [2, 7]], [[-2, 3], [1, 5], [-2, 7]]]}
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gps_subgroup_search • Show schema
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id: 68657262
{'Agroup': False, 'Zgroup': False, 'abelian': False, 'ambient': '80.31', 'ambient_counter': 31, 'ambient_order': 80, 'ambient_tex': 'C_{20}:C_4', 'central': False, 'central_factor': False, 'centralizer_order': 2, 'characteristic': True, 'core_order': 80, 'counter': 1, 'cyclic': False, 'direct': True, 'hall': 10, 'label': '80.31.1.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': '80.31', 'subgroup_hash': 31, 'subgroup_order': 80, 'subgroup_tex': 'C_{20}:C_4', 'supersolvable': True, 'sylow': 0}
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id: 68657273
{'Agroup': True, 'Zgroup': False, 'abelian': False, 'ambient': '80.31', 'ambient_counter': 31, 'ambient_order': 80, 'ambient_tex': 'C_{20}:C_4', 'central': False, 'central_factor': False, 'centralizer_order': 4, 'characteristic': True, 'core_order': 40, 'counter': 2, 'cyclic': False, 'direct': False, 'hall': 0, 'label': '80.31.2.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': [], 'split': False, 'standard_generators': False, 'stem': False, 'subgroup': '40.5', 'subgroup_hash': 5, 'subgroup_order': 40, 'subgroup_tex': 'C_4\\times D_5', 'supersolvable': True, 'sylow': 0}
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id: 68657281
{'Agroup': True, 'Zgroup': False, 'abelian': False, 'ambient': '80.31', 'ambient_counter': 31, 'ambient_order': 80, 'ambient_tex': 'C_{20}:C_4', 'central': False, 'central_factor': False, 'centralizer_order': 2, 'characteristic': False, 'core_order': 40, 'counter': 3, 'cyclic': False, 'direct': False, 'hall': 0, 'label': '80.31.2.b1.a1', 'maximal': True, 'maximal_normal': True, 'metabelian': True, 'metacyclic': None, 'minimal': False, 'minimal_normal': False, 'nilpotent': False, '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': False, 'standard_generators': False, 'stem': False, 'subgroup': '40.12', 'subgroup_hash': 12, 'subgroup_order': 40, 'subgroup_tex': 'C_2\\times F_5', 'supersolvable': True, 'sylow': 0}
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id: 68657282
{'Agroup': True, 'Zgroup': False, 'abelian': False, 'ambient': '80.31', 'ambient_counter': 31, 'ambient_order': 80, 'ambient_tex': 'C_{20}:C_4', 'central': False, 'central_factor': False, 'centralizer_order': 2, 'characteristic': False, 'core_order': 40, 'counter': 4, 'cyclic': False, 'direct': False, 'hall': 0, 'label': '80.31.2.b1.b1', 'maximal': True, 'maximal_normal': True, 'metabelian': True, 'metacyclic': None, 'minimal': False, 'minimal_normal': False, 'nilpotent': False, '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': False, 'standard_generators': False, 'stem': False, 'subgroup': '40.12', 'subgroup_hash': 12, 'subgroup_order': 40, 'subgroup_tex': 'C_2\\times F_5', 'supersolvable': True, 'sylow': 0}
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id: 68657283
{'Agroup': True, 'Zgroup': False, 'abelian': False, 'ambient': '80.31', 'ambient_counter': 31, 'ambient_order': 80, 'ambient_tex': 'C_{20}:C_4', 'central': False, 'central_factor': False, 'centralizer_order': 4, 'characteristic': True, 'core_order': 20, 'counter': 5, 'cyclic': False, 'direct': False, 'hall': 0, 'label': '80.31.4.a1.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': None, 'minimal': False, 'minimal_normal': False, 'nilpotent': False, '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': ['C2'], 'split': False, 'standard_generators': False, 'stem': False, 'subgroup': '20.4', 'subgroup_hash': 4, 'subgroup_order': 20, 'subgroup_tex': 'D_{10}', 'supersolvable': True, 'sylow': 0}
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id: 68657284
{'Agroup': True, 'Zgroup': True, 'abelian': True, 'ambient': '80.31', 'ambient_counter': 31, 'ambient_order': 80, 'ambient_tex': 'C_{20}:C_4', 'central': False, 'central_factor': False, 'centralizer_order': 20, 'characteristic': True, 'core_order': 20, 'counter': 6, 'cyclic': True, 'direct': False, 'hall': 0, 'label': '80.31.4.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': ['F'], 'split': True, 'standard_generators': False, 'stem': False, 'subgroup': '20.2', 'subgroup_hash': 2, 'subgroup_order': 20, 'subgroup_tex': 'C_{20}', 'supersolvable': True, 'sylow': 0}
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id: 68657285
{'Agroup': True, 'Zgroup': True, 'abelian': False, 'ambient': '80.31', 'ambient_counter': 31, 'ambient_order': 80, 'ambient_tex': 'C_{20}:C_4', 'central': False, 'central_factor': False, 'centralizer_order': 4, 'characteristic': True, 'core_order': 20, 'counter': 7, 'cyclic': False, 'direct': False, 'hall': 0, 'label': '80.31.4.c1.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': False, 'minimal_normal': False, 'nilpotent': False, 'normal': True, 'old_label': '4.c1.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': '20.1', 'subgroup_hash': 1, 'subgroup_order': 20, 'subgroup_tex': 'C_5:C_4', 'supersolvable': True, 'sylow': 0}
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id: 68657286
{'Agroup': True, 'Zgroup': True, 'abelian': False, 'ambient': '80.31', 'ambient_counter': 31, 'ambient_order': 80, 'ambient_tex': 'C_{20}:C_4', 'central': False, 'central_factor': False, 'centralizer_order': 2, 'characteristic': False, 'core_order': 10, 'counter': 8, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '80.31.4.d1.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': False, 'minimal_normal': False, 'nilpotent': False, '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': '20.3', 'subgroup_hash': 3, 'subgroup_order': 20, 'subgroup_tex': 'F_5', 'supersolvable': True, 'sylow': 0}
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id: 68657287
{'Agroup': True, 'Zgroup': True, 'abelian': False, 'ambient': '80.31', 'ambient_counter': 31, 'ambient_order': 80, 'ambient_tex': 'C_{20}:C_4', 'central': False, 'central_factor': False, 'centralizer_order': 2, 'characteristic': False, 'core_order': 10, 'counter': 9, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '80.31.4.d1.b1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': False, 'minimal_normal': False, 'nilpotent': False, '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': '20.3', 'subgroup_hash': 3, 'subgroup_order': 20, 'subgroup_tex': 'F_5', 'supersolvable': True, 'sylow': 0}
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id: 68657263
{'Agroup': False, 'Zgroup': False, 'abelian': False, 'ambient': '80.31', 'ambient_counter': 31, 'ambient_order': 80, 'ambient_tex': 'C_{20}:C_4', 'central': False, 'central_factor': False, 'centralizer_order': 4, 'characteristic': False, 'core_order': 4, 'counter': 10, 'cyclic': False, 'direct': None, 'hall': 2, 'label': '80.31.5.a1.a1', 'maximal': True, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': False, 'old_label': '5.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': 5, 'quotient_simple': None, 'quotient_solvable': None, 'quotient_supersolvable': None, 'quotient_tex': None, 'simple': False, 'solvable': True, 'special_labels': [], 'split': None, 'standard_generators': False, 'stem': False, 'subgroup': '16.4', 'subgroup_hash': 4, 'subgroup_order': 16, 'subgroup_tex': 'C_4:C_4', 'supersolvable': True, 'sylow': 2}
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id: 68657264
{'Agroup': True, 'Zgroup': True, 'abelian': True, 'ambient': '80.31', 'ambient_counter': 31, 'ambient_order': 80, 'ambient_tex': 'C_{20}:C_4', 'central': False, 'central_factor': False, 'centralizer_order': 20, 'characteristic': True, 'core_order': 10, 'counter': 11, 'cyclic': True, 'direct': False, 'hall': 0, 'label': '80.31.8.a1.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': True, 'old_label': '8.a1.a1', 'outer_equivalence': False, 'perfect': False, 'proper': True, 'quotient': '8.2', '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': False, 'solvable': True, 'special_labels': ['D', 'S', 'L1', 'D1'], 'split': False, 'standard_generators': False, 'stem': False, 'subgroup': '10.2', 'subgroup_hash': 2, 'subgroup_order': 10, 'subgroup_tex': 'C_{10}', 'supersolvable': True, 'sylow': 0}
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id: 68657265
{'Agroup': True, 'Zgroup': True, 'abelian': False, 'ambient': '80.31', 'ambient_counter': 31, 'ambient_order': 80, 'ambient_tex': 'C_{20}:C_4', 'central': False, 'central_factor': False, 'centralizer_order': 4, 'characteristic': True, 'core_order': 10, 'counter': 12, 'cyclic': False, 'direct': False, 'hall': 0, 'label': '80.31.8.b1.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': False, 'minimal_normal': False, 'nilpotent': False, 'normal': True, 'old_label': '8.b1.a1', 'outer_equivalence': False, 'perfect': False, 'proper': True, 'quotient': '8.4', 'quotient_Agroup': False, 'quotient_abelian': False, 'quotient_cyclic': False, 'quotient_hash': 4, 'quotient_metabelian': True, 'quotient_nilpotent': True, 'quotient_order': 8, 'quotient_simple': False, 'quotient_solvable': True, 'quotient_supersolvable': True, 'quotient_tex': 'Q_8', 'simple': False, 'solvable': True, 'special_labels': [], 'split': False, 'standard_generators': False, 'stem': False, 'subgroup': '10.1', 'subgroup_hash': 1, 'subgroup_order': 10, 'subgroup_tex': 'D_5', 'supersolvable': True, 'sylow': 0}
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id: 68657266
{'Agroup': True, 'Zgroup': True, 'abelian': False, 'ambient': '80.31', 'ambient_counter': 31, 'ambient_order': 80, 'ambient_tex': 'C_{20}:C_4', 'central': False, 'central_factor': False, 'centralizer_order': 4, 'characteristic': True, 'core_order': 10, 'counter': 13, 'cyclic': False, 'direct': False, 'hall': 0, 'label': '80.31.8.c1.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': False, 'minimal_normal': False, 'nilpotent': False, 'normal': True, 'old_label': '8.c1.a1', 'outer_equivalence': False, 'perfect': False, 'proper': True, 'quotient': '8.3', 'quotient_Agroup': False, 'quotient_abelian': False, 'quotient_cyclic': False, 'quotient_hash': 3, 'quotient_metabelian': True, 'quotient_nilpotent': True, 'quotient_order': 8, 'quotient_simple': False, 'quotient_solvable': True, 'quotient_supersolvable': True, 'quotient_tex': 'D_4', 'simple': False, 'solvable': True, 'special_labels': ['C3'], 'split': False, 'standard_generators': False, 'stem': False, 'subgroup': '10.1', 'subgroup_hash': 1, 'subgroup_order': 10, 'subgroup_tex': 'D_5', 'supersolvable': True, 'sylow': 0}
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id: 68657267
{'Agroup': True, 'Zgroup': False, 'abelian': True, 'ambient': '80.31', 'ambient_counter': 31, 'ambient_order': 80, 'ambient_tex': 'C_{20}:C_4', 'central': False, 'central_factor': False, 'centralizer_order': 8, 'characteristic': False, 'core_order': 4, 'counter': 14, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '80.31.10.a1.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': False, 'old_label': '10.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': 10, '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': 0}
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id: 68657268
{'Agroup': True, 'Zgroup': False, 'abelian': True, 'ambient': '80.31', 'ambient_counter': 31, 'ambient_order': 80, 'ambient_tex': 'C_{20}:C_4', 'central': False, 'central_factor': False, 'centralizer_order': 8, 'characteristic': False, 'core_order': 2, 'counter': 15, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '80.31.10.b1.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': False, 'old_label': '10.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': 10, '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': 0}
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id: 68657269
{'Agroup': True, 'Zgroup': False, 'abelian': True, 'ambient': '80.31', 'ambient_counter': 31, 'ambient_order': 80, 'ambient_tex': 'C_{20}:C_4', 'central': False, 'central_factor': False, 'centralizer_order': 8, 'characteristic': False, 'core_order': 2, 'counter': 16, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '80.31.10.b1.b1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': False, 'old_label': '10.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': 10, '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': 0}
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id: 68657270
{'Agroup': True, 'Zgroup': True, 'abelian': True, 'ambient': '80.31', 'ambient_counter': 31, 'ambient_order': 80, 'ambient_tex': 'C_{20}:C_4', 'central': False, 'central_factor': False, 'centralizer_order': 20, 'characteristic': True, 'core_order': 5, 'counter': 17, 'cyclic': True, 'direct': False, 'hall': 5, 'label': '80.31.16.a1.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': True, 'minimal_normal': True, 'nilpotent': True, 'normal': True, 'old_label': '16.a1.a1', 'outer_equivalence': False, 'perfect': False, 'proper': True, 'quotient': '16.4', 'quotient_Agroup': False, 'quotient_abelian': False, 'quotient_cyclic': False, 'quotient_hash': 4, 'quotient_metabelian': True, 'quotient_nilpotent': True, 'quotient_order': 16, 'quotient_simple': False, 'quotient_solvable': True, 'quotient_supersolvable': True, 'quotient_tex': 'C_4:C_4', 'simple': True, 'solvable': True, 'special_labels': ['L2', 'C4'], 'split': True, 'standard_generators': False, 'stem': False, 'subgroup': '5.1', 'subgroup_hash': 1, 'subgroup_order': 5, 'subgroup_tex': 'C_5', 'supersolvable': True, 'sylow': 5}
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id: 68657271
{'Agroup': True, 'Zgroup': True, 'abelian': True, 'ambient': '80.31', 'ambient_counter': 31, 'ambient_order': 80, 'ambient_tex': 'C_{20}:C_4', 'central': False, 'central_factor': False, 'centralizer_order': 40, 'characteristic': True, 'core_order': 4, 'counter': 18, 'cyclic': True, 'direct': False, 'hall': 0, 'label': '80.31.20.a1.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': True, 'old_label': '20.a1.a1', 'outer_equivalence': False, 'perfect': False, 'proper': True, 'quotient': '20.3', 'quotient_Agroup': True, 'quotient_abelian': False, 'quotient_cyclic': False, 'quotient_hash': 3, 'quotient_metabelian': True, 'quotient_nilpotent': False, 'quotient_order': 20, 'quotient_simple': False, 'quotient_solvable': True, 'quotient_supersolvable': True, 'quotient_tex': 'F_5', 'simple': False, 'solvable': True, 'special_labels': ['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: 68657272
{'Agroup': True, 'Zgroup': False, 'abelian': True, 'ambient': '80.31', 'ambient_counter': 31, 'ambient_order': 80, 'ambient_tex': 'C_{20}:C_4', 'central': False, 'central_factor': False, 'centralizer_order': 16, 'characteristic': False, 'core_order': 2, 'counter': 19, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '80.31.20.b1.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': False, 'old_label': '20.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': 20, '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: 68657274
{'Agroup': True, 'Zgroup': True, 'abelian': True, 'ambient': '80.31', 'ambient_counter': 31, 'ambient_order': 80, 'ambient_tex': 'C_{20}:C_4', 'central': False, 'central_factor': False, 'centralizer_order': 8, 'characteristic': False, 'core_order': 2, 'counter': 20, 'cyclic': True, 'direct': None, 'hall': 0, 'label': '80.31.20.c1.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': False, 'old_label': '20.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': 20, '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: 68657275
{'Agroup': True, 'Zgroup': True, 'abelian': True, 'ambient': '80.31', 'ambient_counter': 31, 'ambient_order': 80, 'ambient_tex': 'C_{20}:C_4', 'central': False, 'central_factor': False, 'centralizer_order': 8, 'characteristic': False, 'core_order': 1, 'counter': 21, 'cyclic': True, 'direct': None, 'hall': 0, 'label': '80.31.20.d1.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': False, 'old_label': '20.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': 20, '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: 68657276
{'Agroup': True, 'Zgroup': True, 'abelian': True, 'ambient': '80.31', 'ambient_counter': 31, 'ambient_order': 80, 'ambient_tex': 'C_{20}:C_4', 'central': False, 'central_factor': False, 'centralizer_order': 8, 'characteristic': False, 'core_order': 1, 'counter': 22, 'cyclic': True, 'direct': None, 'hall': 0, 'label': '80.31.20.d1.b1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': False, 'old_label': '20.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': 20, '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: 68657277
{'Agroup': True, 'Zgroup': True, 'abelian': True, 'ambient': '80.31', 'ambient_counter': 31, 'ambient_order': 80, 'ambient_tex': 'C_{20}:C_4', 'central': True, 'central_factor': False, 'centralizer_order': 80, 'characteristic': True, 'core_order': 2, 'counter': 23, 'cyclic': True, 'direct': False, 'hall': 0, 'label': '80.31.40.a1.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': True, 'minimal_normal': True, 'nilpotent': True, 'normal': True, 'old_label': '40.a1.a1', 'outer_equivalence': False, 'perfect': False, 'proper': True, 'quotient': '40.12', 'quotient_Agroup': True, 'quotient_abelian': False, 'quotient_cyclic': False, 'quotient_hash': 12, 'quotient_metabelian': True, 'quotient_nilpotent': False, 'quotient_order': 40, 'quotient_simple': False, 'quotient_solvable': True, 'quotient_supersolvable': True, 'quotient_tex': 'C_2\\times F_5', 'simple': True, 'solvable': True, 'special_labels': ['Z', 'Phi', 'U1'], '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: 68657278
{'Agroup': True, 'Zgroup': True, 'abelian': True, 'ambient': '80.31', 'ambient_counter': 31, 'ambient_order': 80, 'ambient_tex': 'C_{20}:C_4', 'central': False, 'central_factor': False, 'centralizer_order': 16, 'characteristic': False, 'core_order': 1, 'counter': 24, 'cyclic': True, 'direct': None, 'hall': 0, 'label': '80.31.40.b1.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': True, 'minimal_normal': False, 'nilpotent': True, 'normal': False, 'old_label': '40.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': 40, '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: 68657279
{'Agroup': True, 'Zgroup': True, 'abelian': True, 'ambient': '80.31', 'ambient_counter': 31, 'ambient_order': 80, 'ambient_tex': 'C_{20}:C_4', 'central': False, 'central_factor': False, 'centralizer_order': 16, 'characteristic': False, 'core_order': 1, 'counter': 25, 'cyclic': True, 'direct': None, 'hall': 0, 'label': '80.31.40.c1.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': True, 'minimal_normal': False, 'nilpotent': True, 'normal': False, 'old_label': '40.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': 40, '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: 68657280
{'Agroup': True, 'Zgroup': True, 'abelian': True, 'ambient': '80.31', 'ambient_counter': 31, 'ambient_order': 80, 'ambient_tex': 'C_{20}:C_4', 'central': True, 'central_factor': False, 'centralizer_order': 80, 'characteristic': True, 'core_order': 1, 'counter': 26, 'cyclic': True, 'direct': True, 'hall': 1, 'label': '80.31.80.a1.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': True, 'old_label': '80.a1.a1', 'outer_equivalence': False, 'perfect': True, 'proper': False, 'quotient': '80.31', 'quotient_Agroup': False, 'quotient_abelian': False, 'quotient_cyclic': False, 'quotient_hash': 31, 'quotient_metabelian': True, 'quotient_nilpotent': False, 'quotient_order': 80, 'quotient_simple': False, 'quotient_solvable': True, 'quotient_supersolvable': True, 'quotient_tex': 'C_{20}:C_4', 'simple': False, 'solvable': True, 'special_labels': ['PC', 'U2', 'D2', 'C5'], 'split': True, 'standard_generators': False, 'stem': True, 'subgroup': '1.1', 'subgroup_hash': 1, 'subgroup_order': 1, 'subgroup_tex': 'C_1', 'supersolvable': True, 'sylow': 1}
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gps_subgroup_data • Show schema
Hide schema
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id: 68657262
{'ambient': '80.31', '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': '40.a1.a1', 'complements': ['80.a1.a1'], 'conjugacy_class_count': 1, 'contained_in': [], 'contains': ['2.a1.a1', '2.b1.a1', '2.b1.b1', '5.a1.a1'], 'core': '1.a1.a1', 'coset_action_label': None, 'count': 1, 'diagramx': [7012, 4394, 4138, 4470, 4111, 4352, 4913, 4725], 'generators': [1, 2, 40, 20, 16], 'label': '80.31.1.a1.a1', 'mobius_quo': 0, 'mobius_sub': 1, 'normal_closure': '1.a1.a1', 'normal_contained_in': [], 'normal_contains': ['2.a1.a1', '2.b1.b1', '2.b1.a1'], 'normalizer': '1.a1.a1', 'old_label': '1.a1.a1', 'projective_image': '40.12', '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': '40.12'}
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id: 68657263
{'ambient': '80.31', 'aut_centralizer_order': 4, 'aut_label': '5.a1', 'aut_quo_index': None, 'aut_stab_index': 5, 'aut_weyl_group': '8.3', 'aut_weyl_index': 20, 'centralizer': '20.b1.a1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['1.a1.a1'], 'contains': ['10.a1.a1', '10.b1.a1', '10.b1.b1'], 'core': '20.a1.a1', 'coset_action_label': None, 'count': 5, 'diagramx': [9550, -1, 4662, -1, 4111, -1, 5622, -1], 'generators': [1, 20], 'label': '80.31.5.a1.a1', 'mobius_quo': None, 'mobius_sub': -1, 'normal_closure': '1.a1.a1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '5.a1.a1', 'old_label': '5.a1.a1', 'projective_image': '40.12', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '5.a1.a1', 'subgroup_fusion': None, 'weyl_group': '4.2'}
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id: 68657264
{'ambient': '80.31', 'aut_centralizer_order': 40, 'aut_label': '8.a1', 'aut_quo_index': 4, 'aut_stab_index': 1, 'aut_weyl_group': '4.1', 'aut_weyl_index': 40, 'centralizer': '4.b1.a1', 'complements': [], 'conjugacy_class_count': 1, 'contained_in': ['4.a1.a1', '4.b1.a1', '4.c1.a1'], 'contains': ['16.a1.a1', '40.a1.a1'], 'core': '8.a1.a1', 'coset_action_label': None, 'count': 1, 'diagramx': [5527, 6193, 7713, 6732, 7869, 5947, 8022, 7021], 'generators': [40, 16], 'label': '80.31.8.a1.a1', 'mobius_quo': 1, 'mobius_sub': 0, 'normal_closure': '8.a1.a1', 'normal_contained_in': ['4.a1.a1', '4.b1.a1', '4.c1.a1'], 'normal_contains': ['16.a1.a1', '40.a1.a1'], 'normalizer': '1.a1.a1', 'old_label': '8.a1.a1', 'projective_image': '40.12', 'quotient_action_image': '4.1', 'quotient_action_kernel': '2.1', 'quotient_action_kernel_order': 2, 'quotient_fusion': None, 'short_label': '8.a1.a1', 'subgroup_fusion': None, 'weyl_group': '4.1'}
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id: 68657265
{'ambient': '80.31', 'aut_centralizer_order': 8, 'aut_label': '8.b1', 'aut_quo_index': 3, 'aut_stab_index': 1, 'aut_weyl_group': '20.3', 'aut_weyl_index': 8, 'centralizer': '20.a1.a1', 'complements': [], 'conjugacy_class_count': 1, 'contained_in': ['4.a1.a1'], 'contains': ['16.a1.a1', '40.b1.a1'], 'core': '8.b1.a1', 'coset_action_label': None, 'count': 1, 'diagramx': [2288, 3425, 6207, 3222, 3943, 3289, 6132, 3309], 'generators': [10, 16], 'label': '80.31.8.b1.a1', 'mobius_quo': 0, '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': '80.31', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '8.b1.a1', 'subgroup_fusion': None, 'weyl_group': '20.3'}
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id: 68657266
{'ambient': '80.31', 'aut_centralizer_order': 8, 'aut_label': '8.c1', 'aut_quo_index': 1, 'aut_stab_index': 1, 'aut_weyl_group': '20.3', 'aut_weyl_index': 8, 'centralizer': '20.a1.a1', 'complements': [], 'conjugacy_class_count': 1, 'contained_in': ['4.a1.a1', '4.d1.a1', '4.d1.b1'], 'contains': ['16.a1.a1', '40.c1.a1'], 'core': '8.c1.a1', 'coset_action_label': None, 'count': 1, 'diagramx': [450, 935, 2580, 4885, 2735, 897, 3321, 5067], 'generators': [2, 16], 'label': '80.31.8.c1.a1', 'mobius_quo': 0, 'mobius_sub': 0, 'normal_closure': '8.c1.a1', 'normal_contained_in': ['4.a1.a1'], 'normal_contains': ['16.a1.a1'], 'normalizer': '1.a1.a1', 'old_label': '8.c1.a1', 'projective_image': '80.31', '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': '20.3'}
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id: 68657267
{'ambient': '80.31', 'aut_centralizer_order': 16, 'aut_label': '10.a1', 'aut_quo_index': None, 'aut_stab_index': 5, 'aut_weyl_group': '2.1', 'aut_weyl_index': 80, 'centralizer': '10.a1.a1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['2.a1.a1', '5.a1.a1'], 'contains': ['20.a1.a1', '20.b1.a1', '20.c1.a1'], 'core': '20.a1.a1', 'coset_action_label': None, 'count': 5, 'diagramx': [9283, -1, 9630, -1, 5990, -1, 7027, -1], 'generators': [2, 20], 'label': '80.31.10.a1.a1', 'mobius_quo': None, 'mobius_sub': 1, 'normal_closure': '2.a1.a1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '5.a1.a1', 'old_label': '10.a1.a1', 'projective_image': '40.12', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '10.a1.a1', 'subgroup_fusion': None, 'weyl_group': '2.1'}
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id: 68657268
{'ambient': '80.31', 'aut_centralizer_order': 8, 'aut_label': '10.b1', 'aut_quo_index': None, 'aut_stab_index': 10, 'aut_weyl_group': '2.1', 'aut_weyl_index': 80, 'centralizer': '10.b1.a1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['2.b1.a1', '5.a1.a1'], 'contains': ['20.b1.a1', '20.d1.a1'], 'core': '40.a1.a1', 'coset_action_label': None, 'count': 5, 'diagramx': [3807, -1, 2324, -1, 2198, -1, 2525, -1], 'generators': [1, 40], 'label': '80.31.10.b1.a1', 'mobius_quo': None, 'mobius_sub': 1, 'normal_closure': '2.b1.a1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '5.a1.a1', 'old_label': '10.b1.a1', 'projective_image': '40.12', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '10.b1.a1', 'subgroup_fusion': None, 'weyl_group': '2.1'}
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id: 68657269
{'ambient': '80.31', 'aut_centralizer_order': 8, 'aut_label': '10.b1', 'aut_quo_index': None, 'aut_stab_index': 10, 'aut_weyl_group': '2.1', 'aut_weyl_index': 80, 'centralizer': '10.b1.b1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['2.b1.b1', '5.a1.a1'], 'contains': ['20.b1.a1', '20.d1.b1'], 'core': '40.a1.a1', 'coset_action_label': None, 'count': 5, 'diagramx': [3807, -1, 4547, -1, 2198, -1, 4216, -1], 'generators': [5, 40], 'label': '80.31.10.b1.b1', 'mobius_quo': None, 'mobius_sub': 1, 'normal_closure': '2.b1.b1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '5.a1.a1', 'old_label': '10.b1.b1', 'projective_image': '40.12', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '10.b1.b1', 'subgroup_fusion': None, 'weyl_group': '2.1'}
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id: 68657270
{'ambient': '80.31', 'aut_centralizer_order': 40, 'aut_label': '16.a1', 'aut_quo_index': 4, 'aut_stab_index': 1, 'aut_weyl_group': '4.1', 'aut_weyl_index': 40, 'centralizer': '4.b1.a1', 'complements': [], 'conjugacy_class_count': 1, 'contained_in': ['8.a1.a1', '8.b1.a1', '8.c1.a1'], 'contains': ['80.a1.a1'], 'core': '16.a1.a1', 'coset_action_label': None, 'count': 1, 'diagramx': [902, 3667, 6845, 5047, 3943, 3422, 6132, 5165], 'generators': [16], 'label': '80.31.16.a1.a1', 'mobius_quo': -1, 'mobius_sub': 0, 'normal_closure': '16.a1.a1', 'normal_contained_in': ['8.a1.a1', '8.b1.a1', '8.c1.a1'], 'normal_contains': ['80.a1.a1'], 'normalizer': '1.a1.a1', 'old_label': '16.a1.a1', 'projective_image': '80.31', 'quotient_action_image': '4.1', 'quotient_action_kernel': '4.1', 'quotient_action_kernel_order': 4, 'quotient_fusion': None, 'short_label': '16.a1.a1', 'subgroup_fusion': None, 'weyl_group': '4.1'}
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id: 68657271
{'ambient': '80.31', 'aut_centralizer_order': 80, 'aut_label': '20.a1', 'aut_quo_index': 1, 'aut_stab_index': 1, 'aut_weyl_group': '2.1', 'aut_weyl_index': 80, 'centralizer': '2.a1.a1', 'complements': ['4.d1.a1', '4.d1.b1'], 'conjugacy_class_count': 1, 'contained_in': ['4.b1.a1', '10.a1.a1'], 'contains': ['40.a1.a1'], 'core': '20.a1.a1', 'coset_action_label': None, 'count': 1, 'diagramx': [8481, 8686, 8736, 8395, 6527, 8771, 9017, 9096], 'generators': [20], 'label': '80.31.20.a1.a1', 'mobius_quo': 0, 'mobius_sub': 0, 'normal_closure': '20.a1.a1', 'normal_contained_in': ['4.b1.a1'], 'normal_contains': ['40.a1.a1'], 'normalizer': '1.a1.a1', 'old_label': '20.a1.a1', 'projective_image': '40.12', 'quotient_action_image': '2.1', 'quotient_action_kernel': '10.1', 'quotient_action_kernel_order': 10, 'quotient_fusion': None, 'short_label': '20.a1.a1', 'subgroup_fusion': None, 'weyl_group': '2.1'}
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id: 68657272
{'ambient': '80.31', 'aut_centralizer_order': 32, 'aut_label': '20.b1', 'aut_quo_index': None, 'aut_stab_index': 5, 'aut_weyl_group': '1.1', 'aut_weyl_index': 160, 'centralizer': '5.a1.a1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['4.a1.a1', '10.a1.a1', '10.b1.a1', '10.b1.b1'], 'contains': ['40.a1.a1', '40.b1.a1', '40.c1.a1'], 'core': '40.a1.a1', 'coset_action_label': None, 'count': 5, 'diagramx': [4542, -1, 5377, -1, 5235, -1, 5174, -1], 'generators': [2, 40], 'label': '80.31.20.b1.a1', 'mobius_quo': None, 'mobius_sub': -2, 'normal_closure': '4.a1.a1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '5.a1.a1', 'old_label': '20.b1.a1', 'projective_image': '40.12', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '20.b1.a1', 'subgroup_fusion': None, 'weyl_group': '1.1'}
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id: 68657273
{'ambient': '80.31', 'aut_centralizer_order': 4, 'aut_label': '2.a1', 'aut_quo_index': 1, 'aut_stab_index': 1, 'aut_weyl_group': '40.12', 'aut_weyl_index': 4, 'centralizer': '20.a1.a1', 'complements': [], 'conjugacy_class_count': 1, 'contained_in': ['1.a1.a1'], 'contains': ['4.a1.a1', '4.b1.a1', '4.c1.a1', '10.a1.a1'], 'core': '2.a1.a1', 'coset_action_label': None, 'count': 1, 'diagramx': [7012, 6747, 7548, 7610, 6040, 6611, 7052, 8046], 'generators': [2, 40, 16, 20], 'label': '80.31.2.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.a1', '4.c1.a1'], 'normalizer': '1.a1.a1', 'old_label': '2.a1.a1', 'projective_image': '40.12', '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': '20.3'}
-
id: 68657274
{'ambient': '80.31', 'aut_centralizer_order': 16, 'aut_label': '20.c1', 'aut_quo_index': None, 'aut_stab_index': 5, 'aut_weyl_group': '2.1', 'aut_weyl_index': 80, 'centralizer': '10.a1.a1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['4.c1.a1', '10.a1.a1'], 'contains': ['40.a1.a1'], 'core': '40.a1.a1', 'coset_action_label': None, 'count': 5, 'diagramx': [6812, -1, 9655, -1, 9211, -1, 7027, -1], 'generators': [38], 'label': '80.31.20.c1.a1', 'mobius_quo': None, 'mobius_sub': 0, 'normal_closure': '4.c1.a1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '5.a1.a1', 'old_label': '20.c1.a1', 'projective_image': '40.12', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '20.c1.a1', 'subgroup_fusion': None, 'weyl_group': '2.1'}
-
id: 68657275
{'ambient': '80.31', 'aut_centralizer_order': 8, 'aut_label': '20.d1', 'aut_quo_index': None, 'aut_stab_index': 20, 'aut_weyl_group': '1.1', 'aut_weyl_index': 160, 'centralizer': '10.b1.a1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['4.d1.a1', '10.b1.a1'], 'contains': ['40.c1.a1'], 'core': '80.a1.a1', 'coset_action_label': '20T18', 'count': 10, 'diagramx': [1303, -1, 2324, -1, 1527, -1, 2425, -1], 'generators': [1], 'label': '80.31.20.d1.a1', 'mobius_quo': None, 'mobius_sub': 0, 'normal_closure': '2.b1.a1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '10.b1.a1', 'old_label': '20.d1.a1', 'projective_image': '80.31', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '20.d1.a1', 'subgroup_fusion': None, 'weyl_group': '1.1'}
-
id: 68657276
{'ambient': '80.31', 'aut_centralizer_order': 8, 'aut_label': '20.d1', 'aut_quo_index': None, 'aut_stab_index': 20, 'aut_weyl_group': '1.1', 'aut_weyl_index': 160, 'centralizer': '10.b1.b1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['4.d1.b1', '10.b1.b1'], 'contains': ['40.c1.a1'], 'core': '80.a1.a1', 'coset_action_label': '20T18', 'count': 10, 'diagramx': [1303, -1, 3819, -1, 1527, -1, 4216, -1], 'generators': [5], 'label': '80.31.20.d1.b1', 'mobius_quo': None, 'mobius_sub': 0, 'normal_closure': '2.b1.b1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '10.b1.b1', 'old_label': '20.d1.b1', 'projective_image': '80.31', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '20.d1.b1', 'subgroup_fusion': None, 'weyl_group': '1.1'}
-
id: 68657277
{'ambient': '80.31', 'aut_centralizer_order': 160, 'aut_label': '40.a1', 'aut_quo_index': 1, 'aut_stab_index': 1, 'aut_weyl_group': '1.1', 'aut_weyl_index': 160, 'centralizer': '1.a1.a1', 'complements': [], 'conjugacy_class_count': 1, 'contained_in': ['8.a1.a1', '20.a1.a1', '20.b1.a1', '20.c1.a1'], 'contains': ['80.a1.a1'], 'core': '40.a1.a1', 'coset_action_label': None, 'count': 1, 'diagramx': [5894, 6193, 8034, 6732, 6527, 5947, 7027, 7021], 'generators': [40], 'label': '80.31.40.a1.a1', 'mobius_quo': -1, 'mobius_sub': 0, 'normal_closure': '40.a1.a1', 'normal_contained_in': ['8.a1.a1', '20.a1.a1'], 'normal_contains': ['80.a1.a1'], 'normalizer': '1.a1.a1', 'old_label': '40.a1.a1', 'projective_image': '40.12', 'quotient_action_image': '1.1', 'quotient_action_kernel': '40.12', 'quotient_action_kernel_order': 40, 'quotient_fusion': None, 'short_label': '40.a1.a1', 'subgroup_fusion': None, 'weyl_group': '1.1'}
-
id: 68657278
{'ambient': '80.31', 'aut_centralizer_order': 32, 'aut_label': '40.b1', 'aut_quo_index': None, 'aut_stab_index': 5, 'aut_weyl_group': '1.1', 'aut_weyl_index': 160, 'centralizer': '5.a1.a1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['8.b1.a1', '20.b1.a1'], 'contains': ['80.a1.a1'], 'core': '80.a1.a1', 'coset_action_label': '40T52', 'count': 5, 'diagramx': [2504, -1, 6143, -1, 5151, -1, 5236, -1], 'generators': [10], 'label': '80.31.40.b1.a1', 'mobius_quo': None, 'mobius_sub': 0, 'normal_closure': '8.b1.a1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '5.a1.a1', 'old_label': '40.b1.a1', 'projective_image': '80.31', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '40.b1.a1', 'subgroup_fusion': None, 'weyl_group': '1.1'}
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id: 68657279
{'ambient': '80.31', 'aut_centralizer_order': 32, 'aut_label': '40.c1', 'aut_quo_index': None, 'aut_stab_index': 5, 'aut_weyl_group': '1.1', 'aut_weyl_index': 160, 'centralizer': '5.a1.a1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['8.c1.a1', '20.b1.a1', '20.d1.a1', '20.d1.b1'], 'contains': ['80.a1.a1'], 'core': '80.a1.a1', 'coset_action_label': '40T54', 'count': 5, 'diagramx': [1303, -1, 3819, -1, 2735, -1, 4216, -1], 'generators': [2], 'label': '80.31.40.c1.a1', 'mobius_quo': None, 'mobius_sub': 0, 'normal_closure': '8.c1.a1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '5.a1.a1', 'old_label': '40.c1.a1', 'projective_image': '80.31', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '40.c1.a1', 'subgroup_fusion': None, 'weyl_group': '1.1'}
-
id: 68657280
{'ambient': '80.31', 'aut_centralizer_order': None, 'aut_label': '80.a1', 'aut_quo_index': None, 'aut_stab_index': None, 'aut_weyl_group': None, 'aut_weyl_index': None, 'centralizer': '1.a1.a1', 'complements': ['1.a1.a1'], 'conjugacy_class_count': 1, 'contained_in': ['16.a1.a1', '40.a1.a1', '40.b1.a1', '40.c1.a1'], 'contains': [], 'core': '80.a1.a1', 'coset_action_label': None, 'count': 1, 'diagramx': [1904, 5053, 6500, 5970, 4547, 4618, 5684, 6044], 'generators': [], 'label': '80.31.80.a1.a1', 'mobius_quo': 1, 'mobius_sub': 0, 'normal_closure': '80.a1.a1', 'normal_contained_in': ['16.a1.a1', '40.a1.a1'], 'normal_contains': [], 'normalizer': '1.a1.a1', 'old_label': '80.a1.a1', 'projective_image': '80.31', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '80.a1.a1', 'subgroup_fusion': None, 'weyl_group': '1.1'}
-
id: 68657281
{'ambient': '80.31', 'aut_centralizer_order': 2, 'aut_label': '2.b1', 'aut_quo_index': 1, 'aut_stab_index': 2, 'aut_weyl_group': '40.12', 'aut_weyl_index': 4, 'centralizer': '40.a1.a1', 'complements': [], 'conjugacy_class_count': 1, 'contained_in': ['1.a1.a1'], 'contains': ['4.a1.a1', '4.d1.a1', '10.b1.a1'], 'core': '2.b1.a1', 'coset_action_label': None, 'count': 1, 'diagramx': [3773, 2041, 1405, 4470, 2198, 2093, 2525, 4725], 'generators': [1, 2, 16, 40], 'label': '80.31.2.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': '40.12', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '2.b1.a1', 'subgroup_fusion': None, 'weyl_group': '40.12'}
-
id: 68657282
{'ambient': '80.31', 'aut_centralizer_order': 2, 'aut_label': '2.b1', 'aut_quo_index': 1, 'aut_stab_index': 2, 'aut_weyl_group': '40.12', 'aut_weyl_index': 4, 'centralizer': '40.a1.a1', 'complements': [], 'conjugacy_class_count': 1, 'contained_in': ['1.a1.a1'], 'contains': ['4.a1.a1', '4.d1.b1', '10.b1.b1'], 'core': '2.b1.b1', 'coset_action_label': None, 'count': 1, 'diagramx': [3773, 2041, 3575, 1351, 2198, 2093, 4204, 1429], 'generators': [5, 18, 16, 40], 'label': '80.31.2.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': '40.12', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '2.b1.b1', 'subgroup_fusion': None, 'weyl_group': '40.12'}
-
id: 68657283
{'ambient': '80.31', 'aut_centralizer_order': 8, 'aut_label': '4.a1', 'aut_quo_index': 3, 'aut_stab_index': 1, 'aut_weyl_group': '20.3', 'aut_weyl_index': 8, 'centralizer': '20.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.c1.a1', '20.b1.a1'], 'core': '4.a1.a1', 'coset_action_label': None, 'count': 1, 'diagramx': [4341, 2802, 5505, 4470, 4094, 2824, 5622, 4725], 'generators': [2, 16, 40], 'label': '80.31.4.a1.a1', 'mobius_quo': 0, 'mobius_sub': 2, 'normal_closure': '4.a1.a1', 'normal_contained_in': ['2.a1.a1', '2.b1.b1', '2.b1.a1'], 'normal_contains': ['8.a1.a1', '8.b1.a1', '8.c1.a1'], 'normalizer': '1.a1.a1', 'old_label': '4.a1.a1', 'projective_image': '40.12', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '4.a1.a1', 'subgroup_fusion': None, 'weyl_group': '20.3'}
-
id: 68657284
{'ambient': '80.31', 'aut_centralizer_order': 20, 'aut_label': '4.b1', 'aut_quo_index': 2, 'aut_stab_index': 1, 'aut_weyl_group': '8.2', 'aut_weyl_index': 20, 'centralizer': '4.b1.a1', 'complements': ['20.d1.a1', '20.d1.b1'], 'conjugacy_class_count': 1, 'contained_in': ['2.a1.a1'], 'contains': ['8.a1.a1', '20.a1.a1'], 'core': '4.b1.a1', 'coset_action_label': None, 'count': 1, 'diagramx': [7396, 9065, 7548, 8649, 7869, 8837, 9565, 9145], 'generators': [20, 40, 16], 'label': '80.31.4.b1.a1', 'mobius_quo': 0, 'mobius_sub': 0, 'normal_closure': '4.b1.a1', 'normal_contained_in': ['2.a1.a1'], 'normal_contains': ['8.a1.a1', '20.a1.a1'], 'normalizer': '1.a1.a1', 'old_label': '4.b1.a1', 'projective_image': '40.12', 'quotient_action_image': '4.1', 'quotient_action_kernel': '1.1', 'quotient_action_kernel_order': 1, 'quotient_fusion': None, 'short_label': '4.b1.a1', 'subgroup_fusion': None, 'weyl_group': '4.1'}
-
id: 68657285
{'ambient': '80.31', 'aut_centralizer_order': 4, 'aut_label': '4.c1', 'aut_quo_index': 1, 'aut_stab_index': 1, 'aut_weyl_group': '40.12', 'aut_weyl_index': 4, 'centralizer': '20.a1.a1', 'complements': ['20.d1.a1', '20.d1.b1'], 'conjugacy_class_count': 1, 'contained_in': ['2.a1.a1'], 'contains': ['8.a1.a1', '20.c1.a1'], 'core': '4.c1.a1', 'coset_action_label': None, 'count': 1, 'diagramx': [5894, 5952, 8697, 6572, 9379, 5847, 8445, 6948], 'generators': [38, 16, 40], 'label': '80.31.4.c1.a1', 'mobius_quo': 0, 'mobius_sub': 0, 'normal_closure': '4.c1.a1', 'normal_contained_in': ['2.a1.a1'], 'normal_contains': ['8.a1.a1'], 'normalizer': '1.a1.a1', 'old_label': '4.c1.a1', 'projective_image': '40.12', 'quotient_action_image': '4.1', 'quotient_action_kernel': '1.1', 'quotient_action_kernel_order': 1, 'quotient_fusion': None, 'short_label': '4.c1.a1', 'subgroup_fusion': None, 'weyl_group': '20.3'}
-
id: 68657286
{'ambient': '80.31', 'aut_centralizer_order': 2, 'aut_label': '4.d1', 'aut_quo_index': None, 'aut_stab_index': 4, 'aut_weyl_group': '20.3', 'aut_weyl_index': 8, 'centralizer': '40.a1.a1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['2.b1.a1'], 'contains': ['8.c1.a1', '20.d1.a1'], 'core': '8.c1.a1', 'coset_action_label': None, 'count': 2, 'diagramx': [450, -1, 345, -1, 453, -1, 336, -1], 'generators': [1, 16, 2], 'label': '80.31.4.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': '80.31', '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': '20.3'}
-
id: 68657287
{'ambient': '80.31', 'aut_centralizer_order': 2, 'aut_label': '4.d1', 'aut_quo_index': None, 'aut_stab_index': 4, 'aut_weyl_group': '20.3', 'aut_weyl_index': 8, 'centralizer': '40.a1.a1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['2.b1.b1'], 'contains': ['8.c1.a1', '20.d1.b1'], 'core': '8.c1.a1', 'coset_action_label': None, 'count': 2, 'diagramx': [450, -1, 2580, -1, 453, -1, 1231, -1], 'generators': [5, 16, 18], 'label': '80.31.4.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': '80.31', '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': '20.3'}
