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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': 20, 'aut_gen_orders': [2, 4, 10], 'aut_gens': [[1, 2], [21, 2], [1, 26], [17, 22]], 'aut_group': '80.50', 'aut_hash': 50, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 80, 'aut_permdeg': 9, 'aut_perms': [7, 15120, 97336], 'aut_phi_ratio': 5.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 5, 2, 1], [4, 1, 2, 1], [4, 5, 2, 1], [5, 2, 2, 1], [10, 2, 2, 1], [20, 2, 4, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_2^2\\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': '20.3', 'autcentquo_hash': 3, 'autcentquo_nilpotent': False, 'autcentquo_order': 20, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'F_5', 'cc_stats': [[1, 1, 1], [2, 1, 1], [2, 5, 2], [4, 1, 2], [4, 5, 2], [5, 2, 2], [10, 2, 2], [20, 2, 4]], 'center_label': '4.1', 'center_order': 4, 'central_product': True, 'central_quotient': '10.1', 'commutator_count': 1, 'commutator_label': '5.1', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '5.1'], 'composition_length': 4, 'conjugacy_classes_known': True, 'counter': 5, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['10.1', 1], ['4.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 5, 1, 2], [4, 1, 2, 1], [4, 5, 2, 1], [5, 2, 2, 1], [10, 2, 2, 1], [20, 2, 4, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 6, 'exponent': 20, 'exponents_of_order': [3, 1], 'factors_of_aut_order': [2, 5], 'factors_of_order': [2, 5], 'faithful_reps': [[2, 0, 4]], 'familial': False, 'frattini_label': '2.1', 'frattini_quotient': '20.4', 'hash': 5, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 10, 'inner_gen_orders': [2, 5], 'inner_gens': [[1, 18], [25, 2]], 'inner_hash': 1, 'inner_nilpotent': False, 'inner_order': 10, 'inner_split': False, 'inner_tex': 'D_5', 'inner_used': [1, 2], 'irrC_degree': 2, 'irrQ_degree': 8, 'irrQ_dim': 8, 'irrR_degree': 4, 'irrep_stats': [[1, 8], [2, 8]], 'label': '40.5', 'linC_count': 4, 'linC_degree': 2, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 6, 'linQ_degree_count': 4, 'linQ_dim': 6, 'linQ_dim_count': 4, 'linR_count': 10, 'linR_degree': 4, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C4*D5', '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': 16, 'number_divisions': 9, 'number_normal_subgroups': 11, 'number_subgroup_autclasses': 14, 'number_subgroup_classes': 16, 'number_subgroups': 36, 'old_label': None, 'order': 40, 'order_factorization_type': 31, 'order_stats': [[1, 1], [2, 11], [4, 12], [5, 4], [10, 4], [20, 8]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 2, 'outer_gen_orders': [2, 2, 2], 'outer_gen_pows': [1, 0, 0], 'outer_gens': [[1, 34], [1, 22], [21, 2]], 'outer_group': '8.5', 'outer_hash': 5, 'outer_nilpotent': True, 'outer_order': 8, 'outer_permdeg': 6, 'outer_perms': [1, 6, 120], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^3', '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, 2], [4, 2], [8, 1]], 'representations': {'PC': {'code': 1040612736515, 'gens': [1, 2], 'pres': [4, -2, -2, -2, -5, 145, 21, 434, 34, 515]}, 'GLZ': {'b': 3, 'd': 6, 'gens': [125101743041058377, 58438765960015333]}, 'GLFp': {'d': 2, 'p': 5, 'gens': [131, 377, 501]}, 'Perm': {'d': 9, 'gens': [5167, 17, 7, 50520]}}, '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_5', 'transitive_degree': 20, 'wreath_data': None, 'wreath_product': False}
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gps_conj_classes • Show schema
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id: 110108211
{'aut_label': '1A', 'centralizer': '1.a1.a1', 'counter': 1, 'group_counter': 5, 'group_order': 40, 'label': '1A', 'order': 1, 'powers': [1, 1], 'representative': 0, 'size': 1}
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id: 110108219
{'aut_label': '2A', 'centralizer': '1.a1.a1', 'counter': 2, 'group_counter': 5, 'group_order': 40, 'label': '2A', 'order': 2, 'powers': [1, 2], 'representative': 20, 'size': 1}
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id: 110108220
{'aut_label': '2B', 'centralizer': '5.a1.a1', 'counter': 3, 'group_counter': 5, 'group_order': 40, 'label': '2B', 'order': 2, 'powers': [1, 3], 'representative': 1, 'size': 5}
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id: 110108221
{'aut_label': '2B', 'centralizer': '5.a1.a1', 'counter': 4, 'group_counter': 5, 'group_order': 40, 'label': '2C', 'order': 2, 'powers': [1, 4], 'representative': 5, 'size': 5}
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id: 110108222
{'aut_label': '4A1', 'centralizer': '1.a1.a1', 'counter': 5, 'group_counter': 5, 'group_order': 40, 'label': '4A1', 'order': 4, 'powers': [2, 5], 'representative': 10, 'size': 1}
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id: 110108223
{'aut_label': '4A1', 'centralizer': '1.a1.a1', 'counter': 6, 'group_counter': 5, 'group_order': 40, 'label': '4A-1', 'order': 4, 'powers': [2, 6], 'representative': 30, 'size': 1}
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id: 110108224
{'aut_label': '4B1', 'centralizer': '5.a1.a1', 'counter': 7, 'group_counter': 5, 'group_order': 40, 'label': '4B1', 'order': 4, 'powers': [2, 7], 'representative': 3, 'size': 5}
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id: 110108225
{'aut_label': '4B1', 'centralizer': '5.a1.a1', 'counter': 8, 'group_counter': 5, 'group_order': 40, 'label': '4B-1', 'order': 4, 'powers': [2, 8], 'representative': 7, 'size': 5}
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id: 110108226
{'aut_label': '5A1', 'centralizer': '2.b1.a1', 'counter': 9, 'group_counter': 5, 'group_order': 40, 'label': '5A1', 'order': 5, 'powers': [10, 1], 'representative': 8, 'size': 2}
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id: 110108212
{'aut_label': '5A1', 'centralizer': '2.b1.a1', 'counter': 10, 'group_counter': 5, 'group_order': 40, 'label': '5A2', 'order': 5, 'powers': [9, 1], 'representative': 16, 'size': 2}
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id: 110108213
{'aut_label': '10A1', 'centralizer': '2.b1.a1', 'counter': 11, 'group_counter': 5, 'group_order': 40, 'label': '10A1', 'order': 10, 'powers': [9, 2], 'representative': 4, 'size': 2}
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id: 110108214
{'aut_label': '10A1', 'centralizer': '2.b1.a1', 'counter': 12, 'group_counter': 5, 'group_order': 40, 'label': '10A3', 'order': 10, 'powers': [10, 2], 'representative': 12, 'size': 2}
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id: 110108215
{'aut_label': '20A1', 'centralizer': '2.b1.a1', 'counter': 13, 'group_counter': 5, 'group_order': 40, 'label': '20A1', 'order': 20, 'powers': [11, 5], 'representative': 2, 'size': 2}
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id: 110108216
{'aut_label': '20A1', 'centralizer': '2.b1.a1', 'counter': 14, 'group_counter': 5, 'group_order': 40, 'label': '20A-1', 'order': 20, 'powers': [11, 6], 'representative': 22, 'size': 2}
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id: 110108217
{'aut_label': '20A1', 'centralizer': '2.b1.a1', 'counter': 15, 'group_counter': 5, 'group_order': 40, 'label': '20A3', 'order': 20, 'powers': [12, 6], 'representative': 6, 'size': 2}
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id: 110108218
{'aut_label': '20A1', 'centralizer': '2.b1.a1', 'counter': 16, 'group_counter': 5, 'group_order': 40, 'label': '20A-3', 'order': 20, 'powers': [12, 5], 'representative': 26, 'size': 2}
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gps_qchar • Show schema
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id: 4520626
{'cdim': 1, 'counter': 1, 'faithful': False, 'group': '40.5', 'group_order': 40, 'image': None, 'label': '40.5.1a', 'multiplicity': 1, 'nt': [1, 1], 'qdim': 1, 'qvalues': [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1], 'schur_index': 1}
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id: 4520627
{'cdim': 1, 'counter': 2, 'faithful': False, 'group': '40.5', 'group_order': 40, 'image': None, 'label': '40.5.1b', 'multiplicity': 1, 'nt': [2, 1], 'qdim': 1, 'qvalues': [1, 1, -1, -1, -1, -1, 1, 1, 1, 1, 1, 1, -1, -1, -1, -1], 'schur_index': 1}
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id: 4520628
{'cdim': 1, 'counter': 3, 'faithful': False, 'group': '40.5', 'group_order': 40, 'image': None, 'label': '40.5.1c', 'multiplicity': 1, 'nt': [2, 1], 'qdim': 1, 'qvalues': [1, 1, -1, -1, 1, 1, -1, -1, 1, 1, 1, 1, 1, 1, 1, 1], 'schur_index': 1}
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id: 4520629
{'cdim': 1, 'counter': 4, 'faithful': False, 'group': '40.5', 'group_order': 40, 'image': None, 'label': '40.5.1d', 'multiplicity': 1, 'nt': [2, 1], 'qdim': 1, 'qvalues': [1, 1, 1, 1, -1, -1, -1, -1, 1, 1, 1, 1, -1, -1, -1, -1], 'schur_index': 1}
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id: 4520630
{'cdim': 1, 'counter': 5, 'faithful': False, 'group': '40.5', 'group_order': 40, 'image': None, 'label': '40.5.1e', 'multiplicity': 2, 'nt': [4, 1], 'qdim': 2, 'qvalues': [2, -2, -2, 2, 0, 0, 0, 0, 2, 2, -2, -2, 0, 0, 0, 0], 'schur_index': 1}
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id: 4520631
{'cdim': 1, 'counter': 6, 'faithful': False, 'group': '40.5', 'group_order': 40, 'image': None, 'label': '40.5.1f', 'multiplicity': 2, 'nt': [4, 1], 'qdim': 2, 'qvalues': [2, -2, 2, -2, 0, 0, 0, 0, 2, 2, -2, -2, 0, 0, 0, 0], 'schur_index': 1}
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id: 4520632
{'cdim': 2, 'counter': 7, 'faithful': False, 'group': '40.5', 'group_order': 40, 'image': None, 'label': '40.5.2a', 'multiplicity': 2, 'nt': [5, 2], 'qdim': 4, 'qvalues': [4, 4, 0, 0, 4, 4, 0, 0, -1, -1, -1, -1, -1, -1, -1, -1], 'schur_index': 1}
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id: 4520633
{'cdim': 2, 'counter': 8, 'faithful': False, 'group': '40.5', 'group_order': 40, 'image': None, 'label': '40.5.2b', 'multiplicity': 2, 'nt': [10, 3], 'qdim': 4, 'qvalues': [4, 4, 0, 0, -4, -4, 0, 0, -1, -1, -1, -1, 1, 1, 1, 1], 'schur_index': 1}
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id: 4520634
{'cdim': 2, 'counter': 9, 'faithful': True, 'group': '40.5', 'group_order': 40, 'image': None, 'label': '40.5.2c', 'multiplicity': 4, 'nt': [20, 6], 'qdim': 8, 'qvalues': [8, -8, 0, 0, 0, 0, 0, 0, -2, -2, 2, 2, 0, 0, 0, 0], 'schur_index': 1}
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gps_char • Show schema
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id: 7015670
{'center': '1.a1.a1', 'center_index': 1, 'center_order': 40, 'counter': 1, 'cyclotomic_n': 1, 'dim': 1, 'faithful': False, 'field': [0, 1], 'group': '40.5', 'group_counter': 5, 'group_order': 40, 'image': None, 'image_isoclass': '1.1', 'image_order': 1, 'indicator': 1, 'kernel': '1.a1.a1', 'kernel_order': 40, 'label': '40.5.1a', '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]]]}
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id: 7015678
{'center': '1.a1.a1', 'center_index': 1, 'center_order': 40, 'counter': 2, 'cyclotomic_n': 1, 'dim': 1, 'faithful': False, 'field': [0, 1], 'group': '40.5', 'group_counter': 5, 'group_order': 40, 'image': None, 'image_isoclass': '2.1', 'image_order': 2, 'indicator': 1, 'kernel': '2.c1.a1', 'kernel_order': 20, 'label': '40.5.1b', '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]]]}
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id: 7015679
{'center': '1.a1.a1', 'center_index': 1, 'center_order': 40, 'counter': 3, 'cyclotomic_n': 1, 'dim': 1, 'faithful': False, 'field': [0, 1], 'group': '40.5', 'group_counter': 5, 'group_order': 40, 'image': None, 'image_isoclass': '2.1', 'image_order': 2, 'indicator': 1, 'kernel': '2.b1.a1', 'kernel_order': 20, 'label': '40.5.1c', '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]]]}
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id: 7015680
{'center': '1.a1.a1', 'center_index': 1, 'center_order': 40, 'counter': 4, 'cyclotomic_n': 1, 'dim': 1, 'faithful': False, 'field': [0, 1], 'group': '40.5', 'group_counter': 5, 'group_order': 40, 'image': None, 'image_isoclass': '2.1', 'image_order': 2, 'indicator': 1, 'kernel': '2.a1.a1', 'kernel_order': 20, 'label': '40.5.1d', '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]]]}
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id: 7015681
{'center': '1.a1.a1', 'center_index': 1, 'center_order': 40, 'counter': 5, 'cyclotomic_n': 4, 'dim': 1, 'faithful': False, 'field': [1, 0, 1], 'group': '40.5', 'group_counter': 5, 'group_order': 40, 'image': None, 'image_isoclass': '4.1', 'image_order': 4, 'indicator': 0, 'kernel': '4.b1.b1', 'kernel_order': 10, 'label': '40.5.1e1', '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, 1]], [[1, 1]], [[1, 1]], [[-1, 1]]]}
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id: 7015682
{'center': '1.a1.a1', 'center_index': 1, 'center_order': 40, 'counter': 6, 'cyclotomic_n': 4, 'dim': 1, 'faithful': False, 'field': [1, 0, 1], 'group': '40.5', 'group_counter': 5, 'group_order': 40, 'image': None, 'image_isoclass': '4.1', 'image_order': 4, 'indicator': 0, 'kernel': '4.b1.b1', 'kernel_order': 10, 'label': '40.5.1e2', '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, 1]], [[-1, 1]], [[-1, 1]], [[1, 1]]]}
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id: 7015683
{'center': '1.a1.a1', 'center_index': 1, 'center_order': 40, 'counter': 7, 'cyclotomic_n': 4, 'dim': 1, 'faithful': False, 'field': [1, 0, 1], 'group': '40.5', 'group_counter': 5, 'group_order': 40, 'image': None, 'image_isoclass': '4.1', 'image_order': 4, 'indicator': 0, 'kernel': '4.b1.a1', 'kernel_order': 10, 'label': '40.5.1f1', '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, 1]], [[1, 1]], [[1, 1]], [[-1, 1]]]}
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id: 7015684
{'center': '1.a1.a1', 'center_index': 1, 'center_order': 40, 'counter': 8, 'cyclotomic_n': 4, 'dim': 1, 'faithful': False, 'field': [1, 0, 1], 'group': '40.5', 'group_counter': 5, 'group_order': 40, 'image': None, 'image_isoclass': '4.1', 'image_order': 4, 'indicator': 0, 'kernel': '4.b1.a1', 'kernel_order': 10, 'label': '40.5.1f2', '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, 1]], [[-1, 1]], [[-1, 1]], [[1, 1]]]}
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id: 7015685
{'center': '10.a1.a1', 'center_index': 10, 'center_order': 4, 'counter': 9, 'cyclotomic_n': 5, 'dim': 2, 'faithful': False, 'field': [-1, -1, 1], 'group': '40.5', 'group_counter': 5, 'group_order': 40, 'image': None, 'image_isoclass': '10.1', 'image_order': 10, 'indicator': 1, 'kernel': '10.a1.a1', 'kernel_order': 4, 'label': '40.5.2a1', 'nt': [5, 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, 1], [1, -1]], [[1, 1], [1, -1]], [[1, 2], [1, -2]], [[1, 2], [1, -2]], [[1, 2], [1, -2]], [[1, 1], [1, -1]], [[1, 1], [1, -1]]]}
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id: 7015671
{'center': '10.a1.a1', 'center_index': 10, 'center_order': 4, 'counter': 10, 'cyclotomic_n': 5, 'dim': 2, 'faithful': False, 'field': [-1, -1, 1], 'group': '40.5', 'group_counter': 5, 'group_order': 40, 'image': None, 'image_isoclass': '10.1', 'image_order': 10, 'indicator': 1, 'kernel': '10.a1.a1', 'kernel_order': 4, 'label': '40.5.2a2', 'nt': [5, 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, 2], [1, -2]], [[1, 1], [1, -1]], [[1, 1], [1, -1]], [[1, 1], [1, -1]], [[1, 2], [1, -2]], [[1, 2], [1, -2]]]}
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id: 7015672
{'center': '10.a1.a1', 'center_index': 10, 'center_order': 4, 'counter': 11, 'cyclotomic_n': 5, 'dim': 2, 'faithful': False, 'field': [-1, -1, 1], 'group': '40.5', 'group_counter': 5, 'group_order': 40, 'image': None, 'image_isoclass': '20.4', 'image_order': 20, 'indicator': 1, 'kernel': '20.a1.a1', 'kernel_order': 2, 'label': '40.5.2b1', 'nt': [10, 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, 1], [1, -1]], [[1, 1], [1, -1]], [[1, 2], [1, -2]], [[-1, 2], [-1, -2]], [[-1, 2], [-1, -2]], [[-1, 1], [-1, -1]], [[-1, 1], [-1, -1]]]}
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id: 7015673
{'center': '10.a1.a1', 'center_index': 10, 'center_order': 4, 'counter': 12, 'cyclotomic_n': 5, 'dim': 2, 'faithful': False, 'field': [-1, -1, 1], 'group': '40.5', 'group_counter': 5, 'group_order': 40, 'image': None, 'image_isoclass': '20.4', 'image_order': 20, 'indicator': 1, 'kernel': '20.a1.a1', 'kernel_order': 2, 'label': '40.5.2b2', 'nt': [10, 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, 2], [1, -2]], [[1, 1], [1, -1]], [[-1, 1], [-1, -1]], [[-1, 1], [-1, -1]], [[-1, 2], [-1, -2]], [[-1, 2], [-1, -2]]]}
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id: 7015674
{'center': '10.a1.a1', 'center_index': 10, 'center_order': 4, 'counter': 13, 'cyclotomic_n': 20, 'dim': 2, 'faithful': True, 'field': [1, 0, 3, 0, 1], 'group': '40.5', 'group_counter': 5, 'group_order': 40, 'image': None, 'image_isoclass': '40.5', 'image_order': 40, 'indicator': 0, 'kernel': '40.a1.a1', 'kernel_order': 1, 'label': '40.5.2c1', 'nt': [20, 6], 'q_character': None, 'values': [[[2, 0]], [[-2, 0]], [[0, 0]], [[0, 0]], [[-2, 5]], [[2, 5]], [[0, 0]], [[0, 0]], [[-1, 2], [-1, -2]], [[1, 4], [1, -4]], [[-1, 4], [-1, -4]], [[1, 2], [1, -2]], [[1, 3], [1, 7]], [[-1, 3], [-1, 7]], [[1, 3], [-1, 5], [1, 7]], [[-1, 3], [1, 5], [-1, 7]]]}
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id: 7015675
{'center': '10.a1.a1', 'center_index': 10, 'center_order': 4, 'counter': 14, 'cyclotomic_n': 20, 'dim': 2, 'faithful': True, 'field': [1, 0, 3, 0, 1], 'group': '40.5', 'group_counter': 5, 'group_order': 40, 'image': None, 'image_isoclass': '40.5', 'image_order': 40, 'indicator': 0, 'kernel': '40.a1.a1', 'kernel_order': 1, 'label': '40.5.2c2', 'nt': [20, 4], 'q_character': None, 'values': [[[2, 0]], [[-2, 0]], [[0, 0]], [[0, 0]], [[2, 5]], [[-2, 5]], [[0, 0]], [[0, 0]], [[-1, 2], [-1, -2]], [[1, 4], [1, -4]], [[-1, 4], [-1, -4]], [[1, 2], [1, -2]], [[-1, 3], [-1, 7]], [[1, 3], [1, 7]], [[-1, 3], [1, 5], [-1, 7]], [[1, 3], [-1, 5], [1, 7]]]}
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id: 7015676
{'center': '10.a1.a1', 'center_index': 10, 'center_order': 4, 'counter': 15, 'cyclotomic_n': 20, 'dim': 2, 'faithful': True, 'field': [1, 0, 3, 0, 1], 'group': '40.5', 'group_counter': 5, 'group_order': 40, 'image': None, 'image_isoclass': '40.5', 'image_order': 40, 'indicator': 0, 'kernel': '40.a1.a1', 'kernel_order': 1, 'label': '40.5.2c3', 'nt': [20, 6], 'q_character': None, 'values': [[[2, 0]], [[-2, 0]], [[0, 0]], [[0, 0]], [[-2, 5]], [[2, 5]], [[0, 0]], [[0, 0]], [[1, 4], [1, -4]], [[-1, 2], [-1, -2]], [[1, 2], [1, -2]], [[-1, 4], [-1, -4]], [[-1, 3], [1, 5], [-1, 7]], [[1, 3], [-1, 5], [1, 7]], [[-1, 3], [-1, 7]], [[1, 3], [1, 7]]]}
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id: 7015677
{'center': '10.a1.a1', 'center_index': 10, 'center_order': 4, 'counter': 16, 'cyclotomic_n': 20, 'dim': 2, 'faithful': True, 'field': [1, 0, 3, 0, 1], 'group': '40.5', 'group_counter': 5, 'group_order': 40, 'image': None, 'image_isoclass': '40.5', 'image_order': 40, 'indicator': 0, 'kernel': '40.a1.a1', 'kernel_order': 1, 'label': '40.5.2c4', 'nt': [20, 4], 'q_character': None, 'values': [[[2, 0]], [[-2, 0]], [[0, 0]], [[0, 0]], [[2, 5]], [[-2, 5]], [[0, 0]], [[0, 0]], [[1, 4], [1, -4]], [[-1, 2], [-1, -2]], [[1, 2], [1, -2]], [[-1, 4], [-1, -4]], [[1, 3], [-1, 5], [1, 7]], [[-1, 3], [1, 5], [-1, 7]], [[1, 3], [1, 7]], [[-1, 3], [-1, 7]]]}
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gps_subgroup_search • Show schema
Hide schema
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id: 66334603
{'Agroup': True, 'Zgroup': False, 'abelian': False, 'ambient': '40.5', 'ambient_counter': 5, 'ambient_order': 40, 'ambient_tex': 'C_4\\times D_5', 'central': False, 'central_factor': False, 'centralizer_order': 4, 'characteristic': True, 'core_order': 40, 'counter': 1, 'cyclic': False, 'direct': True, 'hall': 10, 'label': '40.5.1.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': '40.5', 'subgroup_hash': 5, 'subgroup_order': 40, 'subgroup_tex': 'C_4\\times D_5', 'supersolvable': True, 'sylow': 0}
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id: 66334611
{'Agroup': True, 'Zgroup': False, 'abelian': False, 'ambient': '40.5', 'ambient_counter': 5, 'ambient_order': 40, 'ambient_tex': 'C_4\\times D_5', 'central': False, 'central_factor': True, 'centralizer_order': 4, 'characteristic': True, 'core_order': 20, 'counter': 2, 'cyclic': False, 'direct': False, 'hall': 0, 'label': '40.5.2.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': '20.4', 'subgroup_hash': 4, 'subgroup_order': 20, 'subgroup_tex': 'D_{10}', 'supersolvable': True, 'sylow': 0}
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id: 66334612
{'Agroup': True, 'Zgroup': True, 'abelian': True, 'ambient': '40.5', 'ambient_counter': 5, 'ambient_order': 40, 'ambient_tex': 'C_4\\times D_5', 'central': False, 'central_factor': False, 'centralizer_order': 20, 'characteristic': True, 'core_order': 20, 'counter': 3, 'cyclic': True, 'direct': False, 'hall': 0, 'label': '40.5.2.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': '20.2', 'subgroup_hash': 2, 'subgroup_order': 20, 'subgroup_tex': 'C_{20}', 'supersolvable': True, 'sylow': 0}
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id: 66334613
{'Agroup': True, 'Zgroup': True, 'abelian': False, 'ambient': '40.5', 'ambient_counter': 5, 'ambient_order': 40, 'ambient_tex': 'C_4\\times D_5', 'central': False, 'central_factor': True, 'centralizer_order': 4, 'characteristic': True, 'core_order': 20, 'counter': 4, 'cyclic': False, 'direct': False, 'hall': 0, 'label': '40.5.2.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': '20.1', 'subgroup_hash': 1, 'subgroup_order': 20, 'subgroup_tex': 'C_5:C_4', 'supersolvable': True, 'sylow': 0}
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id: 66334614
{'Agroup': True, 'Zgroup': True, 'abelian': True, 'ambient': '40.5', 'ambient_counter': 5, 'ambient_order': 40, 'ambient_tex': 'C_4\\times D_5', 'central': False, 'central_factor': False, 'centralizer_order': 20, 'characteristic': True, 'core_order': 10, 'counter': 5, 'cyclic': True, 'direct': False, 'hall': 0, 'label': '40.5.4.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': '10.2', 'subgroup_hash': 2, 'subgroup_order': 10, 'subgroup_tex': 'C_{10}', 'supersolvable': True, 'sylow': 0}
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id: 66334615
{'Agroup': True, 'Zgroup': True, 'abelian': False, 'ambient': '40.5', 'ambient_counter': 5, 'ambient_order': 40, 'ambient_tex': 'C_4\\times D_5', 'central': False, 'central_factor': True, 'centralizer_order': 4, 'characteristic': False, 'core_order': 10, 'counter': 6, 'cyclic': False, 'direct': True, 'hall': 0, 'label': '40.5.4.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': '10.1', 'subgroup_hash': 1, 'subgroup_order': 10, 'subgroup_tex': 'D_5', 'supersolvable': True, 'sylow': 0}
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id: 66334616
{'Agroup': True, 'Zgroup': True, 'abelian': False, 'ambient': '40.5', 'ambient_counter': 5, 'ambient_order': 40, 'ambient_tex': 'C_4\\times D_5', 'central': False, 'central_factor': True, 'centralizer_order': 4, 'characteristic': False, 'core_order': 10, 'counter': 7, 'cyclic': False, 'direct': True, 'hall': 0, 'label': '40.5.4.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': '10.1', 'subgroup_hash': 1, 'subgroup_order': 10, 'subgroup_tex': 'D_5', 'supersolvable': True, 'sylow': 0}
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id: 66334617
{'Agroup': True, 'Zgroup': False, 'abelian': True, 'ambient': '40.5', 'ambient_counter': 5, 'ambient_order': 40, 'ambient_tex': 'C_4\\times D_5', 'central': False, 'central_factor': False, 'centralizer_order': 8, 'characteristic': False, 'core_order': 4, 'counter': 8, 'cyclic': False, 'direct': None, 'hall': 2, 'label': '40.5.5.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': '8.2', 'subgroup_hash': 2, 'subgroup_order': 8, 'subgroup_tex': 'C_2\\times C_4', 'supersolvable': True, 'sylow': 2}
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id: 66334618
{'Agroup': True, 'Zgroup': True, 'abelian': True, 'ambient': '40.5', 'ambient_counter': 5, 'ambient_order': 40, 'ambient_tex': 'C_4\\times D_5', 'central': False, 'central_factor': False, 'centralizer_order': 20, 'characteristic': True, 'core_order': 5, 'counter': 9, 'cyclic': True, 'direct': False, 'hall': 5, 'label': '40.5.8.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': '5.1', 'subgroup_hash': 1, 'subgroup_order': 5, 'subgroup_tex': 'C_5', 'supersolvable': True, 'sylow': 5}
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id: 66334604
{'Agroup': True, 'Zgroup': True, 'abelian': True, 'ambient': '40.5', 'ambient_counter': 5, 'ambient_order': 40, 'ambient_tex': 'C_4\\times D_5', 'central': True, 'central_factor': False, 'centralizer_order': 40, 'characteristic': True, 'core_order': 4, 'counter': 10, 'cyclic': True, 'direct': True, 'hall': 0, 'label': '40.5.10.a1.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': True, 'old_label': '10.a1.a1', 'outer_equivalence': False, 'perfect': False, 'proper': True, 'quotient': '10.1', 'quotient_Agroup': True, 'quotient_abelian': False, 'quotient_cyclic': False, 'quotient_hash': 1, 'quotient_metabelian': True, 'quotient_nilpotent': False, 'quotient_order': 10, 'quotient_simple': False, 'quotient_solvable': True, 'quotient_supersolvable': True, 'quotient_tex': 'D_5', '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: 66334605
{'Agroup': True, 'Zgroup': False, 'abelian': True, 'ambient': '40.5', 'ambient_counter': 5, 'ambient_order': 40, 'ambient_tex': 'C_4\\times D_5', 'central': False, 'central_factor': False, 'centralizer_order': 8, 'characteristic': False, 'core_order': 2, 'counter': 11, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '40.5.10.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': '4.2', 'subgroup_hash': 2, 'subgroup_order': 4, 'subgroup_tex': 'C_2^2', 'supersolvable': True, 'sylow': 0}
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id: 66334606
{'Agroup': True, 'Zgroup': True, 'abelian': True, 'ambient': '40.5', 'ambient_counter': 5, 'ambient_order': 40, 'ambient_tex': 'C_4\\times D_5', 'central': False, 'central_factor': False, 'centralizer_order': 8, 'characteristic': False, 'core_order': 2, 'counter': 12, 'cyclic': True, 'direct': None, 'hall': 0, 'label': '40.5.10.c1.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': False, 'old_label': '10.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': 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': '4.1', 'subgroup_hash': 1, 'subgroup_order': 4, 'subgroup_tex': 'C_4', 'supersolvable': True, 'sylow': 0}
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id: 66334607
{'Agroup': True, 'Zgroup': True, 'abelian': True, 'ambient': '40.5', 'ambient_counter': 5, 'ambient_order': 40, 'ambient_tex': 'C_4\\times D_5', 'central': True, 'central_factor': False, 'centralizer_order': 40, 'characteristic': True, 'core_order': 2, 'counter': 13, 'cyclic': True, 'direct': False, 'hall': 0, 'label': '40.5.20.a1.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': True, 'minimal_normal': True, 'nilpotent': True, 'normal': True, 'old_label': '20.a1.a1', 'outer_equivalence': False, 'perfect': False, 'proper': True, 'quotient': '20.4', 'quotient_Agroup': True, 'quotient_abelian': False, 'quotient_cyclic': False, 'quotient_hash': 4, 'quotient_metabelian': True, 'quotient_nilpotent': False, 'quotient_order': 20, 'quotient_simple': False, 'quotient_solvable': True, 'quotient_supersolvable': True, 'quotient_tex': 'D_{10}', '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: 66334608
{'Agroup': True, 'Zgroup': True, 'abelian': True, 'ambient': '40.5', 'ambient_counter': 5, 'ambient_order': 40, 'ambient_tex': 'C_4\\times D_5', 'central': False, 'central_factor': False, 'centralizer_order': 8, 'characteristic': False, 'core_order': 1, 'counter': 14, 'cyclic': True, 'direct': None, 'hall': 0, 'label': '40.5.20.b1.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': True, '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': 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: 66334609
{'Agroup': True, 'Zgroup': True, 'abelian': True, 'ambient': '40.5', 'ambient_counter': 5, 'ambient_order': 40, 'ambient_tex': 'C_4\\times D_5', 'central': False, 'central_factor': False, 'centralizer_order': 8, 'characteristic': False, 'core_order': 1, 'counter': 15, 'cyclic': True, 'direct': None, 'hall': 0, 'label': '40.5.20.b1.b1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': True, 'minimal_normal': False, 'nilpotent': True, 'normal': False, 'old_label': '20.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': 20, '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: 66334610
{'Agroup': True, 'Zgroup': True, 'abelian': True, 'ambient': '40.5', 'ambient_counter': 5, 'ambient_order': 40, 'ambient_tex': 'C_4\\times D_5', 'central': True, 'central_factor': False, 'centralizer_order': 40, 'characteristic': True, 'core_order': 1, 'counter': 16, 'cyclic': True, 'direct': True, 'hall': 1, 'label': '40.5.40.a1.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': True, 'old_label': '40.a1.a1', 'outer_equivalence': False, 'perfect': True, 'proper': False, 'quotient': '40.5', 'quotient_Agroup': True, 'quotient_abelian': False, 'quotient_cyclic': False, 'quotient_hash': 5, 'quotient_metabelian': True, 'quotient_nilpotent': False, 'quotient_order': 40, 'quotient_simple': False, 'quotient_solvable': True, 'quotient_supersolvable': True, 'quotient_tex': 'C_4\\times D_5', '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
-
id: 66334603
{'ambient': '40.5', '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': '10.a1.a1', 'complements': ['40.a1.a1'], 'conjugacy_class_count': 1, 'contained_in': [], 'contains': ['2.a1.a1', '2.b1.a1', '2.c1.a1', '5.a1.a1'], 'core': '1.a1.a1', 'coset_action_label': None, 'count': 1, 'diagramx': [8025, 5020, 3581, 5952, 4447, 5030, 4207, 5847], 'generators': [1, 20, 8, 10], 'label': '40.5.1.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': '10.1', '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': '10.1'}
-
id: 66334604
{'ambient': '40.5', 'aut_centralizer_order': 40, 'aut_label': '10.a1', 'aut_quo_index': 1, 'aut_stab_index': 1, 'aut_weyl_group': '2.1', 'aut_weyl_index': 40, 'centralizer': '1.a1.a1', 'complements': ['4.b1.a1', '4.b1.b1'], 'conjugacy_class_count': 1, 'contained_in': ['2.b1.a1', '5.a1.a1'], 'contains': ['20.a1.a1'], 'core': '10.a1.a1', 'coset_action_label': None, 'count': 1, 'diagramx': [7115, 7560, 2947, 8478, 8766, 8409, 3069, 8771], 'generators': [10], 'label': '40.5.10.a1.a1', 'mobius_quo': 0, 'mobius_sub': 5, 'normal_closure': '10.a1.a1', 'normal_contained_in': ['2.b1.a1'], 'normal_contains': ['20.a1.a1'], 'normalizer': '1.a1.a1', 'old_label': '10.a1.a1', 'projective_image': '10.1', 'quotient_action_image': '1.1', 'quotient_action_kernel': '10.1', 'quotient_action_kernel_order': 10, 'quotient_fusion': None, 'short_label': '10.a1.a1', 'subgroup_fusion': None, 'weyl_group': '1.1'}
-
id: 66334605
{'ambient': '40.5', 'aut_centralizer_order': 8, 'aut_label': '10.b1', 'aut_quo_index': None, 'aut_stab_index': 5, 'aut_weyl_group': '2.1', 'aut_weyl_index': 40, 'centralizer': '5.a1.a1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['2.a1.a1', '5.a1.a1'], 'contains': ['20.a1.a1', '20.b1.a1', '20.b1.b1'], 'core': '20.a1.a1', 'coset_action_label': None, 'count': 5, 'diagramx': [3671, -1, 5459, -1, 4936, -1, 6423, -1], 'generators': [1, 20], 'label': '40.5.10.b1.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.b1.a1', 'projective_image': '20.4', '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': '1.1'}
-
id: 66334606
{'ambient': '40.5', 'aut_centralizer_order': 8, 'aut_label': '10.c1', 'aut_quo_index': None, 'aut_stab_index': 5, 'aut_weyl_group': '2.1', 'aut_weyl_index': 40, 'centralizer': '5.a1.a1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['2.c1.a1', '5.a1.a1'], 'contains': ['20.a1.a1'], 'core': '20.a1.a1', 'coset_action_label': None, 'count': 5, 'diagramx': [9316, -1, 1288, -1, 6915, -1, 1606, -1], 'generators': [3], 'label': '40.5.10.c1.a1', 'mobius_quo': None, 'mobius_sub': 1, 'normal_closure': '2.c1.a1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '5.a1.a1', 'old_label': '10.c1.a1', 'projective_image': '20.4', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '10.c1.a1', 'subgroup_fusion': None, 'weyl_group': '1.1'}
-
id: 66334607
{'ambient': '40.5', 'aut_centralizer_order': 80, 'aut_label': '20.a1', 'aut_quo_index': 2, 'aut_stab_index': 1, 'aut_weyl_group': '1.1', 'aut_weyl_index': 80, 'centralizer': '1.a1.a1', 'complements': [], 'conjugacy_class_count': 1, 'contained_in': ['4.a1.a1', '10.a1.a1', '10.b1.a1', '10.c1.a1'], 'contains': ['40.a1.a1'], 'core': '20.a1.a1', 'coset_action_label': None, 'count': 1, 'diagramx': [5190, 5179, 4148, 6193, 5681, 6461, 4695, 5947], 'generators': [20], 'label': '40.5.20.a1.a1', 'mobius_quo': -1, 'mobius_sub': -10, 'normal_closure': '20.a1.a1', 'normal_contained_in': ['4.a1.a1', '10.a1.a1'], 'normal_contains': ['40.a1.a1'], 'normalizer': '1.a1.a1', 'old_label': '20.a1.a1', 'projective_image': '20.4', 'quotient_action_image': '1.1', 'quotient_action_kernel': '20.4', 'quotient_action_kernel_order': 20, 'quotient_fusion': None, 'short_label': '20.a1.a1', 'subgroup_fusion': None, 'weyl_group': '1.1'}
-
id: 66334608
{'ambient': '40.5', 'aut_centralizer_order': 8, 'aut_label': '20.b1', 'aut_quo_index': None, 'aut_stab_index': 10, 'aut_weyl_group': '1.1', 'aut_weyl_index': 80, 'centralizer': '5.a1.a1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['4.b1.a1', '10.b1.a1'], 'contains': ['40.a1.a1'], 'core': '40.a1.a1', 'coset_action_label': '20T6', 'count': 5, 'diagramx': [2835, -1, 7816, -1, 3830, -1, 6463, -1], 'generators': [1], 'label': '40.5.20.b1.a1', 'mobius_quo': None, 'mobius_sub': 0, 'normal_closure': '4.b1.a1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '5.a1.a1', 'old_label': '20.b1.a1', 'projective_image': '40.5', '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'}
-
id: 66334609
{'ambient': '40.5', 'aut_centralizer_order': 8, 'aut_label': '20.b1', 'aut_quo_index': None, 'aut_stab_index': 10, 'aut_weyl_group': '1.1', 'aut_weyl_index': 80, 'centralizer': '5.a1.a1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['4.b1.b1', '10.b1.a1'], 'contains': ['40.a1.a1'], 'core': '40.a1.a1', 'coset_action_label': '20T6', 'count': 5, 'diagramx': [2835, -1, 6244, -1, 3830, -1, 9390, -1], 'generators': [5], 'label': '40.5.20.b1.b1', 'mobius_quo': None, 'mobius_sub': 0, 'normal_closure': '4.b1.b1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '5.a1.a1', 'old_label': '20.b1.b1', 'projective_image': '40.5', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '20.b1.b1', 'subgroup_fusion': None, 'weyl_group': '1.1'}
-
id: 66334610
{'ambient': '40.5', 'aut_centralizer_order': None, 'aut_label': '40.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', '20.a1.a1', '20.b1.a1', '20.b1.b1'], 'contains': [], 'core': '40.a1.a1', 'coset_action_label': '40T9', 'count': 1, 'diagramx': [2835, 4107, 7031, 5053, 3830, 5030, 7195, 4618], 'generators': [], 'label': '40.5.40.a1.a1', 'mobius_quo': 1, 'mobius_sub': 0, 'normal_closure': '40.a1.a1', 'normal_contained_in': ['8.a1.a1', '20.a1.a1'], 'normal_contains': [], 'normalizer': '1.a1.a1', 'old_label': '40.a1.a1', 'projective_image': '40.5', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '40.a1.a1', 'subgroup_fusion': None, 'weyl_group': '1.1'}
-
id: 66334611
{'ambient': '40.5', 'aut_centralizer_order': 2, 'aut_label': '2.a1', 'aut_quo_index': 1, 'aut_stab_index': 1, 'aut_weyl_group': '40.12', 'aut_weyl_index': 2, 'centralizer': '10.a1.a1', 'complements': [], 'conjugacy_class_count': 1, 'contained_in': ['1.a1.a1'], 'contains': ['4.a1.a1', '4.b1.a1', '4.b1.b1', '10.b1.a1'], 'core': '2.a1.a1', 'coset_action_label': None, 'count': 1, 'diagramx': [4177, 1409, 6507, 2802, 951, 1412, 7155, 2824], 'generators': [1, 8, 20], 'label': '40.5.2.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': '20.4', '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': '10.1'}
-
id: 66334612
{'ambient': '40.5', 'aut_centralizer_order': 10, 'aut_label': '2.b1', 'aut_quo_index': 1, 'aut_stab_index': 1, 'aut_weyl_group': '8.2', 'aut_weyl_index': 10, 'centralizer': '2.b1.a1', 'complements': ['20.b1.a1', '20.b1.b1'], 'conjugacy_class_count': 1, 'contained_in': ['1.a1.a1'], 'contains': ['4.a1.a1', '10.a1.a1'], 'core': '2.b1.a1', 'coset_action_label': None, 'count': 1, 'diagramx': [6886, 8591, 4564, 9065, 5604, 8608, 5346, 8837], 'generators': [10, 20, 8], 'label': '40.5.2.b1.a1', 'mobius_quo': 0, 'mobius_sub': -1, 'normal_closure': '2.b1.a1', 'normal_contained_in': ['1.a1.a1'], 'normal_contains': ['4.a1.a1', '10.a1.a1'], 'normalizer': '1.a1.a1', 'old_label': '2.b1.a1', 'projective_image': '10.1', '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'}
-
id: 66334613
{'ambient': '40.5', 'aut_centralizer_order': 2, 'aut_label': '2.c1', 'aut_quo_index': 1, 'aut_stab_index': 1, 'aut_weyl_group': '40.12', 'aut_weyl_index': 2, 'centralizer': '10.a1.a1', 'complements': ['20.b1.a1', '20.b1.b1'], 'conjugacy_class_count': 1, 'contained_in': ['1.a1.a1'], 'contains': ['4.a1.a1', '10.c1.a1'], 'core': '2.c1.a1', 'coset_action_label': None, 'count': 1, 'diagramx': [9164, 5020, 2598, 5952, 3291, 5030, 752, 5847], 'generators': [3, 8, 20], 'label': '40.5.2.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': '20.4', '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': '10.1'}
-
id: 66334614
{'ambient': '40.5', 'aut_centralizer_order': 20, 'aut_label': '4.a1', 'aut_quo_index': 6, 'aut_stab_index': 1, 'aut_weyl_group': '4.1', 'aut_weyl_index': 20, 'centralizer': '2.b1.a1', 'complements': [], 'conjugacy_class_count': 1, 'contained_in': ['2.a1.a1', '2.b1.a1', '2.c1.a1'], 'contains': ['8.a1.a1', '20.a1.a1'], 'core': '4.a1.a1', 'coset_action_label': None, 'count': 1, 'diagramx': [5519, 5020, 4564, 6193, 2751, 5030, 4695, 5947], 'generators': [20, 8], 'label': '40.5.4.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', '20.a1.a1'], 'normalizer': '1.a1.a1', 'old_label': '4.a1.a1', 'projective_image': '20.4', 'quotient_action_image': '2.1', 'quotient_action_kernel': '2.1', 'quotient_action_kernel_order': 2, 'quotient_fusion': None, 'short_label': '4.a1.a1', 'subgroup_fusion': None, 'weyl_group': '2.1'}
-
id: 66334615
{'ambient': '40.5', 'aut_centralizer_order': 2, 'aut_label': '4.b1', 'aut_quo_index': 1, 'aut_stab_index': 2, 'aut_weyl_group': '20.3', 'aut_weyl_index': 4, 'centralizer': '10.a1.a1', 'complements': ['10.a1.a1', '10.c1.a1'], 'conjugacy_class_count': 1, 'contained_in': ['2.a1.a1'], 'contains': ['8.a1.a1', '20.b1.a1'], 'core': '4.b1.a1', 'coset_action_label': None, 'count': 1, 'diagramx': [2000, 1647, 8712, 3425, 694, 1650, 7988, 3289], 'generators': [1, 8], 'label': '40.5.4.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': '40.5', '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': '10.1'}
-
id: 66334616
{'ambient': '40.5', 'aut_centralizer_order': 2, 'aut_label': '4.b1', 'aut_quo_index': 1, 'aut_stab_index': 2, 'aut_weyl_group': '20.3', 'aut_weyl_index': 4, 'centralizer': '10.a1.a1', 'complements': ['10.a1.a1', '10.c1.a1'], 'conjugacy_class_count': 1, 'contained_in': ['2.a1.a1'], 'contains': ['8.a1.a1', '20.b1.b1'], 'core': '4.b1.b1', 'coset_action_label': None, 'count': 1, 'diagramx': [2000, 1647, 7140, 935, 694, 1650, 9451, 897], 'generators': [5, 8], 'label': '40.5.4.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': '40.5', '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': '10.1'}
-
id: 66334617
{'ambient': '40.5', 'aut_centralizer_order': 4, 'aut_label': '5.a1', 'aut_quo_index': None, 'aut_stab_index': 5, 'aut_weyl_group': '4.2', 'aut_weyl_index': 20, 'centralizer': '5.a1.a1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['1.a1.a1'], 'contains': ['10.a1.a1', '10.b1.a1', '10.c1.a1'], 'core': '10.a1.a1', 'coset_action_label': None, 'count': 5, 'diagramx': [9316, -1, 1288, -1, 8483, -1, 3069, -1], 'generators': [1, 10], 'label': '40.5.5.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': '10.1', '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': '1.1'}
-
id: 66334618
{'ambient': '40.5', 'aut_centralizer_order': 20, 'aut_label': '8.a1', 'aut_quo_index': 2, 'aut_stab_index': 1, 'aut_weyl_group': '4.1', 'aut_weyl_index': 20, 'centralizer': '2.b1.a1', 'complements': ['5.a1.a1'], 'conjugacy_class_count': 1, 'contained_in': ['4.a1.a1', '4.b1.a1', '4.b1.b1'], 'contains': ['40.a1.a1'], 'core': '8.a1.a1', 'coset_action_label': None, 'count': 1, 'diagramx': [684, 2996, 8014, 3667, 1979, 3598, 7927, 3422], 'generators': [8], 'label': '40.5.8.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': ['40.a1.a1'], 'normalizer': '1.a1.a1', 'old_label': '8.a1.a1', 'projective_image': '40.5', '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'}
