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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': True, 'abelian_quotient': '20.2', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': True, 'aut_cyclic': False, 'aut_derived_length': 1, 'aut_exponent': 4, 'aut_gen_orders': [2, 4], 'aut_gens': [[1], [11], [17]], 'aut_group': '8.2', 'aut_hash': 2, 'aut_nilpotency_class': 1, 'aut_nilpotent': True, 'aut_order': 8, 'aut_permdeg': 6, 'aut_perms': [120, 9], 'aut_phi_ratio': 1.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [4, 1, 2, 1], [5, 1, 4, 1], [10, 1, 4, 1], [20, 1, 8, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_2\\times C_4', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 4, 'autcent_group': '8.2', 'autcent_hash': 2, 'autcent_nilpotent': True, 'autcent_order': 8, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2\\times C_4', 'autcentquo_abelian': True, 'autcentquo_cyclic': True, 'autcentquo_exponent': 1, 'autcentquo_group': '1.1', 'autcentquo_hash': 1, 'autcentquo_nilpotent': True, 'autcentquo_order': 1, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'C_1', 'cc_stats': [[1, 1, 1], [2, 1, 1], [4, 1, 2], [5, 1, 4], [10, 1, 4], [20, 1, 8]], 'center_label': '20.2', 'center_order': 20, 'central_product': True, 'central_quotient': '1.1', 'commutator_count': 0, 'commutator_label': '1.1', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '5.1'], 'composition_length': 3, 'conjugacy_classes_known': True, 'counter': 2, 'cyclic': True, 'derived_length': 1, 'dihedral': False, 'direct_factorization': [['4.1', 1], ['5.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [4, 1, 2, 1], [5, 1, 4, 1], [10, 1, 4, 1], [20, 1, 8, 1]], 'element_repr_type': 'PC', 'elementary': 10, 'eulerian_function': 1, 'exponent': 20, 'exponents_of_order': [2, 1], 'factors_of_aut_order': [2], 'factors_of_order': [2, 5], 'faithful_reps': [[1, 0, 8]], 'familial': True, 'frattini_label': '2.1', 'frattini_quotient': '10.2', 'hash': 2, 'hyperelementary': 10, 'inner_abelian': True, 'inner_cyclic': True, 'inner_exponent': 1, 'inner_gen_orders': [1], 'inner_gens': [[1]], 'inner_hash': 1, 'inner_nilpotent': True, 'inner_order': 1, 'inner_split': True, 'inner_tex': 'C_1', 'inner_used': [], 'irrC_degree': 1, 'irrQ_degree': 8, 'irrQ_dim': 8, 'irrR_degree': 2, 'irrep_stats': [[1, 20]], 'label': '20.2', 'linC_count': 8, 'linC_degree': 1, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 6, 'linQ_degree_count': 2, 'linQ_dim': 6, 'linQ_dim_count': 2, 'linR_count': 4, 'linR_degree': 2, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C20', 'ngens': 3, 'nilpotency_class': 1, 'nilpotent': True, 'normal_counts': [0, 0, 0, 0, 0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 6, 'number_characteristic_subgroups': 6, 'number_conjugacy_classes': 20, 'number_divisions': 6, 'number_normal_subgroups': 6, 'number_subgroup_autclasses': 6, 'number_subgroup_classes': 6, 'number_subgroups': 6, 'old_label': None, 'order': 20, 'order_factorization_type': 22, 'order_stats': [[1, 1], [2, 1], [4, 2], [5, 4], [10, 4], [20, 8]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 4, 'outer_gen_orders': [2, 4], 'outer_gen_pows': [0, 0], 'outer_gens': [[11], [17]], 'outer_group': '8.2', 'outer_hash': 2, 'outer_nilpotent': True, 'outer_order': 8, 'outer_permdeg': 6, 'outer_perms': [120, 9], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2\\times C_4', 'pc_rank': 1, 'perfect': False, 'permutation_degree': 9, 'pgroup': 0, 'primary_abelian_invariants': [4, 5], 'quasisimple': False, 'rank': 1, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 2], [2, 1], [4, 2], [8, 1]], 'representations': {'PC': {'code': 51395, 'gens': [1], 'pres': [3, -2, -2, -5, 6, 16]}, 'GLZ': {'b': 3, 'd': 6, 'gens': [58438750843733785]}, 'GLFp': {'d': 2, 'p': 5, 'gens': [131, 504, 252]}, 'Perm': {'d': 9, 'gens': [131040, 96, 41040]}}, 'schur_multiplier': [], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [20], 'solvability_type': 0, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_{20}', 'transitive_degree': 20, 'wreath_data': None, 'wreath_product': False}
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gps_conj_classes • Show schema
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id: 109174462
{'aut_label': '1A', 'centralizer': '1.a1.a1', 'counter': 1, 'group_counter': 2, 'group_order': 20, 'label': '1A', 'order': 1, 'powers': [1, 1], 'representative': 0, 'size': 1}
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id: 109174473
{'aut_label': '2A', 'centralizer': '1.a1.a1', 'counter': 2, 'group_counter': 2, 'group_order': 20, 'label': '2A', 'order': 2, 'powers': [1, 2], 'representative': 10, 'size': 1}
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id: 109174475
{'aut_label': '4A1', 'centralizer': '1.a1.a1', 'counter': 3, 'group_counter': 2, 'group_order': 20, 'label': '4A1', 'order': 4, 'powers': [2, 3], 'representative': 5, 'size': 1}
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id: 109174476
{'aut_label': '4A1', 'centralizer': '1.a1.a1', 'counter': 4, 'group_counter': 2, 'group_order': 20, 'label': '4A-1', 'order': 4, 'powers': [2, 4], 'representative': 15, 'size': 1}
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id: 109174477
{'aut_label': '5A1', 'centralizer': '1.a1.a1', 'counter': 5, 'group_counter': 2, 'group_order': 20, 'label': '5A1', 'order': 5, 'powers': [7, 1], 'representative': 4, 'size': 1}
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id: 109174478
{'aut_label': '5A1', 'centralizer': '1.a1.a1', 'counter': 6, 'group_counter': 2, 'group_order': 20, 'label': '5A-1', 'order': 5, 'powers': [8, 1], 'representative': 16, 'size': 1}
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id: 109174479
{'aut_label': '5A1', 'centralizer': '1.a1.a1', 'counter': 7, 'group_counter': 2, 'group_order': 20, 'label': '5A2', 'order': 5, 'powers': [6, 1], 'representative': 8, 'size': 1}
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id: 109174480
{'aut_label': '5A1', 'centralizer': '1.a1.a1', 'counter': 8, 'group_counter': 2, 'group_order': 20, 'label': '5A-2', 'order': 5, 'powers': [5, 1], 'representative': 12, 'size': 1}
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id: 109174481
{'aut_label': '10A1', 'centralizer': '1.a1.a1', 'counter': 9, 'group_counter': 2, 'group_order': 20, 'label': '10A1', 'order': 10, 'powers': [5, 2], 'representative': 2, 'size': 1}
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id: 109174463
{'aut_label': '10A1', 'centralizer': '1.a1.a1', 'counter': 10, 'group_counter': 2, 'group_order': 20, 'label': '10A-1', 'order': 10, 'powers': [6, 2], 'representative': 18, 'size': 1}
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id: 109174464
{'aut_label': '10A1', 'centralizer': '1.a1.a1', 'counter': 11, 'group_counter': 2, 'group_order': 20, 'label': '10A3', 'order': 10, 'powers': [8, 2], 'representative': 6, 'size': 1}
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id: 109174465
{'aut_label': '10A1', 'centralizer': '1.a1.a1', 'counter': 12, 'group_counter': 2, 'group_order': 20, 'label': '10A-3', 'order': 10, 'powers': [7, 2], 'representative': 14, 'size': 1}
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id: 109174466
{'aut_label': '20A1', 'centralizer': '1.a1.a1', 'counter': 13, 'group_counter': 2, 'group_order': 20, 'label': '20A1', 'order': 20, 'powers': [9, 3], 'representative': 1, 'size': 1}
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id: 109174467
{'aut_label': '20A1', 'centralizer': '1.a1.a1', 'counter': 14, 'group_counter': 2, 'group_order': 20, 'label': '20A-1', 'order': 20, 'powers': [10, 4], 'representative': 19, 'size': 1}
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id: 109174468
{'aut_label': '20A1', 'centralizer': '1.a1.a1', 'counter': 15, 'group_counter': 2, 'group_order': 20, 'label': '20A3', 'order': 20, 'powers': [11, 4], 'representative': 3, 'size': 1}
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id: 109174469
{'aut_label': '20A1', 'centralizer': '1.a1.a1', 'counter': 16, 'group_counter': 2, 'group_order': 20, 'label': '20A-3', 'order': 20, 'powers': [12, 3], 'representative': 17, 'size': 1}
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id: 109174470
{'aut_label': '20A1', 'centralizer': '1.a1.a1', 'counter': 17, 'group_counter': 2, 'group_order': 20, 'label': '20A7', 'order': 20, 'powers': [12, 4], 'representative': 7, 'size': 1}
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id: 109174471
{'aut_label': '20A1', 'centralizer': '1.a1.a1', 'counter': 18, 'group_counter': 2, 'group_order': 20, 'label': '20A-7', 'order': 20, 'powers': [11, 3], 'representative': 13, 'size': 1}
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id: 109174472
{'aut_label': '20A1', 'centralizer': '1.a1.a1', 'counter': 19, 'group_counter': 2, 'group_order': 20, 'label': '20A9', 'order': 20, 'powers': [10, 3], 'representative': 9, 'size': 1}
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id: 109174474
{'aut_label': '20A1', 'centralizer': '1.a1.a1', 'counter': 20, 'group_counter': 2, 'group_order': 20, 'label': '20A-9', 'order': 20, 'powers': [9, 4], 'representative': 11, 'size': 1}
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gps_qchar • Show schema
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id: 4171496
{'cdim': 1, 'counter': 1, 'faithful': False, 'group': '20.2', 'group_order': 20, 'image': None, 'label': '20.2.1a', 'multiplicity': 1, 'nt': [1, 1], 'qdim': 1, 'qvalues': [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1], 'schur_index': 1}
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id: 4171497
{'cdim': 1, 'counter': 2, 'faithful': False, 'group': '20.2', 'group_order': 20, 'image': None, 'label': '20.2.1b', 'multiplicity': 1, 'nt': [2, 1], 'qdim': 1, 'qvalues': [1, 1, -1, -1, 1, 1, 1, 1, 1, 1, 1, 1, -1, -1, -1, -1, -1, -1, -1, -1], 'schur_index': 1}
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id: 4171498
{'cdim': 1, 'counter': 3, 'faithful': False, 'group': '20.2', 'group_order': 20, 'image': None, 'label': '20.2.1c', 'multiplicity': 2, 'nt': [4, 1], 'qdim': 2, 'qvalues': [2, -2, 0, 0, 2, 2, 2, 2, -2, -2, -2, -2, 0, 0, 0, 0, 0, 0, 0, 0], 'schur_index': 1}
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id: 4171499
{'cdim': 1, 'counter': 4, 'faithful': False, 'group': '20.2', 'group_order': 20, 'image': None, 'label': '20.2.1d', 'multiplicity': 4, 'nt': [5, 1], 'qdim': 4, 'qvalues': [4, 4, 4, 4, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1], 'schur_index': 1}
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id: 4171500
{'cdim': 1, 'counter': 5, 'faithful': False, 'group': '20.2', 'group_order': 20, 'image': None, 'label': '20.2.1e', 'multiplicity': 4, 'nt': [10, 1], 'qdim': 4, 'qvalues': [4, 4, -4, -4, -1, -1, -1, -1, -1, -1, -1, -1, 1, 1, 1, 1, 1, 1, 1, 1], 'schur_index': 1}
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id: 4171501
{'cdim': 1, 'counter': 6, 'faithful': True, 'group': '20.2', 'group_order': 20, 'image': None, 'label': '20.2.1f', 'multiplicity': 8, 'nt': [20, 1], 'qdim': 8, 'qvalues': [8, -8, 0, 0, -2, -2, -2, -2, 2, 2, 2, 2, 0, 0, 0, 0, 0, 0, 0, 0], 'schur_index': 1}
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gps_char • Show schema
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id: 6082159
{'center': '1.a1.a1', 'center_index': 1, 'center_order': 20, 'counter': 1, 'cyclotomic_n': 1, 'dim': 1, 'faithful': False, 'field': [0, 1], 'group': '20.2', 'group_counter': 2, 'group_order': 20, 'image': None, 'image_isoclass': '1.1', 'image_order': 1, 'indicator': 1, 'kernel': '1.a1.a1', 'kernel_order': 20, 'label': '20.2.1a', 'nt': [1, 1], 'q_character': None, 'values': [[[1, 0]], [[1, 0]], [[1, 0]], [[1, 0]], [[1, 0]], [[1, 0]], [[1, 0]], [[1, 0]], [[1, 0]], [[1, 0]], [[1, 0]], [[1, 0]], [[1, 0]], [[1, 0]], [[1, 0]], [[1, 0]], [[1, 0]], [[1, 0]], [[1, 0]], [[1, 0]]]}
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id: 6082170
{'center': '1.a1.a1', 'center_index': 1, 'center_order': 20, 'counter': 2, 'cyclotomic_n': 1, 'dim': 1, 'faithful': False, 'field': [0, 1], 'group': '20.2', 'group_counter': 2, 'group_order': 20, 'image': None, 'image_isoclass': '2.1', 'image_order': 2, 'indicator': 1, 'kernel': '2.a1.a1', 'kernel_order': 10, 'label': '20.2.1b', 'nt': [2, 1], 'q_character': None, 'values': [[[1, 0]], [[1, 0]], [[-1, 0]], [[-1, 0]], [[1, 0]], [[1, 0]], [[1, 0]], [[1, 0]], [[1, 0]], [[1, 0]], [[1, 0]], [[1, 0]], [[-1, 0]], [[-1, 0]], [[-1, 0]], [[-1, 0]], [[-1, 0]], [[-1, 0]], [[-1, 0]], [[-1, 0]]]}
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id: 6082172
{'center': '1.a1.a1', 'center_index': 1, 'center_order': 20, 'counter': 3, 'cyclotomic_n': 4, 'dim': 1, 'faithful': False, 'field': [1, 0, 1], 'group': '20.2', 'group_counter': 2, 'group_order': 20, 'image': None, 'image_isoclass': '4.1', 'image_order': 4, 'indicator': 0, 'kernel': '4.a1.a1', 'kernel_order': 5, 'label': '20.2.1c1', 'nt': [4, 1], 'q_character': None, 'values': [[[1, 0]], [[-1, 0]], [[-1, 1]], [[1, 1]], [[1, 0]], [[1, 0]], [[1, 0]], [[1, 0]], [[-1, 0]], [[-1, 0]], [[-1, 0]], [[-1, 0]], [[-1, 1]], [[1, 1]], [[1, 1]], [[-1, 1]], [[1, 1]], [[-1, 1]], [[-1, 1]], [[1, 1]]]}
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id: 6082173
{'center': '1.a1.a1', 'center_index': 1, 'center_order': 20, 'counter': 4, 'cyclotomic_n': 4, 'dim': 1, 'faithful': False, 'field': [1, 0, 1], 'group': '20.2', 'group_counter': 2, 'group_order': 20, 'image': None, 'image_isoclass': '4.1', 'image_order': 4, 'indicator': 0, 'kernel': '4.a1.a1', 'kernel_order': 5, 'label': '20.2.1c2', 'nt': [4, 2], 'q_character': None, 'values': [[[1, 0]], [[-1, 0]], [[1, 1]], [[-1, 1]], [[1, 0]], [[1, 0]], [[1, 0]], [[1, 0]], [[-1, 0]], [[-1, 0]], [[-1, 0]], [[-1, 0]], [[1, 1]], [[-1, 1]], [[-1, 1]], [[1, 1]], [[-1, 1]], [[1, 1]], [[1, 1]], [[-1, 1]]]}
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id: 6082174
{'center': '1.a1.a1', 'center_index': 1, 'center_order': 20, 'counter': 5, 'cyclotomic_n': 5, 'dim': 1, 'faithful': False, 'field': [1, -1, 1, -1, 1], 'group': '20.2', 'group_counter': 2, 'group_order': 20, 'image': None, 'image_isoclass': '5.1', 'image_order': 5, 'indicator': 0, 'kernel': '5.a1.a1', 'kernel_order': 4, 'label': '20.2.1d1', 'nt': [5, 1], 'q_character': None, 'values': [[[1, 0]], [[1, 0]], [[1, 0]], [[1, 0]], [[1, -2]], [[1, 2]], [[1, 1]], [[1, -1]], [[1, -1]], [[1, 1]], [[1, 2]], [[1, -2]], [[1, 2]], [[1, -2]], [[1, 1]], [[1, -1]], [[1, -1]], [[1, 1]], [[1, -2]], [[1, 2]]]}
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id: 6082175
{'center': '1.a1.a1', 'center_index': 1, 'center_order': 20, 'counter': 6, 'cyclotomic_n': 5, 'dim': 1, 'faithful': False, 'field': [1, -1, 1, -1, 1], 'group': '20.2', 'group_counter': 2, 'group_order': 20, 'image': None, 'image_isoclass': '5.1', 'image_order': 5, 'indicator': 0, 'kernel': '5.a1.a1', 'kernel_order': 4, 'label': '20.2.1d2', 'nt': [5, 4], 'q_character': None, 'values': [[[1, 0]], [[1, 0]], [[1, 0]], [[1, 0]], [[1, 2]], [[1, -2]], [[1, -1]], [[1, 1]], [[1, 1]], [[1, -1]], [[1, -2]], [[1, 2]], [[1, -2]], [[1, 2]], [[1, -1]], [[1, 1]], [[1, 1]], [[1, -1]], [[1, 2]], [[1, -2]]]}
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id: 6082176
{'center': '1.a1.a1', 'center_index': 1, 'center_order': 20, 'counter': 7, 'cyclotomic_n': 5, 'dim': 1, 'faithful': False, 'field': [1, -1, 1, -1, 1], 'group': '20.2', 'group_counter': 2, 'group_order': 20, 'image': None, 'image_isoclass': '5.1', 'image_order': 5, 'indicator': 0, 'kernel': '5.a1.a1', 'kernel_order': 4, 'label': '20.2.1d3', 'nt': [5, 1], 'q_character': None, 'values': [[[1, 0]], [[1, 0]], [[1, 0]], [[1, 0]], [[1, -1]], [[1, 1]], [[1, -2]], [[1, 2]], [[1, 2]], [[1, -2]], [[1, 1]], [[1, -1]], [[1, 1]], [[1, -1]], [[1, -2]], [[1, 2]], [[1, 2]], [[1, -2]], [[1, -1]], [[1, 1]]]}
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id: 6082177
{'center': '1.a1.a1', 'center_index': 1, 'center_order': 20, 'counter': 8, 'cyclotomic_n': 5, 'dim': 1, 'faithful': False, 'field': [1, -1, 1, -1, 1], 'group': '20.2', 'group_counter': 2, 'group_order': 20, 'image': None, 'image_isoclass': '5.1', 'image_order': 5, 'indicator': 0, 'kernel': '5.a1.a1', 'kernel_order': 4, 'label': '20.2.1d4', 'nt': [5, 4], 'q_character': None, 'values': [[[1, 0]], [[1, 0]], [[1, 0]], [[1, 0]], [[1, 1]], [[1, -1]], [[1, 2]], [[1, -2]], [[1, -2]], [[1, 2]], [[1, -1]], [[1, 1]], [[1, -1]], [[1, 1]], [[1, 2]], [[1, -2]], [[1, -2]], [[1, 2]], [[1, 1]], [[1, -1]]]}
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id: 6082178
{'center': '1.a1.a1', 'center_index': 1, 'center_order': 20, 'counter': 9, 'cyclotomic_n': 5, 'dim': 1, 'faithful': False, 'field': [1, -1, 1, -1, 1], 'group': '20.2', 'group_counter': 2, 'group_order': 20, 'image': None, 'image_isoclass': '10.2', 'image_order': 10, 'indicator': 0, 'kernel': '10.a1.a1', 'kernel_order': 2, 'label': '20.2.1e1', 'nt': [10, 1], 'q_character': None, 'values': [[[1, 0]], [[1, 0]], [[-1, 0]], [[-1, 0]], [[1, -2]], [[1, 2]], [[1, 1]], [[1, -1]], [[1, -1]], [[1, 1]], [[1, 2]], [[1, -2]], [[-1, 2]], [[-1, -2]], [[-1, 1]], [[-1, -1]], [[-1, -1]], [[-1, 1]], [[-1, -2]], [[-1, 2]]]}
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id: 6082160
{'center': '1.a1.a1', 'center_index': 1, 'center_order': 20, 'counter': 10, 'cyclotomic_n': 5, 'dim': 1, 'faithful': False, 'field': [1, -1, 1, -1, 1], 'group': '20.2', 'group_counter': 2, 'group_order': 20, 'image': None, 'image_isoclass': '10.2', 'image_order': 10, 'indicator': 0, 'kernel': '10.a1.a1', 'kernel_order': 2, 'label': '20.2.1e2', 'nt': [10, 4], 'q_character': None, 'values': [[[1, 0]], [[1, 0]], [[-1, 0]], [[-1, 0]], [[1, 2]], [[1, -2]], [[1, -1]], [[1, 1]], [[1, 1]], [[1, -1]], [[1, -2]], [[1, 2]], [[-1, -2]], [[-1, 2]], [[-1, -1]], [[-1, 1]], [[-1, 1]], [[-1, -1]], [[-1, 2]], [[-1, -2]]]}
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id: 6082161
{'center': '1.a1.a1', 'center_index': 1, 'center_order': 20, 'counter': 11, 'cyclotomic_n': 5, 'dim': 1, 'faithful': False, 'field': [1, -1, 1, -1, 1], 'group': '20.2', 'group_counter': 2, 'group_order': 20, 'image': None, 'image_isoclass': '10.2', 'image_order': 10, 'indicator': 0, 'kernel': '10.a1.a1', 'kernel_order': 2, 'label': '20.2.1e3', 'nt': [10, 1], 'q_character': None, 'values': [[[1, 0]], [[1, 0]], [[-1, 0]], [[-1, 0]], [[1, -1]], [[1, 1]], [[1, -2]], [[1, 2]], [[1, 2]], [[1, -2]], [[1, 1]], [[1, -1]], [[-1, 1]], [[-1, -1]], [[-1, -2]], [[-1, 2]], [[-1, 2]], [[-1, -2]], [[-1, -1]], [[-1, 1]]]}
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id: 6082162
{'center': '1.a1.a1', 'center_index': 1, 'center_order': 20, 'counter': 12, 'cyclotomic_n': 5, 'dim': 1, 'faithful': False, 'field': [1, -1, 1, -1, 1], 'group': '20.2', 'group_counter': 2, 'group_order': 20, 'image': None, 'image_isoclass': '10.2', 'image_order': 10, 'indicator': 0, 'kernel': '10.a1.a1', 'kernel_order': 2, 'label': '20.2.1e4', 'nt': [10, 4], 'q_character': None, 'values': [[[1, 0]], [[1, 0]], [[-1, 0]], [[-1, 0]], [[1, 1]], [[1, -1]], [[1, 2]], [[1, -2]], [[1, -2]], [[1, 2]], [[1, -1]], [[1, 1]], [[-1, -1]], [[-1, 1]], [[-1, 2]], [[-1, -2]], [[-1, -2]], [[-1, 2]], [[-1, 1]], [[-1, -1]]]}
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id: 6082163
{'center': '1.a1.a1', 'center_index': 1, 'center_order': 20, 'counter': 13, 'cyclotomic_n': 20, 'dim': 1, 'faithful': True, 'field': [1, 0, 1], 'group': '20.2', 'group_counter': 2, 'group_order': 20, 'image': None, 'image_isoclass': '20.2', 'image_order': 20, 'indicator': 0, 'kernel': '20.a1.a1', 'kernel_order': 1, 'label': '20.2.1f1', 'nt': [20, 1], 'q_character': None, 'values': [[[1, 0]], [[-1, 0]], [[-1, 5]], [[1, 5]], [[-1, 2]], [[1, 8]], [[1, 4]], [[-1, 6]], [[1, 6]], [[-1, 4]], [[-1, 8]], [[1, 2]], [[1, 3]], [[-1, 7]], [[1, 9]], [[-1, 1]], [[1, 1]], [[-1, 9]], [[1, 7]], [[-1, 3]]]}
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id: 6082164
{'center': '1.a1.a1', 'center_index': 1, 'center_order': 20, 'counter': 14, 'cyclotomic_n': 20, 'dim': 1, 'faithful': True, 'field': [1, 0, 1], 'group': '20.2', 'group_counter': 2, 'group_order': 20, 'image': None, 'image_isoclass': '20.2', 'image_order': 20, 'indicator': 0, 'kernel': '20.a1.a1', 'kernel_order': 1, 'label': '20.2.1f2', 'nt': [20, 8], 'q_character': None, 'values': [[[1, 0]], [[-1, 0]], [[1, 5]], [[-1, 5]], [[1, 8]], [[-1, 2]], [[-1, 6]], [[1, 4]], [[-1, 4]], [[1, 6]], [[1, 2]], [[-1, 8]], [[-1, 7]], [[1, 3]], [[-1, 1]], [[1, 9]], [[-1, 9]], [[1, 1]], [[-1, 3]], [[1, 7]]]}
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id: 6082165
{'center': '1.a1.a1', 'center_index': 1, 'center_order': 20, 'counter': 15, 'cyclotomic_n': 20, 'dim': 1, 'faithful': True, 'field': [1, 0, 1], 'group': '20.2', 'group_counter': 2, 'group_order': 20, 'image': None, 'image_isoclass': '20.2', 'image_order': 20, 'indicator': 0, 'kernel': '20.a1.a1', 'kernel_order': 1, 'label': '20.2.1f3', 'nt': [20, 1], 'q_character': None, 'values': [[[1, 0]], [[-1, 0]], [[-1, 5]], [[1, 5]], [[1, 8]], [[-1, 2]], [[-1, 6]], [[1, 4]], [[-1, 4]], [[1, 6]], [[1, 2]], [[-1, 8]], [[1, 7]], [[-1, 3]], [[1, 1]], [[-1, 9]], [[1, 9]], [[-1, 1]], [[1, 3]], [[-1, 7]]]}
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id: 6082166
{'center': '1.a1.a1', 'center_index': 1, 'center_order': 20, 'counter': 16, 'cyclotomic_n': 20, 'dim': 1, 'faithful': True, 'field': [1, 0, 1], 'group': '20.2', 'group_counter': 2, 'group_order': 20, 'image': None, 'image_isoclass': '20.2', 'image_order': 20, 'indicator': 0, 'kernel': '20.a1.a1', 'kernel_order': 1, 'label': '20.2.1f4', 'nt': [20, 8], 'q_character': None, 'values': [[[1, 0]], [[-1, 0]], [[1, 5]], [[-1, 5]], [[-1, 2]], [[1, 8]], [[1, 4]], [[-1, 6]], [[1, 6]], [[-1, 4]], [[-1, 8]], [[1, 2]], [[-1, 3]], [[1, 7]], [[-1, 9]], [[1, 1]], [[-1, 1]], [[1, 9]], [[-1, 7]], [[1, 3]]]}
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id: 6082167
{'center': '1.a1.a1', 'center_index': 1, 'center_order': 20, 'counter': 17, 'cyclotomic_n': 20, 'dim': 1, 'faithful': True, 'field': [1, 0, 1], 'group': '20.2', 'group_counter': 2, 'group_order': 20, 'image': None, 'image_isoclass': '20.2', 'image_order': 20, 'indicator': 0, 'kernel': '20.a1.a1', 'kernel_order': 1, 'label': '20.2.1f5', 'nt': [20, 1], 'q_character': None, 'values': [[[1, 0]], [[-1, 0]], [[-1, 5]], [[1, 5]], [[-1, 6]], [[1, 4]], [[-1, 2]], [[1, 8]], [[-1, 8]], [[1, 2]], [[-1, 4]], [[1, 6]], [[-1, 9]], [[1, 1]], [[-1, 7]], [[1, 3]], [[-1, 3]], [[1, 7]], [[-1, 1]], [[1, 9]]]}
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id: 6082168
{'center': '1.a1.a1', 'center_index': 1, 'center_order': 20, 'counter': 18, 'cyclotomic_n': 20, 'dim': 1, 'faithful': True, 'field': [1, 0, 1], 'group': '20.2', 'group_counter': 2, 'group_order': 20, 'image': None, 'image_isoclass': '20.2', 'image_order': 20, 'indicator': 0, 'kernel': '20.a1.a1', 'kernel_order': 1, 'label': '20.2.1f6', 'nt': [20, 8], 'q_character': None, 'values': [[[1, 0]], [[-1, 0]], [[1, 5]], [[-1, 5]], [[1, 4]], [[-1, 6]], [[1, 8]], [[-1, 2]], [[1, 2]], [[-1, 8]], [[1, 6]], [[-1, 4]], [[1, 1]], [[-1, 9]], [[1, 3]], [[-1, 7]], [[1, 7]], [[-1, 3]], [[1, 9]], [[-1, 1]]]}
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id: 6082169
{'center': '1.a1.a1', 'center_index': 1, 'center_order': 20, 'counter': 19, 'cyclotomic_n': 20, 'dim': 1, 'faithful': True, 'field': [1, 0, 1], 'group': '20.2', 'group_counter': 2, 'group_order': 20, 'image': None, 'image_isoclass': '20.2', 'image_order': 20, 'indicator': 0, 'kernel': '20.a1.a1', 'kernel_order': 1, 'label': '20.2.1f7', 'nt': [20, 1], 'q_character': None, 'values': [[[1, 0]], [[-1, 0]], [[-1, 5]], [[1, 5]], [[1, 4]], [[-1, 6]], [[1, 8]], [[-1, 2]], [[1, 2]], [[-1, 8]], [[1, 6]], [[-1, 4]], [[-1, 1]], [[1, 9]], [[-1, 3]], [[1, 7]], [[-1, 7]], [[1, 3]], [[-1, 9]], [[1, 1]]]}
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id: 6082171
{'center': '1.a1.a1', 'center_index': 1, 'center_order': 20, 'counter': 20, 'cyclotomic_n': 20, 'dim': 1, 'faithful': True, 'field': [1, 0, 1], 'group': '20.2', 'group_counter': 2, 'group_order': 20, 'image': None, 'image_isoclass': '20.2', 'image_order': 20, 'indicator': 0, 'kernel': '20.a1.a1', 'kernel_order': 1, 'label': '20.2.1f8', 'nt': [20, 8], 'q_character': None, 'values': [[[1, 0]], [[-1, 0]], [[1, 5]], [[-1, 5]], [[-1, 6]], [[1, 4]], [[-1, 2]], [[1, 8]], [[-1, 8]], [[1, 2]], [[-1, 4]], [[1, 6]], [[1, 9]], [[-1, 1]], [[1, 7]], [[-1, 3]], [[1, 3]], [[-1, 7]], [[1, 1]], [[-1, 9]]]}
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gps_subgroup_search • Show schema
Hide schema
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id: 63654256
{'Agroup': True, 'Zgroup': True, 'abelian': True, 'ambient': '20.2', 'ambient_counter': 2, 'ambient_order': 20, 'ambient_tex': 'C_{20}', 'central': True, 'central_factor': False, 'centralizer_order': 20, 'characteristic': True, 'core_order': 20, 'counter': 1, 'cyclic': True, 'direct': True, 'hall': 10, 'label': '20.2.1.a1.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': True, 'old_label': '1.a1.a1', 'outer_equivalence': False, 'perfect': False, 'proper': False, 'quotient': '1.1', 'quotient_Agroup': True, 'quotient_abelian': True, 'quotient_cyclic': True, 'quotient_hash': 1, 'quotient_metabelian': True, 'quotient_nilpotent': True, 'quotient_order': 1, 'quotient_simple': False, 'quotient_solvable': True, 'quotient_supersolvable': True, 'quotient_tex': 'C_1', 'simple': False, 'solvable': True, 'special_labels': ['Z', 'F', 'R', 'U0', 'L0', 'D0', 'C0'], '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: 63654257
{'Agroup': True, 'Zgroup': True, 'abelian': True, 'ambient': '20.2', 'ambient_counter': 2, 'ambient_order': 20, 'ambient_tex': 'C_{20}', 'central': True, 'central_factor': False, 'centralizer_order': 20, 'characteristic': True, 'core_order': 10, 'counter': 2, 'cyclic': True, 'direct': False, 'hall': 0, 'label': '20.2.2.a1.a1', 'maximal': True, 'maximal_normal': True, 'metabelian': True, 'metacyclic': True, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': True, 'old_label': '2.a1.a1', 'outer_equivalence': False, 'perfect': False, 'proper': True, 'quotient': '2.1', 'quotient_Agroup': True, 'quotient_abelian': True, 'quotient_cyclic': True, 'quotient_hash': 1, 'quotient_metabelian': True, 'quotient_nilpotent': True, 'quotient_order': 2, 'quotient_simple': True, 'quotient_solvable': True, 'quotient_supersolvable': True, 'quotient_tex': 'C_2', 'simple': False, 'solvable': True, 'special_labels': ['S', 'C1'], '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: 63654258
{'Agroup': True, 'Zgroup': True, 'abelian': True, 'ambient': '20.2', 'ambient_counter': 2, 'ambient_order': 20, 'ambient_tex': 'C_{20}', 'central': True, 'central_factor': False, 'centralizer_order': 20, 'characteristic': True, 'core_order': 5, 'counter': 3, 'cyclic': True, 'direct': True, 'hall': 5, 'label': '20.2.4.a1.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': True, 'minimal_normal': True, 'nilpotent': True, 'normal': True, 'old_label': '4.a1.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': True, 'solvable': True, 'special_labels': ['C2'], '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: 63654259
{'Agroup': True, 'Zgroup': True, 'abelian': True, 'ambient': '20.2', 'ambient_counter': 2, 'ambient_order': 20, 'ambient_tex': 'C_{20}', 'central': True, 'central_factor': False, 'centralizer_order': 20, 'characteristic': True, 'core_order': 4, 'counter': 4, 'cyclic': True, 'direct': True, 'hall': 2, 'label': '20.2.5.a1.a1', 'maximal': True, 'maximal_normal': True, 'metabelian': True, 'metacyclic': True, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': True, 'old_label': '5.a1.a1', 'outer_equivalence': False, 'perfect': False, 'proper': True, 'quotient': '5.1', 'quotient_Agroup': True, 'quotient_abelian': True, 'quotient_cyclic': True, 'quotient_hash': 1, 'quotient_metabelian': True, 'quotient_nilpotent': True, 'quotient_order': 5, 'quotient_simple': True, 'quotient_solvable': True, 'quotient_supersolvable': True, 'quotient_tex': 'C_5', 'simple': False, 'solvable': True, 'special_labels': [], 'split': True, 'standard_generators': False, 'stem': False, 'subgroup': '4.1', 'subgroup_hash': 1, 'subgroup_order': 4, 'subgroup_tex': 'C_4', 'supersolvable': True, 'sylow': 2}
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id: 63654260
{'Agroup': True, 'Zgroup': True, 'abelian': True, 'ambient': '20.2', 'ambient_counter': 2, 'ambient_order': 20, 'ambient_tex': 'C_{20}', 'central': True, 'central_factor': False, 'centralizer_order': 20, 'characteristic': True, 'core_order': 2, 'counter': 5, 'cyclic': True, 'direct': False, 'hall': 0, 'label': '20.2.10.a1.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': True, 'minimal_normal': True, 'nilpotent': True, 'normal': True, 'old_label': '10.a1.a1', 'outer_equivalence': False, 'perfect': False, 'proper': True, 'quotient': '10.2', 'quotient_Agroup': True, 'quotient_abelian': True, 'quotient_cyclic': True, 'quotient_hash': 2, 'quotient_metabelian': True, 'quotient_nilpotent': True, 'quotient_order': 10, 'quotient_simple': False, 'quotient_solvable': True, 'quotient_supersolvable': True, 'quotient_tex': 'C_{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: 63654261
{'Agroup': True, 'Zgroup': True, 'abelian': True, 'ambient': '20.2', 'ambient_counter': 2, 'ambient_order': 20, 'ambient_tex': 'C_{20}', 'central': True, 'central_factor': False, 'centralizer_order': 20, 'characteristic': True, 'core_order': 1, 'counter': 6, 'cyclic': True, 'direct': True, 'hall': 1, 'label': '20.2.20.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': True, 'proper': False, 'quotient': '20.2', 'quotient_Agroup': True, 'quotient_abelian': True, 'quotient_cyclic': True, 'quotient_hash': 2, 'quotient_metabelian': True, 'quotient_nilpotent': True, 'quotient_order': 20, 'quotient_simple': False, 'quotient_solvable': True, 'quotient_supersolvable': True, 'quotient_tex': 'C_{20}', 'simple': False, 'solvable': True, 'special_labels': ['D', 'PC', 'U1', 'L1', 'D1', 'C3'], '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: 63654256
{'ambient': '20.2', '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': '1.a1.a1', 'complements': ['20.a1.a1'], 'conjugacy_class_count': 1, 'contained_in': [], 'contains': ['2.a1.a1', '5.a1.a1'], 'core': '1.a1.a1', 'coset_action_label': None, 'count': 1, 'diagramx': [5976, 5976, 3536, 3536, 4718, 4718, 4718, 4718], 'generators': [5, 10, 4], 'label': '20.2.1.a1.a1', 'mobius_quo': 0, 'mobius_sub': 1, 'normal_closure': '1.a1.a1', 'normal_contained_in': [], 'normal_contains': ['2.a1.a1', '5.a1.a1'], 'normalizer': '1.a1.a1', 'old_label': '1.a1.a1', 'projective_image': '1.1', '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': '1.1'}
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id: 63654257
{'ambient': '20.2', 'aut_centralizer_order': 2, 'aut_label': '2.a1', 'aut_quo_index': 1, 'aut_stab_index': 1, 'aut_weyl_group': '4.1', 'aut_weyl_index': 2, 'centralizer': '1.a1.a1', 'complements': [], 'conjugacy_class_count': 1, 'contained_in': ['1.a1.a1'], 'contains': ['4.a1.a1', '10.a1.a1'], 'core': '2.a1.a1', 'coset_action_label': None, 'count': 1, 'diagramx': [4513, 4513, 5489, 5489, 7535, 7535, 7535, 7535], 'generators': [10, 4], 'label': '20.2.2.a1.a1', 'mobius_quo': 1, 'mobius_sub': -1, 'normal_closure': '2.a1.a1', 'normal_contained_in': ['1.a1.a1'], 'normal_contains': ['4.a1.a1', '10.a1.a1'], 'normalizer': '1.a1.a1', 'old_label': '2.a1.a1', 'projective_image': '2.1', 'quotient_action_image': '1.1', 'quotient_action_kernel': '2.1', 'quotient_action_kernel_order': 2, 'quotient_fusion': None, 'short_label': '2.a1.a1', 'subgroup_fusion': None, 'weyl_group': '1.1'}
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id: 63654258
{'ambient': '20.2', 'aut_centralizer_order': 2, 'aut_label': '4.a1', 'aut_quo_index': 1, 'aut_stab_index': 1, 'aut_weyl_group': '4.1', 'aut_weyl_index': 2, 'centralizer': '1.a1.a1', 'complements': ['5.a1.a1'], 'conjugacy_class_count': 1, 'contained_in': ['2.a1.a1'], 'contains': ['20.a1.a1'], 'core': '4.a1.a1', 'coset_action_label': None, 'count': 1, 'diagramx': [1646, 1646, 8354, 8354, 7535, 7535, 7535, 7535], 'generators': [4], 'label': '20.2.4.a1.a1', 'mobius_quo': -1, 'mobius_sub': 0, 'normal_closure': '4.a1.a1', 'normal_contained_in': ['2.a1.a1'], 'normal_contains': ['20.a1.a1'], 'normalizer': '1.a1.a1', 'old_label': '4.a1.a1', 'projective_image': '4.1', 'quotient_action_image': '1.1', 'quotient_action_kernel': '4.1', 'quotient_action_kernel_order': 4, 'quotient_fusion': None, 'short_label': '4.a1.a1', 'subgroup_fusion': None, 'weyl_group': '1.1'}
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id: 63654259
{'ambient': '20.2', 'aut_centralizer_order': 4, 'aut_label': '5.a1', 'aut_quo_index': 1, 'aut_stab_index': 1, 'aut_weyl_group': '2.1', 'aut_weyl_index': 4, 'centralizer': '1.a1.a1', 'complements': ['4.a1.a1'], 'conjugacy_class_count': 1, 'contained_in': ['1.a1.a1'], 'contains': ['10.a1.a1'], 'core': '5.a1.a1', 'coset_action_label': None, 'count': 1, 'diagramx': [8354, 8354, 1646, 1646, 1901, 1901, 1901, 1901], 'generators': [5], 'label': '20.2.5.a1.a1', 'mobius_quo': 0, 'mobius_sub': -1, 'normal_closure': '5.a1.a1', 'normal_contained_in': ['1.a1.a1'], 'normal_contains': ['10.a1.a1'], 'normalizer': '1.a1.a1', 'old_label': '5.a1.a1', 'projective_image': '5.1', 'quotient_action_image': '1.1', 'quotient_action_kernel': '5.1', 'quotient_action_kernel_order': 5, 'quotient_fusion': None, 'short_label': '5.a1.a1', 'subgroup_fusion': None, 'weyl_group': '1.1'}
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id: 63654260
{'ambient': '20.2', 'aut_centralizer_order': 8, 'aut_label': '10.a1', 'aut_quo_index': 1, 'aut_stab_index': 1, 'aut_weyl_group': '1.1', 'aut_weyl_index': 8, 'centralizer': '1.a1.a1', 'complements': [], 'conjugacy_class_count': 1, 'contained_in': ['2.a1.a1', '5.a1.a1'], 'contains': ['20.a1.a1'], 'core': '10.a1.a1', 'coset_action_label': None, 'count': 1, 'diagramx': [5001, 5001, 5000, 5000, 2465, 2465, 2465, 2465], 'generators': [10], 'label': '20.2.10.a1.a1', 'mobius_quo': -1, 'mobius_sub': 1, 'normal_closure': '10.a1.a1', 'normal_contained_in': ['2.a1.a1', '5.a1.a1'], 'normal_contains': ['20.a1.a1'], 'normalizer': '1.a1.a1', 'old_label': '10.a1.a1', 'projective_image': '10.2', 'quotient_action_image': '1.1', 'quotient_action_kernel': '10.2', 'quotient_action_kernel_order': 10, 'quotient_fusion': None, 'short_label': '10.a1.a1', 'subgroup_fusion': None, 'weyl_group': '1.1'}
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id: 63654261
{'ambient': '20.2', 'aut_centralizer_order': None, 'aut_label': '20.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': ['4.a1.a1', '10.a1.a1'], 'contains': [], 'core': '20.a1.a1', 'coset_action_label': '20T1', 'count': 1, 'diagramx': [3353, 3353, 7500, 7500, 5000, 5000, 5000, 5000], 'generators': [], 'label': '20.2.20.a1.a1', 'mobius_quo': 1, 'mobius_sub': 0, 'normal_closure': '20.a1.a1', 'normal_contained_in': ['4.a1.a1', '10.a1.a1'], 'normal_contains': [], 'normalizer': '1.a1.a1', 'old_label': '20.a1.a1', 'projective_image': '20.2', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '20.a1.a1', 'subgroup_fusion': None, 'weyl_group': '1.1'}
