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gps_st • Show schema
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{'character_diagonal': [1, 1, 2, 4, 6, 5, 7, 19, 20, 11], 'character_matrix': [[1, 0, 0, 0, 0, 1, 0, 1, 0, 1], [0, 1, 0, 0, 2, 0, 2, 0, 3, 0], [0, 0, 2, 1, 0, 2, 0, 3, 0, 3], [0, 0, 1, 4, 0, 2, 0, 7, 0, 3], [0, 2, 0, 0, 6, 0, 6, 0, 10, 0], [1, 0, 2, 2, 0, 5, 0, 7, 0, 6], [0, 2, 0, 0, 6, 0, 7, 0, 11, 0], [1, 0, 3, 7, 0, 7, 0, 19, 0, 12], [0, 3, 0, 0, 10, 0, 11, 0, 20, 0], [1, 0, 3, 3, 0, 6, 0, 12, 0, 11]], 'component_group': '4.1', 'component_group_number': 1, 'components': 4, 'counts': [['a_1', [[0, 3]]], ['a_2', [[0, 2], [2, 1]]]], 'degree': 4, 'first_a2_moment': 1, 'fourth_trace_moment': 9, 'gens': [[['0', '0', '0', '1'], ['0', '0', '-1', '0'], ['-1', '0', '0', '0'], ['0', '-1', '0', '0']]], 'identity_component': 'U(1)xU(1)', 'label': '1.4.D.4.1a', 'label_components': [1, 4, 3, 4, 1, 0], 'maximal': True, 'moments': [['a_1', 1, 0, 1, 0, 9, 0, 100, 0, 1225, 0, 15876, 0, 213444], ['a_2', 1, 1, 3, 10, 41, 186, 912, 4656, 24425, 130402, 705368, 3854016, 21230420]], 'name': 'F_{ac}', 'old_label': '1.4.2.4.1a', 'pretty': 'F_{ac}', 'rational': True, 'real_dimension': 2, 'second_trace_moment': 1, 'simplex': [1, 1, 3, 4, 9, 10, 18, 42, 100, 41, 86, 206, 500, 1225, 186, 428, 1044, 2570, 6370, 15876, 912, 2192, 5418, 13480, 33712, 84672, 213444], 'st0_label': '1.4.D', 'subgroup_multiplicities': [1], 'subgroups': ['1.4.D.2.1a'], 'supgroups': [], 'trace_histogram': 'data:image/png;base64,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', 'trace_zero_density': '3/4', 'weight': 1, 'zvector': [3, 0, 0, 2, 0, 1, 0, 0, 2, 0, 1]}
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gps_st0 • Show schema
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{'degree': 4, 'description': '\\left\\{\\begin{bmatrix}A&0\\\\0&B\\end{bmatrix}:A,B\\in \\mathrm{U}(1)\\subseteq\\mathrm{SU}(2)\\right\\}', 'hodge_circle': 'u\\mapsto\\mathrm{diag}(u,\\bar u,\\bar u,u)', 'label': '1.4.D', 'label_components': [1, 4, 3], 'name': 'U(1)xU(1)', 'pretty': '\\mathrm{U}(1)\\times\\mathrm{U}(1)', 'real_dimension': 2, 'symplectic_form': '\\begin{bmatrix}J_2&0\\\\0&J_2\\end{bmatrix}, J_2:=\\begin{bmatrix}0&1\\\\-1&0\\end{bmatrix}', 'weight': 1}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': True, 'abelian_quotient': '4.1', 'all_subgroups_known': True, 'almost_simple': False, 'aut_gens': [[1], [3]], 'aut_group': '2.1', 'aut_order': 2, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [4, 1, 2, 1]], 'cc_stats': [[1, 1, 1], [2, 1, 1], [4, 1, 2]], 'center_label': '4.1', 'central_product': False, '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'], 'composition_length': 2, 'counter': 1, 'cyclic': True, 'derived_length': 1, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [4, 1, 2, 1]], 'element_repr_type': 'PC', 'elementary': 2, 'eulerian_function': 1, 'exponent': 4, 'exponents_of_order': [2], 'factors_of_aut_order': [2], 'factors_of_order': [2], 'faithful_reps': [[1, 0, 2]], 'frattini_label': '2.1', 'frattini_quotient': '2.1', 'hash': 1, 'hyperelementary': 2, 'irrC_degree': 1, 'irrQ_degree': 2, 'irrep_stats': [[1, 4]], 'label': '4.1', 'linC_degree': 1, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 2, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C4', 'ngens': 1, 'nilpotency_class': 1, 'nilpotent': True, 'normal_counts': None, 'normal_index_bound': None, 'normal_order_bound': None, 'normal_subgroups_known': True, 'number_autjugacy_classes': 3, 'number_characteristic_subgroups': 3, 'number_conjugacy_classes': 4, 'number_divisions': 3, 'number_normal_subgroups': 3, 'number_subgroup_autclasses': 3, 'number_subgroup_classes': 3, 'number_subgroups': 3, 'old_label': None, 'order': 4, 'order_factorization_type': 2, 'order_stats': [[1, 1], [2, 1], [4, 2]], 'outer_equivalence': False, 'outer_group': '2.1', 'outer_order': 2, 'pc_rank': 1, 'perfect': False, 'permutation_degree': 4, 'pgroup': 2, 'primary_abelian_invariants': [4], 'quasisimple': False, 'rank': 1, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 2], [2, 1]], 'representations': {'PC': {'code': 5, 'gens': [1], 'pres': [2, -2, -2, 4]}, 'GLZ': {'b': 3, 'd': 2, 'gens': [46]}, 'Lie': [{'d': 2, 'q': 3, 'gens': [56, 15], 'family': 'CSOPlus'}], 'GLFp': {'d': 2, 'p': 3, 'gens': [21]}, 'Perm': {'d': 4, 'gens': [22, 7]}}, 'schur_multiplier': [], 'semidirect_product': False, 'simple': False, 'smith_abelian_invariants': [4], 'solvability_type': 0, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_4', 'transitive_degree': 4, 'wreath_data': None, 'wreath_product': False}