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gps_st • Show schema
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{'character_diagonal': [1, 4, 10, 19, 40, 46, 52, 163, 184, 146], 'character_matrix': [[1, 0, 0, 3, 0, 6, 0, 3, 0, 0], [0, 4, 0, 0, 12, 0, 12, 0, 24, 0], [0, 0, 10, 6, 0, 3, 0, 33, 0, 36], [3, 0, 6, 19, 0, 27, 0, 44, 0, 28], [0, 12, 0, 0, 40, 0, 44, 0, 84, 0], [6, 0, 3, 27, 0, 46, 0, 48, 0, 21], [0, 12, 0, 0, 44, 0, 52, 0, 96, 0], [3, 0, 33, 44, 0, 48, 0, 163, 0, 143], [0, 24, 0, 0, 84, 0, 96, 0, 184, 0], [0, 0, 36, 28, 0, 21, 0, 143, 0, 146]], 'component_group': '2.1', 'component_group_number': 1, 'components': 2, 'counts': [['a_1', [[0, 1]]], ['a_2', [[-2, 1]]]], 'degree': 4, 'first_a2_moment': 1, 'fourth_trace_moment': 48, 'gens': [[['0', '0', '0', '1'], ['0', '0', '-1', '0'], ['0', '-1', '0', '0'], ['1', '0', '0', '0']]], 'identity_component': 'U(1)_2', 'label': '1.4.F.2.1a', 'label_components': [1, 4, 5, 2, 1, 0], 'maximal': False, 'moments': [['a_1', 1, 0, 4, 0, 48, 0, 640, 0, 8960, 0, 129024, 0, 1892352], ['a_2', 1, 1, 11, 40, 235, 1196, 6650, 36632, 205859, 1162732, 6617326, 37840232, 217333390]], 'name': 'J(C_1)', 'old_label': '1.4.1.2.1b', 'pretty': 'J(C_1)', 'rational': True, 'real_dimension': 1, 'second_trace_moment': 4, 'simplex': [1, 4, 11, 20, 48, 40, 104, 256, 640, 235, 556, 1392, 3520, 8960, 1196, 3032, 7680, 19584, 50176, 129024, 6650, 16776, 42848, 109952, 283136, 731136, 1892352], 'st0_label': '1.4.F', 'subgroup_multiplicities': [1], 'subgroups': ['1.4.F.1.1a'], 'supgroups': ['1.4.F.4.2b', '1.4.F.6.2a'], 'trace_histogram': 'data:image/png;base64,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', 'trace_zero_density': '1/2', 'weight': 1, 'zvector': [1, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0]}
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gps_st0 • Show schema
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{'degree': 4, 'description': '\\left\\{\\begin{bmatrix}\\alpha I_2&0\\\\0&\\bar\\alpha I_2\\end{bmatrix}: \\alpha\\bar\\alpha = 1,\\ \\alpha\\in\\mathbb{C}\\right\\}', 'hodge_circle': 'u\\mapsto\\mathrm{diag}(u, u,\\bar u,\\bar u)', 'label': '1.4.F', 'label_components': [1, 4, 5], 'name': 'U(1)_2', 'pretty': '\\mathrm{U}(1)_2', 'real_dimension': 1, 'symplectic_form': '\\begin{bmatrix}0&I_2\\\\-I_2&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': '2.1', 'all_subgroups_known': True, 'almost_simple': False, 'aut_gens': [[1]], 'aut_group': '1.1', 'aut_order': 1, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1]], 'cc_stats': [[1, 1, 1], [2, 1, 1]], 'center_label': '2.1', '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'], 'composition_length': 1, 'counter': 1, 'cyclic': True, 'derived_length': 1, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1]], 'element_repr_type': 'PC', 'elementary': 2, 'eulerian_function': 1, 'exponent': 2, 'exponents_of_order': [1], 'factors_of_aut_order': [], 'factors_of_order': [2], 'faithful_reps': [[1, 1, 1]], 'frattini_label': '1.1', 'frattini_quotient': '2.1', 'hash': 1, 'hyperelementary': 2, 'irrC_degree': 1, 'irrQ_degree': 1, 'irrep_stats': [[1, 2]], 'label': '2.1', 'linC_degree': 1, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 2, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C2', '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': 2, 'number_characteristic_subgroups': 2, 'number_conjugacy_classes': 2, 'number_divisions': 2, 'number_normal_subgroups': 2, 'number_subgroup_autclasses': 2, 'number_subgroup_classes': 2, 'number_subgroups': 2, 'old_label': None, 'order': 2, 'order_factorization_type': 1, 'order_stats': [[1, 1], [2, 1]], 'outer_equivalence': False, 'outer_group': '1.1', 'outer_order': 1, 'pc_rank': 1, 'perfect': False, 'permutation_degree': 2, 'pgroup': 2, 'primary_abelian_invariants': [2], 'quasisimple': False, 'rank': 1, 'rational': True, 'rational_characters_known': True, 'ratrep_stats': [[1, 2]], 'representations': {'PC': {'code': 0, 'gens': [1], 'pres': [1, -2]}, 'GLZ': {'b': 3, 'd': 2, 'gens': [12]}, 'Lie': [{'d': 2, 'q': 2, 'gens': [6], 'family': 'CSOPlus'}, {'d': 2, 'q': 2, 'gens': [6], 'family': 'COPlus'}, {'d': 1, 'q': 2, 'gens': [1], 'family': 'AGL'}, {'d': 1, 'q': 2, 'gens': [1], 'family': 'AGammaL'}, {'d': 1, 'q': 2, 'gens': [1], 'family': 'ASigmaL'}], 'GLFp': {'d': 2, 'p': 2, 'gens': [11]}, 'Perm': {'d': 2, 'gens': [1]}}, 'schur_multiplier': [], 'semidirect_product': False, 'simple': True, 'smith_abelian_invariants': [2], 'solvability_type': 0, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_2', 'transitive_degree': 2, 'wreath_data': None, 'wreath_product': False}