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gps_st • Show schema
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{'character_diagonal': [1, 3, 10, 7, 18, 76, 79, 120, 171, 74, 46, 375, 426, 943, 1800, 843, 888, 1978, 1623, 431], 'character_matrix': [[1, 0, 2, 0, 0, 0, 1, 6, 0, 0, 0, 3, 0, 0, 8], [0, 3, 0, 3, 0, 12, 0, 0, 15, 0, 9, 0, 21, 33, 0], [2, 0, 10, 0, 6, 0, 20, 30, 0, 12, 0, 42, 0, 0, 88], [0, 3, 0, 7, 0, 20, 0, 0, 27, 0, 13, 0, 45, 61, 0], [0, 0, 6, 0, 18, 0, 30, 24, 0, 30, 0, 66, 0, 0, 120], [0, 12, 0, 20, 0, 76, 0, 0, 108, 0, 56, 0, 168, 248, 0], [1, 0, 20, 0, 30, 0, 79, 78, 0, 66, 0, 165, 0, 0, 344], [6, 0, 30, 0, 24, 0, 78, 120, 0, 66, 0, 180, 0, 0, 408], [0, 15, 0, 27, 0, 108, 0, 0, 171, 0, 81, 0, 261, 393, 0], [0, 0, 12, 0, 30, 0, 66, 66, 0, 74, 0, 150, 0, 0, 328], [0, 9, 0, 13, 0, 56, 0, 0, 81, 0, 46, 0, 126, 193, 0], [3, 0, 42, 0, 66, 0, 165, 180, 0, 150, 0, 375, 0, 0, 792], [0, 21, 0, 45, 0, 168, 0, 0, 261, 0, 126, 0, 426, 621, 0], [0, 33, 0, 61, 0, 248, 0, 0, 393, 0, 193, 0, 621, 943, 0], [8, 0, 88, 0, 120, 0, 344, 408, 0, 328, 0, 792, 0, 0, 1800]], 'component_group': '2.1', 'component_group_number': 1, 'components': 2, 'counts': [['a_1', [[0, 1]]], ['a_2', [[3, 1]]], ['a_3', [[0, 1]]], ['(a_1,a_2)', [['(0,3)', 1]]], ['(a_1,a_3)', [['(0,0)', 1]]], ['(a_2,a_3)', [['(3,0)', 1]]], ['(a_1,a_2,a_3)', [['(0,3,0)', 1]]]], 'degree': 6, 'first_a2_moment': 3, 'fourth_trace_moment': 45, 'gens': '\\begin{bmatrix}0 & 1 & 0 & 0 & 0 & 0 \\\\-1 & 0 & 0 & 0 & 0 & 0 \\\\0 & 0 & 0 & 1 & 0 & 0 \\\\0 & 0 & -1 & 0 & 0 & 0 \\\\0 & 0 & 0 & 0 & 0 & 1 \\\\0 & 0 & 0 & 0 & -1 & 0 \\\\\\end{bmatrix}', 'identity_component': 'U(1)^3', 'label': '1.6.H.2.1a', 'label_components': [1, 6, 7, 2, 1, 0], 'maximal': False, 'moments': [['a_1', 1, 0, 3, 0, 45, 0, 930, 0, 22365, 0, 586278, 0, 16248078], ['a_2', 1, 3, 15, 105, 963, 10233, 117429, 1409355, 17432235, 220488369, 2838075345, 37053415215, 489491331921], ['a_3', 1, 0, 16, 0, 2460, 0, 554560, 0, 146654060, 0, 42421671936, 0, 12999626647392]], 'name': 'H_{abc}', 'pretty': 'H_{abc}', 'rational': True, 'real_dimension': 3, 'second_trace_moment': 3, 'simplex': [3, 3, 15, 6, 21, 45, 16, 105, 54, 195, 114, 423, 930, 144, 963, 534, 312, 2025, 1170, 4485, 2580, 9990, 22365, 1500, 10233, 864, 5778, 3312, 22599, 12882, 7368, 50751, 28860, 114390, 64890, 258615, 586278, 2460, 16740, 117429, 9540, 66222, 37548, 264729, 21336, 149454, 84552, 600453, 338340, 191040, 1365330, 767970, 3111297, 1747116, 7103754, 16248078], 'st0_label': '1.6.H', 'subgroup_multiplicities': [1], 'subgroups': ['1.6.H.1.1a'], 'supgroup_multiplicities': [1, 1], 'supgroups': ['1.6.H.6.2a', '1.6.H.4.2b'], 'trace_histogram': 'data:image/png;base64,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', 'trace_zero_density': '1/2', 'weight': 1, 'zvector': [1, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1]}
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gps_st0 • Show schema
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{'degree': 6, 'description': '\\left\\{\\begin{bmatrix}A&0&0\\\\0&B&0\\\\0&0&C\\end{bmatrix}: A,B,C\\in\\mathrm{U}(1)\\subseteq\\mathrm{SU}(2)\\right\\}', 'hodge_circle': 'u\\mapsto\\mathrm{diag}(u,\\bar u, u, \\bar u, u, \\bar u)', 'label': '1.6.H', 'label_components': [1, 6, 7], 'name': 'U(1)^3', 'pretty': '\\mathrm{U}(1)^3', 'real_dimension': 3, 'symplectic_form': '\\begin{bmatrix}J_2&0&0\\\\0&J_2&0\\\\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': '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}