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gps_st • Show schema
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{'character_diagonal': [1, 2, 4, 7, 12, 12, 14, 39, 40, 29], 'character_matrix': [[1, 0, 0, 1, 0, 2, 0, 1, 0, 0], [0, 2, 0, 0, 4, 0, 4, 0, 6, 0], [0, 0, 4, 2, 0, 1, 0, 9, 0, 8], [1, 0, 2, 7, 0, 7, 0, 12, 0, 6], [0, 4, 0, 0, 12, 0, 12, 0, 20, 0], [2, 0, 1, 7, 0, 12, 0, 12, 0, 5], [0, 4, 0, 0, 12, 0, 14, 0, 22, 0], [1, 0, 9, 12, 0, 12, 0, 39, 0, 29], [0, 6, 0, 0, 20, 0, 22, 0, 40, 0], [0, 0, 8, 6, 0, 5, 0, 29, 0, 29]], 'component_group': '12.4', 'component_group_number': 4, 'components': 12, 'counts': [['a_1', [[0, 7]]]], 'degree': 4, 'first_a2_moment': 1, 'fourth_trace_moment': 18, 'gens': [[['\\zeta_{12}', '0', '0', '0'], ['0', '\\zeta_{12}^{11}', '0', '0'], ['0', '0', '\\zeta_{12}^{11}', '0'], ['0', '0', '0', '\\zeta_{12}']], [['0', '1', '0', '0'], ['-1', '0', '0', '0'], ['0', '0', '0', '1'], ['0', '0', '-1', '0']]], 'identity_component': 'U(1)_2', 'label': '1.4.F.12.4c', 'label_components': [1, 4, 5, 12, 4, 2], 'maximal': False, 'moments': [['a_1', 1, 0, 2, 0, 18, 0, 200, 0, 2450, 0, 31752, 0, 427812], ['a_2', 1, 1, 5, 16, 77, 356, 1803, 9262, 48933, 262372, 1427255, 7850822, 43608271]], 'name': 'D_6', 'old_label': '1.4.1.12.4a', 'pretty': 'D_6', 'rational': True, 'real_dimension': 1, 'second_trace_moment': 2, 'simplex': [1, 2, 5, 8, 18, 16, 36, 84, 200, 77, 172, 412, 1000, 2450, 356, 856, 2088, 5140, 12740, 31752, 1803, 4386, 10842, 26980, 67494, 169596, 427812], 'st0_label': '1.4.F', 'subgroup_multiplicities': [1, 2, 1], 'subgroups': ['1.4.F.4.2a', '1.4.F.6.1a', '1.4.F.6.2c'], 'supgroups': ['1.4.F.24.14a'], 'trace_histogram': 'data:image/png;base64,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', 'trace_zero_density': '7/12', 'weight': 1, 'zvector': [7, 0, 0, 0, 0, 0, 0, 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': False, 'abelian': False, 'abelian_quotient': '4.2', 'all_subgroups_known': True, 'almost_simple': False, 'aut_gens': [[1, 2], [7, 10], [1, 10], [5, 2]], 'aut_group': '12.4', 'aut_order': 12, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 3, 2, 1], [3, 2, 1, 1], [6, 2, 1, 1]], 'cc_stats': [[1, 1, 1], [2, 1, 1], [2, 3, 2], [3, 2, 1], [6, 2, 1]], 'center_label': '2.1', 'central_product': True, 'central_quotient': '6.1', 'commutator_count': 1, 'commutator_label': '3.1', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '3.1'], 'composition_length': 3, 'counter': 4, 'cyclic': False, 'derived_length': 2, 'direct_factorization': [['2.1', 1], ['6.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 3, 1, 2], [3, 2, 1, 1], [6, 2, 1, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 3, 'exponent': 6, 'exponents_of_order': [2, 1], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [[2, 1, 1]], 'frattini_label': '1.1', 'frattini_quotient': '12.4', 'hash': 4, 'hyperelementary': 2, 'irrC_degree': 2, 'irrQ_degree': 2, 'irrep_stats': [[1, 4], [2, 2]], 'label': '12.4', 'linC_degree': None, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 2, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'D6', 'ngens': 3, 'nilpotency_class': -1, 'nilpotent': False, 'normal_counts': None, 'normal_index_bound': None, 'normal_order_bound': None, 'normal_subgroups_known': True, 'number_autjugacy_classes': 5, 'number_characteristic_subgroups': 5, 'number_conjugacy_classes': 6, 'number_divisions': 6, 'number_normal_subgroups': 7, 'number_subgroup_autclasses': 8, 'number_subgroup_classes': 10, 'number_subgroups': 16, 'old_label': None, 'order': 12, 'order_factorization_type': 22, 'order_stats': [[1, 1], [2, 7], [3, 2], [6, 2]], 'outer_equivalence': False, 'outer_group': '2.1', 'outer_order': 2, 'pc_rank': 2, 'perfect': False, 'permutation_degree': 5, 'pgroup': 0, 'primary_abelian_invariants': [2, 2], 'quasisimple': False, 'rank': 2, 'rational': True, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 2]], 'representations': {'PC': {'code': 43377, 'gens': [1, 2], 'pres': [3, -2, -2, -3, 61, 16, 74]}, 'GLZ': {'b': 3, 'd': 2, 'gens': [17, 35]}, 'GLFp': {'d': 2, 'p': 3, 'gens': [31, 55, 56]}, 'Perm': {'d': 5, 'gens': [6, 1, 30]}}, 'schur_multiplier': [2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2], 'solvability_type': 6, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'D_6', 'transitive_degree': 6, 'wreath_data': None, 'wreath_product': False}