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gps_st • Show schema
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{'character_diagonal': [1, 1, 3, 5, 6, 9, 7, 21, 20, 20], 'character_matrix': [[1, 0, 0, 0, 0, 2, 0, 0, 0, 0], [0, 1, 0, 0, 2, 0, 2, 0, 3, 0], [0, 0, 3, 0, 0, 0, 0, 5, 0, 6], [0, 0, 0, 5, 0, 3, 0, 6, 0, 1], [0, 2, 0, 0, 6, 0, 6, 0, 10, 0], [2, 0, 0, 3, 0, 9, 0, 4, 0, 1], [0, 2, 0, 0, 6, 0, 7, 0, 11, 0], [0, 0, 5, 6, 0, 4, 0, 21, 0, 16], [0, 3, 0, 0, 10, 0, 11, 0, 20, 0], [0, 0, 6, 1, 0, 1, 0, 16, 0, 20]], 'component_group': '24.14', 'component_group_number': 14, 'components': 24, 'counts': [['a_1', [[0, 19]]], ['a_2', [[-2, 1], [-1, 2], [1, 2], [2, 7]]]], 'degree': 4, 'first_a2_moment': 1, 'fourth_trace_moment': 9, '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']], [['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.24.14a', 'label_components': [1, 4, 5, 24, 14, 0], 'maximal': True, 'moments': [['a_1', 1, 0, 1, 0, 9, 0, 100, 0, 1225, 0, 15876, 0, 213906], ['a_2', 1, 1, 4, 10, 44, 186, 923, 4663, 24552, 131314, 713969, 3925923, 21805501]], 'name': 'J(D_6)', 'old_label': '1.4.1.24.14a', 'pretty': 'J(D_6)', 'rational': True, 'real_dimension': 1, 'second_trace_moment': 1, 'simplex': [1, 1, 4, 4, 9, 10, 18, 42, 100, 44, 86, 206, 500, 1225, 186, 428, 1044, 2570, 6370, 15876, 923, 2193, 5421, 13490, 33747, 84798, 213906], 'st0_label': '1.4.F', 'subgroup_multiplicities': [1, 1, 2, 1, 2, 1], 'subgroups': ['1.4.F.8.5a', '1.4.F.12.4c', '1.4.F.12.4a', '1.4.F.12.4b', '1.4.F.12.4d', '1.4.F.12.5a'], 'supgroups': [], 'trace_histogram': 'data:image/png;base64,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', 'trace_zero_density': '19/24', 'weight': 1, 'zvector': [19, 1, 2, 0, 2, 7, 1, 2, 0, 2, 7]}
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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': '8.5', 'all_subgroups_known': True, 'almost_simple': False, 'aut_gens': [[1, 2, 4], [1, 14, 20], [13, 14, 20], [1, 2, 21], [1, 2, 20], [1, 10, 4]], 'aut_group': '144.183', 'aut_order': 144, 'aut_stats': [[1, 1, 1, 1], [2, 1, 3, 1], [2, 3, 4, 1], [3, 2, 1, 1], [6, 2, 3, 1]], 'cc_stats': [[1, 1, 1], [2, 1, 3], [2, 3, 4], [3, 2, 1], [6, 2, 3]], 'center_label': '4.2', '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', '2.1', '3.1'], 'composition_length': 4, 'counter': 14, 'cyclic': False, 'derived_length': 2, 'direct_factorization': [['2.1', 2], ['6.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [2, 3, 1, 4], [3, 2, 1, 1], [6, 2, 1, 3]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 28, 'exponent': 6, 'exponents_of_order': [3, 1], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [], 'frattini_label': '1.1', 'frattini_quotient': '24.14', 'hash': 14, 'hyperelementary': 2, 'irrC_degree': -1, 'irrQ_degree': -1, 'irrep_stats': [[1, 8], [2, 4]], 'label': '24.14', 'linC_degree': None, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 3, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'C2*D6', 'ngens': 4, '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': 12, 'number_divisions': 12, 'number_normal_subgroups': 21, 'number_subgroup_autclasses': 12, 'number_subgroup_classes': 32, 'number_subgroups': 54, 'old_label': None, 'order': 24, 'order_factorization_type': 31, 'order_stats': [[1, 1], [2, 15], [3, 2], [6, 6]], 'outer_equivalence': False, 'outer_group': '24.12', 'outer_order': 24, 'pc_rank': 3, 'perfect': False, 'permutation_degree': 7, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 2], 'quasisimple': False, 'rank': 3, 'rational': True, 'rational_characters_known': True, 'ratrep_stats': [[1, 8], [2, 4]], 'representations': {'PC': {'code': 5123137, 'gens': [1, 2, 3], 'pres': [4, -2, -2, -2, -3, 126, 34, 135]}, 'GLZ': {'b': 3, 'd': 3, 'gens': [16325, 16295, 3362]}, 'GLFp': {'d': 3, 'p': 3, 'gens': [7426, 8156, 16849, 13286]}, 'Perm': {'d': 7, 'gens': [127, 7, 16, 840]}}, 'schur_multiplier': [2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 2], 'solvability_type': 7, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_2\\times D_6', 'transitive_degree': 12, 'wreath_data': None, 'wreath_product': False}