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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': False, 'ambient': '540.58', 'ambient_counter': 58, 'ambient_order': 540, 'ambient_tex': 'C_{45}:A_4', 'central': False, 'central_factor': False, 'centralizer_order': 15, 'characteristic': False, 'core_order': 20, 'counter': 10, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '540.58.9.b1.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': False, 'minimal': False, 'minimal_normal': False, 'nilpotent': False, 'normal': False, 'old_label': '9.b1.a1', 'outer_equivalence': False, 'perfect': False, 'proper': True, 'quotient': None, 'quotient_Agroup': None, 'quotient_abelian': None, 'quotient_cyclic': None, 'quotient_hash': None, 'quotient_metabelian': None, 'quotient_nilpotent': None, 'quotient_order': 9, 'quotient_simple': None, 'quotient_solvable': None, 'quotient_supersolvable': None, 'quotient_tex': None, 'simple': False, 'solvable': True, 'special_labels': [], 'split': None, 'standard_generators': False, 'stem': False, 'subgroup': '60.9', 'subgroup_hash': 9, 'subgroup_order': 60, 'subgroup_tex': 'C_5\\times A_4', 'supersolvable': False, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '540.58', 'aut_centralizer_order': 6, 'aut_label': '9.b1', 'aut_quo_index': None, 'aut_stab_index': 3, 'aut_weyl_group': '48.31', 'aut_weyl_index': 18, 'centralizer': '36.a1.a1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['3.a1.a1'], 'contains': ['27.a1.a1', '36.b1.a1', '45.b1.a1'], 'core': '27.a1.a1', 'coset_action_label': None, 'count': 3, 'diagramx': [9442, -1, 9725, -1, 762, -1, 992, -1], 'generators': [1, 285, 270, 6], 'label': '540.58.9.b1.a1', 'mobius_quo': None, 'mobius_sub': 0, 'normal_closure': '3.a1.a1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '3.a1.a1', 'old_label': '9.b1.a1', 'projective_image': '108.19', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '9.b1.a1', 'subgroup_fusion': None, 'weyl_group': '12.3'}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '15.1', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 3, 'aut_exponent': 12, 'aut_gen_orders': [2, 2, 4, 4, 2, 6], 'aut_gens': [[1, 3, 6], [1, 3, 54], [34, 3, 54], [4, 3, 18], [1, 3, 42], [35, 33, 54], [31, 30, 57]], 'aut_group': '96.186', 'aut_hash': 186, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 96, 'aut_permdeg': 8, 'aut_perms': [16, 5176, 16689, 18, 15960, 10936], 'aut_phi_ratio': 6.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 3, 1, 1], [3, 4, 2, 1], [5, 1, 4, 1], [10, 3, 4, 1], [15, 4, 8, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_4\\times S_4', 'autcent_abelian': True, 'autcent_cyclic': True, 'autcent_exponent': 4, 'autcent_group': '4.1', 'autcent_hash': 1, 'autcent_nilpotent': True, 'autcent_order': 4, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_4', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 12, 'autcentquo_group': '24.12', 'autcentquo_hash': 12, 'autcentquo_nilpotent': False, 'autcentquo_order': 24, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'S_4', 'cc_stats': [[1, 1, 1], [2, 3, 1], [3, 4, 2], [5, 1, 4], [10, 3, 4], [15, 4, 8]], 'center_label': '5.1', 'center_order': 5, 'central_product': True, 'central_quotient': '12.3', 'commutator_count': 1, 'commutator_label': '4.2', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '3.1', '5.1'], 'composition_length': 4, 'conjugacy_classes_known': True, 'counter': 9, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['12.3', 1], ['5.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 3, 1, 1], [3, 4, 2, 1], [5, 1, 4, 1], [10, 3, 4, 1], [15, 4, 8, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 24, 'exponent': 30, 'exponents_of_order': [2, 1, 1], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3, 5], 'faithful_reps': [[3, 0, 4]], 'familial': False, 'frattini_label': '1.1', 'frattini_quotient': '60.9', 'hash': 9, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 6, 'inner_gen_orders': [3, 2, 2], 'inner_gens': [[1, 30, 9], [34, 3, 6], [4, 3, 6]], 'inner_hash': 3, 'inner_nilpotent': False, 'inner_order': 12, 'inner_split': True, 'inner_tex': 'A_4', 'inner_used': [1, 2], 'irrC_degree': 3, 'irrQ_degree': 12, 'irrQ_dim': 12, 'irrR_degree': 6, 'irrep_stats': [[1, 15], [3, 5]], 'label': '60.9', 'linC_count': 4, 'linC_degree': 3, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 7, 'linQ_degree_count': 1, 'linQ_dim': 7, 'linQ_dim_count': 1, 'linR_count': 6, 'linR_degree': 5, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'C5*A4', 'ngens': 4, 'nilpotency_class': -1, 'nilpotent': False, 'normal_counts': [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 6, 'number_characteristic_subgroups': 6, 'number_conjugacy_classes': 20, 'number_divisions': 6, 'number_normal_subgroups': 6, 'number_subgroup_autclasses': 10, 'number_subgroup_classes': 10, 'number_subgroups': 20, 'old_label': None, 'order': 60, 'order_factorization_type': 222, 'order_stats': [[1, 1], [2, 3], [3, 8], [5, 4], [10, 12], [15, 32]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 4, 'outer_gen_orders': [2, 4], 'outer_gen_pows': [0, 0], 'outer_gens': [[2, 30, 27], [1, 3, 18]], 'outer_group': '8.2', 'outer_hash': 2, 'outer_nilpotent': True, 'outer_order': 8, 'outer_permdeg': 6, 'outer_perms': [120, 9], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2\\times C_4', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 9, 'pgroup': 0, 'primary_abelian_invariants': [3, 5], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 1], [2, 1], [3, 1], [4, 1], [8, 1], [12, 1]], 'representations': {'PC': {'code': 4852423747, 'gens': [1, 2, 3], 'pres': [4, -3, -2, 2, -5, 241, 110, 34]}, 'GLFp': {'d': 3, 'p': 11, 'gens': [656967451, 1691722594, 857494092, 1907820797]}, 'Perm': {'d': 9, 'gens': [5760, 33, 41040, 90720]}}, 'schur_multiplier': [2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [15], 'solvability_type': 8, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': 'C_5\\times A_4', 'transitive_degree': 20, 'wreath_data': None, 'wreath_product': False}
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gps_groups • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '45.2', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 12, 'aut_gen_orders': [4, 6, 6, 6], 'aut_gens': [[1, 3, 30], [1, 9, 30], [1, 273, 255], [271, 3, 210], [196, 3, 30]], 'aut_group': '864.4442', 'aut_hash': 4442, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 864, 'aut_permdeg': 22, 'aut_perms': [704172679344, 104629399848818058855, 908472515748305444675, 22101012625669989300], 'aut_phi_ratio': 6.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 3, 1, 1], [3, 1, 2, 1], [3, 12, 1, 2], [5, 1, 4, 1], [6, 3, 2, 1], [9, 3, 2, 1], [9, 12, 2, 2], [10, 3, 4, 1], [15, 1, 8, 1], [15, 12, 4, 2], [18, 3, 6, 1], [30, 3, 8, 1], [45, 3, 8, 1], [45, 12, 8, 2], [90, 3, 24, 1]], 'aut_supersolvable': False, 'aut_tex': 'S_3\\times C_{12}\\times A_4', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 12, 'autcent_group': '36.8', 'autcent_hash': 8, 'autcent_nilpotent': True, 'autcent_order': 36, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_3\\times C_{12}', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 6, 'autcentquo_group': '24.13', 'autcentquo_hash': 13, 'autcentquo_nilpotent': False, 'autcentquo_order': 24, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'C_2\\times A_4', 'cc_stats': [[1, 1, 1], [2, 3, 1], [3, 1, 2], [3, 12, 2], [5, 1, 4], [6, 3, 2], [9, 3, 2], [9, 12, 4], [10, 3, 4], [15, 1, 8], [15, 12, 8], [18, 3, 6], [30, 3, 8], [45, 3, 8], [45, 12, 16], [90, 3, 24]], 'center_label': '15.1', 'center_order': 15, 'central_product': True, 'central_quotient': '36.11', 'commutator_count': 1, 'commutator_label': '12.5', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '3.1', '3.1', '3.1', '5.1'], 'composition_length': 6, 'conjugacy_classes_known': True, 'counter': 58, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['108.19', 1], ['5.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 3, 1, 1], [3, 1, 2, 1], [3, 12, 2, 1], [5, 1, 4, 1], [6, 3, 2, 1], [9, 3, 2, 1], [9, 12, 2, 2], [10, 3, 4, 1], [15, 1, 8, 1], [15, 12, 8, 1], [18, 3, 6, 1], [30, 3, 8, 1], [45, 3, 8, 1], [45, 12, 8, 2], [90, 3, 24, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 144, 'exponent': 90, 'exponents_of_order': [3, 2, 1], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3, 5], 'faithful_reps': [[3, 0, 24]], 'familial': False, 'frattini_label': '3.1', 'frattini_quotient': '180.31', 'hash': 58, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 6, 'inner_gen_orders': [3, 2, 6], 'inner_gens': [[1, 288, 405], [286, 3, 30], [196, 3, 30]], 'inner_hash': 11, 'inner_nilpotent': False, 'inner_order': 36, 'inner_split': True, 'inner_tex': 'C_3\\times A_4', 'inner_used': [1, 2, 3], 'irrC_degree': 3, 'irrQ_degree': 72, 'irrQ_dim': 72, 'irrR_degree': 6, 'irrep_stats': [[1, 45], [3, 55]], 'label': '540.58', 'linC_count': 24, 'linC_degree': 3, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 13, 'linQ_degree_count': 1, 'linQ_dim': 13, 'linQ_dim_count': 1, 'linR_count': 12, 'linR_degree': 6, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'C45:A4', 'ngens': 6, 'nilpotency_class': -1, 'nilpotent': False, 'normal_counts': [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 20, 'number_characteristic_subgroups': 16, 'number_conjugacy_classes': 100, 'number_divisions': 18, 'number_normal_subgroups': 20, 'number_subgroup_autclasses': 34, 'number_subgroup_classes': 38, 'number_subgroups': 100, 'old_label': None, 'order': 540, 'order_factorization_type': 321, 'order_stats': [[1, 1], [2, 3], [3, 26], [5, 4], [6, 6], [9, 54], [10, 12], [15, 104], [18, 18], [30, 24], [45, 216], [90, 72]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 12, 'outer_gen_orders': [2, 12], 'outer_gen_pows': [0, 0], 'outer_gens': [[1, 3, 510], [1, 21, 150]], 'outer_group': '24.9', 'outer_hash': 9, 'outer_nilpotent': True, 'outer_order': 24, 'outer_permdeg': 9, 'outer_perms': [42000, 867], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2\\times C_{12}', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 18, 'pgroup': 0, 'primary_abelian_invariants': [3, 3, 5], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 1], [2, 4], [3, 1], [4, 1], [6, 2], [8, 4], [12, 1], [18, 1], [24, 2], [72, 1]], 'representations': {'PC': {'code': 244049928211816880252101, 'gens': [1, 2, 4], 'pres': [6, 3, 2, 5, 2, 3, 3, 3457, 31, 9723, 69, 7204, 118]}, 'Perm': {'d': 18, 'gens': [22230464267268, 57401, 6749568000, 93313, 356995102464000, 753220435968000]}}, 'schur_multiplier': [2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [3, 15], 'solvability_type': 8, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': 'C_{45}:A_4', 'transitive_degree': 180, 'wreath_data': None, 'wreath_product': False}
