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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': True, 'ambient': '528.41', 'ambient_counter': 41, 'ambient_order': 528, 'ambient_tex': 'D_{22}:C_{12}', 'central': False, 'central_factor': False, 'centralizer_order': 24, 'characteristic': False, 'core_order': 2, 'counter': 58, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '528.41.132.d1.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': False, 'old_label': '132.d1.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': 132, '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': '4.2', 'subgroup_hash': 2, 'subgroup_order': 4, 'subgroup_tex': 'C_2^2', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '528.41', 'aut_centralizer_order': 80, 'aut_label': '132.d1', 'aut_quo_index': None, 'aut_stab_index': 22, 'aut_weyl_group': '2.1', 'aut_weyl_index': 1760, 'centralizer': '22.b1.a1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['12.b1.a1', '44.d1.a1', '66.b1.a1'], 'contains': ['264.b1.a1', '264.d1.a1'], 'core': '264.b1.a1', 'coset_action_label': None, 'count': 11, 'diagramx': [735, -1, 9265, -1, 1511, -1, 1974, -1], 'generators': [1, 2], 'label': '528.41.132.d1.a1', 'mobius_quo': None, 'mobius_sub': 0, 'normal_closure': '12.b1.a1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '11.a1.a1', 'old_label': '132.d1.a1', 'projective_image': '264.14', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '132.d1.a1', 'subgroup_fusion': None, 'weyl_group': '2.1'}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': True, 'abelian_quotient': '4.2', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 6, 'aut_gen_orders': [2, 3], 'aut_gens': [[1, 2], [3, 2], [2, 3]], 'aut_group': '6.1', 'aut_hash': 1, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 6, 'aut_permdeg': 3, 'aut_perms': [1, 4], 'aut_phi_ratio': 3.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 3, 1]], 'aut_supersolvable': True, 'aut_tex': 'S_3', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 6, 'autcent_group': '6.1', 'autcent_hash': 1, 'autcent_nilpotent': False, 'autcent_order': 6, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'S_3', 'autcentquo_abelian': True, 'autcentquo_cyclic': True, 'autcentquo_exponent': 1, 'autcentquo_group': '1.1', 'autcentquo_hash': 1, 'autcentquo_nilpotent': True, 'autcentquo_order': 1, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'C_1', 'cc_stats': [[1, 1, 1], [2, 1, 3]], 'center_label': '4.2', 'center_order': 4, 'central_product': True, 'central_quotient': '1.1', 'commutator_count': 0, 'commutator_label': '1.1', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1'], 'composition_length': 2, 'conjugacy_classes_known': True, 'counter': 2, 'cyclic': False, 'derived_length': 1, 'dihedral': True, 'direct_factorization': [['2.1', 2]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3]], 'element_repr_type': 'PC', 'elementary': 2, 'eulerian_function': 1, 'exponent': 2, 'exponents_of_order': [2], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2], 'faithful_reps': [], 'familial': True, 'frattini_label': '1.1', 'frattini_quotient': '4.2', 'hash': 2, 'hyperelementary': 2, 'inner_abelian': True, 'inner_cyclic': True, 'inner_exponent': 1, 'inner_gen_orders': [1, 1], 'inner_gens': [[1, 2], [1, 2]], 'inner_hash': 1, 'inner_nilpotent': True, 'inner_order': 1, 'inner_split': True, 'inner_tex': 'C_1', 'inner_used': [], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 4]], 'label': '4.2', 'linC_count': 3, 'linC_degree': 2, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 2, 'linQ_degree_count': 3, 'linQ_dim': 2, 'linQ_dim_count': 3, 'linR_count': 3, 'linR_degree': 2, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C2^2', 'ngens': 2, 'nilpotency_class': 1, 'nilpotent': True, 'normal_counts': [0, 0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 2, 'number_characteristic_subgroups': 2, 'number_conjugacy_classes': 4, 'number_divisions': 4, 'number_normal_subgroups': 5, 'number_subgroup_autclasses': 3, 'number_subgroup_classes': 5, 'number_subgroups': 5, 'old_label': None, 'order': 4, 'order_factorization_type': 2, 'order_stats': [[1, 1], [2, 3]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 6, 'outer_gen_orders': [2, 3], 'outer_gen_pows': [0, 0], 'outer_gens': [[3, 2], [2, 3]], 'outer_group': '6.1', 'outer_hash': 1, 'outer_nilpotent': False, 'outer_order': 6, 'outer_permdeg': 3, 'outer_perms': [1, 4], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'S_3', 'pc_rank': 2, 'perfect': False, 'permutation_degree': 4, 'pgroup': 2, 'primary_abelian_invariants': [2, 2], 'quasisimple': False, 'rank': 2, 'rational': True, 'rational_characters_known': True, 'ratrep_stats': [[1, 4]], 'representations': {'PC': {'code': 0, 'gens': [1, 2], 'pres': [2, -2, 2]}, 'GLZ': {'b': 3, 'd': 2, 'gens': [14, 12]}, 'GLFp': {'d': 2, 'p': 3, 'gens': [55, 56]}, 'Perm': {'d': 4, 'gens': [6, 1]}}, 'schur_multiplier': [2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2], 'solvability_type': 1, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_2^2', 'transitive_degree': 4, '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': '24.9', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 110, 'aut_gen_orders': [2, 2, 2, 2, 2, 10, 11], 'aut_gens': [[1, 2, 4], [3, 2, 436], [265, 2, 436], [1, 2, 438], [1, 2, 172], [1, 2, 260], [1, 2, 340], [49, 2, 4]], 'aut_group': None, 'aut_hash': 2440700417464387638, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 3520, 'aut_permdeg': 50, 'aut_perms': [7603388260560293601249055475692531897323226107948198292397312495, 5579398585731067816422482964535358107562496462813445352856225500, 202400787952586245411168378681899705449862765534082913191412429, 202400717750050565770771359391918623024711166574525567911412429, 202400688990912252296876725755243065046066074274632287796186063, 344954127997840854803423963770005418315715797914304630762019298, 22975526565046682342772266098280531678590028709579317544045342858], 'aut_phi_ratio': 22.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [2, 22, 2, 1], [3, 1, 2, 1], [4, 2, 2, 1], [4, 22, 2, 1], [6, 1, 2, 3], [6, 22, 4, 1], [11, 2, 5, 1], [12, 2, 4, 1], [12, 22, 4, 1], [22, 2, 5, 3], [33, 2, 10, 1], [44, 2, 20, 1], [66, 2, 10, 3], [132, 2, 40, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_{11}:C_5.C_2^6', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 2, 'autcent_group': '32.51', 'autcent_hash': 51, 'autcent_nilpotent': True, 'autcent_order': 32, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^5', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 110, 'autcentquo_group': '110.1', 'autcentquo_hash': 1, 'autcentquo_nilpotent': False, 'autcentquo_order': 110, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'F_{11}', 'cc_stats': [[1, 1, 1], [2, 1, 3], [2, 22, 2], [3, 1, 2], [4, 2, 2], [4, 22, 2], [6, 1, 6], [6, 22, 4], [11, 2, 5], [12, 2, 4], [12, 22, 4], [22, 2, 15], [33, 2, 10], [44, 2, 20], [66, 2, 30], [132, 2, 40]], 'center_label': '12.5', 'center_order': 12, 'central_product': True, 'central_quotient': '44.3', 'commutator_count': 1, 'commutator_label': '22.2', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '3.1', '11.1'], 'composition_length': 6, 'conjugacy_classes_known': True, 'counter': 41, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['176.13', 1], ['3.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [2, 22, 1, 2], [3, 1, 2, 1], [4, 2, 2, 1], [4, 22, 2, 1], [6, 1, 2, 3], [6, 22, 2, 2], [11, 2, 5, 1], [12, 2, 4, 1], [12, 22, 4, 1], [22, 2, 5, 3], [33, 2, 10, 1], [44, 2, 20, 1], [66, 2, 10, 3], [132, 2, 40, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 24, 'exponent': 132, 'exponents_of_order': [4, 1, 1], 'factors_of_aut_order': [2, 5, 11], 'factors_of_order': [2, 3, 11], 'faithful_reps': [], 'familial': False, 'frattini_label': '4.2', 'frattini_quotient': '132.7', 'hash': 41, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 22, 'inner_gen_orders': [2, 1, 22], 'inner_gens': [[1, 2, 438], [1, 2, 4], [99, 2, 4]], 'inner_hash': 3, 'inner_nilpotent': False, 'inner_order': 44, 'inner_split': True, 'inner_tex': 'D_{22}', 'inner_used': [1, 3], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 24], [2, 126]], 'label': '528.41', 'linC_count': 640, 'linC_degree': 3, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 16, 'linQ_degree_count': 60, 'linQ_dim': 16, 'linQ_dim_count': 60, 'linR_count': 40, 'linR_degree': 4, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'D22:C12', '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], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 24, 'number_characteristic_subgroups': 30, 'number_conjugacy_classes': 150, 'number_divisions': 26, 'number_normal_subgroups': 34, 'number_subgroup_autclasses': 60, 'number_subgroup_classes': 68, 'number_subgroups': 392, 'old_label': None, 'order': 528, 'order_factorization_type': 311, 'order_stats': [[1, 1], [2, 47], [3, 2], [4, 48], [6, 94], [11, 10], [12, 96], [22, 30], [33, 20], [44, 40], [66, 60], [132, 80]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 10, 'outer_gen_orders': [2, 2, 2, 10], 'outer_gen_pows': [0, 0, 0, 0], 'outer_gens': [[3, 2, 436], [265, 2, 436], [1, 2, 172], [3, 2, 164]], 'outer_group': '80.52', 'outer_hash': 52, 'outer_nilpotent': True, 'outer_order': 80, 'outer_permdeg': 13, 'outer_perms': [720, 479001600, 3628800, 40353], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^3\\times C_{10}', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 22, 'pgroup': 0, 'primary_abelian_invariants': [2, 4, 3], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 8], [4, 4], [10, 2], [20, 5], [40, 3]], 'representations': {'PC': {'code': 204607384722946170815452000019, 'gens': [1, 2, 3], 'pres': [6, -2, -2, 2, -2, -3, -11, 7886, 50, 8259, 69, 4804, 118, 17285]}, 'GLZN': {'d': 2, 'p': 115, 'gens': [16804976, 142356546, 36501024, 93510556, 138399716, 125310427]}, 'Perm': {'d': 22, 'gens': [87657660847, 53645489293227955200, 376610217984000, 107047688359772160000, 186810624000, 4364893]}}, 'schur_multiplier': [2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 12], 'solvability_type': 7, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'D_{22}:C_{12}', 'transitive_degree': 264, 'wreath_data': None, 'wreath_product': False}
