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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': True, 'ambient': '1760.61', 'ambient_counter': 61, 'ambient_order': 1760, 'ambient_tex': 'C_{88}.C_{20}', 'central': False, 'central_factor': False, 'centralizer_order': 880, 'characteristic': True, 'core_order': 4, 'counter': 55, 'cyclic': False, 'direct': False, 'hall': 0, 'label': '1760.61.440.a1.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': True, 'old_label': '440.a1.a1', 'outer_equivalence': False, 'perfect': False, 'proper': True, 'quotient': '440.14', 'quotient_Agroup': False, 'quotient_abelian': False, 'quotient_cyclic': False, 'quotient_hash': 14, 'quotient_metabelian': True, 'quotient_nilpotent': False, 'quotient_order': 440, 'quotient_simple': False, 'quotient_solvable': True, 'quotient_supersolvable': True, 'quotient_tex': 'C_{44}.C_{10}', 'simple': False, 'solvable': True, 'special_labels': [], 'split': False, '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': '1760.61', 'aut_centralizer_order': 3520, 'aut_label': '440.a1', 'aut_quo_index': 3, 'aut_stab_index': 1, 'aut_weyl_group': '2.1', 'aut_weyl_index': 3520, 'centralizer': '2.a1.a1', 'complements': [], 'conjugacy_class_count': 1, 'contained_in': ['40.a1.a1', '88.a1.a1', '220.a1.a1'], 'contains': ['880.a1.a1', '880.b1.a1'], 'core': '440.a1.a1', 'coset_action_label': None, 'count': 1, 'diagramx': [2015, 4900, 1941, 3365, 5423, 5914, 6213, 6896], 'generators': [450, 880], 'label': '1760.61.440.a1.a1', 'mobius_quo': 0, 'mobius_sub': 0, 'normal_closure': '440.a1.a1', 'normal_contained_in': ['40.a1.a1', '220.a1.a1'], 'normal_contains': ['880.a1.a1'], 'normalizer': '1.a1.a1', 'old_label': '440.a1.a1', 'projective_image': '880.25', 'quotient_action_image': '2.1', 'quotient_action_kernel': '220.2', 'quotient_action_kernel_order': 220, 'quotient_fusion': None, 'short_label': '440.a1.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': '40.9', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 440, 'aut_gen_orders': [10, 10, 20, 10, 20, 10], 'aut_gens': [[1, 20], [1661, 940], [1021, 780], [391, 380], [601, 1580], [1631, 380], [1051, 980]], 'aut_group': None, 'aut_hash': 5249673490414127957, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 7040, 'aut_permdeg': 27, 'aut_perms': [65755183279555494639743721, 9358841512882030359112039407, 5168228772964585534787197131, 3823919013029100125620871353, 3724444232401146138377707028, 9995396706278448593432075210], 'aut_phi_ratio': 11.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 2, 1, 1], [4, 1, 2, 1], [4, 2, 1, 1], [5, 11, 1, 4], [8, 2, 2, 2], [8, 4, 4, 1], [10, 11, 1, 4], [10, 22, 1, 4], [11, 5, 2, 1], [20, 11, 2, 4], [20, 22, 1, 4], [22, 5, 2, 1], [22, 10, 2, 1], [40, 22, 2, 8], [40, 44, 4, 4], [44, 5, 4, 1], [44, 10, 2, 1], [88, 10, 4, 2], [88, 20, 8, 1]], 'aut_supersolvable': True, 'aut_tex': '(C_4\\times F_{11}).C_2^4', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 2, 'autcent_group': '8.5', 'autcent_hash': 5, 'autcent_nilpotent': True, 'autcent_order': 8, 'autcent_solvable': True, 'autcent_split': False, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^3', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 220, 'autcentquo_group': '880.118', 'autcentquo_hash': 118, 'autcentquo_nilpotent': False, 'autcentquo_order': 880, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'D_4\\times F_{11}', 'cc_stats': [[1, 1, 1], [2, 1, 1], [2, 2, 1], [4, 1, 2], [4, 2, 1], [5, 11, 4], [8, 2, 4], [8, 4, 4], [10, 11, 4], [10, 22, 4], [11, 5, 2], [20, 11, 8], [20, 22, 4], [22, 5, 2], [22, 10, 2], [40, 22, 16], [40, 44, 16], [44, 5, 4], [44, 10, 2], [88, 10, 8], [88, 20, 8]], 'center_label': '4.1', 'center_order': 4, 'central_product': True, 'central_quotient': '440.13', 'commutator_count': 1, 'commutator_label': '44.2', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '2.1', '5.1', '11.1'], 'composition_length': 7, 'conjugacy_classes_known': True, 'counter': 61, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['32.15', 1], ['55.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 2, 1, 1], [4, 1, 2, 1], [4, 2, 1, 1], [5, 11, 4, 1], [8, 2, 2, 2], [8, 4, 2, 2], [10, 11, 4, 1], [10, 22, 4, 1], [11, 5, 2, 1], [20, 11, 8, 1], [20, 22, 4, 1], [22, 5, 2, 1], [22, 10, 2, 1], [40, 22, 8, 2], [40, 44, 8, 2], [44, 5, 4, 1], [44, 10, 2, 1], [88, 10, 4, 2], [88, 20, 4, 2]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 144, 'exponent': 440, 'exponents_of_order': [5, 1, 1], 'factors_of_aut_order': [2, 5, 11], 'factors_of_order': [2, 5, 11], 'faithful_reps': [[10, 0, 8]], 'familial': False, 'frattini_label': '8.2', 'frattini_quotient': '220.8', 'hash': 61, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 220, 'inner_gen_orders': [10, 44], 'inner_gens': [[1, 620], [1161, 20]], 'inner_hash': 13, 'inner_nilpotent': False, 'inner_order': 440, 'inner_split': False, 'inner_tex': 'C_{44}:C_{10}', 'inner_used': [1, 2], 'irrC_degree': 10, 'irrQ_degree': 80, 'irrQ_dim': 80, 'irrR_degree': 20, 'irrep_stats': [[1, 40], [2, 30], [5, 16], [10, 12]], 'label': '1760.61', 'linC_count': 320, 'linC_degree': 7, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 18, 'linQ_degree_count': 4, 'linQ_dim': 18, 'linQ_dim_count': 4, 'linR_count': 80, 'linR_degree': 14, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C88.C20', 'ngens': 7, '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': 48, 'number_characteristic_subgroups': 30, 'number_conjugacy_classes': 98, 'number_divisions': 27, 'number_normal_subgroups': 36, 'number_subgroup_autclasses': 52, 'number_subgroup_classes': 60, 'number_subgroups': 252, 'old_label': None, 'order': 1760, 'order_factorization_type': 311, 'order_stats': [[1, 1], [2, 3], [4, 4], [5, 44], [8, 24], [10, 132], [11, 10], [20, 176], [22, 30], [40, 1056], [44, 40], [88, 240]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 2, 'outer_gen_orders': [2, 2, 2, 2], 'outer_gen_pows': [0, 0, 0, 1100], 'outer_gens': [[881, 1300], [1, 420], [11, 1300], [221, 1300]], 'outer_group': '16.14', 'outer_hash': 14, 'outer_nilpotent': True, 'outer_order': 16, 'outer_permdeg': 8, 'outer_perms': [5040, 120, 6, 1], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^4', 'pc_rank': 2, 'perfect': False, 'permutation_degree': 27, 'pgroup': 0, 'primary_abelian_invariants': [2, 4, 5], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 4], [4, 4], [8, 5], [10, 4], [20, 4], [32, 1], [80, 1]], 'representations': {'PC': {'code': 80114038322766087888723706430144200975616145372186891903429466230030009, 'gens': [1, 4], 'pres': [7, -2, -2, -5, -2, -2, -2, -11, 14, 36, 4636, 17363, 22690, 6877, 80, 43404, 25911, 1768, 102, 30245, 25212, 4219, 124, 70566, 15693, 9820]}, 'Perm': {'d': 27, 'gens': [1621638512130, 3067916261725, 16833093532491537100800000, 4504061280061, 5897117537101, 7300107214540, 436283137921489848557568000]}}, 'schur_multiplier': [], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 20], 'solvability_type': 6, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_{88}.C_{20}', 'transitive_degree': 176, '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': '20.5', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 3, 'aut_exponent': 660, 'aut_gen_orders': [20, 30, 2, 22], 'aut_gens': [[1, 10], [211, 410], [251, 65], [221, 10], [21, 230]], 'aut_group': '2640.bv', 'aut_hash': 5844094576270439126, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 2640, 'aut_permdeg': 17, 'aut_perms': [19494048661727, 207791732463532, 704176305845, 144672623319248], 'aut_phi_ratio': 16.5, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [4, 2, 3, 1], [5, 11, 1, 4], [10, 11, 1, 4], [11, 5, 2, 1], [20, 22, 3, 4], [22, 5, 2, 1], [44, 10, 6, 1]], 'aut_supersolvable': False, 'aut_tex': 'S_4\\times F_{11}', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 2, 'autcent_group': '4.2', 'autcent_hash': 2, 'autcent_nilpotent': True, 'autcent_order': 4, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^2', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 330, 'autcentquo_group': '660.15', 'autcentquo_hash': 15, 'autcentquo_nilpotent': False, 'autcentquo_order': 660, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'S_3\\times F_{11}', 'cc_stats': [[1, 1, 1], [2, 1, 1], [4, 2, 3], [5, 11, 4], [10, 11, 4], [11, 5, 2], [20, 22, 12], [22, 5, 2], [44, 10, 6]], 'center_label': '2.1', 'center_order': 2, 'central_product': True, 'central_quotient': '220.8', 'commutator_count': 1, 'commutator_label': '22.2', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '5.1', '11.1'], 'composition_length': 5, 'conjugacy_classes_known': True, 'counter': 14, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['55.1', 1], ['8.4', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [4, 2, 1, 3], [5, 11, 4, 1], [10, 11, 4, 1], [11, 5, 2, 1], [20, 22, 4, 3], [22, 5, 2, 1], [44, 10, 2, 3]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 24, 'exponent': 220, 'exponents_of_order': [3, 1, 1], 'factors_of_aut_order': [2, 3, 5, 11], 'factors_of_order': [2, 5, 11], 'faithful_reps': [[10, 0, 2]], 'familial': False, 'frattini_label': '2.1', 'frattini_quotient': '220.8', 'hash': 14, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 110, 'inner_gen_orders': [10, 22], 'inner_gens': [[1, 150], [301, 10]], 'inner_hash': 8, 'inner_nilpotent': False, 'inner_order': 220, 'inner_split': True, 'inner_tex': 'C_{22}:C_{10}', 'inner_used': [1, 2], 'irrC_degree': 10, 'irrQ_degree': 20, 'irrQ_dim': 20, 'irrR_degree': 20, 'irrep_stats': [[1, 20], [2, 5], [5, 8], [10, 2]], 'label': '440.14', 'linC_count': 40, 'linC_degree': 7, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 12, 'linQ_degree_count': 4, 'linQ_dim': 14, 'linQ_dim_count': 4, 'linR_count': 12, 'linR_degree': 14, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C44.C10', 'ngens': 5, 'nilpotency_class': -1, 'nilpotent': False, 'normal_counts': [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': 18, 'number_characteristic_subgroups': 9, 'number_conjugacy_classes': 35, 'number_divisions': 15, 'number_normal_subgroups': 18, 'number_subgroup_autclasses': 16, 'number_subgroup_classes': 24, 'number_subgroups': 84, 'old_label': None, 'order': 440, 'order_factorization_type': 311, 'order_stats': [[1, 1], [2, 1], [4, 6], [5, 44], [10, 44], [11, 10], [20, 264], [22, 10], [44, 60]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 6, 'outer_gen_orders': [2, 6], 'outer_gen_pows': [330, 0], 'outer_gens': [[111, 210], [111, 105]], 'outer_group': '12.4', 'outer_hash': 4, 'outer_nilpotent': False, 'outer_order': 12, 'outer_permdeg': 5, 'outer_perms': [6, 31], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'D_6', 'pc_rank': 2, 'perfect': False, 'permutation_degree': 19, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 5], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 1], [4, 4], [8, 1], [10, 4], [20, 1]], 'representations': {'PC': {'code': 4055531876230862578582624868703009, 'gens': [1, 3], 'pres': [5, -2, -5, -2, -2, -11, 10, 1106, 2252, 382, 42, 6003, 1008, 58, 4004, 2509]}, 'GLZN': {'d': 2, 'p': 33, 'gens': [35962, 826574, 36037, 384086, 48302]}, 'Perm': {'d': 19, 'gens': [7039, 12593, 400434839844480, 18619, 7159885125120000]}}, 'schur_multiplier': [], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 10], 'solvability_type': 6, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_{44}.C_{10}', 'transitive_degree': 88, 'wreath_data': None, 'wreath_product': False}
