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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': True, 'ambient': '1760.418', 'ambient_counter': 418, 'ambient_order': 1760, 'ambient_tex': '(C_{11}\\times Q_{16}):C_{10}', 'central': False, 'central_factor': False, 'centralizer_order': 176, 'characteristic': True, 'core_order': 44, 'counter': 59, 'cyclic': True, 'direct': False, 'hall': 0, 'label': '1760.418.40.b1.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': True, 'old_label': '40.b1.a1', 'outer_equivalence': False, 'perfect': False, 'proper': True, 'quotient': '40.10', 'quotient_Agroup': False, 'quotient_abelian': False, 'quotient_cyclic': False, 'quotient_hash': 10, 'quotient_metabelian': True, 'quotient_nilpotent': True, 'quotient_order': 40, 'quotient_simple': False, 'quotient_solvable': True, 'quotient_supersolvable': True, 'quotient_tex': 'C_5\\times D_4', 'simple': False, 'solvable': True, 'special_labels': [], 'split': False, 'standard_generators': False, 'stem': False, 'subgroup': '44.2', 'subgroup_hash': 2, 'subgroup_order': 44, 'subgroup_tex': 'C_{44}', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '1760.418', 'aut_centralizer_order': 352, 'aut_label': '40.b1', 'aut_quo_index': 8, 'aut_stab_index': 1, 'aut_weyl_group': '20.5', 'aut_weyl_index': 352, 'centralizer': '10.a1.a1', 'complements': [], 'conjugacy_class_count': 1, 'contained_in': ['8.b1.a1', '20.a1.a1', '20.h1.a1', '20.i1.a1'], 'contains': ['80.a1.a1', '440.b1.a1'], 'core': '40.b1.a1', 'coset_action_label': None, 'count': 1, 'diagramx': [4066, 1495, 2971, 1266, 3263, 4766, 2988, 4809], 'generators': [1330, 880, 160], 'label': '1760.418.40.b1.a1', 'mobius_quo': 0, 'mobius_sub': 0, 'normal_closure': '40.b1.a1', 'normal_contained_in': ['8.b1.a1', '20.a1.a1'], 'normal_contains': ['80.a1.a1', '440.b1.a1'], 'normalizer': '1.a1.a1', 'old_label': '40.b1.a1', 'projective_image': '880.132', 'quotient_action_image': '10.2', 'quotient_action_kernel': '4.2', 'quotient_action_kernel_order': 4, 'quotient_fusion': None, 'short_label': '40.b1.a1', 'subgroup_fusion': None, 'weyl_group': '10.2'}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': True, 'abelian_quotient': '44.2', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': True, 'aut_cyclic': False, 'aut_derived_length': 1, 'aut_exponent': 10, 'aut_gen_orders': [2, 10], 'aut_gens': [[1], [21], [31]], 'aut_group': '20.5', 'aut_hash': 5, 'aut_nilpotency_class': 1, 'aut_nilpotent': True, 'aut_order': 20, 'aut_permdeg': 9, 'aut_perms': [720, 40353], 'aut_phi_ratio': 1.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [4, 1, 2, 1], [11, 1, 10, 1], [22, 1, 10, 1], [44, 1, 20, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_2\\times C_{10}', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 10, 'autcent_group': '20.5', 'autcent_hash': 5, 'autcent_nilpotent': True, 'autcent_order': 20, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2\\times C_{10}', '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, 1], [4, 1, 2], [11, 1, 10], [22, 1, 10], [44, 1, 20]], 'center_label': '44.2', 'center_order': 44, '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', '11.1'], 'composition_length': 3, 'conjugacy_classes_known': True, 'counter': 2, 'cyclic': True, 'derived_length': 1, 'dihedral': False, 'direct_factorization': [['11.1', 1], ['4.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [4, 1, 2, 1], [11, 1, 10, 1], [22, 1, 10, 1], [44, 1, 20, 1]], 'element_repr_type': 'PC', 'elementary': 22, 'eulerian_function': 1, 'exponent': 44, 'exponents_of_order': [2, 1], 'factors_of_aut_order': [2, 5], 'factors_of_order': [2, 11], 'faithful_reps': [[1, 0, 20]], 'familial': True, 'frattini_label': '2.1', 'frattini_quotient': '22.2', 'hash': 2, 'hyperelementary': 22, 'inner_abelian': True, 'inner_cyclic': True, 'inner_exponent': 1, 'inner_gen_orders': [1], 'inner_gens': [[1]], 'inner_hash': 1, 'inner_nilpotent': True, 'inner_order': 1, 'inner_split': True, 'inner_tex': 'C_1', 'inner_used': [], 'irrC_degree': 1, 'irrQ_degree': 20, 'irrQ_dim': 20, 'irrR_degree': 2, 'irrep_stats': [[1, 44]], 'label': '44.2', 'linC_count': 20, 'linC_degree': 1, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 12, 'linQ_degree_count': 2, 'linQ_dim': 12, 'linQ_dim_count': 2, 'linR_count': 10, 'linR_degree': 2, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C44', 'ngens': 3, 'nilpotency_class': 1, 'nilpotent': True, 'normal_counts': [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': 44, 'number_divisions': 6, 'number_normal_subgroups': 6, 'number_subgroup_autclasses': 6, 'number_subgroup_classes': 6, 'number_subgroups': 6, 'old_label': None, 'order': 44, 'order_factorization_type': 22, 'order_stats': [[1, 1], [2, 1], [4, 2], [11, 10], [22, 10], [44, 20]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 10, 'outer_gen_orders': [2, 10], 'outer_gen_pows': [0, 0], 'outer_gens': [[21], [31]], 'outer_group': '20.5', 'outer_hash': 5, 'outer_nilpotent': True, 'outer_order': 20, 'outer_permdeg': 9, 'outer_perms': [720, 40353], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2\\times C_{10}', 'pc_rank': 1, 'perfect': False, 'permutation_degree': 15, 'pgroup': 0, 'primary_abelian_invariants': [4, 11], 'quasisimple': False, 'rank': 1, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 2], [2, 1], [10, 2], [20, 1]], 'representations': {'PC': {'code': 1763009, 'gens': [1], 'pres': [3, -2, -2, -11, 6, 16]}, 'GLFp': {'d': 2, 'p': 23, 'gens': [3335]}, 'Perm': {'d': 15, 'gens': [273988915200, 36288000, 87657292800]}}, 'schur_multiplier': [], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [44], 'solvability_type': 0, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_{44}', 'transitive_degree': 44, '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': '40.14', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 220, 'aut_gen_orders': [2, 20, 20, 10, 10, 10], 'aut_gens': [[1, 2, 20], [881, 1282, 1740], [441, 1202, 1590], [1321, 722, 100], [1, 402, 1140], [1321, 402, 1660], [441, 1522, 1180]], 'aut_group': None, 'aut_hash': 2609713509392042030, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 7040, 'aut_permdeg': 23, 'aut_perms': [21009707227327692753963, 13205767901623387011487, 125493105322493934930, 10479748454258682517019, 20932878672245288136612, 20320754940603165672614], 'aut_phi_ratio': 11.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 2, 1, 1], [2, 4, 1, 1], [4, 2, 1, 2], [4, 4, 1, 1], [4, 4, 2, 1], [5, 11, 1, 4], [8, 4, 2, 1], [10, 11, 1, 4], [10, 22, 1, 4], [10, 44, 1, 4], [11, 5, 2, 1], [20, 22, 1, 8], [20, 44, 1, 4], [20, 44, 2, 4], [22, 5, 2, 1], [22, 10, 2, 1], [22, 20, 2, 1], [40, 44, 2, 4], [44, 10, 2, 2], [44, 20, 2, 1], [44, 20, 4, 1], [88, 20, 4, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_{11}.(C_5\\times D_4^2).C_2', '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': 110, 'autcentquo_group': '880.210', 'autcentquo_hash': 210, 'autcentquo_nilpotent': False, 'autcentquo_order': 880, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'C_2^3\\times F_{11}', 'cc_stats': [[1, 1, 1], [2, 1, 1], [2, 2, 1], [2, 4, 1], [4, 2, 2], [4, 4, 3], [5, 11, 4], [8, 4, 2], [10, 11, 4], [10, 22, 4], [10, 44, 4], [11, 5, 2], [20, 22, 8], [20, 44, 12], [22, 5, 2], [22, 10, 2], [22, 20, 2], [40, 44, 8], [44, 10, 4], [44, 20, 6], [88, 20, 4]], 'center_label': '2.1', 'center_order': 2, 'central_product': True, 'central_quotient': '880.132', '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': 418, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['32.44', 1], ['55.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 2, 1, 1], [2, 4, 1, 1], [4, 2, 1, 2], [4, 4, 1, 3], [5, 11, 4, 1], [8, 4, 1, 2], [10, 11, 4, 1], [10, 22, 4, 1], [10, 44, 4, 1], [11, 5, 2, 1], [20, 22, 4, 2], [20, 44, 4, 3], [22, 5, 2, 1], [22, 10, 2, 1], [22, 20, 2, 1], [40, 44, 4, 2], [44, 10, 2, 2], [44, 20, 2, 3], [88, 20, 2, 2]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 249984, 'exponent': 440, 'exponents_of_order': [5, 1, 1], 'factors_of_aut_order': [2, 5, 11], 'factors_of_order': [2, 5, 11], 'faithful_reps': [[20, 0, 2]], 'familial': False, 'frattini_label': '4.1', 'frattini_quotient': '440.44', 'hash': 418, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 220, 'inner_gen_orders': [2, 10, 44], 'inner_gens': [[1, 882, 1340], [881, 2, 740], [441, 1042, 20]], 'inner_hash': 132, 'inner_nilpotent': False, 'inner_order': 880, 'inner_split': True, 'inner_tex': 'C_2\\times C_{44}:C_{10}', 'inner_used': [1, 2, 3], 'irrC_degree': 20, 'irrQ_degree': 40, 'irrQ_dim': 40, 'irrR_degree': 40, 'irrep_stats': [[1, 40], [2, 10], [4, 5], [5, 16], [10, 4], [20, 2]], 'label': '1760.418', 'linC_count': 80, 'linC_degree': 9, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 14, 'linQ_degree_count': 8, 'linQ_dim': 18, 'linQ_dim_count': 8, 'linR_count': 24, 'linR_degree': 18, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': '(C11*Q16):C10', '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': 54, 'number_characteristic_subgroups': 36, 'number_conjugacy_classes': 77, 'number_divisions': 33, 'number_normal_subgroups': 60, 'number_subgroup_autclasses': 100, 'number_subgroup_classes': 120, 'number_subgroups': 588, 'old_label': None, 'order': 1760, 'order_factorization_type': 311, 'order_stats': [[1, 1], [2, 7], [4, 16], [5, 44], [8, 8], [10, 308], [11, 10], [20, 704], [22, 70], [40, 352], [44, 160], [88, 80]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 2, 'outer_gen_orders': [2, 2, 2], 'outer_gen_pows': [0, 0, 0], 'outer_gens': [[1, 2, 430], [1, 882, 1300], [881, 2, 1300]], 'outer_group': '8.5', 'outer_hash': 5, 'outer_nilpotent': True, 'outer_order': 8, 'outer_permdeg': 6, 'outer_perms': [1, 6, 120], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^3', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 27, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 2, 5], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 8], [2, 2], [4, 9], [8, 2], [10, 8], [16, 1], [20, 2], [40, 1]], 'representations': {'PC': {'code': 14695034851535735857218935997617905179861788353898668740030009, 'gens': [1, 2, 4], 'pres': [7, -2, -2, -5, -2, -2, -2, -11, 12349, 36, 37523, 10370, 6877, 80, 32204, 25911, 1768, 102, 25212, 4219, 124, 15693, 9820]}, 'Perm': {'d': 27, 'gens': [37914513688, 1616856149013, 2968576096997, 16807387368228042313728000, 4410550437631, 5795954564346, 435636986031685323325440000]}}, 'schur_multiplier': [2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 10], 'solvability_type': 7, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': '(C_{11}\\times Q_{16}):C_{10}', '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': 2, 'aut_exponent': 4, 'aut_gen_orders': [2, 4, 4], 'aut_gens': [[1, 2], [1, 22], [1, 34], [11, 2]], 'aut_group': '32.25', 'aut_hash': 25, 'aut_nilpotency_class': 2, 'aut_nilpotent': True, 'aut_order': 32, 'aut_permdeg': 8, 'aut_perms': [10800, 11529, 5880], 'aut_phi_ratio': 2.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 2, 2, 1], [4, 2, 1, 1], [5, 1, 4, 1], [10, 1, 4, 1], [10, 2, 8, 1], [20, 2, 4, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_4\\times D_4', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 4, 'autcent_group': '16.10', 'autcent_hash': 10, 'autcent_nilpotent': True, 'autcent_order': 16, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^2\\times C_4', 'autcentquo_abelian': True, 'autcentquo_cyclic': True, 'autcentquo_exponent': 2, 'autcentquo_group': '2.1', 'autcentquo_hash': 1, 'autcentquo_nilpotent': True, 'autcentquo_order': 2, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'C_2', 'cc_stats': [[1, 1, 1], [2, 1, 1], [2, 2, 2], [4, 2, 1], [5, 1, 4], [10, 1, 4], [10, 2, 8], [20, 2, 4]], 'center_label': '10.2', 'center_order': 10, 'central_product': True, 'central_quotient': '4.2', 'commutator_count': 1, 'commutator_label': '2.1', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '5.1'], 'composition_length': 4, 'conjugacy_classes_known': True, 'counter': 10, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['5.1', 1], ['8.3', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 2, 1, 2], [4, 2, 1, 1], [5, 1, 4, 1], [10, 1, 4, 1], [10, 2, 4, 2], [20, 2, 4, 1]], 'element_repr_type': 'PC', 'elementary': 2, 'eulerian_function': 18, 'exponent': 20, 'exponents_of_order': [3, 1], 'factors_of_aut_order': [2], 'factors_of_order': [2, 5], 'faithful_reps': [[2, 0, 4]], 'familial': False, 'frattini_label': '2.1', 'frattini_quotient': '20.5', 'hash': 10, 'hyperelementary': 2, 'inner_abelian': True, 'inner_cyclic': False, 'inner_exponent': 2, 'inner_gen_orders': [2, 2], 'inner_gens': [[1, 22], [21, 2]], 'inner_hash': 2, 'inner_nilpotent': True, 'inner_order': 4, 'inner_split': True, 'inner_tex': 'C_2^2', 'inner_used': [1, 2], 'irrC_degree': 2, 'irrQ_degree': 8, 'irrQ_dim': 8, 'irrR_degree': 4, 'irrep_stats': [[1, 20], [2, 5]], 'label': '40.10', 'linC_count': 4, 'linC_degree': 2, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 6, 'linQ_degree_count': 4, 'linQ_dim': 6, 'linQ_dim_count': 4, 'linR_count': 10, 'linR_degree': 4, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C5*D4', 'ngens': 4, 'nilpotency_class': 2, 'nilpotent': True, 'normal_counts': [0, 0, 0, 0, 0, 0, 0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 8, 'number_characteristic_subgroups': 8, 'number_conjugacy_classes': 25, 'number_divisions': 10, 'number_normal_subgroups': 12, 'number_subgroup_autclasses': 12, 'number_subgroup_classes': 16, 'number_subgroups': 20, 'old_label': None, 'order': 40, 'order_factorization_type': 31, 'order_stats': [[1, 1], [2, 5], [4, 2], [5, 4], [10, 20], [20, 8]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 4, 'outer_gen_orders': [2, 4], 'outer_gen_pows': [2, 0], 'outer_gens': [[11, 2], [1, 34]], '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': 2, 'perfect': False, 'permutation_degree': 9, '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]], 'representations': {'PC': {'code': 1473908227, 'gens': [1, 2], 'pres': [4, -2, -2, -2, -5, 177, 21, 34]}, 'GLZ': {'b': 3, 'd': 6, 'gens': [25038659807093174, 58438750843733785]}, 'GLFp': {'d': 2, 'p': 11, 'gens': [13320, 132, 4001]}, 'Perm': {'d': 9, 'gens': [40320, 90720, 96, 41040]}}, 'schur_multiplier': [2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 10], 'solvability_type': 3, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_5\\times D_4', 'transitive_degree': 20, 'wreath_data': None, 'wreath_product': False}
