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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': True, 'ambient': '320.1569', 'ambient_counter': 1569, 'ambient_order': 320, 'ambient_tex': 'C_{40}.C_2^3', 'central': True, 'central_factor': False, 'centralizer_order': 320, 'characteristic': True, 'core_order': 20, 'counter': 32, 'cyclic': True, 'direct': False, 'hall': 0, 'label': '320.1569.16.d1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': True, 'old_label': '16.d1', 'outer_equivalence': True, 'perfect': False, 'proper': True, 'quotient': '16.14', 'quotient_Agroup': True, 'quotient_abelian': True, 'quotient_cyclic': False, 'quotient_hash': 14, 'quotient_metabelian': True, 'quotient_nilpotent': True, 'quotient_order': 16, 'quotient_simple': False, 'quotient_solvable': True, 'quotient_supersolvable': True, 'quotient_tex': 'C_2^4', 'simple': False, 'solvable': True, 'special_labels': ['C4'], 'split': False, 'standard_generators': False, 'stem': False, 'subgroup': '20.2', 'subgroup_hash': 2, 'subgroup_order': 20, 'subgroup_tex': 'C_{20}', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '320.1569', 'aut_centralizer_order': 768, 'aut_label': '16.d1', 'aut_quo_index': 420, 'aut_stab_index': 1, 'aut_weyl_group': '8.2', 'aut_weyl_index': 768, 'centralizer': '1.a1', 'complements': [], 'conjugacy_class_count': 1, 'contained_in': ['8.b1', '8.e1', '8.g1', '8.h1'], 'contains': ['32.b1', '80.d1'], 'core': '16.d1', 'coset_action_label': None, 'count': 1, 'diagramx': [3608, 6396, 6531, 8314], 'generators': [80, 160, 64], 'label': '320.1569.16.d1', 'mobius_quo': 0, 'mobius_sub': 64, 'normal_closure': '16.d1', 'normal_contained_in': ['8.b1', '8.e1', '8.g1', '8.h1'], 'normal_contains': ['32.b1', '80.d1'], 'normalizer': '1.a1', 'old_label': '16.d1', 'projective_image': '16.14', 'quotient_action_image': '1.1', 'quotient_action_kernel': '16.14', 'quotient_action_kernel_order': 16, 'quotient_fusion': None, 'short_label': '16.d1', 'subgroup_fusion': None, 'weyl_group': '1.1'}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': True, 'abelian_quotient': '20.2', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': True, 'aut_cyclic': False, 'aut_derived_length': 1, 'aut_exponent': 4, 'aut_gen_orders': [2, 4], 'aut_gens': [[1], [11], [17]], 'aut_group': '8.2', 'aut_hash': 2, 'aut_nilpotency_class': 1, 'aut_nilpotent': True, 'aut_order': 8, 'aut_permdeg': 6, 'aut_perms': [120, 9], 'aut_phi_ratio': 1.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [4, 1, 2, 1], [5, 1, 4, 1], [10, 1, 4, 1], [20, 1, 8, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_2\\times C_4', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 4, 'autcent_group': '8.2', 'autcent_hash': 2, 'autcent_nilpotent': True, 'autcent_order': 8, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2\\times C_4', '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], [5, 1, 4], [10, 1, 4], [20, 1, 8]], 'center_label': '20.2', 'center_order': 20, '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', '5.1'], 'composition_length': 3, 'conjugacy_classes_known': True, 'counter': 2, 'cyclic': True, 'derived_length': 1, 'dihedral': False, 'direct_factorization': [['4.1', 1], ['5.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [4, 1, 2, 1], [5, 1, 4, 1], [10, 1, 4, 1], [20, 1, 8, 1]], 'element_repr_type': 'PC', 'elementary': 10, 'eulerian_function': 1, 'exponent': 20, 'exponents_of_order': [2, 1], 'factors_of_aut_order': [2], 'factors_of_order': [2, 5], 'faithful_reps': [[1, 0, 8]], 'familial': True, 'frattini_label': '2.1', 'frattini_quotient': '10.2', 'hash': 2, 'hyperelementary': 10, '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': 8, 'irrQ_dim': 8, 'irrR_degree': 2, 'irrep_stats': [[1, 20]], 'label': '20.2', 'linC_count': 8, 'linC_degree': 1, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 6, 'linQ_degree_count': 2, 'linQ_dim': 6, 'linQ_dim_count': 2, 'linR_count': 4, 'linR_degree': 2, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C20', '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': 20, 'number_divisions': 6, 'number_normal_subgroups': 6, 'number_subgroup_autclasses': 6, 'number_subgroup_classes': 6, 'number_subgroups': 6, 'old_label': None, 'order': 20, 'order_factorization_type': 22, 'order_stats': [[1, 1], [2, 1], [4, 2], [5, 4], [10, 4], [20, 8]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 4, 'outer_gen_orders': [2, 4], 'outer_gen_pows': [0, 0], 'outer_gens': [[11], [17]], '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': 1, 'perfect': False, 'permutation_degree': 9, 'pgroup': 0, 'primary_abelian_invariants': [4, 5], 'quasisimple': False, 'rank': 1, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 2], [2, 1], [4, 2], [8, 1]], 'representations': {'PC': {'code': 51395, 'gens': [1], 'pres': [3, -2, -2, -5, 6, 16]}, 'GLZ': {'b': 3, 'd': 6, 'gens': [58438750843733785]}, 'GLFp': {'d': 2, 'p': 5, 'gens': [131, 504, 252]}, 'Perm': {'d': 9, 'gens': [131040, 96, 41040]}}, 'schur_multiplier': [], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [20], 'solvability_type': 0, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_{20}', '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': '160.228', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 3, 'aut_exponent': 12, 'aut_gen_orders': [12, 4, 12, 6, 4, 4], 'aut_gens': [[1, 2, 4, 8], [161, 86, 162, 217], [1, 5, 3, 57], [1, 86, 162, 185], [1, 164, 86, 72], [161, 162, 246, 312], [161, 2, 164, 73]], 'aut_group': None, 'aut_hash': 5020595138065828458, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 6144, 'aut_permdeg': 24, 'aut_perms': [85596428256966259304371, 59621471298480057119975, 193544399623723239167994, 151689141391693626225219, 279995029932549161698421, 21420019452335272041869], 'aut_phi_ratio': 48.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 1, 2, 1], [2, 2, 6, 1], [4, 1, 2, 2], [4, 2, 6, 1], [5, 1, 4, 1], [8, 1, 8, 1], [8, 2, 12, 1], [10, 1, 4, 1], [10, 1, 8, 1], [10, 2, 24, 1], [20, 1, 8, 2], [20, 2, 24, 1], [40, 1, 32, 1], [40, 2, 48, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_2^4.(C_{12}\\times D_4).C_2^2', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 4, 'autcent_group': None, 'autcent_hash': 4843724316645053051, 'autcent_nilpotent': True, 'autcent_order': 1024, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2\\times C_4\\times C_2^4.C_2^3', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 6, 'autcentquo_group': '6.1', 'autcentquo_hash': 1, 'autcentquo_nilpotent': False, 'autcentquo_order': 6, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'S_3', 'cc_stats': [[1, 1, 1], [2, 1, 3], [2, 2, 6], [4, 1, 4], [4, 2, 6], [5, 1, 4], [8, 1, 8], [8, 2, 12], [10, 1, 12], [10, 2, 24], [20, 1, 16], [20, 2, 24], [40, 1, 32], [40, 2, 48]], 'center_label': '80.23', 'center_order': 80, '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', '2.1', '2.1', '2.1', '5.1'], 'composition_length': 7, 'conjugacy_classes_known': True, 'counter': 1569, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['2.1', 1], ['32.38', 1], ['5.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [2, 2, 1, 6], [4, 1, 2, 2], [4, 2, 1, 6], [5, 1, 4, 1], [8, 1, 4, 2], [8, 2, 2, 6], [10, 1, 4, 3], [10, 2, 4, 6], [20, 1, 8, 2], [20, 2, 4, 6], [40, 1, 16, 2], [40, 2, 8, 6]], 'element_repr_type': 'PC', 'elementary': 2, 'eulerian_function': 524160, 'exponent': 40, 'exponents_of_order': [6, 1], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 5], 'faithful_reps': [], 'familial': False, 'frattini_label': '4.1', 'frattini_quotient': '80.52', 'hash': 1569, 'hyperelementary': 2, 'inner_abelian': True, 'inner_cyclic': False, 'inner_exponent': 2, 'inner_gen_orders': [1, 2, 2, 1], 'inner_gens': [[1, 2, 4, 8], [1, 2, 164, 8], [1, 162, 4, 8], [1, 2, 4, 8]], 'inner_hash': 2, 'inner_nilpotent': True, 'inner_order': 4, 'inner_split': False, 'inner_tex': 'C_2^2', 'inner_used': [2, 3], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 160], [2, 40]], 'label': '320.1569', 'linC_count': None, 'linC_degree': None, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': None, 'linQ_degree_count': None, 'linQ_dim': None, 'linQ_dim_count': None, 'linR_count': None, 'linR_degree': None, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'C40.C2^3', 'ngens': 7, 'nilpotency_class': 2, 'nilpotent': True, 'normal_counts': [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': 20, 'number_conjugacy_classes': 200, 'number_divisions': 52, 'number_normal_subgroups': 242, 'number_subgroup_autclasses': 66, 'number_subgroup_classes': 266, 'number_subgroups': 290, 'old_label': None, 'order': 320, 'order_factorization_type': 51, 'order_stats': [[1, 1], [2, 15], [4, 16], [5, 4], [8, 32], [10, 60], [20, 64], [40, 128]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': True, 'outer_exponent': 12, 'outer_gen_orders': [4, 2, 6, 4, 2, 2, 2], 'outer_gen_pows': [0, 0, 0, 0, 0, 0, 0], 'outer_gens': [[1, 2, 4, 24], [1, 164, 162, 168], [1, 86, 162, 9], [161, 2, 4, 169], [1, 163, 5, 8], [1, 2, 5, 248], [1, 163, 5, 248]], 'outer_group': '1536.408633772', 'outer_hash': 3864209259142932085, 'outer_nilpotent': False, 'outer_order': 1536, 'outer_permdeg': 14, 'outer_perms': [17, 20237817600, 8263140600, 13898315640, 12940663680, 12933043207, 1480550407], 'outer_solvable': True, 'outer_supersolvable': False, 'outer_tex': 'C_4\\times \\GL(2,\\mathbb{Z}/4):C_2^2', 'pc_rank': 4, 'perfect': False, 'permutation_degree': 23, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 2, 4, 5], 'quasisimple': False, 'rank': 4, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 16], [2, 8], [4, 16], [8, 10], [32, 2]], 'representations': {'PC': {'code': 504631878674373541891, 'gens': [1, 2, 3, 4], 'pres': [7, -2, -2, -2, -2, -2, -2, -5, 1731, 80, 102, 124]}, 'GLZN': {'d': 2, 'p': 82, 'gens': [43558151, 551449, 20400653, 4962321, 44660889, 554731, 22620889]}, 'Perm': {'d': 23, 'gens': [1233866929367995413840, 14727638140776208080, 2453127238877280860880, 720, 96, 3644893139689418400000, 4817999772507852806400]}}, 'schur_multiplier': [2, 2, 2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 2, 20], 'solvability_type': 4, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_{40}.C_2^3', 'transitive_degree': 160, 'wreath_data': None, 'wreath_product': False}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': True, 'abelian_quotient': '16.14', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 0, 'aut_exponent': 420, 'aut_gen_orders': [6, 3], 'aut_gens': [[1, 2, 4, 8], [12, 4, 5, 3], [10, 14, 7, 5]], 'aut_group': '20160.a', 'aut_hash': 3764836782182912467, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 20160, 'aut_permdeg': 8, 'aut_perms': [5193, 5760], 'aut_phi_ratio': 2520.0, 'aut_solvable': False, 'aut_stats': [[1, 1, 1, 1], [2, 1, 15, 1]], 'aut_supersolvable': False, 'aut_tex': 'A_8', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 420, 'autcent_group': '20160.a', 'autcent_hash': 3764836782182912467, 'autcent_nilpotent': False, 'autcent_order': 20160, 'autcent_solvable': False, 'autcent_split': True, 'autcent_supersolvable': False, 'autcent_tex': 'A_8', '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, 15]], 'center_label': '16.14', 'center_order': 16, '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', '2.1', '2.1'], 'composition_length': 4, 'conjugacy_classes_known': True, 'counter': 14, 'cyclic': False, 'derived_length': 1, 'dihedral': False, 'direct_factorization': [['2.1', 4]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 15]], 'element_repr_type': 'PC', 'elementary': 2, 'eulerian_function': 1, 'exponent': 2, 'exponents_of_order': [4], 'factors_of_aut_order': [2, 3, 5, 7], 'factors_of_order': [2], 'faithful_reps': [], 'familial': False, 'frattini_label': '1.1', 'frattini_quotient': '16.14', 'hash': 14, 'hyperelementary': 2, 'inner_abelian': True, 'inner_cyclic': True, 'inner_exponent': 1, 'inner_gen_orders': [1, 1, 1, 1], 'inner_gens': [[1, 2, 4, 8], [1, 2, 4, 8], [1, 2, 4, 8], [1, 2, 4, 8]], '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, 16]], 'label': '16.14', 'linC_count': 840, 'linC_degree': 4, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 4, 'linQ_degree_count': 840, 'linQ_dim': 4, 'linQ_dim_count': 840, 'linR_count': 840, 'linR_degree': 4, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'C2^4', 'ngens': 4, 'nilpotency_class': 1, 'nilpotent': True, 'normal_counts': [0, 0, 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': 16, 'number_divisions': 16, 'number_normal_subgroups': 67, 'number_subgroup_autclasses': 5, 'number_subgroup_classes': 67, 'number_subgroups': 67, 'old_label': None, 'order': 16, 'order_factorization_type': 3, 'order_stats': [[1, 1], [2, 15]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 420, 'outer_gen_orders': [6, 3], 'outer_gen_pows': [0, 0], 'outer_gens': [[12, 4, 5, 3], [10, 14, 7, 5]], 'outer_group': '20160.a', 'outer_hash': 3764836782182912467, 'outer_nilpotent': False, 'outer_order': 20160, 'outer_permdeg': 8, 'outer_perms': [5193, 5760], 'outer_solvable': False, 'outer_supersolvable': False, 'outer_tex': 'A_8', 'pc_rank': 4, 'perfect': False, 'permutation_degree': 8, 'pgroup': 2, 'primary_abelian_invariants': [2, 2, 2, 2], 'quasisimple': False, 'rank': 4, 'rational': True, 'rational_characters_known': True, 'ratrep_stats': [[1, 16]], 'representations': {'PC': {'code': 0, 'gens': [1, 2, 3, 4], 'pres': [4, -2, 2, 2, 2]}, 'GLZ': {'b': 3, 'd': 4, 'gens': [7233746, 7115648, 35812976, 7115160]}, 'GLFp': {'d': 4, 'p': 2, 'gens': [33837, 18465, 27183, 18467]}, 'Perm': {'d': 8, 'gens': [5040, 120, 6, 1]}}, 'schur_multiplier': [2, 2, 2, 2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 2, 2], 'solvability_type': 2, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_2^4', 'transitive_degree': 16, 'wreath_data': None, 'wreath_product': False}
