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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': True, 'ambient': '688.42', 'ambient_counter': 42, 'ambient_order': 688, 'ambient_tex': 'C_2^3\\times C_{86}', 'central': True, 'central_factor': False, 'centralizer_order': 688, 'characteristic': True, 'core_order': 16, 'counter': 6, 'cyclic': False, 'direct': True, 'hall': 2, 'label': '688.42.43.a1', 'maximal': True, 'maximal_normal': True, 'metabelian': True, 'metacyclic': False, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': True, 'old_label': '43.a1', 'outer_equivalence': True, 'perfect': False, 'proper': True, 'quotient': '43.1', 'quotient_Agroup': True, 'quotient_abelian': True, 'quotient_cyclic': True, 'quotient_hash': 1, 'quotient_metabelian': True, 'quotient_nilpotent': True, 'quotient_order': 43, 'quotient_simple': True, 'quotient_solvable': True, 'quotient_supersolvable': True, 'quotient_tex': 'C_{43}', 'simple': False, 'solvable': True, 'special_labels': [], 'split': True, 'standard_generators': False, 'stem': False, 'subgroup': '16.14', 'subgroup_hash': 14, 'subgroup_order': 16, 'subgroup_tex': 'C_2^4', 'supersolvable': True, 'sylow': 2}
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gps_subgroup_data • Show schema
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{'ambient': '688.42', 'aut_centralizer_order': 42, 'aut_label': '43.a1', 'aut_quo_index': 1, 'aut_stab_index': 1, 'aut_weyl_group': '20160.a', 'aut_weyl_index': 42, 'centralizer': '1.a1', 'complements': ['16.a1'], 'conjugacy_class_count': 1, 'contained_in': ['1.a1'], 'contains': ['86.a1'], 'core': '43.a1', 'coset_action_label': None, 'count': 1, 'diagramx': [8797, 8797, 1290, 1290], 'generators': [1, 2, 4, 344], 'label': '688.42.43.a1', 'mobius_quo': 64, 'mobius_sub': -1, 'normal_closure': '43.a1', 'normal_contained_in': ['1.a1'], 'normal_contains': ['86.a1'], 'normalizer': '1.a1', 'old_label': '43.a1', 'projective_image': '43.1', 'quotient_action_image': '1.1', 'quotient_action_kernel': '43.1', 'quotient_action_kernel_order': 43, 'quotient_fusion': None, 'short_label': '43.a1', 'subgroup_fusion': None, 'weyl_group': '1.1'}
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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}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': True, 'abelian_quotient': '688.42', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 1, 'aut_exponent': 420, 'aut_gen_orders': [28, 84], 'aut_gens': [[1, 2, 4, 8], [1, 347, 3, 135], [3, 344, 350, 370]], 'aut_group': None, 'aut_hash': 2915338221183693103, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 846720, 'aut_permdeg': 57, 'aut_perms': [167626732703393498146014444341264629054224695241132589953135836544368640000, 762938985925403668759388808546049551866666756812282934032823257624420940313], 'aut_phi_ratio': 2520.0, 'aut_solvable': False, 'aut_stats': [[1, 1, 1, 1], [2, 1, 15, 1], [43, 1, 42, 1], [86, 1, 630, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_{42}\\times A_8', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 420, 'autcent_group': None, 'autcent_hash': 2915338221183693103, 'autcent_nilpotent': False, 'autcent_order': 846720, 'autcent_solvable': False, 'autcent_split': True, 'autcent_supersolvable': False, 'autcent_tex': 'C_{42}\\times 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], [43, 1, 42], [86, 1, 630]], 'center_label': '688.42', 'center_order': 688, 'central_product': True, 'central_quotient': '1.1', 'commutator_count': 0, 'commutator_label': '1.1', 'complements_known': True, 'complete': False, 'complex_characters_known': False, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '43.1'], 'composition_length': 5, 'conjugacy_classes_known': True, 'counter': 42, 'cyclic': False, 'derived_length': 1, 'dihedral': False, 'direct_factorization': [['2.1', 4], ['43.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 15], [43, 1, 42, 1], [86, 1, 42, 15]], 'element_repr_type': 'PC', 'elementary': 2, 'eulerian_function': 81400, 'exponent': 86, 'exponents_of_order': [4, 1], 'factors_of_aut_order': [2, 3, 5, 7], 'factors_of_order': [2, 43], 'faithful_reps': [], 'familial': False, 'frattini_label': '1.1', 'frattini_quotient': '688.42', 'hash': 42, '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': None, 'irrep_stats': [[1, 688]], 'label': '688.42', 'linC_count': None, 'linC_degree': 4, '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': 'C2^3*C86', 'ngens': 5, 'nilpotency_class': 1, 'nilpotent': True, 'normal_counts': [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': 4, 'number_characteristic_subgroups': 4, 'number_conjugacy_classes': 688, 'number_divisions': 32, 'number_normal_subgroups': 134, 'number_subgroup_autclasses': 10, 'number_subgroup_classes': 134, 'number_subgroups': 134, 'old_label': None, 'order': 688, 'order_factorization_type': 31, 'order_stats': [[1, 1], [2, 15], [43, 42], [86, 630]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': True, 'outer_exponent': 420, 'outer_gen_orders': [42, 42], 'outer_gen_pows': [0, 0], 'outer_gens': [[346, 351, 344, 604], [5, 4, 344, 274]], 'outer_group': None, 'outer_hash': 2915338221183693103, 'outer_nilpotent': False, 'outer_order': 846720, 'outer_permdeg': 57, 'outer_perms': [10120452898598474648486821837824756040011893391876252416311514159835788046720, 9244409533382259713748624130956381303228038939444307137333589104418816000000], 'outer_solvable': False, 'outer_supersolvable': False, 'outer_tex': 'C_{42}\\times A_8', 'pc_rank': 4, 'perfect': False, 'permutation_degree': 51, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 2, 2, 43], 'quasisimple': False, 'rank': 4, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 16], [42, 16]], 'representations': {'PC': {'code': 2142290674985, 'gens': [1, 2, 3, 4], 'pres': [5, -2, -2, -2, -2, -43, 58]}, 'Perm': {'d': 51, 'gens': [30414093201713378043612608166064768844377641568960512000000000000, 12413915592536072670862289047373375038521486354677760000000000, 5502622159812088949850305428800254892961651752960000000000, 2658271574788448768043625811014615890319638528000000000, 59010256945620955738811989462269485937328128000000000]}}, 'schur_multiplier': [2, 2, 2, 2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 2, 86], 'solvability_type': 2, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_2^3\\times C_{86}', 'transitive_degree': 688, 'wreath_data': None, 'wreath_product': False}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': True, 'abelian_quotient': '43.1', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': True, 'aut_cyclic': True, 'aut_derived_length': 1, 'aut_exponent': 42, 'aut_gen_orders': [42], 'aut_gens': [[1], [3]], 'aut_group': '42.6', 'aut_hash': 6, 'aut_nilpotency_class': 1, 'aut_nilpotent': True, 'aut_order': 42, 'aut_permdeg': 12, 'aut_perms': [40320873], 'aut_phi_ratio': 1.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [43, 1, 42, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_{42}', 'autcent_abelian': True, 'autcent_cyclic': True, 'autcent_exponent': 42, 'autcent_group': '42.6', 'autcent_hash': 6, 'autcent_nilpotent': True, 'autcent_order': 42, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_{42}', '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], [43, 1, 42]], 'center_label': '43.1', 'center_order': 43, 'central_product': False, 'central_quotient': '1.1', 'commutator_count': 0, 'commutator_label': '1.1', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['43.1'], 'composition_length': 1, 'conjugacy_classes_known': True, 'counter': 1, 'cyclic': True, 'derived_length': 1, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [43, 1, 42, 1]], 'element_repr_type': 'PC', 'elementary': 43, 'eulerian_function': 1, 'exponent': 43, 'exponents_of_order': [1], 'factors_of_aut_order': [2, 3, 7], 'factors_of_order': [43], 'faithful_reps': [[1, 0, 42]], 'familial': True, 'frattini_label': '1.1', 'frattini_quotient': '43.1', 'hash': 1, 'hyperelementary': 43, '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': 42, 'irrQ_dim': 42, 'irrR_degree': 2, 'irrep_stats': [[1, 43]], 'label': '43.1', 'linC_count': 42, 'linC_degree': 1, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 42, 'linQ_degree_count': 1, 'linQ_dim': 42, 'linQ_dim_count': 1, 'linR_count': 21, 'linR_degree': 2, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C43', 'ngens': 1, 'nilpotency_class': 1, 'nilpotent': True, 'normal_counts': [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': 43, 'number_divisions': 2, 'number_normal_subgroups': 2, 'number_subgroup_autclasses': 2, 'number_subgroup_classes': 2, 'number_subgroups': 2, 'old_label': None, 'order': 43, 'order_factorization_type': 1, 'order_stats': [[1, 1], [43, 42]], 'outer_abelian': True, 'outer_cyclic': True, 'outer_equivalence': False, 'outer_exponent': 42, 'outer_gen_orders': [42], 'outer_gen_pows': [0], 'outer_gens': [[3]], 'outer_group': '42.6', 'outer_hash': 6, 'outer_nilpotent': True, 'outer_order': 42, 'outer_permdeg': 12, 'outer_perms': [40320873], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_{42}', 'pc_rank': 1, 'perfect': False, 'permutation_degree': 43, 'pgroup': 43, 'primary_abelian_invariants': [43], 'quasisimple': False, 'rank': 1, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 1], [42, 1]], 'representations': {'PC': {'code': 0, 'gens': [1], 'pres': [1, -43]}, 'Lie': [{'d': 1, 'q': 43, 'gens': [1439295494700374021157505910939096377494040420940313], 'family': 'ASigmaL'}], 'GLFp': {'d': 2, 'p': 43, 'gens': [79551]}, 'Perm': {'d': 43, 'gens': [59010256945620955738811989462269485937328128000000000]}}, 'schur_multiplier': [], 'semidirect_product': False, 'simple': True, 'smith_abelian_invariants': [43], 'solvability_type': 0, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_{43}', 'transitive_degree': 43, 'wreath_data': None, 'wreath_product': False}
