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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': True, 'ambient': '996.4', 'ambient_counter': 4, 'ambient_order': 996, 'ambient_tex': 'C_{996}', 'central': True, 'central_factor': False, 'centralizer_order': 996, 'characteristic': True, 'core_order': 332, 'counter': 3, 'cyclic': True, 'direct': True, 'hall': 166, 'label': '996.4.3.a1.a1', 'maximal': True, 'maximal_normal': True, 'metabelian': True, 'metacyclic': True, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': True, 'old_label': '3.a1.a1', 'outer_equivalence': False, 'perfect': False, 'proper': True, 'quotient': '3.1', 'quotient_Agroup': True, 'quotient_abelian': True, 'quotient_cyclic': True, 'quotient_hash': 1, 'quotient_metabelian': True, 'quotient_nilpotent': True, 'quotient_order': 3, 'quotient_simple': True, 'quotient_solvable': True, 'quotient_supersolvable': True, 'quotient_tex': 'C_3', 'simple': False, 'solvable': True, 'special_labels': [], 'split': True, 'standard_generators': False, 'stem': False, 'subgroup': '332.2', 'subgroup_hash': 2, 'subgroup_order': 332, 'subgroup_tex': 'C_{332}', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '996.4', 'aut_centralizer_order': 2, 'aut_label': '3.a1', 'aut_quo_index': 1, 'aut_stab_index': 1, 'aut_weyl_group': '164.5', 'aut_weyl_index': 2, 'centralizer': '1.a1.a1', 'complements': ['332.a1.a1'], 'conjugacy_class_count': 1, 'contained_in': ['1.a1.a1'], 'contains': ['6.a1.a1', '249.a1.a1'], 'core': '3.a1.a1', 'coset_action_label': None, 'count': 1, 'diagramx': [9109, 9109, 7273, 7273, 6103, 6103, 8817, 8817], 'generators': [249, 12, 498], 'label': '996.4.3.a1.a1', 'mobius_quo': 0, 'mobius_sub': -1, 'normal_closure': '3.a1.a1', 'normal_contained_in': ['1.a1.a1'], 'normal_contains': ['6.a1.a1', '249.a1.a1'], 'normalizer': '1.a1.a1', 'old_label': '3.a1.a1', 'projective_image': '3.1', 'quotient_action_image': '1.1', 'quotient_action_kernel': '3.1', 'quotient_action_kernel_order': 3, 'quotient_fusion': None, 'short_label': '3.a1.a1', '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': '332.2', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': True, 'aut_cyclic': False, 'aut_derived_length': 1, 'aut_exponent': 82, 'aut_gen_orders': [2, 82], 'aut_gens': [[1], [165], [207]], 'aut_group': '164.5', 'aut_hash': 5, 'aut_nilpotency_class': 1, 'aut_nilpotent': True, 'aut_order': 164, 'aut_permdeg': 84, 'aut_perms': [240653664262646398824364291566343783056841073003064150375214511720418715778479274250420219239686814127869506093056000000000, 39690023759901851291238314391404848408565128844719606421931883881267721041837538408892844648735826359864255861227520000000000], 'aut_phi_ratio': 1.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [4, 1, 2, 1], [83, 1, 82, 1], [166, 1, 82, 1], [332, 1, 164, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_2\\times C_{82}', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 82, 'autcent_group': '164.5', 'autcent_hash': 5, 'autcent_nilpotent': True, 'autcent_order': 164, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2\\times C_{82}', '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], [83, 1, 82], [166, 1, 82], [332, 1, 164]], 'center_label': '332.2', 'center_order': 332, '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', '83.1'], 'composition_length': 3, 'conjugacy_classes_known': True, 'counter': 2, 'cyclic': True, 'derived_length': 1, 'dihedral': False, 'direct_factorization': [['4.1', 1], ['83.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [4, 1, 2, 1], [83, 1, 82, 1], [166, 1, 82, 1], [332, 1, 164, 1]], 'element_repr_type': 'PC', 'elementary': 166, 'eulerian_function': 1, 'exponent': 332, 'exponents_of_order': [2, 1], 'factors_of_aut_order': [2, 41], 'factors_of_order': [2, 83], 'faithful_reps': [[1, 0, 164]], 'familial': True, 'frattini_label': '2.1', 'frattini_quotient': '166.2', 'hash': 2, 'hyperelementary': 166, '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': 164, 'irrQ_dim': 164, 'irrR_degree': 2, 'irrep_stats': [[1, 332]], 'label': '332.2', 'linC_count': 164, 'linC_degree': 1, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 84, 'linQ_degree_count': 2, 'linQ_dim': 84, 'linQ_dim_count': 2, 'linR_count': 82, 'linR_degree': 2, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C332', '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': 332, 'number_divisions': 6, 'number_normal_subgroups': 6, 'number_subgroup_autclasses': 6, 'number_subgroup_classes': 6, 'number_subgroups': 6, 'old_label': None, 'order': 332, 'order_factorization_type': 22, 'order_stats': [[1, 1], [2, 1], [4, 2], [83, 82], [166, 82], [332, 164]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 82, 'outer_gen_orders': [2, 82], 'outer_gen_pows': [0, 0], 'outer_gens': [[165], [207]], 'outer_group': '164.5', 'outer_hash': 5, 'outer_nilpotent': True, 'outer_order': 164, 'outer_permdeg': 84, 'outer_perms': [240653664262646398824364291566343783056841073003064150375214511720418715778479274250420219239686814127869506093056000000000, 39690023759901851291238314391404848408565128844719606421931883881267721041837538408892844648735826359864255861227520000000000], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2\\times C_{82}', 'pc_rank': 1, 'perfect': False, 'permutation_degree': 87, 'pgroup': 0, 'primary_abelian_invariants': [4, 83], 'quasisimple': False, 'rank': 1, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 2], [2, 1], [82, 2], [164, 1]], 'representations': {'PC': {'code': 7323821225, 'gens': [1], 'pres': [3, -2, -2, -83, 6, 16]}, 'GLFp': {'d': 2, 'p': 167, 'gens': [201068]}, 'Perm': {'d': 87, 'gens': [73244706973894307200686465498775901464814724925949293665326497315665606801546942697270474477608640244751532032000000000000000000000, 38979875363505302337055332973131205624792690113575452946607196594293007568261730186266584808295291703787520000000000000000000, 24230409623807297735032522591020389846572871218715623800325883342752545308873741993653594520624288182324816773120000000000000000000]}}, 'schur_multiplier': [], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [332], 'solvability_type': 0, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_{332}', 'transitive_degree': 332, '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': '996.4', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': True, 'aut_cyclic': False, 'aut_derived_length': 1, 'aut_exponent': 82, 'aut_gen_orders': [2, 2, 82], 'aut_gens': [[1], [499], [665], [457]], 'aut_group': '328.15', 'aut_hash': 15, 'aut_nilpotency_class': 1, 'aut_nilpotent': True, 'aut_order': 328, 'aut_permdeg': 86, 'aut_perms': [281710411438055027694947944226061159480056634330574206405101912752560026159795933451040286452340924018275123200000000000000000000, 39455239697206586511897471180120610571436503407643446275224357528369751562996629334879591940103770870906880000000000000000000, 469567207680536806856258472757826837538400936823261702658415069027027604555205256556750947274229421178880000000000000000000], 'aut_phi_ratio': 1.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [3, 1, 2, 1], [4, 1, 2, 1], [6, 1, 2, 1], [12, 1, 4, 1], [83, 1, 82, 1], [166, 1, 82, 1], [249, 1, 164, 1], [332, 1, 164, 1], [498, 1, 164, 1], [996, 1, 328, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_2^2\\times C_{82}', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 82, 'autcent_group': '328.15', 'autcent_hash': 15, 'autcent_nilpotent': True, 'autcent_order': 328, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^2\\times C_{82}', '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], [3, 1, 2], [4, 1, 2], [6, 1, 2], [12, 1, 4], [83, 1, 82], [166, 1, 82], [249, 1, 164], [332, 1, 164], [498, 1, 164], [996, 1, 328]], 'center_label': '996.4', 'center_order': 996, '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', '3.1', '83.1'], 'composition_length': 4, 'conjugacy_classes_known': True, 'counter': 4, 'cyclic': True, 'derived_length': 1, 'dihedral': False, 'direct_factorization': [['3.1', 1], ['4.1', 1], ['83.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [3, 1, 2, 1], [4, 1, 2, 1], [6, 1, 2, 1], [12, 1, 4, 1], [83, 1, 82, 1], [166, 1, 82, 1], [249, 1, 164, 1], [332, 1, 164, 1], [498, 1, 164, 1], [996, 1, 328, 1]], 'element_repr_type': 'PC', 'elementary': 498, 'eulerian_function': 1, 'exponent': 996, 'exponents_of_order': [2, 1, 1], 'factors_of_aut_order': [2, 41], 'factors_of_order': [2, 3, 83], 'faithful_reps': [[1, 0, 328]], 'familial': True, 'frattini_label': '2.1', 'frattini_quotient': '498.4', 'hash': 4, 'hyperelementary': 498, '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': 328, 'irrQ_dim': 328, 'irrR_degree': None, 'irrep_stats': [[1, 996]], 'label': '996.4', 'linC_count': None, 'linC_degree': 1, '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': True, 'monomial': True, 'name': 'C996', 'ngens': 4, 'nilpotency_class': 1, 'nilpotent': True, 'normal_counts': [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': 12, 'number_characteristic_subgroups': 12, 'number_conjugacy_classes': 996, 'number_divisions': 12, 'number_normal_subgroups': 12, 'number_subgroup_autclasses': 12, 'number_subgroup_classes': 12, 'number_subgroups': 12, 'old_label': None, 'order': 996, 'order_factorization_type': 222, 'order_stats': [[1, 1], [2, 1], [3, 2], [4, 2], [6, 2], [12, 4], [83, 82], [166, 82], [249, 164], [332, 164], [498, 164], [996, 328]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 82, 'outer_gen_orders': [2, 2, 82], 'outer_gen_pows': [0, 0, 0], 'outer_gens': [[499], [665], [457]], 'outer_group': '328.15', 'outer_hash': 15, 'outer_nilpotent': True, 'outer_order': 328, 'outer_permdeg': 86, 'outer_perms': [281710411438055027694947944226061159480056634330574206405101912752560026159795933451040286452340924018275123200000000000000000000, 39455239697206586511897471180120610571436503407643446275224357528369751562996629334879591940103770870906880000000000000000000, 469567207680536806856258472757826837538400936823261702658415069027027604555205256556750947274229421178880000000000000000000], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^2\\times C_{82}', 'pc_rank': 1, 'perfect': False, 'permutation_degree': 90, 'pgroup': 0, 'primary_abelian_invariants': [4, 3, 83], 'quasisimple': False, 'rank': 1, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 2], [2, 3], [4, 1], [82, 2], [164, 3], [328, 1]], 'representations': {'PC': {'code': 24883231028399891, 'gens': [1], 'pres': [4, -2, -2, -3, -83, 8, 21, 46]}, 'GLFp': {'d': 2, 'p': 167, 'gens': [226374894]}, 'Perm': {'d': 90, 'gens': [49894830767240180121880351478651950498279440647388424258109405285473292227503585616837937994852410991844074416032972800000000000000000000, 563420822876110055389895888452122318960113268661148412810203825505120052319591866902080572904681848036550246400000000000000000000, 38979875363505302337055332973131205624792690113575452946607196594293007568261730186266584808295291703787520000000000000000000, 16510062918206840608934132862972318699434896020234603211125843680344385688494237332573993254210837077522585117471539200000000000000000000]}}, 'schur_multiplier': [], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [996], 'solvability_type': 0, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_{996}', 'transitive_degree': 996, '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': '3.1', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': True, 'aut_cyclic': True, 'aut_derived_length': 1, 'aut_exponent': 2, 'aut_gen_orders': [2], 'aut_gens': [[1], [2]], 'aut_group': '2.1', 'aut_hash': 1, 'aut_nilpotency_class': 1, 'aut_nilpotent': True, 'aut_order': 2, 'aut_permdeg': 2, 'aut_perms': [1], 'aut_phi_ratio': 1.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [3, 1, 2, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_2', 'autcent_abelian': True, 'autcent_cyclic': True, 'autcent_exponent': 2, 'autcent_group': '2.1', 'autcent_hash': 1, 'autcent_nilpotent': True, 'autcent_order': 2, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2', '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], [3, 1, 2]], 'center_label': '3.1', 'center_order': 3, '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': ['3.1'], '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], [3, 1, 2, 1]], 'element_repr_type': 'PC', 'elementary': 3, 'eulerian_function': 1, 'exponent': 3, 'exponents_of_order': [1], 'factors_of_aut_order': [2], 'factors_of_order': [3], 'faithful_reps': [[1, 0, 2]], 'familial': True, 'frattini_label': '1.1', 'frattini_quotient': '3.1', 'hash': 1, 'hyperelementary': 3, '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': 2, 'irrQ_dim': 2, 'irrR_degree': 2, 'irrep_stats': [[1, 3]], 'label': '3.1', 'linC_count': 2, 'linC_degree': 1, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 2, 'linQ_degree_count': 1, 'linQ_dim': 2, 'linQ_dim_count': 1, 'linR_count': 1, 'linR_degree': 2, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C3', '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': 3, 'number_divisions': 2, 'number_normal_subgroups': 2, 'number_subgroup_autclasses': 2, 'number_subgroup_classes': 2, 'number_subgroups': 2, 'old_label': None, 'order': 3, 'order_factorization_type': 1, 'order_stats': [[1, 1], [3, 2]], 'outer_abelian': True, 'outer_cyclic': True, 'outer_equivalence': False, 'outer_exponent': 2, 'outer_gen_orders': [2], 'outer_gen_pows': [0], 'outer_gens': [[2]], 'outer_group': '2.1', 'outer_hash': 1, 'outer_nilpotent': True, 'outer_order': 2, 'outer_permdeg': 2, 'outer_perms': [1], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2', 'pc_rank': 1, 'perfect': False, 'permutation_degree': 3, 'pgroup': 3, 'primary_abelian_invariants': [3], 'quasisimple': False, 'rank': 1, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 1], [2, 1]], 'representations': {'PC': {'code': 0, 'gens': [1], 'pres': [1, -3]}, 'GLZ': {'b': 3, 'd': 2, 'gens': [7]}, 'Lie': [{'d': 1, 'q': 3, 'gens': [3], 'family': 'ASigmaL'}], 'GLFp': {'d': 2, 'p': 2, 'gens': [7]}, 'Perm': {'d': 3, 'gens': [4]}}, 'schur_multiplier': [], 'semidirect_product': False, 'simple': True, 'smith_abelian_invariants': [3], 'solvability_type': 0, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_3', 'transitive_degree': 3, 'wreath_data': None, 'wreath_product': False}
