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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': True, 'ambient': '396.9', 'ambient_counter': 9, 'ambient_order': 396, 'ambient_tex': 'D_{198}', 'central': False, 'central_factor': False, 'centralizer_order': 198, 'characteristic': True, 'core_order': 3, 'counter': 26, 'cyclic': True, 'direct': False, 'hall': 0, 'label': '396.9.132.a1.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': True, 'minimal_normal': True, 'nilpotent': True, 'normal': True, 'old_label': '132.a1.a1', 'outer_equivalence': False, 'perfect': False, 'proper': True, 'quotient': '132.9', 'quotient_Agroup': True, 'quotient_abelian': False, 'quotient_cyclic': False, 'quotient_hash': 9, 'quotient_metabelian': True, 'quotient_nilpotent': False, 'quotient_order': 132, 'quotient_simple': False, 'quotient_solvable': True, 'quotient_supersolvable': True, 'quotient_tex': 'D_{66}', 'simple': True, 'solvable': True, 'special_labels': ['Phi'], 'split': False, 'standard_generators': False, 'stem': False, 'subgroup': '3.1', 'subgroup_hash': 1, 'subgroup_order': 3, 'subgroup_tex': 'C_3', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '396.9', 'aut_centralizer_order': 5940, 'aut_label': '132.a1', 'aut_quo_index': 1, 'aut_stab_index': 1, 'aut_weyl_group': '2.1', 'aut_weyl_index': 5940, 'centralizer': '2.a1.a1', 'complements': [], 'conjugacy_class_count': 1, 'contained_in': ['12.a1.a1', '44.a1.a1', '66.a1.a1', '66.b1.a1', '66.b1.b1'], 'contains': ['396.a1.a1'], 'core': '132.a1.a1', 'coset_action_label': None, 'count': 1, 'diagramx': [4068, 5001, 5920, 5955, 4408, 6240, 4471, 5119], 'generators': [264], 'label': '396.9.132.a1.a1', 'mobius_quo': -1, 'mobius_sub': 66, 'normal_closure': '132.a1.a1', 'normal_contained_in': ['12.a1.a1', '44.a1.a1', '66.a1.a1'], 'normal_contains': ['396.a1.a1'], 'normalizer': '1.a1.a1', 'old_label': '132.a1.a1', 'projective_image': '396.9', 'quotient_action_image': '2.1', 'quotient_action_kernel': '66.4', 'quotient_action_kernel_order': 66, 'quotient_fusion': None, 'short_label': '132.a1.a1', 'subgroup_fusion': None, 'weyl_group': '2.1'}
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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}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '4.2', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 990, 'aut_gen_orders': [30, 30, 30, 6], 'aut_gens': [[1, 2], [49, 94], [95, 238], [101, 382], [109, 262]], 'aut_group': None, 'aut_hash': 4676639020669913352, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 11880, 'aut_permdeg': 198, 'aut_perms': [5472016787768452843848087026436369699809196834011976419326714317015425008377424481334962170464217514508561968842044448941854282395247242361231244589749803147691392297100597192403784343277723733748839389775944116750389967941902047578616727762939071634116760692447647639222291435060375744034530604019172297850149496204636373623882633571860184739776524022957278230759078193, 3983713445047449321170146108588407152529085132713279173321954786780828100858511937283653796579462113122193005787185511170689900924843812803512759849500743614810196370348916312123491309676449269483706509973217522293063824822306467245426110708181075676079541371123599407047225935727653588767676666200024041239645648593203581579263704285688163013128547410947751117933132751, 18621095868459628348654863474698836961523279219559163372431563970903258291122088581979142784794290030067362792264572542129919349270250257256951759166306987168264308046656152357443752372224598911778819697576730923128881948462189683468412377881707654585450657088817916369927035137376031599249920511536743767631046085426349025050868008065379294798124834696889991630262172215, 16023925158070106671457671140016145742431213762158477269431055484045528872707778707997076139340811125490379780340280887639457901998516396625555479985781131946536120501947123764171633627965120261576346410041050058113677796489443914685618583713863520275067480472912609445917337573630256308248131130243966398984862810629989232751007480480387385135924145143801598829893315851], 'aut_phi_ratio': 99.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 99, 2, 1], [3, 2, 1, 1], [6, 2, 1, 1], [9, 2, 3, 1], [11, 2, 5, 1], [18, 2, 3, 1], [22, 2, 5, 1], [33, 2, 10, 1], [66, 2, 10, 1], [99, 2, 30, 1], [198, 2, 30, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_{99}.C_{30}.C_2^2', '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': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 990, 'autcentquo_group': '5940.d', 'autcentquo_hash': 4717361004838527357, 'autcentquo_nilpotent': False, 'autcentquo_order': 5940, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'D_{99}:C_{30}', 'cc_stats': [[1, 1, 1], [2, 1, 1], [2, 99, 2], [3, 2, 1], [6, 2, 1], [9, 2, 3], [11, 2, 5], [18, 2, 3], [22, 2, 5], [33, 2, 10], [66, 2, 10], [99, 2, 30], [198, 2, 30]], 'center_label': '2.1', 'center_order': 2, 'central_product': True, 'central_quotient': '198.3', 'commutator_count': 1, 'commutator_label': '99.1', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '3.1', '3.1', '11.1'], 'composition_length': 5, 'conjugacy_classes_known': True, 'counter': 9, 'cyclic': False, 'derived_length': 2, 'dihedral': True, 'direct_factorization': [['198.3', 1], ['2.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 99, 1, 2], [3, 2, 1, 1], [6, 2, 1, 1], [9, 2, 3, 1], [11, 2, 5, 1], [18, 2, 3, 1], [22, 2, 5, 1], [33, 2, 10, 1], [66, 2, 10, 1], [99, 2, 30, 1], [198, 2, 30, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 3, 'exponent': 198, 'exponents_of_order': [2, 2, 1], 'factors_of_aut_order': [2, 3, 5, 11], 'factors_of_order': [2, 3, 11], 'faithful_reps': [[2, 1, 30]], 'familial': True, 'frattini_label': '3.1', 'frattini_quotient': '132.9', 'hash': 9, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 198, 'inner_gen_orders': [2, 99], 'inner_gens': [[1, 394], [5, 2]], 'inner_hash': 3, 'inner_nilpotent': False, 'inner_order': 198, 'inner_split': True, 'inner_tex': 'D_{99}', 'inner_used': [1, 2], 'irrC_degree': 2, 'irrQ_degree': 60, 'irrQ_dim': 60, 'irrR_degree': 2, 'irrep_stats': [[1, 4], [2, 98]], 'label': '396.9', 'linC_count': 30, 'linC_degree': 2, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 16, 'linQ_degree_count': 3, 'linQ_dim': 16, 'linQ_dim_count': 3, 'linR_count': 30, 'linR_degree': 2, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'D198', '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, 0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 13, 'number_characteristic_subgroups': 13, 'number_conjugacy_classes': 102, 'number_divisions': 14, 'number_normal_subgroups': 15, 'number_subgroup_autclasses': 24, 'number_subgroup_classes': 30, 'number_subgroups': 480, 'old_label': None, 'order': 396, 'order_factorization_type': 222, 'order_stats': [[1, 1], [2, 199], [3, 2], [6, 2], [9, 6], [11, 10], [18, 6], [22, 10], [33, 20], [66, 20], [99, 60], [198, 60]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 30, 'outer_gen_orders': [2, 30], 'outer_gen_pows': [352, 0], 'outer_gens': [[353, 218], [111, 238]], 'outer_group': '60.13', 'outer_hash': 13, 'outer_nilpotent': True, 'outer_order': 60, 'outer_permdeg': 12, 'outer_perms': [40279680, 368673], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2\\times C_{30}', 'pc_rank': 2, 'perfect': False, 'permutation_degree': 22, 'pgroup': 0, 'primary_abelian_invariants': [2, 2], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 2], [6, 2], [10, 2], [20, 2], [60, 2]], 'representations': {'PC': {'code': 57201180091164480332600119, 'gens': [1, 2], 'pres': [5, -2, -2, -3, -3, -11, 3941, 26, 5882, 57, 7683, 78, 9004]}, 'GLFp': {'d': 2, 'p': 197, 'gens': [1498493109, 1358484794]}, 'Perm': {'d': 22, 'gens': [5353116803292348847, 479001600, 58755318410460672000, 4364893, 112426456502987827200]}}, 'schur_multiplier': [2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2], 'solvability_type': 6, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'D_{198}', 'transitive_degree': 198, 'wreath_data': None, 'wreath_product': False}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '4.2', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 330, 'aut_gen_orders': [2, 10, 66], 'aut_gens': [[1, 2], [67, 46], [67, 122], [99, 2]], 'aut_group': '1320.142', 'aut_hash': 142, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 1320, 'aut_permdeg': 16, 'aut_perms': [6, 834421876561, 2627969828449], 'aut_phi_ratio': 33.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 33, 2, 1], [3, 2, 1, 1], [6, 2, 1, 1], [11, 2, 5, 1], [22, 2, 5, 1], [33, 2, 10, 1], [66, 2, 10, 1]], 'aut_supersolvable': True, 'aut_tex': 'D_6\\times F_{11}', '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': 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], [2, 33, 2], [3, 2, 1], [6, 2, 1], [11, 2, 5], [22, 2, 5], [33, 2, 10], [66, 2, 10]], 'center_label': '2.1', 'center_order': 2, 'central_product': True, 'central_quotient': '66.3', 'commutator_count': 1, 'commutator_label': '33.1', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '3.1', '11.1'], 'composition_length': 4, 'conjugacy_classes_known': True, 'counter': 9, 'cyclic': False, 'derived_length': 2, 'dihedral': True, 'direct_factorization': [['2.1', 1], ['66.3', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 33, 1, 2], [3, 2, 1, 1], [6, 2, 1, 1], [11, 2, 5, 1], [22, 2, 5, 1], [33, 2, 10, 1], [66, 2, 10, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 3, 'exponent': 66, 'exponents_of_order': [2, 1, 1], 'factors_of_aut_order': [2, 3, 5, 11], 'factors_of_order': [2, 3, 11], 'faithful_reps': [[2, 1, 10]], 'familial': True, 'frattini_label': '1.1', 'frattini_quotient': '132.9', 'hash': 9, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 66, 'inner_gen_orders': [2, 33], 'inner_gens': [[1, 130], [5, 2]], 'inner_hash': 3, 'inner_nilpotent': False, 'inner_order': 66, 'inner_split': True, 'inner_tex': 'D_{33}', 'inner_used': [1, 2], 'irrC_degree': 2, 'irrQ_degree': 20, 'irrQ_dim': 20, 'irrR_degree': 2, 'irrep_stats': [[1, 4], [2, 32]], 'label': '132.9', 'linC_count': 10, 'linC_degree': 2, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 12, 'linQ_degree_count': 3, 'linQ_dim': 12, 'linQ_dim_count': 3, 'linR_count': 10, 'linR_degree': 2, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'D66', 'ngens': 4, 'nilpotency_class': -1, 'nilpotent': False, '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': 9, 'number_characteristic_subgroups': 9, 'number_conjugacy_classes': 36, 'number_divisions': 10, 'number_normal_subgroups': 11, 'number_subgroup_autclasses': 16, 'number_subgroup_classes': 20, 'number_subgroups': 152, 'old_label': None, 'order': 132, 'order_factorization_type': 222, 'order_stats': [[1, 1], [2, 67], [3, 2], [6, 2], [11, 10], [22, 10], [33, 20], [66, 20]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 10, 'outer_gen_orders': [2, 10], 'outer_gen_pows': [0, 110], 'outer_gens': [[67, 86], [89, 38]], 'outer_group': '20.5', 'outer_hash': 5, 'outer_nilpotent': True, 'outer_order': 20, 'outer_permdeg': 9, 'outer_perms': [40320, 810], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2\\times C_{10}', 'pc_rank': 2, 'perfect': False, 'permutation_degree': 16, 'pgroup': 0, 'primary_abelian_invariants': [2, 2], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 2], [10, 2], [20, 2]], 'representations': {'PC': {'code': 14239841398380019, 'gens': [1, 2], 'pres': [4, -2, -2, -3, -11, 1041, 21, 1538, 46, 1923]}, 'GLFp': {'d': 2, 'p': 67, 'gens': [4556, 10225944]}, 'Perm': {'d': 16, 'gens': [87660962646, 1, 30, 1482996292560]}}, 'schur_multiplier': [2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2], 'solvability_type': 6, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'D_{66}', 'transitive_degree': 66, 'wreath_data': None, 'wreath_product': False}
