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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': True, 'ambient': '1284.8', 'ambient_counter': 8, 'ambient_order': 1284, 'ambient_tex': 'S_3\\times C_{214}', 'central': False, 'central_factor': False, 'centralizer_order': 642, 'characteristic': True, 'core_order': 642, 'counter': 2, 'cyclic': True, 'direct': False, 'hall': 0, 'label': '1284.8.2.a1.a1', 'maximal': True, 'maximal_normal': True, 'metabelian': True, 'metacyclic': True, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': True, 'old_label': '2.a1.a1', 'outer_equivalence': False, 'perfect': False, 'proper': True, 'quotient': '2.1', 'quotient_Agroup': True, 'quotient_abelian': True, 'quotient_cyclic': True, 'quotient_hash': 1, 'quotient_metabelian': True, 'quotient_nilpotent': True, 'quotient_order': 2, 'quotient_simple': True, 'quotient_solvable': True, 'quotient_supersolvable': True, 'quotient_tex': 'C_2', 'simple': False, 'solvable': True, 'special_labels': ['F', 'S'], 'split': True, 'standard_generators': False, 'stem': False, 'subgroup': '642.4', 'subgroup_hash': 4, 'subgroup_order': 642, 'subgroup_tex': 'C_{642}', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '1284.8', 'aut_centralizer_order': 6, 'aut_label': '2.a1', 'aut_quo_index': 1, 'aut_stab_index': 1, 'aut_weyl_group': '212.5', 'aut_weyl_index': 6, 'centralizer': '2.a1.a1', 'complements': ['642.b1.a1', '642.b1.b1'], 'conjugacy_class_count': 1, 'contained_in': ['1.a1.a1'], 'contains': ['4.a1.a1', '6.a1.a1', '214.a1.a1'], 'core': '2.a1.a1', 'coset_action_label': None, 'count': 1, 'diagramx': [2647, 7034, 771, 2335, 999, 7641, 3219, 9200], 'generators': [642, 856, 12], 'label': '1284.8.2.a1.a1', 'mobius_quo': -1, 'mobius_sub': -1, 'normal_closure': '2.a1.a1', 'normal_contained_in': ['1.a1.a1'], 'normal_contains': ['4.a1.a1', '6.a1.a1', '214.a1.a1'], 'normalizer': '1.a1.a1', 'old_label': '2.a1.a1', 'projective_image': '6.1', 'quotient_action_image': '2.1', 'quotient_action_kernel': '1.1', 'quotient_action_kernel_order': 1, 'quotient_fusion': None, 'short_label': '2.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': '642.4', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': True, 'aut_cyclic': False, 'aut_derived_length': 1, 'aut_exponent': 106, 'aut_gen_orders': [2, 106], 'aut_gens': [[1], [427], [53]], 'aut_group': '212.5', 'aut_hash': 5, 'aut_nilpotency_class': 1, 'aut_nilpotent': True, 'aut_order': 212, 'aut_permdeg': 108, 'aut_perms': [557024662637020771945084889843702490055696591610664751166773942071598365582559610699529229471403417181300693114612878523587371085356187371180171985788916707481021385105407999, 1170453789648879928147644221021994479217309672971741413562818786645778635813084173781499226277378559080115491950416834537733970642052124512531699563254404578118303358619772589], 'aut_phi_ratio': 1.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [3, 1, 2, 1], [6, 1, 2, 1], [107, 1, 106, 1], [214, 1, 106, 1], [321, 1, 212, 1], [642, 1, 212, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_2\\times C_{106}', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 106, 'autcent_group': '212.5', 'autcent_hash': 5, 'autcent_nilpotent': True, 'autcent_order': 212, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2\\times C_{106}', '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], [6, 1, 2], [107, 1, 106], [214, 1, 106], [321, 1, 212], [642, 1, 212]], 'center_label': '642.4', 'center_order': 642, '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', '3.1', '107.1'], 'composition_length': 3, 'conjugacy_classes_known': True, 'counter': 4, 'cyclic': True, 'derived_length': 1, 'dihedral': False, 'direct_factorization': [['107.1', 1], ['2.1', 1], ['3.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [3, 1, 2, 1], [6, 1, 2, 1], [107, 1, 106, 1], [214, 1, 106, 1], [321, 1, 212, 1], [642, 1, 212, 1]], 'element_repr_type': 'PC', 'elementary': 642, 'eulerian_function': 1, 'exponent': 642, 'exponents_of_order': [1, 1, 1], 'factors_of_aut_order': [2, 53], 'factors_of_order': [2, 3, 107], 'faithful_reps': [[1, 0, 212]], 'familial': True, 'frattini_label': '1.1', 'frattini_quotient': '642.4', 'hash': 4, 'hyperelementary': 642, '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': 212, 'irrQ_dim': 212, 'irrR_degree': None, 'irrep_stats': [[1, 642]], 'label': '642.4', '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': 'C642', 'ngens': 3, 'nilpotency_class': 1, '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': 642, 'number_divisions': 8, 'number_normal_subgroups': 8, 'number_subgroup_autclasses': 8, 'number_subgroup_classes': 8, 'number_subgroups': 8, 'old_label': None, 'order': 642, 'order_factorization_type': 11, 'order_stats': [[1, 1], [2, 1], [3, 2], [6, 2], [107, 106], [214, 106], [321, 212], [642, 212]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 106, 'outer_gen_orders': [2, 106], 'outer_gen_pows': [0, 0], 'outer_gens': [[427], [53]], 'outer_group': '212.5', 'outer_hash': 5, 'outer_nilpotent': True, 'outer_order': 212, 'outer_permdeg': 108, 'outer_perms': [557024662637020771945084889843702490055696591610664751166773942071598365582559610699529229471403417181300693114612878523587371085356187371180171985788916707481021385105407999, 1170453789648879928147644221021994479217309672971741413562818786645778635813084173781499226277378559080115491950416834537733970642052124512531699563254404578118303358619772589], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2\\times C_{106}', 'pc_rank': 1, 'perfect': False, 'permutation_degree': 112, 'pgroup': 0, 'primary_abelian_invariants': [2, 3, 107], 'quasisimple': False, 'rank': 1, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 2], [2, 2], [106, 2], [212, 2]], 'representations': {'PC': {'code': 28549844747, 'gens': [1], 'pres': [3, -2, -3, -107, 6, 22]}, 'GLFp': {'d': 2, 'p': 641, 'gens': [127549321384]}, 'Perm': {'d': 112, 'gens': [1762952551090244663872161047107075788761409536026565516041574063347346955087248316436555574598462315773196047662837978913145847497199871623320096254145331200000000000000000000000000, 288771916640498716440976420492559506758625640626792058319668151244446675689966964199272002391230518554168066775239636185609475429516768488668320434749440000000000000000000000000, 12150573975587908558064082271972503944261179666958805379263968688536773312539694302781998252415217388457727618432222125237219155992191776065669038080000000000000000000000000]}}, 'schur_multiplier': [], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [642], 'solvability_type': 0, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_{642}', 'transitive_degree': 642, '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': '428.4', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 318, 'aut_gen_orders': [2, 2, 318], 'aut_gens': [[1, 2], [1, 854], [1, 430], [215, 422]], 'aut_group': '1272.42', 'aut_hash': 42, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 1272, 'aut_permdeg': 112, 'aut_perms': [1378276269579308311041959210487656557616053986477149127881350565071879447209316155919079989926763094632722969268213801422212934731684117913755331674462171538323655805731455631359999, 64791835513190871888452927681661780126567996302198396175948976051400977214153754065840297865411855534436717688732257291976939421183246336000000000, 33079031560989574594910091140803980836426652625725227730415628687407076200692518206432270892153917602254240886604404925886729972607150096185636183457301730449523630388824564447784959], 'aut_phi_ratio': 3.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 3, 2, 1], [3, 2, 1, 1], [6, 2, 1, 1], [107, 1, 106, 1], [214, 1, 106, 1], [214, 3, 212, 1], [321, 2, 106, 1], [642, 2, 106, 1]], 'aut_supersolvable': True, 'aut_tex': 'D_6\\times C_{106}', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 106, 'autcent_group': '212.5', 'autcent_hash': 5, 'autcent_nilpotent': True, 'autcent_order': 212, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2\\times C_{106}', '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, 1], [2, 3, 2], [3, 2, 1], [6, 2, 1], [107, 1, 106], [214, 1, 106], [214, 3, 212], [321, 2, 106], [642, 2, 106]], 'center_label': '214.2', 'center_order': 214, 'central_product': True, 'central_quotient': '6.1', 'commutator_count': 1, 'commutator_label': '3.1', 'complements_known': True, 'complete': False, 'complex_characters_known': False, 'composition_factors': ['2.1', '2.1', '3.1', '107.1'], 'composition_length': 4, 'conjugacy_classes_known': True, 'counter': 8, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['107.1', 1], ['2.1', 1], ['6.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 3, 1, 2], [3, 2, 1, 1], [6, 2, 1, 1], [107, 1, 106, 1], [214, 1, 106, 1], [214, 3, 106, 2], [321, 2, 106, 1], [642, 2, 106, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 324, 'exponent': 642, 'exponents_of_order': [2, 1, 1], 'factors_of_aut_order': [2, 3, 53], 'factors_of_order': [2, 3, 107], 'faithful_reps': [[2, 0, 106]], 'familial': False, 'frattini_label': '1.1', 'frattini_quotient': '1284.8', 'hash': 8, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 6, 'inner_gen_orders': [2, 3], 'inner_gens': [[1, 430], [857, 2]], 'inner_hash': 1, 'inner_nilpotent': False, 'inner_order': 6, 'inner_split': True, 'inner_tex': 'S_3', 'inner_used': [1, 2], 'irrC_degree': 2, 'irrQ_degree': 212, 'irrQ_dim': 212, 'irrR_degree': None, 'irrep_stats': [[1, 428], [2, 214]], 'label': '1284.8', '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': True, 'monomial': True, 'name': 'S3*C214', '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': 10, 'number_characteristic_subgroups': 10, 'number_conjugacy_classes': 642, 'number_divisions': 12, 'number_normal_subgroups': 14, 'number_subgroup_autclasses': 16, 'number_subgroup_classes': 20, 'number_subgroups': 32, 'old_label': None, 'order': 1284, 'order_factorization_type': 222, 'order_stats': [[1, 1], [2, 7], [3, 2], [6, 2], [107, 106], [214, 742], [321, 212], [642, 212]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 106, 'outer_gen_orders': [2, 106], 'outer_gen_pows': [4, 0], 'outer_gens': [[215, 2], [1, 374]], 'outer_group': '212.5', 'outer_hash': 5, 'outer_nilpotent': True, 'outer_order': 212, 'outer_permdeg': 57, 'outer_perms': [710998587804863451854045647463724949736497978881168458687447040000000000000, 235037922448130462157819095799560576965993618854733871579136000000000000], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2\\times C_{106}', 'pc_rank': 2, 'perfect': False, 'permutation_degree': 112, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 107], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 2], [106, 4], [212, 2]], 'representations': {'PC': {'code': 24382677213724686253235, 'gens': [1, 2], 'pres': [4, -2, -2, -3, -107, 3441, 21, 10322, 46]}, 'GLFp': {'d': 2, 'p': 643, 'gens': [47320892312, 11963146860, 414092]}, 'Perm': {'d': 112, 'gens': [6, 1, 1794720111433615139042129505015880909184014701785542626517322669898565102217738672375305160446549314266850740456282898849595419705987604573151881818199523207157487649617276886080560, 30]}}, 'schur_multiplier': [2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 214], 'solvability_type': 6, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'S_3\\times C_{214}', 'transitive_degree': 642, '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': '2.1', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': True, 'aut_cyclic': True, 'aut_derived_length': 0, 'aut_exponent': 1, 'aut_gen_orders': [], 'aut_gens': [[1]], 'aut_group': '1.1', 'aut_hash': 1, 'aut_nilpotency_class': 0, 'aut_nilpotent': True, 'aut_order': 1, 'aut_permdeg': 1, 'aut_perms': [], 'aut_phi_ratio': 1.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_1', 'autcent_abelian': True, 'autcent_cyclic': True, 'autcent_exponent': 1, 'autcent_group': '1.1', 'autcent_hash': 1, 'autcent_nilpotent': True, 'autcent_order': 1, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_1', '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]], 'center_label': '2.1', 'center_order': 2, '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': ['2.1'], 'composition_length': 1, 'conjugacy_classes_known': True, 'counter': 1, 'cyclic': True, 'derived_length': 1, 'dihedral': True, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1]], 'element_repr_type': 'PC', 'elementary': 2, 'eulerian_function': 1, 'exponent': 2, 'exponents_of_order': [1], 'factors_of_aut_order': [], 'factors_of_order': [2], 'faithful_reps': [[1, 1, 1]], 'familial': True, 'frattini_label': '1.1', 'frattini_quotient': '2.1', 'hash': 1, 'hyperelementary': 2, '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': 1, 'irrQ_dim': 1, 'irrR_degree': 1, 'irrep_stats': [[1, 2]], 'label': '2.1', 'linC_count': 1, 'linC_degree': 1, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 1, 'linQ_degree_count': 1, 'linQ_dim': 1, 'linQ_dim_count': 1, 'linR_count': 1, 'linR_degree': 1, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C2', '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': 2, 'number_divisions': 2, 'number_normal_subgroups': 2, 'number_subgroup_autclasses': 2, 'number_subgroup_classes': 2, 'number_subgroups': 2, 'old_label': None, 'order': 2, 'order_factorization_type': 1, 'order_stats': [[1, 1], [2, 1]], 'outer_abelian': True, 'outer_cyclic': True, 'outer_equivalence': False, 'outer_exponent': 1, 'outer_gen_orders': [], 'outer_gen_pows': [], 'outer_gens': [], 'outer_group': '1.1', 'outer_hash': 1, 'outer_nilpotent': True, 'outer_order': 1, 'outer_permdeg': 1, 'outer_perms': [], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_1', 'pc_rank': 1, 'perfect': False, 'permutation_degree': 2, 'pgroup': 2, 'primary_abelian_invariants': [2], 'quasisimple': False, 'rank': 1, 'rational': True, 'rational_characters_known': True, 'ratrep_stats': [[1, 2]], 'representations': {'PC': {'code': 0, 'gens': [1], 'pres': [1, -2]}, 'GLZ': {'b': 3, 'd': 2, 'gens': [12]}, 'Lie': [{'d': 2, 'q': 2, 'gens': [6], 'family': 'CSOPlus'}, {'d': 2, 'q': 2, 'gens': [6], 'family': 'COPlus'}, {'d': 1, 'q': 2, 'gens': [1], 'family': 'AGL'}, {'d': 1, 'q': 2, 'gens': [1], 'family': 'AGammaL'}, {'d': 1, 'q': 2, 'gens': [1], 'family': 'ASigmaL'}], 'GLFp': {'d': 2, 'p': 2, 'gens': [11]}, 'Perm': {'d': 2, 'gens': [1]}}, 'schur_multiplier': [], 'semidirect_product': False, 'simple': True, 'smith_abelian_invariants': [2], 'solvability_type': 0, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_2', 'transitive_degree': 2, 'wreath_data': None, 'wreath_product': False}
