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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': True, 'ambient': '448.244', 'ambient_counter': 244, 'ambient_order': 448, 'ambient_tex': 'C_4^2.D_{14}', 'central': False, 'central_factor': False, 'centralizer_order': 16, 'characteristic': False, 'core_order': 1, 'counter': 119, 'cyclic': True, 'direct': None, 'hall': 0, 'label': '448.244.224.d1.b1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': True, 'minimal_normal': False, 'nilpotent': True, 'normal': False, 'old_label': '224.d1.b1', 'outer_equivalence': False, 'perfect': False, 'proper': True, 'quotient': None, 'quotient_Agroup': None, 'quotient_abelian': None, 'quotient_cyclic': None, 'quotient_hash': None, 'quotient_metabelian': None, 'quotient_nilpotent': None, 'quotient_order': 224, 'quotient_simple': None, 'quotient_solvable': None, 'quotient_supersolvable': None, 'quotient_tex': None, 'simple': True, 'solvable': True, 'special_labels': [], 'split': None, 'standard_generators': False, 'stem': False, 'subgroup': '2.1', 'subgroup_hash': 1, 'subgroup_order': 2, 'subgroup_tex': 'C_2', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '448.244', 'aut_centralizer_order': 192, 'aut_label': '224.d1', 'aut_quo_index': None, 'aut_stab_index': 56, 'aut_weyl_group': '1.1', 'aut_weyl_index': 10752, 'centralizer': '28.d1.a1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['32.d1.b1', '112.d1.a1', '112.h1.a1', '112.j1.b1'], 'contains': ['448.a1.a1'], 'core': '448.a1.a1', 'coset_action_label': None, 'count': 28, 'diagramx': [1118, -1, 1999, -1, 6844, -1, 7493, -1], 'generators': [1], 'label': '448.244.224.d1.b1', 'mobius_quo': None, 'mobius_sub': 0, 'normal_closure': '8.d1.b1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '28.d1.a1', 'old_label': '224.d1.b1', 'projective_image': '448.244', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '224.d1.b1', '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': '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}
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gps_groups • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '16.10', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 84, 'aut_gen_orders': [6, 6, 6, 12, 14, 14, 14, 12, 6], 'aut_gens': [[1, 2, 8], [257, 230, 204], [325, 6, 28], [33, 2, 380], [209, 6, 204], [197, 6, 236], [65, 230, 232], [417, 2, 120], [145, 230, 264], [161, 226, 44]], 'aut_group': None, 'aut_hash': 5229004673012084454, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 10752, 'aut_permdeg': 68, 'aut_perms': [1544084595643385320346870193681582532976070253047324241144119883933173501777933214501022820838791, 862441917082964479018095723487712174182100407054586854400538425383947600704288055831883304863363, 1510409190498044362703572482145581358473836558715334673473862318730186047649294616525681437897328, 1188867323173938972711569979987607741460604068191327502310802343000399506241685418461652049248549, 382989322363970962977127681944351480621395587488710249044875933305065974493226567727563522533764, 2342828386628604353222041087622738825947121492103553513548608697531723732625862811272562072022923, 1295995027966040763360128522862796777126228781733842236719754143227246651444667405425629970654476, 1781880883308290338012806165439311976602243950579636519194924267759311952471700447373323781493230, 266695696832983279463094787060287567175355149195292124897890948223169857023648010158441345669418], 'aut_phi_ratio': 56.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [2, 28, 2, 1], [4, 2, 1, 2], [4, 2, 2, 2], [4, 28, 2, 3], [7, 2, 3, 1], [8, 4, 2, 2], [14, 2, 3, 3], [28, 2, 6, 2], [28, 4, 6, 2], [56, 4, 12, 2]], 'aut_supersolvable': True, 'aut_tex': 'C_7.(C_2^5\\times C_6).C_2^3', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 2, 'autcent_group': '64.267', 'autcent_hash': 267, 'autcent_nilpotent': True, 'autcent_order': 64, 'autcent_solvable': True, 'autcent_split': False, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^6', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 42, 'autcentquo_group': '168.47', 'autcentquo_hash': 47, 'autcentquo_nilpotent': False, 'autcentquo_order': 168, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'C_2^2\\times F_7', 'cc_stats': [[1, 1, 1], [2, 1, 3], [2, 28, 2], [4, 2, 6], [4, 28, 6], [7, 2, 3], [8, 4, 4], [14, 2, 9], [28, 2, 12], [28, 4, 12], [56, 4, 24]], 'center_label': '4.2', 'center_order': 4, 'central_product': False, 'central_quotient': '112.29', 'commutator_count': 1, 'commutator_label': '28.2', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '2.1', '2.1', '7.1'], 'composition_length': 7, 'conjugacy_classes_known': True, 'counter': 244, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [2, 28, 1, 2], [4, 2, 1, 2], [4, 2, 2, 2], [4, 28, 1, 2], [4, 28, 2, 2], [7, 2, 3, 1], [8, 4, 1, 2], [8, 4, 2, 1], [14, 2, 3, 3], [28, 2, 6, 2], [28, 4, 6, 2], [56, 4, 6, 2], [56, 4, 12, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 2688, 'exponent': 56, 'exponents_of_order': [6, 1], 'factors_of_aut_order': [2, 3, 7], 'factors_of_order': [2, 7], 'faithful_reps': [], 'familial': False, 'frattini_label': '8.2', 'frattini_quotient': '56.12', 'hash': 244, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 28, 'inner_gen_orders': [2, 2, 28], 'inner_gens': [[1, 2, 216], [1, 2, 232], [241, 226, 8]], 'inner_hash': 29, 'inner_nilpotent': False, 'inner_order': 112, 'inner_split': None, 'inner_tex': 'C_2\\times D_{28}', 'inner_used': [1, 2, 3], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 16], [2, 52], [4, 14]], 'label': '448.244', 'linC_count': 96, 'linC_degree': 5, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 12, 'linQ_degree_count': 32, 'linQ_dim': 12, 'linQ_dim_count': 16, 'linR_count': 24, 'linR_degree': 6, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'C4^2.D14', 'ngens': 7, 'nilpotency_class': -1, 'nilpotent': False, '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': 24, 'number_characteristic_subgroups': 39, 'number_conjugacy_classes': 82, 'number_divisions': 28, 'number_normal_subgroups': 51, 'number_subgroup_autclasses': 100, 'number_subgroup_classes': 120, 'number_subgroups': 676, 'old_label': None, 'order': 448, 'order_factorization_type': 51, 'order_stats': [[1, 1], [2, 59], [4, 180], [7, 6], [8, 16], [14, 18], [28, 72], [56, 96]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 6, 'outer_gen_orders': [2, 2, 2, 2, 6], 'outer_gen_pows': [400, 192, 0, 192, 336], 'outer_gens': [[177, 6, 236], [197, 6, 236], [161, 2, 444], [417, 2, 120], [241, 230, 136]], 'outer_group': '96.231', 'outer_hash': 231, 'outer_nilpotent': True, 'outer_order': 96, 'outer_permdeg': 96, 'outer_perms': [373740914582254807103027978750459456754894590119561231940756477506607035487443908635698901547987256246234198117623553709186676733749862111940121806699, 42424868060139296843491445511124423515857297872562393482278511423147335551867951200358921660647020438214393220183616984276376320706613128195194752034, 543300300654097092556438557543567162191149721523630138375279791872143555879776996912104244871876924506517464815576131750928699133865862316293821400379, 63304639940175329827575756773775981676374107052861334639764423899533324974159335097911917826693311256845316444363740812661256798612682539638775808000, 414101987395272820879706417919270202713952224910878512007972886796411882979344693450412895241154702044783478943061471582417160826333905168106729330327], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^4\\times C_6', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 19, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 4], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 8], [2, 6], [4, 3], [6, 4], [12, 4], [24, 3]], 'representations': {'PC': {'code': 610349346814309512887090824076738309, 'gens': [1, 2, 4], 'pres': [7, -2, -2, -2, -2, -2, -2, -7, 36, 6051, 3258, 80, 15124, 102, 17477, 124, 18822]}, 'Perm': {'d': 19, 'gens': [357001329487492, 40285737, 87457123, 40279680, 12568, 18498, 7116370137676800]}}, 'schur_multiplier': [2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 4], 'solvability_type': 7, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_4^2.D_{14}', 'transitive_degree': 224, 'wreath_data': None, 'wreath_product': False}
