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gps_subgroup_search • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'ambient': '1344.8738', 'ambient_counter': 8738, 'ambient_order': 1344, 'ambient_tex': 'D_{84}:C_2^3', 'central': False, 'central_factor': False, 'centralizer_order': 8, 'characteristic': False, 'core_order': 84, 'counter': 46, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '1344.8738.8.o1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': None, 'minimal': False, 'minimal_normal': False, 'nilpotent': False, 'normal': False, 'old_label': '8.o1', 'outer_equivalence': True, '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': 8, 'quotient_simple': None, 'quotient_solvable': None, 'quotient_supersolvable': None, 'quotient_tex': None, 'simple': False, 'solvable': True, 'special_labels': [], 'split': None, 'standard_generators': False, 'stem': False, 'subgroup': '168.38', 'subgroup_hash': 38, 'subgroup_order': 168, 'subgroup_tex': 'C_{21}:D_4', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '1344.8738', 'aut_centralizer_order': None, 'aut_label': '8.o1', 'aut_quo_index': None, 'aut_stab_index': None, 'aut_weyl_group': None, 'aut_weyl_index': None, 'centralizer': '168.c1', 'complements': None, 'conjugacy_class_count': 4, 'contained_in': ['4.g1', '4.l1', '4.p1'], 'contains': ['16.b1', '16.j1', '16.k1', '24.l1', '56.l1'], 'core': '16.b1', 'coset_action_label': None, 'count': 8, 'diagramx': None, 'generators': [673, 112, 448, 6, 32], 'label': '1344.8738.8.o1', 'mobius_quo': None, 'mobius_sub': None, 'normal_closure': '4.g1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '2.b1', 'old_label': '8.o1', 'projective_image': None, 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '8.o1', 'subgroup_fusion': None, 'weyl_group': None}
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gps_groups • Show schema
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{'Agroup': False, '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': 42, 'aut_gen_orders': [2, 2, 6, 42], 'aut_gens': [[1892, 2305706, 3339295], [79464, 2305706, 3339295], [1892, 954102, 79549], [1892, 397561, 79549], [33067, 2305706, 79549]], 'aut_group': '1008.906', 'aut_hash': 906, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 1008, 'aut_permdeg': 14, 'aut_perms': [7, 136, 2079302520, 7192322776], 'aut_phi_ratio': 21.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 2, 1, 1], [2, 42, 1, 1], [3, 2, 1, 1], [4, 42, 1, 1], [6, 2, 1, 1], [6, 2, 2, 1], [7, 2, 3, 1], [14, 2, 3, 1], [14, 2, 6, 1], [21, 2, 6, 1], [42, 2, 6, 1], [42, 2, 12, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_2\\times D_6\\times F_7', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 2, 'autcent_group': '4.2', 'autcent_hash': 2, 'autcent_nilpotent': True, 'autcent_order': 4, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^2', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 42, 'autcentquo_group': '252.26', 'autcentquo_hash': 26, 'autcentquo_nilpotent': False, 'autcentquo_order': 252, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'S_3\\times F_7', 'cc_stats': [[1, 1, 1], [2, 1, 1], [2, 2, 1], [2, 42, 1], [3, 2, 1], [4, 42, 1], [6, 2, 3], [7, 2, 3], [14, 2, 9], [21, 2, 6], [42, 2, 18]], 'center_label': '2.1', 'center_order': 2, 'central_product': False, 'central_quotient': '84.14', 'commutator_count': 1, 'commutator_label': '42.6', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '3.1', '7.1'], 'composition_length': 5, 'conjugacy_classes_known': True, 'counter': 38, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 2, 1, 1], [2, 42, 1, 1], [3, 2, 1, 1], [4, 42, 1, 1], [6, 2, 1, 1], [6, 2, 2, 1], [7, 2, 3, 1], [14, 2, 3, 1], [14, 2, 6, 1], [21, 2, 6, 1], [42, 2, 6, 1], [42, 2, 12, 1]], 'element_repr_type': 'GLFp', 'elementary': 1, 'eulerian_function': 6, 'exponent': 84, 'exponents_of_order': [3, 1, 1], 'factors_of_aut_order': [2, 3, 7], 'factors_of_order': [2, 3, 7], 'faithful_reps': [[2, 0, 12]], 'familial': False, 'frattini_label': '2.1', 'frattini_quotient': '84.14', 'hash': 38, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 42, 'inner_gen_orders': [2, 21, 2], 'inner_gens': [[1892, 238550, 79549], [17673, 2305706, 3339295], [79464, 2305706, 3339295]], 'inner_hash': 14, 'inner_nilpotent': False, 'inner_order': 84, 'inner_split': True, 'inner_tex': 'D_{42}', 'inner_used': [1, 2, 3], 'irrC_degree': 2, 'irrQ_degree': 24, 'irrQ_dim': 24, 'irrR_degree': 4, 'irrep_stats': [[1, 4], [2, 41]], 'label': '168.38', 'linC_count': 12, 'linC_degree': 2, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 10, 'linQ_degree_count': 6, 'linQ_dim': 10, 'linQ_dim_count': 6, 'linR_count': 18, 'linR_degree': 4, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'C21:D4', 'ngens': 3, 'nilpotency_class': -1, 'nilpotent': False, 'normal_counts': [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': 14, 'number_characteristic_subgroups': 15, 'number_conjugacy_classes': 45, 'number_divisions': 14, 'number_normal_subgroups': 15, 'number_subgroup_autclasses': 32, 'number_subgroup_classes': 32, 'number_subgroups': 180, 'old_label': None, 'order': 168, 'order_factorization_type': 311, 'order_stats': [[1, 1], [2, 45], [3, 2], [4, 42], [6, 6], [7, 6], [14, 18], [21, 12], [42, 36]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 6, 'outer_gen_orders': [2, 6], 'outer_gen_pows': [79508, 79508], 'outer_gens': [[1892, 238550, 3339295], [1892, 2703257, 3339295]], 'outer_group': '12.5', 'outer_hash': 5, 'outer_nilpotent': True, 'outer_order': 12, 'outer_permdeg': 7, 'outer_perms': [720, 27], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2\\times C_6', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 14, 'pgroup': 0, 'primary_abelian_invariants': [2, 2], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 3], [4, 1], [6, 2], [12, 3], [24, 1]], 'representations': {'PC': {'code': 515514567040958603, 'gens': [1, 2, 3], 'pres': [5, -2, -2, 2, -3, -7, 2492, 42, 3203, 78, 3604]}, 'GLFp': {'d': 2, 'p': 43, 'gens': [1892, 2305706, 3339295]}, 'Perm': {'d': 14, 'gens': [997960447, 6266937600, 13412044800, 403200, 973]}}, 'schur_multiplier': [2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2], 'solvability_type': 7, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_{21}:D_4', 'transitive_degree': 84, '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': '16.14', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 84, 'aut_gen_orders': [14, 14, 6, 2, 2, 4, 6, 84, 12], 'aut_gens': [[1, 2, 4, 224], [1, 130, 4, 1232], [113, 803, 677, 1232], [785, 162, 1076, 224], [113, 2, 620, 1120], [785, 818, 612, 1120], [785, 787, 845, 1232], [113, 890, 44, 1120], [785, 186, 1180, 336], [785, 171, 1213, 1120]], 'aut_group': None, 'aut_hash': 2121436511852193631, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 258048, 'aut_permdeg': 160, 'aut_perms': [342906011369171562033954521938554985601957975504566000030952712549335734636881129851699990485637663445406151759108607235042485605874408943587900661240488275285008461927875433067481064427132138370375749114588227935183564555141272175457433493437365070305453225678268417016079132300134704, 77184662643784264146680829086333738935263263516654173511419733749619874700005739528167835446192067243048102228327192499971715801808787512994699311027798682914135255822844630012040060743064076216072745673597035828342677215189182529617208507257763387612219081204287230551592751125672132, 307536620730170784668664970864475533989642929602135953778519726972538165559157603096316485339070691186583462374948046646250555889882076611364253916816378452026811203832000666824668239007562720264460169419440968406500097845241567658756871148426067902765415254137846167398935067726515979, 356576634353358844403558877012815321726948323161621339512391538448979103878467868821545328175017596280833597448684787691008872309055550164961574033125591028682121026261370131120652244613024315377172108100991704889060818801482085649393071527693951641717951992920725630257580986287804074, 31410878843918519229828966039509603343960339192054635050243389839815251710393614533199967684187466032337740785569515203019655554706066784460396789631004166418178245687343759961221242907145532786969629100790879871338590351146348185685664841788513131553852918181803893821671844554218788, 411248652971741614041229440511284063069642116983738788651132432382316829957293135250450210733058214437590179611004117330643485899862704156486288223018586348928952649798500088108458314375006624463115788640332284244086733873110100019147612455610608352793142736519036841100955620227320036, 385766418702965439033848357895949741145834412302490890055706474471479349345304505093574000534531587997807857458166673290483470881708251189095232400691078404264002282384238745851374883637945195477305884873158665843528211703624737558213223534607336211305966004684405096510497725410263781, 404558514328673547242357920838390694774218535540688956694336210776084906251313672576296778404564855881859471791780476369464869824349656606224897914649513254246459369539512986839639280154946415443991915327227626035556403961009226887890122853337962681523019349358682313131364511091652236, 462991360470428782189704369597715627528693309313673919846844428519275361150502528697624740378077822094384891257729406248290679013081334226767671830182124227501683235690902178502972916852824314317529731416371307788549306560906510418424364468476847716596447121480409459568368416333605751], 'aut_phi_ratio': 672.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 1, 2, 1], [2, 2, 2, 1], [2, 28, 4, 1], [2, 84, 2, 1], [3, 2, 1, 1], [4, 2, 1, 2], [4, 2, 2, 1], [4, 84, 2, 1], [6, 2, 1, 1], [6, 2, 2, 1], [6, 2, 4, 1], [6, 28, 8, 1], [7, 2, 3, 1], [8, 12, 4, 1], [12, 4, 1, 2], [12, 4, 2, 1], [14, 2, 3, 1], [14, 2, 6, 1], [14, 4, 6, 1], [21, 4, 3, 1], [28, 2, 6, 2], [28, 4, 6, 1], [42, 4, 3, 1], [42, 4, 6, 1], [42, 4, 12, 1], [56, 12, 24, 1], [84, 4, 6, 2], [84, 4, 12, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_{42}.(C_2^5\\times C_6).C_2^5', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 4, 'autcent_group': '128.1578', 'autcent_hash': 1578, 'autcent_nilpotent': True, 'autcent_order': 128, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^3\\wr C_2', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 42, 'autcentquo_group': None, 'autcentquo_hash': 5451351369529179689, 'autcentquo_nilpotent': False, 'autcentquo_order': 2016, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'C_2^3\\times S_3\\times F_7', 'cc_stats': [[1, 1, 1], [2, 1, 3], [2, 2, 2], [2, 28, 4], [2, 84, 2], [3, 2, 1], [4, 2, 4], [4, 84, 2], [6, 2, 7], [6, 28, 8], [7, 2, 3], [8, 12, 4], [12, 4, 4], [14, 2, 9], [14, 4, 6], [21, 4, 3], [28, 2, 12], [28, 4, 6], [42, 4, 21], [56, 12, 24], [84, 4, 24]], 'center_label': '4.2', 'center_order': 4, 'central_product': True, 'central_quotient': '336.158', 'commutator_count': 1, 'commutator_label': '84.6', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '2.1', '2.1', '3.1', '7.1'], 'composition_length': 8, 'conjugacy_classes_known': True, 'counter': 8738, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['2.1', 1], ['672.456', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [2, 2, 1, 2], [2, 28, 1, 4], [2, 84, 1, 2], [3, 2, 1, 1], [4, 2, 1, 4], [4, 84, 1, 2], [6, 2, 1, 3], [6, 2, 2, 2], [6, 28, 2, 4], [7, 2, 3, 1], [8, 12, 1, 4], [12, 4, 1, 4], [14, 2, 3, 3], [14, 4, 3, 2], [21, 4, 3, 1], [28, 2, 6, 2], [28, 4, 3, 2], [42, 4, 3, 3], [42, 4, 6, 2], [56, 12, 6, 4], [84, 4, 6, 4]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 3734640, 'exponent': 168, 'exponents_of_order': [6, 1, 1], 'factors_of_aut_order': [2, 3, 7], 'factors_of_order': [2, 3, 7], 'faithful_reps': [], 'familial': False, 'frattini_label': '4.1', 'frattini_quotient': '336.219', 'hash': 8738, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 84, 'inner_gen_orders': [2, 2, 28, 3], 'inner_gens': [[1, 2, 116, 224], [1, 2, 220, 224], [113, 10, 4, 1120], [1, 2, 452, 224]], 'inner_hash': 158, 'inner_nilpotent': False, 'inner_order': 336, 'inner_split': False, 'inner_tex': 'C_6:D_{28}', 'inner_used': [1, 2, 3, 4], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 16], [2, 68], [4, 66]], 'label': '1344.8738', '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': False, 'monomial': True, 'name': 'D84:C2^3', 'ngens': 8, '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, 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': 34, 'number_characteristic_subgroups': 44, 'number_conjugacy_classes': 150, 'number_divisions': 60, 'number_normal_subgroups': 160, 'number_subgroup_autclasses': 244, 'number_subgroup_classes': 596, 'number_subgroups': 5364, 'old_label': None, 'order': 1344, 'order_factorization_type': 311, 'order_stats': [[1, 1], [2, 287], [3, 2], [4, 176], [6, 238], [7, 6], [8, 48], [12, 16], [14, 42], [21, 12], [28, 48], [42, 84], [56, 288], [84, 96]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': True, 'outer_exponent': 12, 'outer_gen_orders': [6, 4, 2, 2, 2, 2, 2], 'outer_gen_pows': [0, 0, 56, 0, 0, 0, 0], 'outer_gens': [[673, 674, 860, 224], [673, 674, 676, 336], [1, 842, 676, 224], [1, 2, 60, 224], [1, 674, 676, 224], [1, 2, 676, 224], [1, 675, 789, 224]], 'outer_group': '768.55643', 'outer_hash': 55643, 'outer_nilpotent': True, 'outer_order': 768, 'outer_permdeg': 17, 'outer_perms': [21204442869868, 23819871398424, 24, 131760, 47089213920000, 90720, 5324585414400], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^7:C_6', 'pc_rank': 4, 'perfect': False, 'permutation_degree': 20, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 2, 2], 'quasisimple': False, 'rank': 4, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 16], [2, 12], [4, 6], [6, 8], [8, 2], [12, 12], [24, 2], [48, 2]], 'representations': {'PC': {'code': 1111489944815756596009013726469293393134705106101, 'gens': [1, 2, 3, 7], 'pres': [8, -2, -2, -2, -2, -2, -7, -2, -3, 2786, 2650, 66, 3467, 91, 4172, 116, 4621, 15702, 166, 14359]}, 'Perm': {'d': 20, 'gens': [6423383674049808, 40331670, 1, 167407, 859248, 1270440, 518918400, 134491780578355200]}}, 'schur_multiplier': [2, 2, 2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 2, 2], 'solvability_type': 7, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'D_{84}:C_2^3', 'transitive_degree': 336, 'wreath_data': None, 'wreath_product': False}
