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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': True, 'ambient': '1944.2458', 'ambient_counter': 2458, 'ambient_order': 1944, 'ambient_tex': 'C_3^3.(C_3\\times S_4)', 'central': False, 'central_factor': False, 'centralizer_order': 648, 'characteristic': False, 'core_order': 1, 'counter': 295, 'cyclic': True, 'direct': None, 'hall': 0, 'label': '1944.2458.972.a1.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': True, 'minimal_normal': False, 'nilpotent': True, 'normal': False, 'old_label': '972.a1.a1', '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': 972, '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': '1944.2458', 'aut_centralizer_order': 69984, 'aut_label': '972.a1', 'aut_quo_index': None, 'aut_stab_index': 3, 'aut_weyl_group': '1.1', 'aut_weyl_index': 209952, 'centralizer': '3.b1.a1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['324.a1.a1', '324.a1.b1', '324.a1.c1', '324.b1.a1', '324.c1.a1', '324.d1.a1', '324.d1.b1', '324.e1.a1', '324.f1.a1', '324.g1.a1', '324.h1.a1', '324.h1.b1', '486.a1.a1', '486.b1.a1', '486.c1.a1'], 'contains': ['1944.a1.a1'], 'core': '1944.a1.a1', 'coset_action_label': None, 'count': 3, 'diagramx': [6944, -1, 6998, -1, 6908, -1, 7666, -1], 'generators': [162], 'label': '1944.2458.972.a1.a1', 'mobius_quo': None, 'mobius_sub': 0, 'normal_closure': '486.a1.a1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '3.b1.a1', 'old_label': '972.a1.a1', 'projective_image': '1944.2458', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '972.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': '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': '6.2', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 3, 'aut_exponent': 36, 'aut_gen_orders': [6, 18, 12, 18], 'aut_gens': [[1, 6, 54, 324], [113, 1236, 270, 486], [1421, 780, 1116, 1818], [55, 1038, 306, 1782], [1163, 816, 1278, 846]], 'aut_group': None, 'aut_hash': 8636023269513274417, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 209952, 'aut_permdeg': 180, 'aut_perms': [87041360560616241832356421051390076648914375838539592469025966133370278889100583364542475835869329025646922770423087054078842595420054502032551393621462397630136316812813340338955833762128478544737867441275444395435145019403760912645239010820156488030639945135543924902443321437097316105141609574529811375329260282736940117217123, 184316525512496513761943627681214198080459088492405184656552418141975324446319350135157540903570353294455856918018161073866697594122311348851013612609451051374458172002435900814973668197701074471260707393693414340304418723739582957363145882939577559302946165058833470236973617272325565186033272521265936036047217480981784674076512, 43534894762237275706910990163672584682452869132605199526228281677593354007034302110851187050024392284450185523762876304774206996195532617988899637271200865829678379697004913317611118108138264085514689831710736858990692392266927875442753530027047127970496805491716335242261570619882426591593199053299790408804459069613137273135949, 23767495854022267123648607381945264287455090804688155224306480719437441900356373369865824370046234518819815604002870686055761357047249368134132618257060114757140198200618548966768599031212522529350201669582296190795641157452398784417956962893735405206402653974601642506783519743486693953558997935991978125908916387216640779688487], 'aut_phi_ratio': 324.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 3, 1, 1], [2, 162, 1, 1], [3, 2, 1, 1], [3, 2, 3, 1], [3, 6, 1, 1], [3, 6, 2, 1], [3, 9, 2, 1], [3, 18, 2, 1], [3, 72, 6, 1], [4, 162, 1, 1], [6, 6, 1, 1], [6, 6, 3, 1], [6, 9, 2, 1], [6, 18, 1, 1], [6, 18, 2, 3], [6, 18, 4, 1], [6, 162, 2, 1], [9, 24, 9, 1], [12, 162, 2, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_3^4.C_6^2.C_3.D_6.C_2', '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': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 36, 'autcentquo_group': None, 'autcentquo_hash': 8636023269513274417, 'autcentquo_nilpotent': False, 'autcentquo_order': 209952, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'C_3^4.C_6^2.C_3.D_6.C_2', 'cc_stats': [[1, 1, 1], [2, 3, 1], [2, 162, 1], [3, 2, 4], [3, 6, 3], [3, 9, 2], [3, 18, 2], [3, 72, 6], [4, 162, 1], [6, 6, 4], [6, 9, 2], [6, 18, 11], [6, 162, 2], [9, 24, 9], [12, 162, 2]], 'center_label': '1.1', 'center_order': 1, 'central_product': False, 'central_quotient': '1944.2458', 'commutator_count': 1, 'commutator_label': '324.133', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '3.1', '3.1', '3.1', '3.1', '3.1'], 'composition_length': 8, 'conjugacy_classes_known': True, 'counter': 2458, 'cyclic': False, 'derived_length': 3, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 3, 1, 1], [2, 162, 1, 1], [3, 2, 1, 4], [3, 6, 1, 3], [3, 9, 2, 1], [3, 18, 2, 1], [3, 72, 2, 3], [4, 162, 1, 1], [6, 6, 1, 4], [6, 9, 2, 1], [6, 18, 1, 3], [6, 18, 2, 4], [6, 162, 2, 1], [9, 24, 3, 3], [12, 162, 2, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 16380, 'exponent': 36, 'exponents_of_order': [5, 3], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [], 'familial': False, 'frattini_label': '9.2', 'frattini_quotient': '216.164', 'hash': 2458, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 36, 'inner_gen_orders': [6, 9, 6, 6], 'inner_gens': [[1, 1326, 270, 1818], [679, 6, 1188, 486], [109, 1140, 54, 324], [829, 168, 54, 324]], 'inner_hash': 2458, 'inner_nilpotent': False, 'inner_order': 1944, 'inner_split': True, 'inner_tex': 'C_3^3.(C_3\\times S_4)', 'inner_used': [1, 2, 3], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 6], [2, 12], [3, 6], [6, 24], [18, 3]], 'label': '1944.2458', 'linC_count': 486, 'linC_degree': 11, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 20, 'linQ_degree_count': 9, 'linQ_dim': 20, 'linQ_dim_count': 9, 'linR_count': 54, 'linR_degree': 11, 'maximal_subgroups_known': True, 'metabelian': False, 'metacyclic': False, 'monomial': True, 'name': 'C3^3.(C3*S4)', '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], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 22, 'number_characteristic_subgroups': 18, 'number_conjugacy_classes': 51, 'number_divisions': 33, 'number_normal_subgroups': 34, 'number_subgroup_autclasses': 179, 'number_subgroup_classes': 297, 'number_subgroups': 5980, 'old_label': None, 'order': 1944, 'order_factorization_type': 33, 'order_stats': [[1, 1], [2, 165], [3, 512], [4, 162], [6, 564], [9, 216], [12, 324]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 6, 'outer_gen_orders': [3, 3, 2, 2, 6], 'outer_gen_pows': [0, 0, 0, 0, 0], 'outer_gens': [[1, 24, 90, 324], [1, 24, 54, 324], [5, 48, 270, 486], [1, 240, 288, 324], [5, 786, 270, 522]], 'outer_group': '108.28', 'outer_hash': 28, 'outer_nilpotent': False, 'outer_order': 108, 'outer_permdeg': 11, 'outer_perms': [7257648, 3997488, 1, 28495, 31818265], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_3^2:D_6', 'pc_rank': 4, 'perfect': False, 'permutation_degree': 34, 'pgroup': 0, 'primary_abelian_invariants': [2, 3], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 2], [2, 6], [3, 2], [4, 4], [6, 9], [12, 4], [18, 6]], 'representations': {'PC': {'code': 2325463000743622450855197797787505006859097327741532901430645, 'gens': [1, 3, 5, 7], 'pres': [8, 2, 3, 3, 3, 2, 3, 2, 3, 16, 31826, 15850, 90, 1155, 10804, 7940, 116, 10373, 101814, 10094, 4558, 166, 84103, 21327]}, 'Perm': {'d': 34, 'gens': [58099643967956820718081562749019785, 9802994283332432459778205216259193600, 18758764712693665258740463202159328147, 27711765226107940722789841891336126083, 1587285462305251557879099929509224960, 36665658661657639071555069539274005760, 744, 1680]}}, 'schur_multiplier': [3, 6], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [6], 'solvability_type': 11, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': 'C_3^3.(C_3\\times S_4)', 'transitive_degree': 162, 'wreath_data': None, 'wreath_product': False}
