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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': True, 'ambient': '512.10481222', 'ambient_counter': 10481222, 'ambient_order': 512, 'ambient_tex': 'C_2^4\\times C_4\\times C_8', 'central': True, 'central_factor': False, 'centralizer_order': 512, 'characteristic': False, 'core_order': 16, 'counter': 37, 'cyclic': False, 'direct': True, 'hall': 0, 'label': '512.10481222.32.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': True, 'old_label': '32.a1', 'outer_equivalence': True, 'perfect': False, 'proper': True, 'quotient': '32.45', 'quotient_Agroup': True, 'quotient_abelian': True, 'quotient_cyclic': False, 'quotient_hash': 45, 'quotient_metabelian': True, 'quotient_nilpotent': True, 'quotient_order': 32, 'quotient_simple': False, 'quotient_solvable': True, 'quotient_supersolvable': True, 'quotient_tex': 'C_2^3\\times C_4', 'simple': False, 'solvable': True, 'special_labels': [], 'split': True, 'standard_generators': False, 'stem': False, 'subgroup': '16.5', 'subgroup_hash': 5, 'subgroup_order': 16, 'subgroup_tex': 'C_2\\times C_8', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '512.10481222', 'aut_centralizer_order': None, 'aut_label': '32.a1', 'aut_quo_index': None, 'aut_stab_index': None, 'aut_weyl_group': None, 'aut_weyl_index': None, 'centralizer': '1.a1', 'complements': ['16.h1'], 'conjugacy_class_count': 960, 'contained_in': ['16.c1', '16.d1'], 'contains': ['64.b1', '64.k1'], 'core': '32.a1', 'coset_action_label': None, 'count': 960, 'diagramx': None, 'generators': [1, 128], 'label': '512.10481222.32.a1', 'mobius_quo': None, 'mobius_sub': None, 'normal_closure': '32.a1', 'normal_contained_in': ['16.c1', '16.d1'], 'normal_contains': ['64.b1', '64.k1'], 'normalizer': '1.a1', 'old_label': '32.a1', 'projective_image': None, 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '32.a1', 'subgroup_fusion': None, 'weyl_group': None}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': True, 'abelian_quotient': '16.5', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 4, 'aut_gen_orders': [2, 2, 2, 2], 'aut_gens': [[1, 2], [1, 3], [9, 14], [1, 6], [1, 10]], 'aut_group': '16.11', 'aut_hash': 11, 'aut_nilpotency_class': 2, 'aut_nilpotent': True, 'aut_order': 16, 'aut_permdeg': 6, 'aut_perms': [126, 55, 289, 288], 'aut_phi_ratio': 2.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 1, 2, 1], [4, 1, 2, 2], [8, 1, 8, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_2\\times D_4', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 4, 'autcent_group': '16.11', 'autcent_hash': 11, 'autcent_nilpotent': True, 'autcent_order': 16, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2\\times D_4', '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, 3], [4, 1, 4], [8, 1, 8]], 'center_label': '16.5', 'center_order': 16, 'central_product': True, 'central_quotient': '1.1', 'commutator_count': 0, 'commutator_label': '1.1', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1'], 'composition_length': 4, 'conjugacy_classes_known': True, 'counter': 5, 'cyclic': False, 'derived_length': 1, 'dihedral': False, 'direct_factorization': [['2.1', 1], ['8.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [4, 1, 2, 2], [8, 1, 4, 2]], 'element_repr_type': 'PC', 'elementary': 2, 'eulerian_function': 6, 'exponent': 8, 'exponents_of_order': [4], 'factors_of_aut_order': [2], 'factors_of_order': [2], 'faithful_reps': [], 'familial': False, 'frattini_label': '4.1', 'frattini_quotient': '4.2', 'hash': 5, 'hyperelementary': 2, 'inner_abelian': True, 'inner_cyclic': True, 'inner_exponent': 1, 'inner_gen_orders': [1, 1], 'inner_gens': [[1, 2], [1, 2]], '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, 16]], 'label': '16.5', 'linC_count': 48, 'linC_degree': 2, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 5, 'linQ_degree_count': 4, 'linQ_dim': 5, 'linQ_dim_count': 4, 'linR_count': 8, 'linR_degree': 3, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C2*C8', 'ngens': 2, 'nilpotency_class': 1, 'nilpotent': True, 'normal_counts': [0, 0, 0, 0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 6, 'number_characteristic_subgroups': 7, 'number_conjugacy_classes': 16, 'number_divisions': 8, 'number_normal_subgroups': 11, 'number_subgroup_autclasses': 9, 'number_subgroup_classes': 11, 'number_subgroups': 11, 'old_label': None, 'order': 16, 'order_factorization_type': 3, 'order_stats': [[1, 1], [2, 3], [4, 4], [8, 8]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 4, 'outer_gen_orders': [2, 2, 2, 2], 'outer_gen_pows': [0, 0, 0, 0], 'outer_gens': [[1, 3], [9, 14], [1, 6], [1, 10]], 'outer_group': '16.11', 'outer_hash': 11, 'outer_nilpotent': True, 'outer_order': 16, 'outer_permdeg': 6, 'outer_perms': [126, 55, 289, 288], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2\\times D_4', 'pc_rank': 2, 'perfect': False, 'permutation_degree': 10, 'pgroup': 2, 'primary_abelian_invariants': [2, 8], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 2], [4, 2]], 'representations': {'PC': {'code': 9222, 'gens': [1, 2], 'pres': [4, -2, 2, -2, -2, 21, 34]}, 'GLZ': {'b': 3, 'd': 5, 'gens': [141602720900, 706304316034]}, 'GLFp': {'d': 2, 'p': 17, 'gens': [19665, 44226]}, 'Perm': {'d': 10, 'gens': [40176, 362880, 16582, 5167]}}, 'schur_multiplier': [2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 8], 'solvability_type': 1, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_2\\times C_8', 'transitive_degree': 16, 'wreath_data': None, 'wreath_product': False}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': True, 'abelian_quotient': '512.10481222', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 840, 'aut_gen_orders': [28, 12, 60], 'aut_gens': [[1, 8, 32, 64, 128, 256], [209, 254, 356, 180, 128, 244], [449, 222, 84, 112, 192, 484], [235, 474, 372, 320, 100, 464]], 'aut_group': None, 'aut_hash': 2357992776964681294, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 169114337280, 'aut_permdeg': 256, 'aut_perms': [554906160603383509701339631319784628500204555354850439756736439205365825335310352822594999834035325066513700356075299945148195313498994062060378275482574886418922434982880311517566036193364746253660662453864035439427785134134810276917542077225105184473731358251490380564528742759132954871011571446466327878767305194786639726101135183937081988472530068437659710671246934894974156970354504156376579802051406561671691465942192959052155926828372092885036442354673164616946918163785367628101580482186485965715157, 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785354248750998387572881876643486259999892458832115053114383131314524675546033266105364295789219040805057503323998958606551097308106349577767221774194075327787410617174175295743628757120059460811670969752923486916248738979295243391579200460077955207648471951213380898209457030710265605818600537490012435912119433438333823224895300104278036196437882870155205758154494467164260977841966119588872367121810267239697050119554279545009738299917426243564428502065663788038293197281844222575426614414232065412503823], 'aut_phi_ratio': 660602880.0, 'aut_solvable': False, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 1, 2, 1], [2, 1, 60, 1], [4, 1, 4, 1], [4, 1, 60, 1], [4, 1, 128, 1], [8, 1, 256, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_2^7.C_2^6.C_2^4.C_2^6.A_8', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 840, 'autcent_group': None, 'autcent_hash': 2357992776964681294, 'autcent_nilpotent': False, 'autcent_order': 169114337280, 'autcent_solvable': False, 'autcent_split': True, 'autcent_supersolvable': False, 'autcent_tex': 'C_2^7.C_2^6.C_2^4.C_2^6.A_8', '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, 63], [4, 1, 192], [8, 1, 256]], 'center_label': '512.10481222', 'center_order': 512, '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', '2.1', '2.1', '2.1', '2.1', '2.1', '2.1', '2.1', '2.1'], 'composition_length': 9, 'conjugacy_classes_known': True, 'counter': 10481222, 'cyclic': False, 'derived_length': 1, 'dihedral': False, 'direct_factorization': [['2.1', 4], ['4.1', 1], ['8.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 63], [4, 1, 2, 96], [8, 1, 4, 64]], 'element_repr_type': 'PC', 'elementary': 2, 'eulerian_function': 31248, 'exponent': 8, 'exponents_of_order': [9], 'factors_of_aut_order': [2, 3, 5, 7], 'factors_of_order': [2], 'faithful_reps': [], 'familial': False, 'frattini_label': '8.2', 'frattini_quotient': '64.267', 'hash': 10481222, 'hyperelementary': 2, 'inner_abelian': True, 'inner_cyclic': True, 'inner_exponent': 1, 'inner_gen_orders': [1, 1, 1, 1, 1, 1], 'inner_gens': [[1, 8, 32, 64, 128, 256], [1, 8, 32, 64, 128, 256], [1, 8, 32, 64, 128, 256], [1, 8, 32, 64, 128, 256], [1, 8, 32, 64, 128, 256], [1, 8, 32, 64, 128, 256]], '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': None, 'irrep_stats': [[1, 512]], 'label': '512.10481222', 'linC_count': None, 'linC_degree': 6, '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': 'C2^4*C4*C8', 'ngens': 6, 'nilpotency_class': 1, 'nilpotent': True, 'normal_counts': [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': 8, 'number_characteristic_subgroups': 8, 'number_conjugacy_classes': 512, 'number_divisions': 224, 'number_normal_subgroups': 18754, 'number_subgroup_autclasses': 72, 'number_subgroup_classes': 18754, 'number_subgroups': 18754, 'old_label': None, 'order': 512, 'order_factorization_type': 7, 'order_stats': [[1, 1], [2, 63], [4, 192], [8, 256]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': True, 'outer_exponent': 840, 'outer_gen_orders': [14, 12, 24], 'outer_gen_pows': [0, 0, 0], 'outer_gens': [[467, 156, 212, 352, 244, 176], [59, 378, 68, 352, 500, 160], [113, 378, 48, 128, 324, 244]], 'outer_group': None, 'outer_hash': 2357992776964681294, 'outer_nilpotent': False, 'outer_order': 169114337280, 'outer_permdeg': 256, 'outer_perms': [780570001503345976510044007375392678139490775015884338064672050850005515737670833600909745957753433115596836110118616093247312815752446112332099624072228044397183576640717441513190910714409073432162005784677928890320691509386837368080693462160305235851824267119681251070004291674133807895915219719833296263578322386382091727274172929195997232259020301525404116071462734317847349199060928855245101864041292039207670663891945264176104409320939257567054876272476961978688295124837691224134635031223556686180442, 317583331614934862371163383979727506646652358350415083142213456416085247570139486430223153130230035368428585240345905127222318537956216286481084628435832593899497143018777225907321347918011742686067581217720497391355523294223383318889425244384529983366744061191348119605954655334471022347948161698477628920387049254139311554552840673708000480997721161564356427587784526621352087700494011095698418972946313187680434897726933262762003236583549394967469730139513291924471915673418254161845271722503345128359533, 112428381110767018452298090402405355758107213534106156261693833201825582494694948915969855579445297473410368977449020924900350620199752090149020273439298866978216974653398995954656647623355451445017433638060830684771495598214539009343149000816528571122739217811040830962541140329663760132525143369039099876773910510566889548489805431289399761922170497281795037962545246347748385651177665888334237789158415009074776279901236837616694569412899745697284176104693166032690365224307159417342827932389918680829053], 'outer_solvable': False, 'outer_supersolvable': False, 'outer_tex': 'C_2^7.C_2^6.C_2^4.C_2^6.A_8', 'pc_rank': None, 'perfect': False, 'permutation_degree': 20, 'pgroup': 2, 'primary_abelian_invariants': [2, 2, 2, 2, 4, 8], 'quasisimple': False, 'rank': 6, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 64], [2, 96], [4, 64]], 'representations': {'PC': {'code': '74365688846955315211', 'gens': [1, 4, 6, 7, 8, 9], 'pres': [9, 2, 2, 2, 2, 2, 2, 2, 2, 2, 18, 46, 102]}, 'Perm': {'d': 20, 'gens': [40176, 127008000, 6227020800, 121645100408832000, 355687428096000, 1307674368000, 16582, 40279680, 5167]}}, 'schur_multiplier': [2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 4], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 2, 2, 4, 8], 'solvability_type': 2, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_2^4\\times C_4\\times C_8', 'transitive_degree': 512, 'wreath_data': None, 'wreath_product': False}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': True, 'abelian_quotient': '32.45', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 0, 'aut_exponent': 168, 'aut_gen_orders': [7, 3], 'aut_gens': [[1, 2, 4, 8], [20, 19, 2, 14], [1, 22, 19, 28]], 'aut_group': '21504.y', 'aut_hash': 2095861522221326572, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 21504, 'aut_permdeg': 16, 'aut_perms': [19981329134312, 7769561399125], 'aut_phi_ratio': 1344.0, 'aut_solvable': False, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 1, 14, 1], [4, 1, 16, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_2^4:C_2^3:\\GL(3,2)', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 168, 'autcent_group': '21504.y', 'autcent_hash': 2095861522221326572, 'autcent_nilpotent': False, 'autcent_order': 21504, 'autcent_solvable': False, 'autcent_split': True, 'autcent_supersolvable': False, 'autcent_tex': 'C_2^4:C_2^3:\\GL(3,2)', '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], [2, 1, 15], [4, 1, 16]], 'center_label': '32.45', 'center_order': 32, 'central_product': True, 'central_quotient': '1.1', 'commutator_count': 0, 'commutator_label': '1.1', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '2.1'], 'composition_length': 5, 'conjugacy_classes_known': True, 'counter': 45, 'cyclic': False, 'derived_length': 1, 'dihedral': False, 'direct_factorization': [['2.1', 3], ['4.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 15], [4, 1, 2, 8]], 'element_repr_type': 'PC', 'elementary': 2, 'eulerian_function': 15, 'exponent': 4, 'exponents_of_order': [5], 'factors_of_aut_order': [2, 3, 7], 'factors_of_order': [2], 'faithful_reps': [], 'familial': False, 'frattini_label': '2.1', 'frattini_quotient': '16.14', 'hash': 45, 'hyperelementary': 2, 'inner_abelian': True, 'inner_cyclic': True, 'inner_exponent': 1, 'inner_gen_orders': [1, 1, 1, 1], 'inner_gens': [[1, 2, 4, 8], [1, 2, 4, 8], [1, 2, 4, 8], [1, 2, 4, 8]], '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, 32]], 'label': '32.45', 'linC_count': 13440, 'linC_degree': 4, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 5, 'linQ_degree_count': 1792, 'linQ_dim': 5, 'linQ_dim_count': 1792, 'linR_count': 1792, 'linR_degree': 5, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'C2^3*C4', 'ngens': 4, 'nilpotency_class': 1, 'nilpotent': True, 'normal_counts': [0, 0, 0, 0, 0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 4, 'number_characteristic_subgroups': 4, 'number_conjugacy_classes': 32, 'number_divisions': 24, 'number_normal_subgroups': 118, 'number_subgroup_autclasses': 12, 'number_subgroup_classes': 118, 'number_subgroups': 118, 'old_label': None, 'order': 32, 'order_factorization_type': 3, 'order_stats': [[1, 1], [2, 15], [4, 16]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 168, 'outer_gen_orders': [7, 3], 'outer_gen_pows': [0, 0], 'outer_gens': [[20, 19, 2, 14], [1, 22, 19, 28]], 'outer_group': '21504.y', 'outer_hash': 2095861522221326572, 'outer_nilpotent': False, 'outer_order': 21504, 'outer_permdeg': 16, 'outer_perms': [19981329134312, 7769561399125], 'outer_solvable': False, 'outer_supersolvable': False, 'outer_tex': 'C_2^4:C_2^3:\\GL(3,2)', 'pc_rank': 4, 'perfect': False, 'permutation_degree': 10, 'pgroup': 2, 'primary_abelian_invariants': [2, 2, 2, 4], 'quasisimple': False, 'rank': 4, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 16], [2, 8]], 'representations': {'PC': {'code': 16392, 'gens': [1, 2, 3, 4], 'pres': [5, -2, 2, 2, 2, -2, 58]}, 'GLZ': {'b': 3, 'd': 5, 'gens': [141602720900, 705686952884, 706461792892, 706460730980]}, 'GLFp': {'d': 4, 'p': 5, 'gens': [122109387504, 122297461254, 30594431879, 30704310001, 30590220001]}, 'Perm': {'d': 10, 'gens': [22, 362880, 5040, 120, 7]}}, 'schur_multiplier': [2, 2, 2, 2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 2, 4], 'solvability_type': 2, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_2^3\\times C_4', 'transitive_degree': 32, 'wreath_data': None, 'wreath_product': False}
