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gps_subgroup_search • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'ambient': '81920.eez', 'ambient_counter': 2834, 'ambient_order': 81920, 'ambient_tex': 'C_2^5.C_2^8:C_{10}', 'central': False, 'central_factor': False, 'centralizer_order': 64, 'characteristic': True, 'core_order': 1024, 'counter': 132, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '81920.eez.80.I', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': False, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': True, 'old_label': '80.i1.N', 'outer_equivalence': True, 'perfect': False, 'proper': True, 'quotient': '80.49', 'quotient_Agroup': True, 'quotient_abelian': False, 'quotient_cyclic': False, 'quotient_hash': 49, 'quotient_metabelian': True, 'quotient_nilpotent': False, 'quotient_order': 80, 'quotient_simple': False, 'quotient_solvable': True, 'quotient_supersolvable': False, 'quotient_tex': 'C_2^4:C_5', 'simple': False, 'solvable': True, 'special_labels': [], 'split': None, 'standard_generators': False, 'stem': False, 'subgroup': None, 'subgroup_hash': 2424603210854798003, 'subgroup_order': 1024, 'subgroup_tex': 'C_2^6.C_2^4', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '81920.eez', 'aut_centralizer_order': None, 'aut_label': '80.I', 'aut_quo_index': None, 'aut_stab_index': None, 'aut_weyl_group': None, 'aut_weyl_index': None, 'centralizer': '1280.A', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': None, 'contains': None, 'core': '80.I', 'coset_action_label': None, 'count': 1, 'diagramx': None, 'generators': [125582002776257369502581541926814927622046479921, 13763761774543974762745257230317365869935422, 13774086247415580734455141016572565613907207, 25903229683158464681647, 5167, 42220633073647645357253607990055537, 13763761774543974762608987699848042092501696, 25903229681844523008000, 8231691320578226211045097771008000, 20411656176119953708643439784883692028203013167], 'label': '81920.eez.80.I', 'mobius_quo': None, 'mobius_sub': None, 'normal_closure': '80.I', 'normal_contained_in': [], 'normal_contains': [], 'normalizer': '1.a1', 'old_label': '80.i1.N', 'projective_image': None, 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '80.I', 'subgroup_fusion': None, 'weyl_group': None}
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label None does not appear in gps_groups
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gps_groups • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '10.2', 'all_subgroups_known': False, 'almost_simple': False, 'aut_abelian': None, 'aut_cyclic': None, 'aut_derived_length': None, 'aut_exponent': None, 'aut_gen_orders': None, 'aut_gens': [[313460272251343925208528588711217506579522665017, 167494204002376887338982319098330629704546091657, 648434853331641096527567534324334624779528217737], [293076143598781273413620379921699289547396205942, 167494204002376887338982345001560310247582198537, 648434853331632864858055743322276627819784222662], [313460272251343925208528588711217506579522665017, 167494193677920461410318910944763441610147662776, 544296809461389872050385534892406085413982150383], [293076153923237699342283679742494379030766671976, 167494193677920461410318936745810527736395593623, 502455021320865534605061018128868938839847585272], [293062379837040727255070380349783623700035344016, 167480440257934998512741464795357226227236021696, 648434842989801589683972079763882010995334355223], [313474025671346373119200636422961611730275998767, 167480429916128904654247107128574867792028763519, 628036971241752112722350139882784314804658083137], [313474036013135436491083858943277509702108872193, 167480440257994177622734513177108575159707959806, 564694681209481560825081248200184423035226874942], [271632247872611099023950501054899778590209132800, 209322228398526746759379610363510819882161964799, 502441257576466379189727307794519533830496193216], [209335981818495781663172955984867562180373724840, 251234365782696349093841181028254067826939750392, 522839129324490090048203986867238395288239350521], [187892075750392384200597862260280773802084849943, 313474036013160680952809413520006648065654684776, 648421089569832554769305535458528809174045270456], [229733863891009898572128039018007493515771440142, 271632247872568887232654520532169267716764106942, 586195172776703893073501004355557219632911713216], [313460272251343925208528562910412999279236205942, 167494204002376887338982345001560311549069104582, 648434853331641096527567534324334623465667113542], [313460272251343925208528562910413000580723102137, 167494204002376887338982345001560310247582198537, 628023197155521142818924120442680614595848222662], [313460272251352139194547388853153749123716201017, 167494204002385119030302897324541674802317099657, 648434853331641096527567508523530117479241758662], [293076143598773059427601553978958539702916201017, 187878312006018455585243267590748484881549099657, 648434853331632864858055743322276627819784222662], [313460272251343925208528588711217506579522665017, 167494204002376887338982319098330629704546091657, 648434853331641096527567534324334624779528217737], [293076143598781273413620379921699290848883107062, 187878312006026687276563819913729846833310203462, 648434853331641096527567534324334624779528217737], [313460272251343925208528588711217506579522665017, 187878312006018455585243241687518803037026096582, 648434853331641096527567508523530117479241758662], [313460272251343925208528588711217506579522665017, 167494204002376887338982345001560310247582198537, 648434853331632864858055717521472120519497753737], [313460272251343925208528588711217506579522665017, 167494204002385119030302897324541673500830203462, 628023197155521142818924094641876107295561753737], [313460272251343925208528588711217506579522665017, 167494204002376887338982319098330629704546091657, 628023197155512911149412303639818109021956644617], [313460272251343925208528588711217506579522665017, 167494204002376887338982319098330629704546091657, 648434853331641096527567534324334623465667113542], [313460272251343925208528588711217506579522665017, 167494204002376887338982319098330629704546091657, 628023197155512911149412329440622616322243113542], [313460272251343925208528588711217506579522665017, 167494204002376887338982319098330629704546091657, 648434853331632864858055717521472120519497753737]], 'aut_group': '3456.oc', 'aut_hash': None, 'aut_nilpotency_class': None, 'aut_nilpotent': None, 'aut_order': 10485760, 'aut_permdeg': None, 'aut_perms': None, 'aut_phi_ratio': 320.0, 'aut_solvable': None, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 5, 1, 6], [2, 16, 2, 1], [2, 80, 2, 1], [2, 160, 1, 10], [4, 40, 4, 5], [4, 80, 2, 26], [4, 160, 1, 60], [5, 4096, 1, 4], [10, 4096, 1, 4], [10, 4096, 2, 4]], 'aut_supersolvable': None, 'aut_tex': 'D_6^2:S_4', 'autcent_abelian': None, 'autcent_cyclic': None, 'autcent_exponent': None, 'autcent_group': None, 'autcent_hash': None, 'autcent_nilpotent': None, 'autcent_order': None, 'autcent_solvable': None, 'autcent_split': None, 'autcent_supersolvable': None, 'autcent_tex': None, 'autcentquo_abelian': None, 'autcentquo_cyclic': None, 'autcentquo_exponent': None, 'autcentquo_group': None, 'autcentquo_hash': None, 'autcentquo_nilpotent': None, 'autcentquo_order': None, 'autcentquo_solvable': None, 'autcentquo_supersolvable': None, 'autcentquo_tex': None, 'cc_stats': [[1, 1, 1], [2, 1, 1], [2, 5, 6], [2, 16, 2], [2, 80, 2], [2, 160, 10], [4, 40, 20], [4, 80, 52], [4, 160, 60], [5, 4096, 4], [10, 4096, 12]], 'center_label': '2.1', 'center_order': 2, 'central_product': False, 'central_quotient': None, 'commutator_count': 1, 'commutator_label': None, 'complements_known': False, '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', '2.1', '2.1', '2.1', '2.1', '2.1', '5.1'], 'composition_length': 15, 'conjugacy_classes_known': True, 'counter': 2834, 'cyclic': False, 'derived_length': 3, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 5, 1, 6], [2, 16, 1, 2], [2, 80, 1, 2], [2, 160, 1, 10], [4, 40, 1, 20], [4, 80, 1, 48], [4, 80, 2, 2], [4, 160, 1, 60], [5, 4096, 4, 1], [10, 4096, 4, 3]], 'element_repr_type': 'Perm', 'elementary': 1, 'eulerian_function': None, 'exponent': 20, 'exponents_of_order': [14, 1], 'factors_of_aut_order': [2, 5], 'factors_of_order': [2, 5], 'faithful_reps': [[20, 1, 32], [80, 0, 2], [80, 1, 2]], 'familial': False, 'frattini_label': '32.51', 'frattini_quotient': '2560.o', 'hash': 6600526450111564660, 'hyperelementary': 1, 'inner_abelian': None, 'inner_cyclic': None, 'inner_exponent': None, 'inner_gen_orders': [10, 10, 5], 'inner_gens': [[313460272251343925208528588711217506579522665017, 167494193677912247413426726710678367907120956783, 522852893068906779897140118684836298432857399758], [293076153923245931022699851455983361821025377983, 167494204002376887338982319098330629704546091657, 522852893068915011577556290398325282698966840262], [251220591696474160323990466549365447316005270977, 229720110471057894133199191866041794760111236542, 648434853331641096527567534324334624779528217737]], 'inner_hash': None, 'inner_nilpotent': None, 'inner_order': 40960, 'inner_split': None, 'inner_tex': None, 'inner_used': None, 'irrC_degree': 20, 'irrQ_degree': 20, 'irrQ_dim': None, 'irrR_degree': None, 'irrep_stats': [[1, 10], [5, 102], [16, 10], [20, 32], [40, 8], [80, 8]], 'label': '81920.eez', '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': False, 'metacyclic': False, 'monomial': None, 'name': 'C2^5.C2^8:C10', '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, 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': 123, 'number_characteristic_subgroups': 43, 'number_conjugacy_classes': 170, 'number_divisions': 156, 'number_normal_subgroups': 45, 'number_subgroup_autclasses': None, 'number_subgroup_classes': None, 'number_subgroups': None, 'old_label': None, 'order': 81920, 'order_factorization_type': 51, 'order_stats': [[1, 1], [2, 1823], [4, 14560], [5, 16384], [10, 49152]], 'outer_abelian': None, 'outer_cyclic': None, 'outer_equivalence': True, 'outer_exponent': None, 'outer_gen_orders': None, 'outer_gen_pows': None, 'outer_gens': None, 'outer_group': '256.56092', 'outer_hash': None, 'outer_nilpotent': None, 'outer_order': 256, 'outer_permdeg': None, 'outer_perms': None, 'outer_solvable': None, 'outer_supersolvable': None, 'outer_tex': 'C_2^8', 'pc_rank': None, 'perfect': False, 'permutation_degree': None, 'pgroup': 0, 'primary_abelian_invariants': [2, 5], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': False, 'ratrep_stats': [[1, 2], [4, 2], [5, 102], [16, 2], [20, 32], [40, 8], [64, 2], [80, 4], [160, 2]], 'representations': {'PC': {'code': '117197684564259494505110931090013900448898722887323090745403238591168051516216947312100010821521442331643105186049101796120316155721057904216242694042014043208407258515498010414620719876669191282138752626459882133255228475643419714296298779356164358438786407496166330211547106333860663171563257009942747238171763186005241401762403895685121663419424616990700004704256', 'gens': [1, 3, 4, 5, 6, 7, 9, 11, 13, 14, 15], 'pres': [15, 2, 5, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 30, 2252252, 739592, 23072, 1339203, 1460718, 255513, 139248, 5856754, 1840894, 292999, 60664, 1155605, 2254970, 3635, 328730, 72965, 4683006, 1392846, 72276, 224331, 108006, 56151, 306, 4454407, 2860822, 1910258, 3805673, 630758, 6533, 127508, 124823, 39518, 398, 6336009, 480024, 10294360, 361375, 47560, 359095, 143950, 103045, 4060, 490, 13881611, 7056026, 13940562, 3023502, 305802, 611577, 324552, 128787, 20382, 7932, 4842, 4260913, 568078, 544363, 551098, 295753, 165568, 35383, 5173, 4363, 10368014, 3492029]}, 'Perm': {'d': 40, 'gens': [648434853331641096527567534324334624779528217737, 313460272251343925208528588711217506579522665017, 167494204002376887338982319098330629704546091657]}}, 'schur_multiplier': [2, 2, 2, 2, 2, 2, 2], 'semidirect_product': None, 'simple': False, 'smith_abelian_invariants': [10], 'solvability_type': 17, 'solvable': True, 'subgroup_inclusions_known': False, 'subgroup_index_bound': 16, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': 'C_2^5.C_2^8:C_{10}', 'transitive_degree': None, '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': '5.1', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 3, 'aut_exponent': 120, 'aut_gen_orders': [4, 2, 3, 5, 2, 2, 2, 2], 'aut_gens': [[3225600, 5167, 367920, 363000, 368046], [2453760, 121, 5160, 5167, 363000], [818224, 368046, 7, 121, 5167], [3225600, 5041, 368041, 7, 362886], [3225600, 367927, 126, 121, 363007], [3225721, 5167, 367920, 363000, 368046], [3230767, 5167, 367920, 363000, 368046], [3588481, 5167, 367920, 363000, 368046], [3588486, 5167, 367920, 363000, 368046]], 'aut_group': '960.11359', 'aut_hash': 11359, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 960, 'aut_permdeg': 16, 'aut_perms': [8504203199768, 9320163852632, 4599784418484, 7549310535630, 1348039567087, 10277271846354, 12062407412682, 16600569329714], 'aut_phi_ratio': 30.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 5, 3, 1], [5, 16, 4, 1]], 'aut_supersolvable': False, 'aut_tex': 'F_{16}:C_4', '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': 120, 'autcentquo_group': '960.11359', 'autcentquo_hash': 11359, 'autcentquo_nilpotent': False, 'autcentquo_order': 960, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'F_{16}:C_4', 'cc_stats': [[1, 1, 1], [2, 5, 3], [5, 16, 4]], 'center_label': '1.1', 'center_order': 1, 'central_product': False, 'central_quotient': '80.49', 'commutator_count': 1, 'commutator_label': '16.14', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '5.1'], 'composition_length': 5, 'conjugacy_classes_known': True, 'counter': 49, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 5, 1, 3], [5, 16, 4, 1]], 'element_repr_type': 'Perm', 'elementary': 1, 'eulerian_function': 6, 'exponent': 10, 'exponents_of_order': [4, 1], 'factors_of_aut_order': [2, 3, 5], 'factors_of_order': [2, 5], 'faithful_reps': [[5, 1, 3]], 'familial': False, 'frattini_label': '1.1', 'frattini_quotient': '80.49', 'hash': 49, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 10, 'inner_gen_orders': [5, 2, 2, 2, 2], 'inner_gens': [[3225600, 368046, 362881, 5041, 368041], [3593520, 5167, 367920, 363000, 368046], [3588600, 5167, 367920, 363000, 368046], [3593646, 5167, 367920, 363000, 368046], [3588481, 5167, 367920, 363000, 368046]], 'inner_hash': 49, 'inner_nilpotent': False, 'inner_order': 80, 'inner_split': True, 'inner_tex': 'C_2^4:C_5', 'inner_used': [1, 2], 'irrC_degree': 5, 'irrQ_degree': 5, 'irrQ_dim': 5, 'irrR_degree': 5, 'irrep_stats': [[1, 5], [5, 3]], 'label': '80.49', 'linC_count': 3, 'linC_degree': 5, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 5, 'linQ_degree_count': 3, 'linQ_dim': 5, 'linQ_dim_count': 3, 'linR_count': 3, 'linR_degree': 5, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'C2^4:C5', 'ngens': 5, 'nilpotency_class': -1, 'nilpotent': False, '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': 3, 'number_characteristic_subgroups': 3, 'number_conjugacy_classes': 8, 'number_divisions': 5, 'number_normal_subgroups': 3, 'number_subgroup_autclasses': 8, 'number_subgroup_classes': 17, 'number_subgroups': 84, 'old_label': None, 'order': 80, 'order_factorization_type': 31, 'order_stats': [[1, 1], [2, 15], [5, 64]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 12, 'outer_gen_orders': [4, 3], 'outer_gen_pows': [0, 2453760], 'outer_gens': [[2453760, 121, 5160, 5167, 363000], [3225600, 7, 362886, 367927, 126]], 'outer_group': '12.1', 'outer_hash': 1, 'outer_nilpotent': False, 'outer_order': 12, 'outer_permdeg': 7, 'outer_perms': [129, 840], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_3:C_4', 'pc_rank': 5, 'perfect': False, 'permutation_degree': 10, 'pgroup': 0, 'primary_abelian_invariants': [5], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 1], [4, 1], [5, 3]], 'representations': {'PC': {'code': 121558530245376, 'gens': [1, 2, 3, 4, 5], 'pres': [5, -5, -2, 2, 2, 2, 501, 227, 703, 1879]}, 'GLZ': {'b': 3, 'd': 5, 'gens': [456187803214, 140827879922]}, 'GLFp': {'d': 5, 'p': 2, 'gens': [10024001, 32746561, 5128257, 33082092, 6210625]}, 'Perm': {'d': 10, 'gens': [3225600, 5167, 367920, 363000, 368046]}}, 'schur_multiplier': [2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [5], 'solvability_type': 8, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': 'C_2^4:C_5', 'transitive_degree': 10, 'wreath_data': None, 'wreath_product': False}
