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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': True, 'ambient': '960.10052', 'ambient_counter': 10052, 'ambient_order': 960, 'ambient_tex': 'C_4^2:S_3\\times C_{10}', 'central': False, 'central_factor': False, 'centralizer_order': 480, 'characteristic': False, 'core_order': 20, 'counter': 90, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '960.10052.24.l1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': False, 'old_label': '24.l1', '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': 24, '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': '40.9', 'subgroup_hash': 9, 'subgroup_order': 40, 'subgroup_tex': 'C_2\\times C_{20}', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '960.10052', 'aut_centralizer_order': 384, 'aut_label': '24.l1', 'aut_quo_index': None, 'aut_stab_index': 8, 'aut_weyl_group': '32.25', 'aut_weyl_index': 3072, 'centralizer': '2.c1', 'complements': None, 'conjugacy_class_count': 4, 'contained_in': ['8.l1', '12.b1'], 'contains': ['48.d1', '48.h1', '120.l1'], 'core': '48.d1', 'coset_action_label': None, 'count': 8, 'diagramx': [1241, -1, 1473, -1], 'generators': [10, 4, 480, 720], 'label': '960.10052.24.l1', 'mobius_quo': None, 'mobius_sub': 0, 'normal_closure': '12.b1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '2.c1', 'old_label': '24.l1', 'projective_image': '48.37', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '24.l1', 'subgroup_fusion': None, 'weyl_group': '1.1'}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': True, 'abelian_quotient': '40.9', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 4, 'aut_gen_orders': [2, 4, 4], 'aut_gens': [[1, 2], [21, 2], [1, 14], [21, 19]], 'aut_group': '32.25', 'aut_hash': 25, 'aut_nilpotency_class': 2, 'aut_nilpotent': True, 'aut_order': 32, 'aut_permdeg': 8, 'aut_perms': [11527, 17, 11647], 'aut_phi_ratio': 2.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 1, 2, 1], [4, 1, 4, 1], [5, 1, 4, 1], [10, 1, 4, 1], [10, 1, 8, 1], [20, 1, 16, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_4\\times D_4', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 4, 'autcent_group': '32.25', 'autcent_hash': 25, 'autcent_nilpotent': True, 'autcent_order': 32, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_4\\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], [5, 1, 4], [10, 1, 12], [20, 1, 16]], 'center_label': '40.9', 'center_order': 40, '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', '5.1'], 'composition_length': 4, 'conjugacy_classes_known': True, 'counter': 9, 'cyclic': False, 'derived_length': 1, 'dihedral': False, 'direct_factorization': [['2.1', 1], ['4.1', 1], ['5.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [4, 1, 2, 2], [5, 1, 4, 1], [10, 1, 4, 3], [20, 1, 8, 2]], 'element_repr_type': 'PC', 'elementary': 2, 'eulerian_function': 18, 'exponent': 20, 'exponents_of_order': [3, 1], 'factors_of_aut_order': [2], 'factors_of_order': [2, 5], 'faithful_reps': [], 'familial': False, 'frattini_label': '2.1', 'frattini_quotient': '20.5', 'hash': 9, '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, 40]], 'label': '40.9', 'linC_count': 288, 'linC_degree': 2, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 6, 'linQ_degree_count': 4, 'linQ_dim': 6, 'linQ_dim_count': 4, 'linR_count': 16, 'linR_degree': 3, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C2*C20', 'ngens': 4, 'nilpotency_class': 1, 'nilpotent': True, 'normal_counts': [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': 40, 'number_divisions': 12, 'number_normal_subgroups': 16, 'number_subgroup_autclasses': 12, 'number_subgroup_classes': 16, 'number_subgroups': 16, 'old_label': None, 'order': 40, 'order_factorization_type': 31, 'order_stats': [[1, 1], [2, 3], [4, 4], [5, 4], [10, 12], [20, 16]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 4, 'outer_gen_orders': [2, 4, 4], 'outer_gen_pows': [0, 0, 0], 'outer_gens': [[21, 2], [1, 14], [21, 19]], 'outer_group': '32.25', 'outer_hash': 25, 'outer_nilpotent': True, 'outer_order': 32, 'outer_permdeg': 8, 'outer_perms': [11527, 17, 11647], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_4\\times D_4', 'pc_rank': 2, 'perfect': False, 'permutation_degree': 11, 'pgroup': 0, 'primary_abelian_invariants': [2, 4, 5], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 2], [4, 4], [8, 2]], 'representations': {'PC': {'code': 5899779, 'gens': [1, 2], 'pres': [4, -2, -2, -2, -5, 21, 34]}, 'GLZ': {'b': 3, 'd': 6, 'gens': [125101729074815868, 91655853488337157]}, 'GLFp': {'d': 2, 'p': 41, 'gens': [1585222, 2756880]}, 'Perm': {'d': 11, 'gens': [131040, 3628800, 96, 41040]}}, 'schur_multiplier': [2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 20], '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_{20}', 'transitive_degree': 40, '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': '160.228', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 3, 'aut_exponent': 24, 'aut_gen_orders': [4, 4, 8, 4, 12, 12], 'aut_gens': [[1, 2, 20, 80], [851, 58, 60, 900], [811, 486, 540, 410], [521, 526, 550, 940], [851, 54, 30, 460], [161, 538, 70, 600], [851, 534, 500, 580]], 'aut_group': None, 'aut_hash': 4741775015152245913, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 98304, 'aut_permdeg': 44, 'aut_perms': [2186748863410372259383286755034091770325609770567878754, 1170935976181801035701944328046195838723281219565495017, 527904328706537419854160969707650194761274480894723957, 2186749067775327149769595444301650445466398101820217962, 179472054180061413719841387545286758501878320942706440, 2228290890191264784110336356454495775608987558313545273], 'aut_phi_ratio': 384.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 1, 2, 1], [2, 1, 4, 1], [2, 6, 4, 1], [3, 2, 1, 1], [4, 1, 8, 1], [4, 2, 8, 1], [4, 6, 4, 1], [4, 6, 8, 1], [5, 1, 4, 1], [6, 2, 1, 1], [6, 2, 2, 1], [6, 2, 4, 1], [10, 1, 4, 1], [10, 1, 8, 1], [10, 1, 16, 1], [10, 6, 16, 1], [12, 2, 8, 1], [12, 2, 16, 1], [15, 2, 4, 1], [20, 1, 32, 1], [20, 2, 32, 1], [20, 6, 16, 1], [20, 6, 32, 1], [30, 2, 4, 1], [30, 2, 8, 1], [30, 2, 16, 1], [60, 2, 32, 1], [60, 2, 64, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_3:(C_2^5.C_2^5.C_2^5)', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 8, 'autcent_group': None, 'autcent_hash': 3230402273819117835, 'autcent_nilpotent': True, 'autcent_order': 16384, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^6.C_2^3.C_2^5', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 6, 'autcentquo_group': '6.1', 'autcentquo_hash': 1, 'autcentquo_nilpotent': False, 'autcentquo_order': 6, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'S_3', 'cc_stats': [[1, 1, 1], [2, 1, 7], [2, 6, 4], [3, 2, 1], [4, 1, 8], [4, 2, 8], [4, 6, 12], [5, 1, 4], [6, 2, 7], [10, 1, 28], [10, 6, 16], [12, 2, 24], [15, 2, 4], [20, 1, 32], [20, 2, 32], [20, 6, 48], [30, 2, 28], [60, 2, 96]], 'center_label': '80.45', 'center_order': 80, 'central_product': True, 'central_quotient': '12.4', 'commutator_count': 1, 'commutator_label': '6.2', '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', '5.1'], 'composition_length': 8, 'conjugacy_classes_known': True, 'counter': 10052, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['2.1', 1], ['5.1', 1], ['96.79', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 7], [2, 6, 1, 4], [3, 2, 1, 1], [4, 1, 2, 4], [4, 2, 2, 4], [4, 6, 1, 4], [4, 6, 2, 4], [5, 1, 4, 1], [6, 2, 1, 7], [10, 1, 4, 7], [10, 6, 4, 4], [12, 2, 2, 4], [12, 2, 4, 4], [15, 2, 4, 1], [20, 1, 8, 4], [20, 2, 8, 4], [20, 6, 4, 4], [20, 6, 8, 4], [30, 2, 4, 7], [60, 2, 8, 4], [60, 2, 16, 4]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 2555280, 'exponent': 60, 'exponents_of_order': [6, 1, 1], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3, 5], 'faithful_reps': [], 'familial': False, 'frattini_label': '4.2', 'frattini_quotient': '240.206', 'hash': 10052, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 6, 'inner_gen_orders': [2, 1, 1, 6], 'inner_gens': [[1, 2, 20, 440], [1, 2, 20, 80], [1, 2, 20, 80], [681, 2, 20, 80]], 'inner_hash': 4, 'inner_nilpotent': False, 'inner_order': 12, 'inner_split': False, 'inner_tex': 'D_6', 'inner_used': [1, 4], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 160], [2, 200]], 'label': '960.10052', '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': 'C4^2:S3*C10', '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': 30, 'number_characteristic_subgroups': 34, 'number_conjugacy_classes': 360, 'number_divisions': 88, 'number_normal_subgroups': 334, 'number_subgroup_autclasses': 184, 'number_subgroup_classes': 660, 'number_subgroups': 1392, 'old_label': None, 'order': 960, 'order_factorization_type': 311, 'order_stats': [[1, 1], [2, 31], [3, 2], [4, 96], [5, 4], [6, 14], [10, 124], [12, 48], [15, 8], [20, 384], [30, 56], [60, 192]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': True, 'outer_exponent': 8, 'outer_gen_orders': [4, 4, 8, 4, 4, 4], 'outer_gen_pows': [0, 0, 0, 0, 320, 640], 'outer_gens': [[851, 58, 60, 900], [811, 486, 540, 410], [521, 526, 550, 940], [851, 54, 30, 460], [161, 538, 70, 600], [851, 534, 500, 580]], 'outer_group': None, 'outer_hash': 6714790469588492835, 'outer_nilpotent': True, 'outer_order': 8192, 'outer_permdeg': 256, 'outer_perms': [3430266600048808168769762230461324084304721731796304436557936479750989055006706905230964183020788839413883561249421806916805436894902230948229053420470200786715120354369643968868273454382107934220806122978808001455966490734076646502184340149169410654473995867580022774518339348921790489728997137100808016613802468845129944246517240898119639186924104133344385540342606507891602545009718853097498181354153873620782359447197317931647615238534425190733779509360513458401080813538250119691796003537244054897768, 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16886439463005935661709002246107990647569551150342948893882124503796329184421579428822425909902267011255634113858432390053956808364916513512072739241123939869827771078801566322977982660548992181774243425564055625697822782323573732993326437228097645095560116056651343438032655002336313935958388062394955401064011725659539269602759803540515439326453453231693803646964251014956031857762793711265974071573755246756289647880937284825270485332018339979803084424022902491396751417299643069512823530536314018702780, 20250482678859983927232622503179758530321372477282221916349748518676851206628001794856039734996374118615444577825135680431480507202401706391335069931705737192572212695838471168525534739742466976049448859763029937600773988989068004727409176032024581115245277899914772918700622421638406756072102546582314782369770407252408125014311379101221868512595299928425363760080570426267442700562393719116646315093367209337932314939803201640811961981135184306102933841017700689963665392159491985225094040603428028021161], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^4.C_2^5.C_2^4', 'pc_rank': 4, 'perfect': False, 'permutation_degree': 22, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 2, 4, 5], 'quasisimple': False, 'rank': 4, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 16], [2, 16], [4, 24], [8, 20], [16, 8], [32, 4]], 'representations': {'PC': {'code': 1766310822433241266427578932661249, 'gens': [1, 2, 4, 6], 'pres': [8, -2, -2, -5, -2, -2, 2, -2, -3, 41, 91, 21125, 141, 44806, 166, 40967]}, 'GLZN': {'d': 2, 'p': 88, 'gens': [1193611, 30666285, 30843451, 6133257, 45658691, 685345, 49875609, 40598257]}, 'Perm': {'d': 22, 'gens': [51347038428091930320, 109506243016029525840, 163626575012832597120, 166059456098223703680, 33, 109506243016029525120, 109506243016025856000, 6706022400]}}, 'schur_multiplier': [2, 2, 2, 2, 4], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 2, 20], 'solvability_type': 7, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_4^2:S_3\\times C_{10}', 'transitive_degree': 480, 'wreath_data': None, 'wreath_product': False}
