-
gps_subgroup_search • Show schema
Hide schema
{'Agroup': True, 'Zgroup': False, 'abelian': True, 'ambient': '1944.3473', 'ambient_counter': 3473, 'ambient_order': 1944, 'ambient_tex': 'C_3^3.F_9', 'central': False, 'central_factor': False, 'centralizer_order': 81, 'characteristic': False, 'core_order': 3, 'counter': 44, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '1944.3473.216.c1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': False, 'old_label': '216.c1', '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': 216, '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': '9.2', 'subgroup_hash': 2, 'subgroup_order': 9, 'subgroup_tex': 'C_3^2', 'supersolvable': True, 'sylow': 0}
-
gps_subgroup_data • Show schema
Hide schema
{'ambient': '1944.3473', 'aut_centralizer_order': 648, 'aut_label': '216.c1', 'aut_quo_index': None, 'aut_stab_index': 4, 'aut_weyl_group': '12.4', 'aut_weyl_index': 2592, 'centralizer': '24.c1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['72.b1', '72.c1', '72.e1', '108.d1'], 'contains': ['648.a1', '648.d1'], 'core': '648.a1', 'coset_action_label': None, 'count': 4, 'diagramx': [3833, -1, 2190, -1], 'generators': [864, 648], 'label': '1944.3473.216.c1', 'mobius_quo': None, 'mobius_sub': 0, 'normal_closure': '72.b1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '4.a1', 'old_label': '216.c1', 'projective_image': '1944.3473', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '216.c1', 'subgroup_fusion': None, 'weyl_group': '6.1'}
-
gps_groups • Show schema
Hide schema
{'Agroup': True, 'Zgroup': False, 'abelian': True, 'abelian_quotient': '9.2', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 4, 'aut_exponent': 24, 'aut_gen_orders': [2, 3], 'aut_gens': [[1, 3], [1, 7], [4, 3]], 'aut_group': '48.29', 'aut_hash': 29, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 48, 'aut_permdeg': 8, 'aut_perms': [31834, 28334], 'aut_phi_ratio': 8.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [3, 1, 8, 1]], 'aut_supersolvable': False, 'aut_tex': '\\GL(2,3)', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 24, 'autcent_group': '48.29', 'autcent_hash': 29, 'autcent_nilpotent': False, 'autcent_order': 48, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': False, 'autcent_tex': '\\GL(2,3)', '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], [3, 1, 8]], 'center_label': '9.2', 'center_order': 9, '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': ['3.1', '3.1'], 'composition_length': 2, 'conjugacy_classes_known': True, 'counter': 2, 'cyclic': False, 'derived_length': 1, 'dihedral': False, 'direct_factorization': [['3.1', 2]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [3, 1, 2, 4]], 'element_repr_type': 'PC', 'elementary': 3, 'eulerian_function': 1, 'exponent': 3, 'exponents_of_order': [2], 'factors_of_aut_order': [2, 3], 'factors_of_order': [3], 'faithful_reps': [], 'familial': False, 'frattini_label': '1.1', 'frattini_quotient': '9.2', 'hash': 2, 'hyperelementary': 3, 'inner_abelian': True, 'inner_cyclic': True, 'inner_exponent': 1, 'inner_gen_orders': [1, 1], 'inner_gens': [[1, 3], [1, 3]], '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, 9]], 'label': '9.2', 'linC_count': 24, 'linC_degree': 2, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 4, 'linQ_degree_count': 6, 'linQ_dim': 4, 'linQ_dim_count': 6, 'linR_count': 6, 'linR_degree': 4, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C3^2', 'ngens': 2, 'nilpotency_class': 1, 'nilpotent': True, 'normal_counts': [0, 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': 9, 'number_divisions': 5, 'number_normal_subgroups': 6, 'number_subgroup_autclasses': 3, 'number_subgroup_classes': 6, 'number_subgroups': 6, 'old_label': None, 'order': 9, 'order_factorization_type': 2, 'order_stats': [[1, 1], [3, 8]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 24, 'outer_gen_orders': [2, 3], 'outer_gen_pows': [0, 0], 'outer_gens': [[1, 7], [4, 3]], 'outer_group': '48.29', 'outer_hash': 29, 'outer_nilpotent': False, 'outer_order': 48, 'outer_permdeg': 8, 'outer_perms': [31834, 28334], 'outer_solvable': True, 'outer_supersolvable': False, 'outer_tex': '\\GL(2,3)', 'pc_rank': 2, 'perfect': False, 'permutation_degree': 6, 'pgroup': 3, 'primary_abelian_invariants': [3, 3], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 1], [2, 4]], 'representations': {'PC': {'code': 0, 'gens': [1, 2], 'pres': [2, -3, 3]}, 'GLZ': {'b': 3, 'd': 4, 'gens': [16858733, 35931237]}, 'GLFp': {'d': 2, 'p': 7, 'gens': [687, 1374]}, 'Perm': {'d': 6, 'gens': [240, 4]}}, 'schur_multiplier': [3], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [3, 3], 'solvability_type': 1, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_3^2', 'transitive_degree': 9, 'wreath_data': None, 'wreath_product': False}
-
gps_groups • Show schema
Hide schema
{'Agroup': False, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '8.1', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 4, 'aut_exponent': 24, 'aut_gen_orders': [12, 8, 24, 4], 'aut_gens': [[1, 8, 24, 72, 216, 648], [1139, 88, 696, 80, 888, 1296], [713, 88, 480, 8, 48, 1296], [1265, 72, 24, 80, 1512, 648], [403, 144, 1752, 8, 432, 1296]], 'aut_group': None, 'aut_hash': 8814442550351613409, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 31104, 'aut_permdeg': 486, 'aut_perms': [16697766646669006554587215894788164986772652778850073920491113374103396164319746491392651064854098854090424809715628166651170447512880283988504611807197703970462206384325730666112547152290819281620069840210719445080303692985432239393279877758948050276736172015827217004448153174505591675341940183288039407830149525570117170516299549701419129050430008042700416730508791771480212549671959934345892083328154832332981428692136794081998678380540595772409460823798010808443272347946044980934473730728291602557004189863203599781333872949286608205774251441445371325246352118103569762172504932345738384461088070401772663157855295005490094742947129045722049005458283731913478772230138414035260264588364679113011070378117670336257153431599051514672350131350542561562711354493240868505864771388309683795701187226403935652283228171278239155638285398375553636717203567070602459191968156289472973783958979582082798383289451664516930344644152998018182105501319639082128569682541913913711988651664448868581005007979432654037987189347952295808082478414584744624384635508122898562576699265897885478476807860245974703, 10131151946603504910124600389869993407551890137561459483047271624071195125208709498084803160713949775919196286293003104444113080062602583404660517769329240950041764066038608029603016487247723903661117846480739298789484698570322448260749582064710346205279045627241044042556269009418086744367268856785798688417288021135118555160950996026399818964956924917301543163368993867251694711816360438234058456057650716052342741444800093602648801784159845010062191285784069543730810520152876433856361490450216802236783681684640998650136003497182106423190427550799057484661336352126085790766599325636242750013826597803997575849854669256183606946128090890209679632723752743459357033956334092268113771711562777848264622965726497894131638202006068256641358097797315047731474290816991673113294617298634494440600020052814536805029832223994462762187985805997980769792409663090237556557647812542235148254856404921241280535129383290494561920017335423780712230608768624374906731351279152771922145373409146137245295986127818754929828027263198892863971225332483218359874957198242416768331781542452219557825266252490445188, 19652688104361367491349278434428575903423686878796927653537503959356671423147100842652644979687324574551740480294835942679685403099588251131868919457460296918392041940878541474343389387835367613701886308478564028184793420422424308632856468372758800787159011434039648406955982861107304267689602800058037834301445691252389602976235089564443579121977614888572258195808289210959774949487094092945349272097175696118870515241705747210593365541582183107669418749975871663328464914757423063928442857853132614334939836487565353219976172295502254163257782488367297885006572733926829369113167640316034704041240728138289642016687996064136073801779364862194554509422509110890248122758485496873868208137379568899106214097648587834913797087580701502899332819775863461049402111565031988883891853767590505835719719498052789758626731996518472646974101686649370522059957836029629697436862999427020529609294642622030599252406495416587493585513837538776882953128279847053783064965793244253624085107658263848691829972729253314425974089410326915986615227516954405775962928015009344214465069647811365093736585053360003800, 11397428749213978662521275830981320359267074302790286815591843293491488350414864069558969458810976174440028699767517860488503047981058206357472722454078664132277807121470325914156600760033807793099714079809269883044705674386017204472173376187146177348065451011716185484698216230569356429415476391368939710769837287279028378173476484798963702643603388524951751651551880529417201233658944784439419790141266834291460234890445158901110500855254515699260823106819789911841160585209861208481094683151487876020935247037338610338207559546972010022560415468712357169994451310326067704426664348112240907826790248507536244704800045909173751687268707530392414489294164867155367532497886173530841853044926581654029688613871315155098850841353414169954569358954028580230302776393163375517320889523523778510767767992108686119880037555881194063540485485020327745777455964088587235560705010326534441031224166937931266389874621753797160914714471740152382176720494477704683932994331041745207848358964549570793257953801441007460554917622570207037134892545156889027899206521609595424627486064467390243050707473876940156], 'aut_phi_ratio': 48.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 9, 1, 1], [3, 2, 1, 1], [3, 4, 2, 1], [3, 4, 4, 1], [3, 24, 1, 1], [3, 24, 8, 1], [4, 81, 2, 1], [6, 18, 1, 1], [6, 36, 2, 1], [6, 36, 4, 1], [8, 243, 2, 2], [12, 162, 2, 1]], 'aut_supersolvable': False, 'aut_tex': '(C_3^2\\times \\He_3).C_8^2.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': 24, 'autcentquo_group': None, 'autcentquo_hash': 8814442550351613409, 'autcentquo_nilpotent': False, 'autcentquo_order': 31104, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'C_3.C_3^4:C_2^2.C_4:D_4', 'cc_stats': [[1, 1, 1], [2, 9, 1], [3, 2, 1], [3, 4, 6], [3, 24, 9], [4, 81, 2], [6, 18, 1], [6, 36, 6], [8, 243, 4], [12, 162, 2]], 'center_label': '1.1', 'center_order': 1, 'central_product': False, 'central_quotient': '1944.3473', 'commutator_count': 1, 'commutator_label': '243.62', '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': 3473, 'cyclic': False, 'derived_length': 3, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 9, 1, 1], [3, 2, 1, 1], [3, 4, 1, 2], [3, 4, 2, 2], [3, 24, 1, 1], [3, 24, 2, 4], [4, 81, 2, 1], [6, 18, 1, 1], [6, 36, 1, 2], [6, 36, 2, 2], [8, 243, 4, 1], [12, 162, 2, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 72, 'exponent': 24, 'exponents_of_order': [5, 3], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [[12, 0, 8]], 'familial': False, 'frattini_label': '3.1', 'frattini_quotient': '648.710', 'hash': 3473, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 24, 'inner_gen_orders': [8, 3, 3, 3, 3, 3], 'inner_gens': [[1, 160, 864, 88, 456, 1296], [89, 8, 24, 72, 216, 648], [457, 8, 24, 72, 1512, 648], [9, 8, 24, 72, 216, 648], [1129, 8, 672, 72, 216, 648], [1297, 8, 24, 72, 216, 648]], 'inner_hash': 3473, 'inner_nilpotent': False, 'inner_order': 1944, 'inner_split': True, 'inner_tex': 'C_3^3.F_9', 'inner_used': [1, 2, 3], 'irrC_degree': 12, 'irrQ_degree': 24, 'irrQ_dim': 24, 'irrR_degree': 24, 'irrep_stats': [[1, 8], [4, 4], [6, 4], [8, 9], [12, 8]], 'label': '1944.3473', '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': 'C3^3.F9', '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': 14, 'number_characteristic_subgroups': 10, 'number_conjugacy_classes': 33, 'number_divisions': 20, 'number_normal_subgroups': 10, 'number_subgroup_autclasses': 63, 'number_subgroup_classes': 144, 'number_subgroups': 2264, 'old_label': None, 'order': 1944, 'order_factorization_type': 33, 'order_stats': [[1, 1], [2, 9], [3, 242], [4, 162], [6, 234], [8, 972], [12, 324]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': True, 'outer_exponent': 8, 'outer_gen_orders': [2, 8], 'outer_gen_pows': [4, 0], 'outer_gens': [[3, 8, 888, 152, 456, 648], [1, 88, 24, 8, 216, 648]], 'outer_group': '16.8', 'outer_hash': 8, 'outer_nilpotent': True, 'outer_order': 16, 'outer_permdeg': 8, 'outer_perms': [4761, 7602], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': '\\SD_{16}', 'pc_rank': 6, 'perfect': False, 'permutation_degree': 33, 'pgroup': 0, 'primary_abelian_invariants': [8], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 2], [2, 1], [4, 5], [6, 2], [8, 1], [12, 1], [16, 4], [24, 4]], 'representations': {'PC': {'code': 99702897758858508457332253025251313008553975453765731103, 'gens': [1, 4, 5, 6, 7, 8], 'pres': [8, -2, -2, -2, -3, 3, 3, 3, -3, 16, 41, 5123, 267, 34564, 35052, 6980, 4229, 3469, 25542, 13454, 6070, 3566, 82951]}, 'Perm': {'d': 33, 'gens': [26623572949910681785411638872819889, 35034452734229988522576174545995456, 43522963801640179201739062199059200, 306954736513841819570653009619429040, 578545468616193147127084197783230640, 59, 304, 847849739698839490270833622484903040]}}, 'schur_multiplier': [3], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [8], 'solvability_type': 12, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': 'C_3^3.F_9', 'transitive_degree': 36, 'wreath_data': None, 'wreath_product': False}
