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gps_subgroup_search • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'ambient': '1434890700.g', 'ambient_counter': 7, 'ambient_order': 1434890700, 'ambient_tex': 'C_3^{12}.C_{15}^2.D_6', 'central': False, 'central_factor': False, 'centralizer_order': None, 'characteristic': False, 'core_order': 9, 'counter': 106, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '1434890700.g.531441._.A', 'maximal': True, 'maximal_normal': False, 'metabelian': False, 'metacyclic': False, 'minimal': False, 'minimal_normal': False, 'nilpotent': False, 'normal': False, 'old_label': '531441.A', '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': 531441, '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': '2700.q', 'subgroup_hash': None, 'subgroup_order': 2700, 'subgroup_tex': 'C_{15}^2:D_6', 'supersolvable': False, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '1434890700.g', 'aut_centralizer_order': None, 'aut_label': None, 'aut_quo_index': None, 'aut_stab_index': None, 'aut_weyl_group': None, 'aut_weyl_index': None, 'centralizer': None, 'complements': None, 'conjugacy_class_count': 1, 'contained_in': None, 'contains': None, 'core': None, 'coset_action_label': None, 'count': 531441, 'diagramx': None, 'generators': [2718686843732590950020475776421955982433938888238920880, 13477264432119708967819002634374675831926338184767470723, 651483107294171852602166839940201297045864020513923, 2718721106830516014595779163968886096789150540884679107, 2719165227245477680238637343195128094439481796133212227, 97924369333700764766441967118295354725393262807020891443, 34269487941646909416098305223398339831441094745748], 'label': '1434890700.g.531441._.A', 'mobius_quo': None, 'mobius_sub': None, 'normal_closure': '1.a1.a1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '531441._.A', 'old_label': '531441.A', 'projective_image': None, 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '531441._.A', 'subgroup_fusion': None, 'weyl_group': None}
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gps_groups • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '4.2', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 3, 'aut_exponent': 120, 'aut_gen_orders': [30, 8, 24, 24], 'aut_gens': [[1, 2, 12, 180], [2261, 1402, 168, 264], [431, 694, 2244, 1608], [2431, 302, 2244, 2484], [2119, 1510, 1644, 2364]], 'aut_group': None, 'aut_hash': 7509908978670635385, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 21600, 'aut_permdeg': 90, 'aut_perms': [1223718560084616643378011208422965467506668793203639699838220593818746746451832335686901635315283618356704229363990232635125174632593337037, 1181674562415681112563614535571498175673032522628224692185802545653006757524643646115437306083046741477899046391560581345618446675707747838, 365485412783665425823287248426271448329414864055784104381614255917074554446401272649710462951220353102249428940907350473684225225077400145, 427414949721339587139027811969459071248455897039281133220302817737683668206477290996457252490043882635733454882249136064436272803425588382], 'aut_phi_ratio': 30.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 25, 1, 1], [2, 45, 2, 1], [3, 1, 2, 1], [3, 6, 1, 1], [3, 150, 3, 1], [5, 6, 4, 1], [6, 25, 2, 1], [6, 45, 4, 1], [6, 150, 1, 1], [6, 150, 3, 1], [10, 90, 4, 1], [15, 6, 8, 1], [15, 12, 4, 1], [15, 12, 8, 1], [30, 90, 8, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_{15}^2.C_{12}.C_2^3', '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': None, 'autcentquo_hash': 7509908978670635385, 'autcentquo_nilpotent': False, 'autcentquo_order': 21600, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'C_{15}^2.C_{12}.C_2^3', 'cc_stats': [[1, 1, 1], [2, 25, 1], [2, 45, 2], [3, 1, 2], [3, 6, 1], [3, 150, 3], [5, 6, 4], [6, 25, 2], [6, 45, 4], [6, 150, 4], [10, 90, 4], [15, 6, 8], [15, 12, 12], [30, 90, 8]], 'center_label': '3.1', 'center_order': 3, 'central_product': False, 'central_quotient': '900.96', 'commutator_count': 1, 'commutator_label': '675.12', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '3.1', '3.1', '3.1', '5.1', '5.1'], 'composition_length': 7, 'conjugacy_classes_known': True, 'counter': 17, 'cyclic': False, 'derived_length': 3, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 25, 1, 1], [2, 45, 1, 2], [3, 1, 2, 1], [3, 6, 1, 1], [3, 150, 1, 3], [5, 6, 2, 2], [6, 25, 2, 1], [6, 45, 2, 2], [6, 150, 1, 4], [10, 90, 2, 2], [15, 6, 4, 2], [15, 12, 2, 2], [15, 12, 4, 2], [30, 90, 4, 2]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 353808, 'exponent': 30, 'exponents_of_order': [3, 2, 2], 'factors_of_aut_order': [2, 3, 5], 'factors_of_order': [2, 3, 5], 'faithful_reps': [[6, 0, 16], [12, 0, 8]], 'familial': False, 'frattini_label': '3.1', 'frattini_quotient': '900.96', 'hash': 8099013394032210383, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 30, 'inner_gen_orders': [10, 6, 5, 15], 'inner_gens': [[1, 2290, 552, 2640], [137, 2, 2280, 204], [2161, 614, 12, 180], [421, 158, 12, 180]], 'inner_hash': 96, 'inner_nilpotent': False, 'inner_order': 900, 'inner_split': False, 'inner_tex': '(C_5\\times C_{15}):D_6', 'inner_used': [1, 2, 4], 'irrC_degree': 6, 'irrQ_degree': 24, 'irrQ_dim': 24, 'irrR_degree': 12, 'irrep_stats': [[1, 4], [2, 8], [3, 8], [6, 24], [12, 12]], 'label': '2700.q', 'linC_count': 16, 'linC_degree': 6, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 18, 'linQ_degree_count': 16, 'linQ_dim': 18, 'linQ_dim_count': 16, 'linR_count': 40, 'linR_degree': 12, 'maximal_subgroups_known': True, 'metabelian': False, 'metacyclic': False, 'monomial': True, 'name': 'C15^2:D6', 'ngens': 7, '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, 0, 0, 0, 0, 0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 16, 'number_characteristic_subgroups': 12, 'number_conjugacy_classes': 56, 'number_divisions': 28, 'number_normal_subgroups': 20, 'number_subgroup_autclasses': 78, 'number_subgroup_classes': 154, 'number_subgroups': 3982, 'old_label': None, 'order': 2700, 'order_factorization_type': 321, 'order_stats': [[1, 1], [2, 115], [3, 458], [5, 24], [6, 830], [10, 360], [15, 192], [30, 720]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 12, 'outer_gen_orders': [2, 12], 'outer_gen_pows': [0, 2274], 'outer_gens': [[1, 1234, 672, 780], [547, 2686, 588, 2028]], 'outer_group': '24.5', 'outer_hash': 5, 'outer_nilpotent': False, 'outer_order': 24, 'outer_permdeg': 7, 'outer_perms': [120, 1457], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_4\\times S_3', 'pc_rank': 4, 'perfect': False, 'permutation_degree': 24, 'pgroup': 0, 'primary_abelian_invariants': [2, 2], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 8], [6, 4], [12, 4], [24, 6], [48, 2]], 'representations': {'PC': {'code': '1612142572079446564926954310253041976083279094358905608142674983885914487', 'gens': [1, 2, 4, 6], 'pres': [7, 2, 2, 3, 3, 5, 3, 5, 4788, 32061, 36, 48302, 15459, 31930, 15473, 108, 57964, 18911, 10728, 110885, 4296, 19801, 166, 105846, 15007, 902]}, 'Perm': {'d': 24, 'gens': [28304768306459701734966, 4817187026792126393844, 55346845900579008695499]}}, 'schur_multiplier': [2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2], 'solvability_type': 11, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': 'C_{15}^2:D_6', 'transitive_degree': 45, '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': '4.2', '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': [[35360028966967316450724827507836661922706227773650253254, 13477399933638919937892418820951354672422602371621516164, 97924026452809295187749374614540365392668536365711253638], [108802849995686647750025059846688879505071945694083676114, 13477708529973966154674704502795795024464408933139425877, 27200439244069133280738946007709331732820507068844627253], [119561052066005546461241084495481217710700948481796814674, 95080669587471357927254013225555270901382146638057012480, 97924575067305163409958675199083985603267481112851246884], [37957917915840487002776087337552002366818864278076067154, 8160824147450671819328506925787000610591089952457147764, 97924026452809295187749374614540365392668536365711253638], [57123265989158597981768961695234904975041022162480502323, 76162195308487067612785290015907309974580679169643213680, 97924575067305152781832817771614430696346314939603738884], [35360028966967316450724827507836661922706227773650253254, 13477399933638919937892418820951354672422602371621516164, 97924026452809295187749374614540365392668536365711253638], [81602961065554724672271047314886943885026004963920916724, 59840615576253235873842202195531908429020496326300101078, 97923784760481420025987577509671226014148304724482428320], [84322023403183332884929260438422415550940652288701529747, 111401050879715940538579992327041256993498707883762405268, 51681332741088918805761125738780631441812061315525093810], [37957917915840487002776096484621170363110825525144163010, 8160856784062001735237898434180574603246159432707963520, 103240636507439419051450833430043390429100083054585311410], [35360061603578625700083352024456589745896241510703757253, 10879543622450621337857463722526671137882268840819233923, 97924060721251201637369393013628099400539116458637308133], [37957885278692391383131080502452141707272745177891597493, 13477333028585672943710916032142934079824271583277004308, 103240636507439419051450842576693728780515898018754040774], [32641342129115483808602409006665099377254844409001212610, 13477367298073614590348960771336246836875591872089979664, 103240569602386192723836626236569597168442009579671045814], [37957851010787281341457803770136557487477265344457955154, 13477399934175685642409239186883516650456409579815355668, 100642780195714333490663797325348218019462503307392462530], [35360028966967316450198565833162667214413491513922893253, 13477333029122479980836047620350975554063593440897074704, 103240636507439419051450833430043390429100082880228728914], [35360028966967316450724818970575232545945781225067817974, 13477333029122479980836029937633519292879417854799880323, 103240603870828089135541468752242893231846649368800760774], [35360028966967316450469910972295665234563145677023843154, 13477399934175706308705171959561156234841647900915698704, 103240569602386202761751784436460047636058587568641567170], [35360028966967316450198565833162640233530220741536435893, 13477399934175706308450255424408016224966236957519880564, 97924026452809295187749383151414014520656630582179836133], [35360028966967316450469919509169341336257114883670774914, 13477399934175706308450263960893833400392125688664117824, 97924093357862521515363581808014158645906258715563388034], [35360028966967316450198565833969250134355780641997581253, 13477399934684965172825630191121707472038565999177804163, 97924026452809295187749383151414014518345373500091733637], [35360028966967316450724827507448881671622241655788749398, 13477399933638919937892418820951354672422978888434153028, 97924026452809295187749374614540365392668536459116565494], [35359994699062185743281879676679751054495092950934298293, 13477333029631738845211413850190992940266390007810018163, 100642780195714344119044580434233495762053084992775566774], [86920670020629934973994419228375798688241602140631472054, 10879202390578902014509364721381874637770524101376520464, 54278642091097932027556475911608261378963557513851184643], [106084130521223495230434687395630953565188640204626604853, 13477675894408691511613682565688434620237373513872417734, 27200506149122370236988827234569572934473989323991507158], [119561052065468750052768072008222310983792795213245051893, 89764059532304458321901736369757193177326308508681674387, 103241083948976950904627566622725387575454070571178779123]], 'aut_group': None, 'aut_hash': None, 'aut_nilpotency_class': None, 'aut_nilpotent': None, 'aut_order': 45916502400, 'aut_permdeg': None, 'aut_perms': None, 'aut_phi_ratio': None, 'aut_solvable': None, 'aut_stats': None, 'aut_supersolvable': None, 'aut_tex': None, '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': None, 'center_label': '3.1', 'center_order': None, 'central_product': None, 'central_quotient': None, 'commutator_count': None, 'commutator_label': '358722675.a', 'complements_known': False, 'complete': False, 'complex_characters_known': False, 'composition_factors': ['2.1', '2.1', '3.1', '3.1', '3.1', '3.1', '3.1', '3.1', '3.1', '3.1', '3.1', '3.1', '3.1', '3.1', '3.1', '3.1', '3.1', '5.1', '5.1'], 'composition_length': 19, 'conjugacy_classes_known': False, 'counter': 7, 'cyclic': False, 'derived_length': 4, 'dihedral': False, 'direct_factorization': None, 'direct_product': None, 'div_stats': None, 'element_repr_type': 'Perm', 'elementary': 1, 'eulerian_function': None, 'exponent': 90, 'exponents_of_order': [15, 2, 2], 'factors_of_aut_order': [2, 3, 5], 'factors_of_order': [2, 3, 5], 'faithful_reps': None, 'familial': False, 'frattini_label': '3.1', 'frattini_quotient': None, 'hash': 7392868699652681735, 'hyperelementary': 1, 'inner_abelian': None, 'inner_cyclic': None, 'inner_exponent': None, 'inner_gen_orders': None, 'inner_gens': None, 'inner_hash': None, 'inner_nilpotent': None, 'inner_order': None, 'inner_split': None, 'inner_tex': None, 'inner_used': None, 'irrC_degree': None, 'irrQ_degree': None, 'irrQ_dim': None, 'irrR_degree': None, 'irrep_stats': None, 'label': '1434890700.g', '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': 'C3^12.C15^2.D6', 'ngens': 3, 'nilpotency_class': -1, 'nilpotent': False, 'normal_counts': [1, 0, 1, 0, 0, 0, 1, 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, 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, 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, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 1, 0, 4, 0, 0, 4, 0, 1, 0, 3, 1], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': None, 'number_characteristic_subgroups': 17, 'number_conjugacy_classes': None, 'number_divisions': None, 'number_normal_subgroups': 23, 'number_subgroup_autclasses': None, 'number_subgroup_classes': None, 'number_subgroups': None, 'old_label': None, 'order': 1434890700, 'order_factorization_type': 321, 'order_stats': None, 'outer_abelian': None, 'outer_cyclic': None, 'outer_equivalence': False, 'outer_exponent': None, 'outer_gen_orders': None, 'outer_gen_pows': None, 'outer_gens': None, 'outer_group': None, 'outer_hash': None, 'outer_nilpotent': None, 'outer_order': 96, 'outer_permdeg': None, 'outer_perms': None, 'outer_solvable': None, 'outer_supersolvable': None, 'outer_tex': None, 'pc_rank': None, 'perfect': False, 'permutation_degree': 45, 'pgroup': 0, 'primary_abelian_invariants': [2, 2], 'quasisimple': False, 'rank': 3, 'rational': None, 'rational_characters_known': False, 'ratrep_stats': None, 'representations': {'PC': {'code': '2796805579265791428079097110960424057991814118824168746425301393857221419018309917659154701394641027676678600590160746631992777204989181560630643997801469749519358261434597585620130917659269816406600831274100012229189024671480040936472304762575058922252760031369494247907662720570969450535324603592065046704563224242853128748128664269882870608904224418183836998180696114349872858371836552805161078140755224730738012343324481763659546551710568086553970312168208256763494572368962663689840202302282088804411827500745462044944728265604989024563166617521318124756004598475152382692534195496941391594292894541873820566991973359173560107320655478781557928472288514799474003161013930610984429744840316906871856525302416327502814973602271998947984630185432692276050819490217831668334155974909130364910306541718009708995221649884690234979722657191987011032139317426331649694542794684857032784846233977901392085639152555784018658564680211679048142563932625893737924606532077192078111328855750113493954214645267665182288773537558545303744067007781077865813', 'gens': [1, 2, 4, 6, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19], 'pres': [19, 2, 2, 3, 3, 5, 3, 5, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 8543197620, 36434055189, 96, 56268075086, 5863513090, 45660925251, 29124523486, 16554721625, 288, 96329922484, 32819592263, 6442782432, 146915115125, 32396337852, 35289971827, 10663322372, 174300951, 442, 13269535026, 12174643933, 46643878493, 15465305898, 80290852, 37961589607, 81351950426, 2107404045, 878324464, 453343883, 1011880062, 396582561, 160262995508, 128814327, 30818521, 982882415, 2026434, 65525593, 135212, 143127009, 91599741028, 20263547, 11927316, 568147585, 161569454, 265173, 239543657110, 100679020229, 2864951148, 1774253317, 502101686, 1381704330, 519263759, 253802030411, 5326479030, 39053664049, 730614668, 161680587, 1177492426, 304683365, 231679059312, 119071993981, 6586137725, 1448062269, 731793163, 1258232927, 339551026, 226763749813, 197505032, 7125621351, 890987020, 1034966189, 390321858, 323994777, 162831586514, 13759429533, 90451647052, 1243191446, 541215090, 630135109, 706469528, 157956393615, 196091582434, 92565720053, 3951604872, 670456891, 809272430, 724027809, 404495677816, 156237151085, 4323871854, 1308295423, 957590117, 2231902656, 770724965, 128429242217, 69408284436, 53975038555, 7007067074, 2257943943, 35694652, 671096471, 329502208518, 100232787187, 37994537081, 19025169375, 4814232919, 1287221423, 34322207]}, 'Perm': {'d': 45, 'gens': [97924026452809295187749374614540365392668536365711253638, 13477399933638919937892418820951354672422602371621516164, 35360028966967316450724827507836661922706227773650253254]}}, 'schur_multiplier': [2], 'semidirect_product': None, 'simple': False, 'smith_abelian_invariants': [2, 2], 'solvability_type': 17, 'solvable': True, 'subgroup_inclusions_known': False, 'subgroup_index_bound': 81, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': 'C_3^{12}.C_{15}^2.D_6', 'transitive_degree': 45, 'wreath_data': None, 'wreath_product': False}
