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gps_subgroup_search • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'ambient': '1024.dhh', 'ambient_counter': 2218, 'ambient_order': 1024, 'ambient_tex': 'C_2^4.C_2\\wr C_4', 'central': False, 'central_factor': False, 'centralizer_order': 4, 'characteristic': False, 'core_order': 32, 'counter': 24, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '1024.dhh.8.m1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': False, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': False, 'old_label': '8.m1', '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': 8, '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': '128.52', 'subgroup_hash': 52, 'subgroup_order': 128, 'subgroup_tex': 'C_2^4.C_8', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '1024.dhh', 'aut_centralizer_order': 8, 'aut_label': '8.m1', 'aut_quo_index': None, 'aut_stab_index': 4, 'aut_weyl_group': '256.26541', 'aut_weyl_index': 32, 'centralizer': '256.i1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['4.g1'], 'contains': ['16.v1', '16.bm1'], 'core': '32.a1', 'coset_action_label': None, 'count': 4, 'diagramx': [8403, -1, 8400, -1], 'generators': [4097506710982, 8402207368918, 18132957163582, 7, 5167, 40279687, 1313941673647], 'label': '1024.dhh.8.m1', 'mobius_quo': None, 'mobius_sub': 0, 'normal_closure': '2.c1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '4.g1', 'old_label': '8.m1', 'projective_image': '512.1822', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '8.m1', 'subgroup_fusion': None, 'weyl_group': '64.32'}
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gps_groups • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '16.5', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 3, 'aut_exponent': 8, 'aut_gen_orders': [2, 2, 4, 2, 4, 4], 'aut_gens': [[1, 2, 16, 32], [1, 102, 16, 112], [65, 2, 16, 32], [1, 127, 16, 48], [81, 82, 16, 96], [41, 118, 80, 96], [65, 10, 16, 32]], 'aut_group': '512.7530064', 'aut_hash': 8918508347790019411, 'aut_nilpotency_class': 4, 'aut_nilpotent': True, 'aut_order': 512, 'aut_permdeg': 32, 'aut_perms': [38297028662849451039630893135369740, 33248077363941092840739464770113360, 185940697339491274260225540440583678, 40869978559660493171246076176905073, 47684336672132333036522658629455317, 108797906604174586657857215415838090], 'aut_phi_ratio': 8.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 2, 1, 1], [2, 4, 1, 1], [2, 4, 2, 1], [4, 1, 2, 1], [4, 2, 1, 1], [4, 4, 1, 1], [4, 4, 2, 1], [8, 4, 4, 1], [8, 8, 2, 1], [16, 8, 8, 1]], 'aut_supersolvable': True, 'aut_tex': 'D_4^2:C_2^3', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 4, 'autcent_group': '8.2', 'autcent_hash': 2, 'autcent_nilpotent': True, 'autcent_order': 8, 'autcent_solvable': True, 'autcent_split': False, 'autcent_supersolvable': True, 'autcent_tex': 'C_2\\times C_4', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 4, 'autcentquo_group': '64.138', 'autcentquo_hash': 138, 'autcentquo_nilpotent': True, 'autcentquo_order': 64, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'C_2\\wr C_2^2', 'cc_stats': [[1, 1, 1], [2, 1, 1], [2, 2, 1], [2, 4, 3], [4, 1, 2], [4, 2, 1], [4, 4, 3], [8, 4, 4], [8, 8, 2], [16, 8, 8]], 'center_label': '4.1', 'center_order': 4, 'central_product': False, 'central_quotient': '32.6', 'commutator_count': 1, 'commutator_label': '8.5', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '2.1', '2.1', '2.1'], 'composition_length': 7, 'conjugacy_classes_known': True, 'counter': 52, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 2, 1, 1], [2, 4, 1, 3], [4, 1, 2, 1], [4, 2, 1, 1], [4, 4, 1, 1], [4, 4, 2, 1], [8, 4, 2, 2], [8, 8, 2, 1], [16, 8, 4, 2]], 'element_repr_type': 'PC', 'elementary': 2, 'eulerian_function': 12, 'exponent': 16, 'exponents_of_order': [7], 'factors_of_aut_order': [2], 'factors_of_order': [2], 'faithful_reps': [[4, 0, 4]], 'familial': False, 'frattini_label': '32.37', 'frattini_quotient': '4.2', 'hash': 52, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 4, 'inner_gen_orders': [2, 4, 2, 2], 'inner_gens': [[1, 122, 16, 32], [121, 2, 80, 48], [1, 66, 16, 32], [1, 18, 16, 32]], 'inner_hash': 6, 'inner_nilpotent': True, 'inner_order': 32, 'inner_split': False, 'inner_tex': 'C_2^3:C_4', 'inner_used': [1, 2], 'irrC_degree': 4, 'irrQ_degree': 8, 'irrQ_dim': 8, 'irrR_degree': 8, 'irrep_stats': [[1, 16], [2, 4], [4, 6]], 'label': '128.52', 'linC_count': 4, 'linC_degree': 4, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 8, 'linQ_degree_count': 2, 'linQ_dim': 8, 'linQ_dim_count': 2, 'linR_count': 2, 'linR_degree': 8, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'C2^4.C8', 'ngens': 2, 'nilpotency_class': 4, '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': 12, 'number_characteristic_subgroups': 16, 'number_conjugacy_classes': 26, 'number_divisions': 15, 'number_normal_subgroups': 20, 'number_subgroup_autclasses': 52, 'number_subgroup_classes': 64, 'number_subgroups': 160, 'old_label': None, 'order': 128, 'order_factorization_type': 7, 'order_stats': [[1, 1], [2, 15], [4, 16], [8, 32], [16, 64]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 2, 'outer_gen_orders': [2, 2, 2, 2], 'outer_gen_pows': [0, 80, 80, 0], 'outer_gens': [[1, 3, 16, 96], [1, 6, 16, 112], [1, 10, 16, 32], [65, 2, 16, 32]], 'outer_group': '16.14', 'outer_hash': 14, 'outer_nilpotent': True, 'outer_order': 16, 'outer_permdeg': 8, 'outer_perms': [5040, 120, 6, 1], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^4', 'pc_rank': 4, 'perfect': False, 'permutation_degree': 16, 'pgroup': 2, 'primary_abelian_invariants': [2, 8], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 4], [4, 5], [8, 2]], 'representations': {'PC': {'code': 9974251906776910405353710, 'gens': [1, 2, 5, 6], 'pres': [7, 2, 2, 2, 2, 2, 2, 2, 1709, 36, 1766, 58, 248, 1411, 1020, 1027, 124]}, 'GLFp': {'d': 4, 'p': 5, 'gens': [122109387504, 77370891596, 91714451877, 61179005627, 27831734212, 61054693752, 149861448128]}, 'Perm': {'d': 16, 'gens': [20916435456000, 1313941673640, 40279687, 9703614452976, 5167, 4097506710982, 1313941673647]}}, 'schur_multiplier': [2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 8], 'solvability_type': 4, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_2^4.C_8', 'transitive_degree': 16, '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': '8.2', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 3, 'aut_exponent': 8, 'aut_gen_orders': [4, 4, 2, 4, 4, 2, 4, 2, 2], 'aut_gens': [[8402254180432, 18132957163582], [9703741807312, 12520495797713], [6920263514743, 11212948074823], [8396027534016, 19608888459023], [5612502050633, 20916562480567], [8402294465272, 20922662851337], [9703741812479, 12520495802878], [9709968481896, 16819143240383], [9703741812479, 19615075574527], [8402294465272, 14002653767736]], 'aut_group': '8192.bbp', 'aut_hash': 6549841272043568797, 'aut_nilpotency_class': 6, 'aut_nilpotent': True, 'aut_order': 8192, 'aut_permdeg': 192, 'aut_perms': [254685658252297114889119936180921247450569289993093169535658996765698543345052886455662824265979759389425242066281156749778861191356307127902371493187198854384095605438652356357195313824285756370202096251823496127659985424764942533766827243055389184684782588175970677578391671120624420283174944540631766944208532707902505924227742549972251268272242369298087, 251553677154727316002175454164112830356304335184065948741086280916887916127740736989672174187870661764883370819289811614987671850414357641074080434694883318117434641771867185816920006101388072245254058581878953430793167377293185796419871426345338504346847998988534798433993656398959676647899329933707422752676152095786427388637546916159977084854489369569833, 132364601355149437433935801338746909380548389424895679077487822602953549668444105669914871651091890156718221837405182645256905413727861634829810744830285799499147455596942901685250733901077669036284757223795823302460306538525128112725674303507053089370821465496409098484255656061344534794118274317614467464820269528653178615883533793410021543202621634474, 9151357438837104493947011211774641736673748158416689024391759959025994414952875291501273641957499125165288568489305384421581017162703143840646101900376355692764909416436070941188880064074436691390345279394697956744473424290078743956070372637624820182303798501749668980886955816907415554988678123594159652557760786235686834930252012789572887164604335784300, 319862051434694339094542913703868692952461860318671271209287589994298362503948255344642412360773319204858910698665404493494481026091844887401382144275793604249673823694450349110561367880525343163364756581871553694422548350741049331489114537416410993614813360060772419648920667999967117302365859963319362950980636592617832520914990959986524978375369309252646, 275594136346822959794907024426628683560122159659824411952715721453603615164202010070518572959120083077661521521481113604541293010960734598485985862693523545808543464146375882410436405313449876195691175327168883139572799078866238290439363329166717132544420587048221646711553038586766827510132025075866445451481472492400990555417915501871665925037725077630721, 336074208125579552872237198197477256701036742748321480839666204652557343277057744110114174687733175387818537901289467120746426687790534275120697345351472318489922336149668551026549538883541890534246966618817294972547234149323648840879067579815041120015648119008419565902916531860254546216950347815286704363810129995660097275556270970784857463398445739211785, 129671935377976976750068257298857976257758799974004475545197583764039485026990158538732818751505711764120002952719557995938962289375783302106118368064267139328791522569100772790363733452276353122379930777849494044452179750125465121298948999487450901015463302036734094791744213324163616246907873393439067262762695491909920525396127950204601646624736789667744, 290743075247589344026082677984927757952819668543128542454738967741642316402551455659555996169211367021651368635633554441412697851113106288990057203480782744356351802942321232686065276088301360059967710459835089877995827666157939802682303418401990152929855582442926751133731258858835451524733209164835092062601272112810262454171173176222705431981830543987412], 'aut_phi_ratio': 16.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 2, 1, 1], [2, 4, 1, 3], [2, 8, 2, 1], [2, 16, 1, 2], [2, 16, 4, 1], [4, 8, 1, 2], [4, 16, 1, 1], [4, 32, 1, 1], [4, 32, 2, 2], [4, 64, 2, 2], [8, 32, 2, 1], [8, 64, 2, 1], [16, 64, 4, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_2^7.C_2\\wr C_2^2', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 2, 'autcent_group': '4.2', 'autcent_hash': 2, 'autcent_nilpotent': True, 'autcent_order': 4, 'autcent_solvable': True, 'autcent_split': False, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^2', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 8, 'autcentquo_group': None, 'autcentquo_hash': 8014087728474443289, 'autcentquo_nilpotent': True, 'autcentquo_order': 2048, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'C_2\\times C_2^4.C_2^5.C_2', 'cc_stats': [[1, 1, 1], [2, 1, 1], [2, 2, 1], [2, 4, 3], [2, 8, 2], [2, 16, 6], [4, 8, 2], [4, 16, 1], [4, 32, 5], [4, 64, 4], [8, 32, 2], [8, 64, 2], [16, 64, 4]], 'center_label': '2.1', 'center_order': 2, 'central_product': False, 'central_quotient': '512.1822', 'commutator_count': 1, 'commutator_label': '128.170', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '2.1', '2.1', '2.1', '2.1', '2.1', '2.1'], 'composition_length': 10, 'conjugacy_classes_known': True, 'counter': 2218, 'cyclic': False, 'derived_length': 3, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 2, 1, 1], [2, 4, 1, 3], [2, 8, 1, 2], [2, 16, 1, 6], [4, 8, 1, 2], [4, 16, 1, 1], [4, 32, 1, 5], [4, 64, 1, 2], [4, 64, 2, 1], [8, 32, 2, 1], [8, 64, 2, 1], [16, 64, 4, 1]], 'element_repr_type': 'Perm', 'elementary': 2, 'eulerian_function': 48, 'exponent': 16, 'exponents_of_order': [10], 'factors_of_aut_order': [2], 'factors_of_order': [2], 'faithful_reps': [[8, 1, 4], [16, 1, 1]], 'familial': False, 'frattini_label': '256.1441', 'frattini_quotient': '4.2', 'hash': 720239556428680473, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 8, 'inner_gen_orders': [4, 8], 'inner_gens': [[8402254180432, 19608761462542], [8402207022718, 18132957163582]], 'inner_hash': 2679906585708388427, 'inner_nilpotent': True, 'inner_order': 512, 'inner_split': True, 'inner_tex': 'C_2^5.D_8', 'inner_used': [1, 2], 'irrC_degree': 8, 'irrQ_degree': 8, 'irrQ_dim': 8, 'irrR_degree': 8, 'irrep_stats': [[1, 8], [2, 10], [4, 5], [8, 10], [16, 1]], 'label': '1024.dhh', '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': True, 'name': 'C2^4.C2wrC4', 'ngens': 2, 'nilpotency_class': 7, 'nilpotent': True, 'normal_counts': [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': 21, 'number_characteristic_subgroups': 23, 'number_conjugacy_classes': 34, 'number_divisions': 28, 'number_normal_subgroups': 23, 'number_subgroup_autclasses': 474, 'number_subgroup_classes': 1112, 'number_subgroups': 11885, 'old_label': None, 'order': 1024, 'order_factorization_type': 7, 'order_stats': [[1, 1], [2, 127], [4, 448], [8, 192], [16, 256]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': True, 'outer_exponent': 4, 'outer_gen_orders': [2, 2, 2, 2], 'outer_gen_pows': [0, 2796059358862, 2796059358862, 0], 'outer_gens': [[8396067087936, 15310241044553], [6913949402753, 18132997438097], [6913949397838, 18132997438097], [9703741812479, 18132957163582]], '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': 6, 'perfect': False, 'permutation_degree': 16, 'pgroup': 2, 'primary_abelian_invariants': [2, 4], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 4], [4, 9], [8, 10], [16, 1]], 'representations': {'PC': {'code': '30621015504110511458824244505973475577956753353371245410256046856418939575723687191', 'gens': [1, 3, 5, 7, 8, 9], 'pres': [10, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 20, 2571, 9002, 13632, 82, 8003, 12973, 29804, 7614, 9824, 4134, 144, 52325, 18255, 1465, 13446, 20176, 2566, 51207, 8668, 4728, 2218, 268]}, 'Perm': {'d': 16, 'gens': [18132957163582, 8402254180432]}}, 'schur_multiplier': [2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 4], 'solvability_type': 5, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_2^4.C_2\\wr C_4', 'transitive_degree': 16, 'wreath_data': None, 'wreath_product': False}
