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gps_subgroup_search • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'ambient': '6718464.bbw', 'ambient_counter': 725, 'ambient_order': 6718464, 'ambient_tex': 'C_3^8:C_4^2.Q_{16}:C_2^2', 'central': False, 'central_factor': False, 'centralizer_order': None, 'characteristic': False, 'core_order': 1679616, 'counter': 26, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '6718464.bbw.4._.J', 'maximal': False, 'maximal_normal': False, 'metabelian': False, 'metacyclic': False, 'minimal': False, 'minimal_normal': False, 'nilpotent': False, 'normal': True, 'old_label': '4.J', 'outer_equivalence': False, 'perfect': False, 'proper': True, 'quotient': '4.2', 'quotient_Agroup': True, 'quotient_abelian': True, 'quotient_cyclic': False, 'quotient_hash': None, 'quotient_metabelian': True, 'quotient_nilpotent': True, 'quotient_order': 4, 'quotient_simple': False, 'quotient_solvable': True, 'quotient_supersolvable': True, 'quotient_tex': 'C_2^2', 'simple': False, 'solvable': True, 'special_labels': [], 'split': None, 'standard_generators': False, 'stem': False, 'subgroup': '1679616.sz', 'subgroup_hash': None, 'subgroup_order': 1679616, 'subgroup_tex': 'C_3^8:C_4^2.(C_2\\times Q_8)', 'supersolvable': False, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '6718464.bbw', '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': 1, 'diagramx': None, 'generators': [376703666980290, 20962878091100235933258515839342491697797, 63822384498269430727129924222762399012290, 21210673760395, 3243086199688471728464077307, 418829649741950392737872884, 233429858751468248293285255275731225592679, 31348116926917302048766235135238543047049, 10628652128250503819036102992460278548480, 822228046644134973649184189, 117636672851497474501150158352061313164001, 276480, 1845584622685684, 219574, 1258350440040947635195968000, 72654371322884864975018568834369420981909], 'label': '6718464.bbw.4._.J', 'mobius_quo': None, 'mobius_sub': None, 'normal_closure': None, 'normal_contained_in': None, 'normal_contains': None, 'normalizer': None, 'old_label': '4.J', 'projective_image': None, 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '4._.J', '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': '16.14', 'all_subgroups_known': False, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 4, 'aut_exponent': 48, 'aut_gen_orders': None, 'aut_gens': [[158351696908009581004324625581167689587782, 298471379949881024792106271419790166683094, 148314035861439568603777610764092181596213, 106998564724047187817468140699014846982324], [361363604906013725856734772214592095620450, 96361229022076646077260760082268086475132, 339204058184572927506301424560324637704841, 360148212314720507440235709098121969408518], [288432421003384534787421280021311141541523, 157753328640016847822207269598084167658000, 299339457343109447503200830364987762816518, 330320744948963290730610877858711189938678], [340645759988878309107579211539381457799766, 117635357188164933461354705220427643216766, 328880646041470671315987414916049664596074, 308509056663442259460358048546025148326820], [370499269349294268515175188001170921877620, 128524508853512567651874751435695687857269, 288432421003385294809737266595163123702749, 340081863169407567137341519107772648542556], [147445210994797250246422664707011536640582, 308509056663441157545122879793600320428455, 137407789453876649265026297194627348957566, 168390664341987809470895621831575681718544]], 'aut_group': None, 'aut_hash': 374530317592635896, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 53747712, 'aut_permdeg': 46980, 'aut_perms': None, 'aut_phi_ratio': 96.0, 'aut_solvable': True, 'aut_stats': None, 'aut_supersolvable': False, 'aut_tex': 'C_3^8.C_2^2.C_4.C_2^5.C_2^4', 'autcent_abelian': True, 'autcent_cyclic': True, 'autcent_exponent': 1, 'autcent_group': None, 'autcent_hash': 1, 'autcent_nilpotent': True, 'autcent_order': 1, 'autcent_solvable': True, 'autcent_split': None, 'autcent_supersolvable': True, 'autcent_tex': 'C_1', 'autcentquo_abelian': None, 'autcentquo_cyclic': None, 'autcentquo_exponent': None, 'autcentquo_group': None, 'autcentquo_hash': None, 'autcentquo_nilpotent': None, 'autcentquo_order': 53747712, 'autcentquo_solvable': None, 'autcentquo_supersolvable': None, 'autcentquo_tex': None, 'cc_stats': [[1, 1, 1], [2, 162, 3], [2, 324, 2], [2, 648, 1], [2, 6561, 1], [3, 32, 3], [3, 64, 5], [3, 128, 8], [3, 256, 20], [4, 324, 1], [4, 2916, 2], [4, 6561, 2], [4, 11664, 4], [4, 13122, 3], [4, 26244, 3], [4, 52488, 1], [6, 2592, 5], [6, 5184, 9], [6, 10368, 8], [8, 13122, 4], [8, 26244, 6], [8, 52488, 12], [12, 2592, 2], [12, 5184, 4], [12, 23328, 2], [12, 46656, 8]], 'center_label': '1.1', 'center_order': 1, 'central_product': False, 'central_quotient': '1679616.sz', 'commutator_count': 1, 'commutator_label': '104976.bq', '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', '3.1', '3.1', '3.1', '3.1', '3.1', '3.1', '3.1', '3.1'], 'composition_length': 16, 'conjugacy_classes_known': True, 'counter': 494, 'cyclic': False, 'derived_length': 3, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 162, 1, 3], [2, 324, 1, 2], [2, 648, 1, 1], [2, 6561, 1, 1], [3, 32, 1, 3], [3, 64, 1, 5], [3, 128, 1, 8], [3, 256, 1, 20], [4, 324, 1, 1], [4, 2916, 2, 1], [4, 6561, 2, 1], [4, 11664, 1, 4], [4, 13122, 1, 3], [4, 26244, 1, 3], [4, 52488, 1, 1], [6, 2592, 1, 5], [6, 5184, 1, 9], [6, 10368, 1, 8], [8, 13122, 2, 2], [8, 26244, 1, 2], [8, 26244, 4, 1], [8, 52488, 1, 2], [8, 52488, 2, 5], [12, 2592, 2, 1], [12, 5184, 1, 2], [12, 5184, 2, 1], [12, 23328, 2, 1], [12, 46656, 2, 4]], 'element_repr_type': 'Perm', 'elementary': 1, 'eulerian_function': None, 'exponent': 24, 'exponents_of_order': [8, 8], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [[32, 0, 4], [32, 1, 12], [64, 0, 10], [64, 1, 12], [128, 1, 16], [256, 1, 20]], 'familial': False, 'frattini_label': '1.1', 'frattini_quotient': '1679616.sz', 'hash': 4978646181862450805, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 24, 'inner_gen_orders': [12, 4, 12, 8], 'inner_gens': [[158351696908009581004324625581167689587782, 370499269349295866792452531838197629664425, 96361229022075392097188449983352373056420, 97229848583613555634510714502531743655634], [149174219303037408270045292452205497754375, 298471379949881024792106271419790166683094, 158621948201554567063597113785138910073743, 97229848583616072173228145801563282195154], [139135515026116868460028960667815062063520, 349536935318946711953187158138623808627622, 148314035861439568603777610764092181596213, 96031262422360917842757415321668553052129], [137693837351286543784848691567200586163446, 330320744948962128731963155730539632556299, 148045125155848819059440985300860477456766, 106998564724047187817468140699014846982324]], 'inner_hash': 4978646181862450805, 'inner_nilpotent': False, 'inner_order': 1679616, 'inner_split': None, 'inner_tex': 'C_3^8:C_4^2.(C_2\\times Q_8)', 'inner_used': [1, 2, 3, 4], 'irrC_degree': 32, 'irrQ_degree': 32, 'irrQ_dim': None, 'irrR_degree': None, 'irrep_stats': [[1, 16], [2, 20], [4, 10], [32, 16], [64, 22], [128, 16], [256, 20]], 'label': '1679616.sz', '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': 'C3^8:C4^2.(C2*Q8)', 'ngens': 4, '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, 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], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': None, 'number_characteristic_subgroups': 42, 'number_conjugacy_classes': 120, 'number_divisions': 101, 'number_normal_subgroups': 118, 'number_subgroup_autclasses': None, 'number_subgroup_classes': None, 'number_subgroups': None, 'old_label': None, 'order': 1679616, 'order_factorization_type': 33, 'order_stats': [[1, 1], [2, 8343], [3, 6560], [4, 236520], [6, 142560], [8, 839808], [12, 445824]], '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': None, 'outer_hash': None, 'outer_nilpotent': None, 'outer_order': 32, 'outer_permdeg': None, 'outer_perms': None, 'outer_solvable': None, 'outer_supersolvable': None, 'outer_tex': None, 'pc_rank': None, 'perfect': False, 'permutation_degree': 36, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 2, 2], 'quasisimple': False, 'rank': 4, 'rational': False, 'rational_characters_known': False, 'ratrep_stats': [[1, 16], [2, 8], [4, 8], [16, 2], [32, 12], [64, 14], [128, 21], [256, 20]], 'representations': {'PC': {'code': '7087293407565244696681943982537238695972684360502118518959994174810784967408811437468051323741032785767313426576465036201848881435905940700494245568780233774617628693023806056051835635379858564608917753734480417170809744130293595573575290987214375955242464502941432009043535358870837857722601197645859830900332219176239179599157341787212427700937781324756321011329879984220074415294220942832776756476292676790213679010556114154726823388489080498252185510066693351355040291410825861717514045732478717667561266086274493148936265408593165594575027978973935333976777561446663246258484176830901994905019722721761797791818067722350261549596927', 'gens': [1, 2, 4, 6, 7, 9, 10, 11, 12, 13, 14, 15, 16], 'pres': [16, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 6755584, 42977857, 81, 33140930, 18737698, 70165251, 47223571, 22692003, 179, 22600964, 39573140, 3481956, 726468, 155349893, 53313045, 8037157, 7599029, 6956805, 3037909, 115340294, 9400854, 25140006, 11227830, 11666438, 3338358, 326, 177586183, 20373527, 2191399, 24365111, 1164359, 141911, 67424264, 61470744, 3732520, 15206456, 1306440, 657880, 328424, 696, 183296009, 368665, 33740841, 2089017, 16663113, 4016729, 2108905, 2041, 148729866, 146634778, 50305066, 5598266, 12576330, 5537754, 1667178, 6458, 226689035, 32759835, 27476011, 17061947, 6706251, 3704923, 2566763, 20859, 214642700, 171245596, 57960492, 4412988, 13125708, 8459868, 4248300, 67516, 14737421, 106975261, 62404653, 43258941, 14128205, 2093149, 1649645, 217853, 25866254, 63252510, 70087726, 44313662, 16412238, 9267934, 4389230, 699966, 142671887, 5013535, 30474287, 3235903, 8261711, 11278431, 1088623, 2239615]}, 'Perm': {'d': 36, 'gens': [298471379949881024792106271419790166683094, 106998564724047187817468140699014846982324, 148314035861439568603777610764092181596213, 158351696908009581004324625581167689587782]}}, 'schur_multiplier': None, 'semidirect_product': None, 'simple': False, 'smith_abelian_invariants': [2, 2, 2, 2], 'solvability_type': 17, 'solvable': True, 'subgroup_inclusions_known': False, 'subgroup_index_bound': 8, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': 'C_3^8:C_4^2.(C_2\\times Q_8)', 'transitive_degree': 36, 'wreath_data': None, 'wreath_product': None}
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gps_groups • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '16.14', 'all_subgroups_known': False, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 4, 'aut_exponent': 48, 'aut_gen_orders': None, 'aut_gens': [[117635899382695670896743841408361219088290, 41385768963436925426803053031375283665226, 21300737289660421092597957053252033099363, 31348116926917302048766235075067323693100], [169892903206436213370601759576862119908620, 33076072994754930527653441167648919823734, 1806928228556912663876426171367267098870, 62616217410280194855565130159644374413438], [370768698898620761773006499343523454790280, 255294427298622306276351395518448527984583, 21587780663367327018626985470200645240600, 938612381543215421184464761842058913270], [340080793413150880529555976021668213227305, 83846140135119008347003917706235027602590, 51710233877449115254444731098873749353609, 10976774490765740574439616348326811189600], [330034746123935881197576249957111636960856, 21587287036339589924225373051551893147916, 62329125761160580019421756689988778468995, 63197481925999922834395242152606643802030], [147445704886478166406124408623764943276241, 21301239404793886179848323231248847617499, 32789523248073306375852531137676299904638, 51710233877447101999417293488623229393562]], 'aut_group': '53747712.ks', 'aut_hash': 5836576575400874840, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 53747712, 'aut_permdeg': 95904, 'aut_perms': None, 'aut_phi_ratio': 24.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 162, 1, 1], [2, 324, 1, 1], [2, 648, 2, 1], [2, 1296, 1, 1], [2, 2592, 1, 1], [2, 6561, 1, 1], [3, 32, 1, 1], [3, 64, 1, 2], [3, 128, 2, 1], [3, 512, 2, 4], [3, 1024, 1, 2], [4, 324, 1, 1], [4, 648, 1, 1], [4, 5832, 1, 1], [4, 5832, 2, 1], [4, 13122, 1, 2], [4, 23328, 1, 2], [4, 23328, 4, 1], [4, 26244, 1, 2], [4, 52488, 1, 1], [4, 52488, 2, 1], [4, 104976, 1, 1], [4, 209952, 1, 1], [6, 2592, 1, 1], [6, 5184, 1, 3], [6, 10368, 1, 2], [6, 10368, 2, 2], [6, 20736, 1, 2], [6, 41472, 2, 4], [8, 13122, 1, 2], [8, 26244, 1, 3], [8, 26244, 2, 1], [8, 46656, 1, 1], [8, 52488, 1, 5], [8, 104976, 2, 1], [8, 104976, 4, 2], [8, 209952, 1, 1], [8, 419904, 1, 1], [12, 2592, 2, 1], [12, 5184, 2, 1], [12, 10368, 1, 2], [12, 20736, 2, 1], [12, 46656, 1, 1], [12, 46656, 2, 2], [12, 93312, 1, 1], [12, 93312, 2, 1], [12, 186624, 4, 1], [16, 209952, 8, 1], [24, 93312, 2, 2]], 'aut_supersolvable': False, 'aut_tex': 'C_3^8:C_4^2.D_4^2:D_4', 'autcent_abelian': True, 'autcent_cyclic': True, 'autcent_exponent': 1, 'autcent_group': None, 'autcent_hash': 1, 'autcent_nilpotent': True, 'autcent_order': 1, 'autcent_solvable': True, 'autcent_split': None, 'autcent_supersolvable': True, 'autcent_tex': 'C_1', 'autcentquo_abelian': None, 'autcentquo_cyclic': None, 'autcentquo_exponent': None, 'autcentquo_group': None, 'autcentquo_hash': None, 'autcentquo_nilpotent': None, 'autcentquo_order': 53747712, 'autcentquo_solvable': None, 'autcentquo_supersolvable': None, 'autcentquo_tex': None, 'cc_stats': [[1, 1, 1], [2, 162, 1], [2, 324, 1], [2, 648, 2], [2, 1296, 1], [2, 2592, 1], [2, 6561, 1], [3, 32, 1], [3, 64, 2], [3, 128, 2], [3, 512, 8], [3, 1024, 2], [4, 324, 1], [4, 648, 1], [4, 5832, 3], [4, 13122, 2], [4, 23328, 6], [4, 26244, 2], [4, 52488, 3], [4, 104976, 1], [4, 209952, 1], [6, 2592, 1], [6, 5184, 3], [6, 10368, 6], [6, 20736, 2], [6, 41472, 8], [8, 13122, 2], [8, 26244, 5], [8, 46656, 1], [8, 52488, 5], [8, 104976, 10], [8, 209952, 1], [8, 419904, 1], [12, 2592, 2], [12, 5184, 2], [12, 10368, 2], [12, 20736, 2], [12, 46656, 5], [12, 93312, 3], [12, 186624, 4], [16, 209952, 8], [24, 93312, 4]], 'center_label': '1.1', 'center_order': 1, 'central_product': None, 'central_quotient': '6718464.bbw', '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', '3.1', '3.1', '3.1', '3.1', '3.1', '3.1', '3.1', '3.1'], 'composition_length': 18, 'conjugacy_classes_known': True, 'counter': 725, 'cyclic': False, 'derived_length': 4, 'dihedral': False, 'direct_factorization': None, 'direct_product': None, 'div_stats': [[1, 1, 1, 1], [2, 162, 1, 1], [2, 324, 1, 1], [2, 648, 1, 2], [2, 1296, 1, 1], [2, 2592, 1, 1], [2, 6561, 1, 1], [3, 32, 1, 1], [3, 64, 1, 2], [3, 128, 1, 2], [3, 512, 1, 8], [3, 1024, 1, 2], [4, 324, 1, 1], [4, 648, 1, 1], [4, 5832, 1, 1], [4, 5832, 2, 1], [4, 13122, 1, 2], [4, 23328, 1, 6], [4, 26244, 1, 2], [4, 52488, 1, 3], [4, 104976, 1, 1], [4, 209952, 1, 1], [6, 2592, 1, 1], [6, 5184, 1, 3], [6, 10368, 1, 6], [6, 20736, 1, 2], [6, 41472, 1, 8], [8, 13122, 2, 1], [8, 26244, 1, 1], [8, 26244, 2, 2], [8, 46656, 1, 1], [8, 52488, 1, 3], [8, 52488, 2, 1], [8, 104976, 1, 2], [8, 104976, 2, 4], [8, 209952, 1, 1], [8, 419904, 1, 1], [12, 2592, 2, 1], [12, 5184, 2, 1], [12, 10368, 1, 2], [12, 20736, 1, 2], [12, 46656, 1, 1], [12, 46656, 2, 2], [12, 93312, 1, 3], [12, 186624, 1, 4], [16, 209952, 2, 4], [24, 93312, 2, 2]], 'element_repr_type': 'Perm', 'elementary': 1, 'eulerian_function': None, 'exponent': 48, 'exponents_of_order': [10, 8], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [[32, 0, 4], [32, 1, 4], [64, 0, 4], [64, 1, 14], [128, 0, 2], [128, 1, 11], [256, -1, 2], [512, 1, 16], [1024, 1, 2]], 'familial': False, 'frattini_label': '1.1', 'frattini_quotient': '6718464.bbw', 'hash': 5091201899506233364, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 48, 'inner_gen_orders': [16, 12, 8, 8], 'inner_gens': [[117635899382695670896743841408361219088290, 255294427298623036621620451368385223295975, 22169045179087132010619946365936015277272, 1225663712837555534467017287866140439249], [299625496460338886717042557233517717060905, 41385768963436925426803053031375283665226, 63197481925999115733209025763754005925479, 938612381545276502442993693455504364284], [168676448551228310263029154788026839690053, 31347064686513514854967107113822395925729, 21300737289660421092597957053252033099363, 22169045179089100051591676890641300024195], [117037515447711747086613086559388880116716, 10976535771738039642479542172144005703634, 32789523248072052395780221099879311091838, 31348116926917302048766235075067323693100]], 'inner_hash': 5091201899506233364, 'inner_nilpotent': False, 'inner_order': 6718464, 'inner_split': None, 'inner_tex': 'C_3^8:C_4^2.Q_{16}:C_2^2', 'inner_used': [1, 2, 3, 4], 'irrC_degree': 32, 'irrQ_degree': 32, 'irrQ_dim': None, 'irrR_degree': None, 'irrep_stats': [[1, 16], [2, 20], [4, 14], [8, 11], [32, 8], [64, 18], [128, 13], [256, 2], [512, 16], [1024, 2]], 'label': '6718464.bbw', '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': 'C3^8:C4^2.Q16:C2^2', 'ngens': 4, 'nilpotency_class': -1, 'nilpotent': False, 'normal_counts': [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, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, 5, 0, 0, 0, 0, 9, 0, 0, 0, 15, 0, 0, 0, 23, 0, 0, 27, 0, 35, 0, 15, 1], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 77, 'number_characteristic_subgroups': 58, 'number_conjugacy_classes': 120, 'number_divisions': 101, 'number_normal_subgroups': 136, 'number_subgroup_autclasses': None, 'number_subgroup_classes': None, 'number_subgroups': None, 'old_label': None, 'order': 6718464, 'order_factorization_type': 33, 'order_stats': [[1, 1], [2, 12231], [3, 6560], [4, 709560], [6, 453600], [8, 2146176], [12, 1337472], [16, 1679616], [24, 373248]], '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': '8.5', 'outer_hash': None, 'outer_nilpotent': None, 'outer_order': 8, 'outer_permdeg': None, 'outer_perms': None, 'outer_solvable': None, 'outer_supersolvable': None, 'outer_tex': 'C_2^3', 'pc_rank': None, 'perfect': False, 'permutation_degree': 36, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 2, 2], 'quasisimple': False, 'rank': 4, 'rational': False, 'rational_characters_known': False, 'ratrep_stats': [[1, 16], [2, 12], [4, 10], [8, 7], [16, 2], [32, 5], [64, 16], [128, 11], [256, 4], [512, 16], [1024, 2]], 'representations': {'PC': {'code': '363684519855727186920687131171428705088377058962746720762658457505216907640549625314287683516749894447815515203264858343203604054585081116220949421437420309903985745761588086527276656184666742549549902742840030839753598064881874023983269548275928658539969708281925389633644572181449600806249825479194255863335308891119965644854826612694415589604934276298203491404110273754961924263332841590487900429119562376739454870061301804306954264928153233002206799168064285705026442920992863814334812619732477580715883352333298389103733680721104924745589221675995376370760153404544509399842291244247520978930721860787955820609749662103765701081465918583819247010076617436340114807002148179034642519055764751420916587443613480865308782983772222986700924575587106546275332730181117290221225449747915372183469925947708209139019716285136300241716852568274322606574705574698768169289676281768221923559079286748941062504703', 'gens': [1, 2, 5, 6, 9, 12, 13, 14, 15, 16, 17, 18], 'pres': [18, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 71748864, 102138841, 91, 311462282, 146, 206448195, 40689849, 203041444, 287318182, 102220240, 64258798, 32619676, 518216837, 324278231, 139130825, 82474475, 8185397, 311, 634886790, 217635288, 55389642, 45127212, 35797686, 366, 768264199, 380224537, 212009515, 5402941, 13604047, 836946440, 309707738, 34494380, 14624126, 14595632, 26989298, 14692220, 6860186, 476, 628439049, 592646427, 189457965, 119174463, 38759121, 2070819, 15671637, 716535, 531, 843651082, 627720220, 322679854, 105481792, 73713106, 26376868, 17132662, 3304, 1042219019, 117780509, 60162095, 144930881, 38679635, 27731045, 13817207, 10505, 5344859, 2676413, 1024358412, 311261214, 346664496, 185193282, 83161812, 27159078, 23149272, 33834, 3386604, 8598, 632733709, 400877599, 385394737, 192052291, 13692757, 8846311, 4564345, 109003, 1596829, 2236423, 281733134, 310625312, 409605170, 34352708, 35493206, 37178024, 3663482, 350060, 4179758, 1603976, 1369571343, 211451937, 426000435, 51425349, 50430039, 14183529, 10368123, 1119885, 5090847, 280113, 869216272, 600523810, 314049076, 201441094, 35819224, 20896234, 3314716, 3569326, 3161752, 1421242, 96878609, 183140387, 368603189, 953927, 95240537, 54483947, 21684797, 11337551, 7546769, 1423187]}, 'Perm': {'d': 36, 'gens': [21300737289660421092597957053252033099363, 31348116926917302048766235075067323693100, 41385768963436925426803053031375283665226, 117635899382695670896743841408361219088290]}}, 'schur_multiplier': [2, 2, 2, 2, 2], 'semidirect_product': None, 'simple': False, 'smith_abelian_invariants': [2, 2, 2, 2], 'solvability_type': 17, 'solvable': True, 'subgroup_inclusions_known': False, 'subgroup_index_bound': 24, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': 'C_3^8:C_4^2.Q_{16}:C_2^2', 'transitive_degree': 36, 'wreath_data': None, 'wreath_product': False}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': True, 'abelian_quotient': '4.2', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 6, 'aut_gen_orders': [2, 3], 'aut_gens': [[1, 2], [3, 2], [2, 3]], 'aut_group': '6.1', 'aut_hash': 1, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 6, 'aut_permdeg': 3, 'aut_perms': [1, 4], 'aut_phi_ratio': 3.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 3, 1]], 'aut_supersolvable': True, 'aut_tex': 'S_3', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 6, 'autcent_group': '6.1', 'autcent_hash': 1, 'autcent_nilpotent': False, 'autcent_order': 6, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'S_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], [2, 1, 3]], 'center_label': '4.2', 'center_order': 4, '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'], 'composition_length': 2, 'conjugacy_classes_known': True, 'counter': 2, 'cyclic': False, 'derived_length': 1, 'dihedral': True, 'direct_factorization': [['2.1', 2]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3]], 'element_repr_type': 'PC', 'elementary': 2, 'eulerian_function': 1, 'exponent': 2, 'exponents_of_order': [2], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2], 'faithful_reps': [], 'familial': True, 'frattini_label': '1.1', 'frattini_quotient': '4.2', 'hash': 2, '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, 4]], 'label': '4.2', 'linC_count': 3, 'linC_degree': 2, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 2, 'linQ_degree_count': 3, 'linQ_dim': 2, 'linQ_dim_count': 3, 'linR_count': 3, 'linR_degree': 2, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C2^2', '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': 4, 'number_divisions': 4, 'number_normal_subgroups': 5, 'number_subgroup_autclasses': 3, 'number_subgroup_classes': 5, 'number_subgroups': 5, 'old_label': None, 'order': 4, 'order_factorization_type': 2, 'order_stats': [[1, 1], [2, 3]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 6, 'outer_gen_orders': [2, 3], 'outer_gen_pows': [0, 0], 'outer_gens': [[3, 2], [2, 3]], 'outer_group': '6.1', 'outer_hash': 1, 'outer_nilpotent': False, 'outer_order': 6, 'outer_permdeg': 3, 'outer_perms': [1, 4], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'S_3', 'pc_rank': 2, 'perfect': False, 'permutation_degree': 4, 'pgroup': 2, 'primary_abelian_invariants': [2, 2], 'quasisimple': False, 'rank': 2, 'rational': True, 'rational_characters_known': True, 'ratrep_stats': [[1, 4]], 'representations': {'PC': {'code': 0, 'gens': [1, 2], 'pres': [2, -2, 2]}, 'GLZ': {'b': 3, 'd': 2, 'gens': [14, 12]}, 'GLFp': {'d': 2, 'p': 3, 'gens': [55, 56]}, 'Perm': {'d': 4, 'gens': [6, 1]}}, 'schur_multiplier': [2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2], 'solvability_type': 1, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_2^2', 'transitive_degree': 4, 'wreath_data': None, 'wreath_product': False}
