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gps_subgroup_search • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'ambient': '768.85027', 'ambient_counter': 85027, 'ambient_order': 768, 'ambient_tex': '(D_6\\times C_2^4):C_4', 'central': False, 'central_factor': False, 'centralizer_order': 48, 'characteristic': True, 'core_order': 96, 'counter': 36, 'cyclic': False, 'direct': False, 'hall': 0, 'label': '768.85027.8.i1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': None, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': True, 'old_label': '8.i1', 'outer_equivalence': True, 'perfect': False, 'proper': True, 'quotient': '8.3', 'quotient_Agroup': False, 'quotient_abelian': False, 'quotient_cyclic': False, 'quotient_hash': 3, 'quotient_metabelian': True, 'quotient_nilpotent': True, 'quotient_order': 8, 'quotient_simple': False, 'quotient_solvable': True, 'quotient_supersolvable': True, 'quotient_tex': 'D_4', 'simple': False, 'solvable': True, 'special_labels': [], 'split': False, 'standard_generators': False, 'stem': False, 'subgroup': '96.221', 'subgroup_hash': 221, 'subgroup_order': 96, 'subgroup_tex': 'C_{12}:C_2^3', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '768.85027', 'aut_centralizer_order': None, 'aut_label': '8.i1', 'aut_quo_index': None, 'aut_stab_index': None, 'aut_weyl_group': None, 'aut_weyl_index': None, 'centralizer': '16.a1', 'complements': [], 'conjugacy_class_count': 1, 'contained_in': ['4.a1', '4.n1', '4.o1'], 'contains': ['16.c1', '16.e1', '16.f1', '16.be1', '16.br1', '16.ck1', '24.g1'], 'core': '8.i1', 'coset_action_label': None, 'count': 1, 'diagramx': [7456, 6835, 2930, 5219], 'generators': [194163338880, 193198078344, 3, 288848891520, 1680, 25434259200], 'label': '768.85027.8.i1', 'mobius_quo': 0, 'mobius_sub': 0, 'normal_closure': '8.i1', 'normal_contained_in': ['4.a1'], 'normal_contains': ['16.c1', '16.e1', '16.f1', '24.g1'], 'normalizer': '1.a1', 'old_label': '8.i1', 'projective_image': None, 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '8.i1', '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': '48.52', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 4, 'aut_exponent': 24, 'aut_gen_orders': [4, 4, 8], 'aut_gens': [[1, 2, 4, 8], [50, 1, 78, 8], [49, 3, 53, 42], [49, 3, 28, 42]], 'aut_group': '6144.biy', 'aut_hash': 4968888535216498975, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 6144, 'aut_permdeg': 18, 'aut_perms': [5290307176470169, 2225254750929486, 5635656648599040], 'aut_phi_ratio': 192.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 1, 6, 1], [2, 2, 8, 1], [3, 1, 2, 1], [4, 2, 4, 1], [6, 1, 2, 1], [6, 1, 12, 1], [6, 2, 16, 1], [12, 2, 8, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_2^7.(C_2\\times S_4)', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 12, 'autcent_group': None, 'autcent_hash': 1930889827755753443, 'autcent_nilpotent': False, 'autcent_order': 3072, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': False, 'autcent_tex': 'C_2^4.C_2^4.D_6', 'autcentquo_abelian': True, 'autcentquo_cyclic': True, 'autcentquo_exponent': 2, 'autcentquo_group': '2.1', 'autcentquo_hash': 1, 'autcentquo_nilpotent': True, 'autcentquo_order': 2, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'C_2', 'cc_stats': [[1, 1, 1], [2, 1, 7], [2, 2, 8], [3, 1, 2], [4, 2, 4], [6, 1, 14], [6, 2, 16], [12, 2, 8]], 'center_label': '24.15', 'center_order': 24, 'central_product': True, 'central_quotient': '4.2', 'commutator_count': 1, 'commutator_label': '2.1', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '2.1', '3.1'], 'composition_length': 6, 'conjugacy_classes_known': True, 'counter': 221, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['2.1', 2], ['3.1', 1], ['8.3', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 7], [2, 2, 1, 8], [3, 1, 2, 1], [4, 2, 1, 4], [6, 1, 2, 7], [6, 2, 2, 8], [12, 2, 2, 4]], 'element_repr_type': 'PC', 'elementary': 2, 'eulerian_function': 4200, 'exponent': 12, 'exponents_of_order': [5, 1], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [], 'familial': False, 'frattini_label': '2.1', 'frattini_quotient': '48.52', 'hash': 221, 'hyperelementary': 2, 'inner_abelian': True, 'inner_cyclic': False, 'inner_exponent': 2, 'inner_gen_orders': [1, 1, 2, 2], 'inner_gens': [[1, 2, 4, 8], [1, 2, 4, 8], [1, 2, 4, 56], [1, 2, 52, 8]], 'inner_hash': 2, 'inner_nilpotent': True, 'inner_order': 4, 'inner_split': False, 'inner_tex': 'C_2^2', 'inner_used': [3, 4], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 48], [2, 12]], 'label': '96.221', 'linC_count': None, 'linC_degree': None, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 5, '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': 'C12:C2^3', 'ngens': 6, 'nilpotency_class': 2, 'nilpotent': True, 'normal_counts': [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': 10, 'number_characteristic_subgroups': 10, 'number_conjugacy_classes': 60, 'number_divisions': 40, 'number_normal_subgroups': 156, 'number_subgroup_autclasses': 36, 'number_subgroup_classes': 236, 'number_subgroups': 316, 'old_label': None, 'order': 96, 'order_factorization_type': 51, 'order_stats': [[1, 1], [2, 23], [3, 2], [4, 8], [6, 46], [12, 16]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': True, 'outer_exponent': 12, 'outer_gen_orders': [2, 3, 2, 2, 2, 2, 2, 2, 2, 2], 'outer_gen_pows': [0, 0, 0, 8, 0, 0, 0, 0, 0, 0], 'outer_gens': [[3, 2, 4, 8], [3, 1, 4, 8], [1, 2, 4, 40], [1, 2, 28, 8], [49, 50, 7, 8], [49, 2, 6, 8], [1, 50, 5, 9], [49, 2, 6, 10], [1, 2, 7, 8], [1, 2, 6, 8]], 'outer_group': '1536.408639927', 'outer_hash': 1736637719712388863, 'outer_nilpotent': False, 'outer_order': 1536, 'outer_permdeg': 14, 'outer_perms': [164465335, 160796341, 316, 6314117356, 314, 82, 1480636349, 1960645762, 40364311680, 20757824640], 'outer_solvable': True, 'outer_supersolvable': False, 'outer_tex': 'C_2^6:S_4', 'pc_rank': 4, 'perfect': False, 'permutation_degree': 11, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 2, 2, 3], 'quasisimple': False, 'rank': 4, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 16], [2, 20], [4, 4]], 'representations': {'PC': {'code': 2480545990145, 'gens': [1, 2, 3, 4], 'pres': [6, -2, -2, -2, -2, -2, -3, 351, 69, 88]}, 'GLZ': {'b': 3, 'd': 5, 'gens': [705943107446, 141602720900, 705686952884, 706461794101]}, 'GLFq': {'d': 4, 'q': 4, 'gens': [1079019009, 2149582850, 1186335877, 1087570945, 3256190595, 18712128]}, 'Perm': {'d': 11, 'gens': [766824, 3669120, 7985040, 2424, 4, 7983360]}}, 'schur_multiplier': [2, 2, 2, 2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 2, 6], 'solvability_type': 4, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_{12}:C_2^3', 'transitive_degree': 48, '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': '16.10', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 12, 'aut_gen_orders': [4, 2, 12, 12, 4, 12, 4, 4, 4], 'aut_gens': [[89214089064, 19200345361, 264, 193198078080, 1680, 2446577280, 25434259200, 288848891520, 3], [386713959264, 31661644442, 1560, 205662967440, 1680, 27390224400, 25434259200, 288848891520, 4], [562986951144, 282611710082, 288848891784, 205662965760, 1680, 268248516480, 25434259200, 288848891520, 4], [562986951144, 282611711642, 264, 205662967440, 1680, 268248516480, 25434259200, 288848891520, 3], [562986952824, 295076597765, 288848893080, 636677354880, 1680, 268248516480, 25434259200, 288848891520, 3], [437328503280, 31661642881, 1560, 649138654080, 1680, 287404590480, 25434259200, 288848891520, 4], [437818712544, 617477012882, 288848893080, 649138654080, 1680, 287404590480, 270160934400, 288848891520, 3], [388162576104, 642911271965, 25434259464, 205662967440, 288848893200, 268248516480, 270160934400, 288848891520, 4], [437818714104, 282611710081, 1560, 205662965760, 1680, 287404588800, 25434259200, 288848891520, 4], [388162576224, 199435259525, 25434259464, 636677354880, 288848893200, 268248518160, 270160934400, 288848891520, 4]], 'aut_group': None, 'aut_hash': 8414629624453799526, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 98304, 'aut_permdeg': 160, 'aut_perms': [22735495313422756785566932289311564585603988548295444015275992324990455068905715111633139962040248141103823467443166972017188582715656157681520296307448695689302660087397125939616213505894116431344112253382184908633238396633976496232777762478145896360657557884182101942442020347595394, 216926413695619706460179178162208769429102956976190035758559915334512358743348020725884823859670256305746662818703973226530439670202723344855041742528573074680701102166930533973024474286794869859961295168058563126983236935861087244367505183137699300784508680015227365855347872410675358, 249409221006867414976124348570691429689631777715024225363146986057447596104123633724196041458407987866547924272881727052578244348659222621489447501586758137721702875053442700830665462949355589470699588497878816644707081664563074510721334167042042970289095932097162140270928036255602776, 264077255786887331740496327063308329622434789262041499157754529217334677691668552171414938090492455258984025061698194375847093840718018028196851304150212659099874805233135655707318684343681052382119574278450505863190167086607211174147770024263424279478580780613495986987257841814267015, 74348443118235962086208189865163647524566946086899147933910464744719496467930884040072876283294597506985999230633304697323599657523558803228039010790768688821771629746817079209666359549254174650483306332954325741359334554270199590911495525628602892815667536020170157457966766784127735, 371415646161606021373888788920228544717360842555822872042822739163851463421044737164733146280935061128025158381315613509683678797593682667446465695081282628695141157527426520225592034948452608875803618191729116486915658096958806344471774517337924598272315599104067894277101857251640364, 5700159565474724312621457295525436914872892903677731744596452971944623435846155859439661917123908831451098021870900509690320119848072765471503949705372240394571903532274956559560635661620009877770285855819826404394204392312120884381466362695109583862767864919473912126273328421333581, 404198918248890996260403234510663267933514499504478585583721694036102616293493377651333969867933372882826494965751113249286239857504125805778834241621363957399609457303615759036824902784501555454694064850876035309161312048084354351526569255888525284552538831420814607536836535081500212, 368563939425543630867275124304626743457685085802994062057817623702714106704472747319741320242175874180028268912164631274472840450741684865302906469292752194070299553068079798553399219003432770991252955655693105205902913819092767355466880705806992366115861434807673619718591909288221497], 'aut_phi_ratio': 384.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 1, 2, 1], [2, 2, 1, 2], [2, 2, 4, 1], [2, 4, 2, 3], [2, 8, 1, 1], [2, 12, 8, 1], [3, 2, 1, 1], [4, 4, 2, 1], [4, 8, 1, 1], [4, 16, 4, 1], [4, 24, 4, 1], [6, 2, 1, 1], [6, 2, 2, 1], [6, 4, 1, 2], [6, 4, 4, 2], [6, 8, 2, 3], [8, 48, 4, 1], [12, 8, 2, 2], [12, 16, 8, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_3:(C_2^7.C_2^6.C_2^2)', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 2, 'autcent_group': '64.267', 'autcent_hash': 267, 'autcent_nilpotent': True, 'autcent_order': 64, 'autcent_solvable': True, 'autcent_split': False, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^6', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 12, 'autcentquo_group': '1536.408498090', 'autcentquo_hash': 631692073767964252, 'autcentquo_nilpotent': False, 'autcentquo_order': 1536, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'S_3\\times C_4^2:C_2^4', 'cc_stats': [[1, 1, 1], [2, 1, 3], [2, 2, 6], [2, 4, 6], [2, 8, 1], [2, 12, 8], [3, 2, 1], [4, 4, 2], [4, 8, 1], [4, 16, 4], [4, 24, 4], [6, 2, 3], [6, 4, 10], [6, 8, 6], [8, 48, 4], [12, 8, 4], [12, 16, 8]], 'center_label': '4.2', 'center_order': 4, 'central_product': False, 'central_quotient': '192.513', 'commutator_count': 1, 'commutator_label': '48.52', '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', '3.1'], 'composition_length': 9, 'conjugacy_classes_known': True, 'counter': 85027, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [2, 2, 1, 6], [2, 4, 1, 6], [2, 8, 1, 1], [2, 12, 1, 8], [3, 2, 1, 1], [4, 4, 1, 2], [4, 8, 1, 1], [4, 16, 2, 2], [4, 24, 1, 4], [6, 2, 1, 3], [6, 4, 1, 6], [6, 4, 2, 2], [6, 8, 1, 4], [6, 8, 2, 1], [8, 48, 2, 2], [12, 8, 1, 2], [12, 8, 2, 1], [12, 16, 4, 2]], 'element_repr_type': 'Perm', 'elementary': 1, 'eulerian_function': 1344, 'exponent': 24, 'exponents_of_order': [8, 1], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [], 'familial': False, 'frattini_label': '32.46', 'frattini_quotient': '24.14', 'hash': 85027, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 12, 'inner_gen_orders': [4, 2, 2, 2, 1, 2, 2, 1, 3], 'inner_gens': [[89214089064, 180747304081, 1560, 205662965760, 1680, 2446577280, 270160934400, 288848891520, 3], [362724728424, 19200345361, 264, 193198078080, 1680, 27390222720, 25434259200, 288848891520, 4], [89214090744, 19200345361, 264, 193198078080, 1680, 2446577280, 25434259200, 288848891520, 3], [437328501864, 19200345361, 264, 193198078080, 1680, 287404588800, 25434259200, 288848891520, 3], [89214089064, 19200345361, 264, 193198078080, 1680, 2446577280, 25434259200, 288848891520, 3], [89214089064, 31661644561, 264, 649138654080, 1680, 2446577280, 25434259200, 288848891520, 3], [456060327144, 19200345361, 264, 193198078080, 1680, 2446577280, 25434259200, 288848891520, 3], [89214089064, 19200345361, 264, 193198078080, 1680, 2446577280, 25434259200, 288848891520, 3], [89214089064, 19200345365, 264, 193198078080, 1680, 2446577280, 25434259200, 288848891520, 3]], 'inner_hash': 513, 'inner_nilpotent': False, 'inner_order': 192, 'inner_split': False, 'inner_tex': 'C_2^3.D_{12}', 'inner_used': [1, 2, 3, 9], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 16], [2, 28], [4, 24], [8, 4]], 'label': '768.85027', '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': '(D6*C2^4):C4', 'ngens': 9, '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], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 29, 'number_characteristic_subgroups': 49, 'number_conjugacy_classes': 72, 'number_divisions': 58, 'number_normal_subgroups': 123, 'number_subgroup_autclasses': 524, 'number_subgroup_classes': 2508, 'number_subgroups': 15156, 'old_label': None, 'order': 768, 'order_factorization_type': 51, 'order_stats': [[1, 1], [2, 143], [3, 2], [4, 176], [6, 94], [8, 192], [12, 160]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': True, 'outer_exponent': 4, 'outer_gen_orders': [2, 4, 4, 2, 2, 2, 2], 'outer_gen_pows': [0, 0, 0, 0, 0, 0, 25434260760], 'outer_gens': [[89214089064, 295076597761, 288848893080, 193198078080, 1680, 2446577280, 25434259200, 288848891520, 3], [89214089064, 295076597761, 270160935960, 193198078080, 288848893200, 2446577280, 25434259200, 288848891520, 3], [89214089064, 295076598025, 25434259464, 193198079760, 288848893200, 2446577280, 25434259200, 288848891520, 3], [88227378024, 19200345361, 264, 636677354880, 1680, 2446577280, 25434259200, 288848891520, 4], [89214089064, 295076599441, 264, 193198078080, 1680, 2446577280, 25434259200, 288848891520, 4], [89214089064, 19200343681, 288848891784, 193198078080, 1680, 2446577280, 25434259200, 288848891520, 4], [88227379440, 19200343681, 288848891784, 636677354880, 1680, 2446578960, 25434259200, 288848891520, 3]], 'outer_group': '512.10482065', 'outer_hash': 2448485374602736457, 'outer_nilpotent': True, 'outer_order': 512, 'outer_permdeg': 14, 'outer_perms': [14964049, 46663465087, 32732870094, 1, 126, 13905647406, 13905647695], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^6:D_4', 'pc_rank': 6, 'perfect': False, 'permutation_degree': 15, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 4], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 8], [2, 20], [4, 22], [8, 8]], 'representations': {'PC': {'code': '1290160561327314358889220094794645375193229314', 'gens': [1, 2, 3, 5, 7, 9], 'pres': [9, -2, -2, 2, -2, -2, -2, 2, -3, -2, 164, 74, 6682, 301, 130, 5639, 1536, 3561, 2814, 186, 1195]}, 'Perm': {'d': 15, 'gens': [89214089064, 19200345361, 264, 193198078080, 1680, 2446577280, 25434259200, 288848891520, 3]}}, 'schur_multiplier': [2, 2, 2, 2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 4], 'solvability_type': 7, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': '(D_6\\times C_2^4):C_4', 'transitive_degree': 48, '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': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 4, 'aut_gen_orders': [2, 4], 'aut_gens': [[1, 2], [1, 6], [3, 2]], 'aut_group': '8.3', 'aut_hash': 3, 'aut_nilpotency_class': 2, 'aut_nilpotent': True, 'aut_order': 8, 'aut_permdeg': 4, 'aut_perms': [5, 9], 'aut_phi_ratio': 2.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 2, 2, 1], [4, 2, 1, 1]], 'aut_supersolvable': True, 'aut_tex': 'D_4', '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': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^2', 'autcentquo_abelian': True, 'autcentquo_cyclic': True, 'autcentquo_exponent': 2, 'autcentquo_group': '2.1', 'autcentquo_hash': 1, 'autcentquo_nilpotent': True, 'autcentquo_order': 2, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'C_2', 'cc_stats': [[1, 1, 1], [2, 1, 1], [2, 2, 2], [4, 2, 1]], 'center_label': '2.1', 'center_order': 2, 'central_product': False, 'central_quotient': '4.2', 'commutator_count': 1, 'commutator_label': '2.1', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1'], 'composition_length': 3, 'conjugacy_classes_known': True, 'counter': 3, 'cyclic': False, 'derived_length': 2, 'dihedral': True, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 2, 1, 2], [4, 2, 1, 1]], 'element_repr_type': 'PC', 'elementary': 2, 'eulerian_function': 3, 'exponent': 4, 'exponents_of_order': [3], 'factors_of_aut_order': [2], 'factors_of_order': [2], 'faithful_reps': [[2, 1, 1]], 'familial': True, 'frattini_label': '2.1', 'frattini_quotient': '4.2', 'hash': 3, 'hyperelementary': 2, 'inner_abelian': True, 'inner_cyclic': False, 'inner_exponent': 2, 'inner_gen_orders': [2, 2], 'inner_gens': [[1, 6], [5, 2]], 'inner_hash': 2, 'inner_nilpotent': True, 'inner_order': 4, 'inner_split': True, 'inner_tex': 'C_2^2', 'inner_used': [1, 2], 'irrC_degree': 2, 'irrQ_degree': 2, 'irrQ_dim': 2, 'irrR_degree': 2, 'irrep_stats': [[1, 4], [2, 1]], 'label': '8.3', 'linC_count': 1, 'linC_degree': 2, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 2, 'linQ_degree_count': 1, 'linQ_dim': 2, 'linQ_dim_count': 1, 'linR_count': 1, 'linR_degree': 2, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'D4', 'ngens': 2, 'nilpotency_class': 2, 'nilpotent': True, 'normal_counts': [0, 0, 0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 4, 'number_characteristic_subgroups': 4, 'number_conjugacy_classes': 5, 'number_divisions': 5, 'number_normal_subgroups': 6, 'number_subgroup_autclasses': 6, 'number_subgroup_classes': 8, 'number_subgroups': 10, 'old_label': None, 'order': 8, 'order_factorization_type': 3, 'order_stats': [[1, 1], [2, 5], [4, 2]], 'outer_abelian': True, 'outer_cyclic': True, 'outer_equivalence': False, 'outer_exponent': 2, 'outer_gen_orders': [2], 'outer_gen_pows': [2], 'outer_gens': [[3, 2]], 'outer_group': '2.1', 'outer_hash': 1, 'outer_nilpotent': True, 'outer_order': 2, 'outer_permdeg': 2, 'outer_perms': [1], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2', '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], [2, 1]], 'representations': {'PC': {'code': 294, 'gens': [1, 2], 'pres': [3, -2, 2, -2, 37, 16]}, 'GLZ': {'b': 3, 'd': 2, 'gens': [14, 46]}, 'Lie': [{'d': 2, 'q': 3, 'gens': [29, 56, 24], 'family': 'COPlus'}, {'d': 1, 'q': 4, 'gens': [7, 16, 1], 'family': 'ASigmaL'}], 'GLFp': {'d': 2, 'p': 3, 'gens': [12, 55, 56]}, 'Perm': {'d': 4, 'gens': [6, 16, 7]}}, 'schur_multiplier': [2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2], 'solvability_type': 3, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'D_4', 'transitive_degree': 4, 'wreath_data': ['C_2', 'C_2', '2T1'], 'wreath_product': True}
