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gps_subgroup_search • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'ambient': '216000.d', 'ambient_counter': 4, 'ambient_order': 216000, 'ambient_tex': 'D_5^3:\\He_3.C_2^3', 'central': False, 'central_factor': False, 'centralizer_order': 1, 'characteristic': True, 'core_order': 108000, 'counter': 8, 'cyclic': False, 'direct': False, 'hall': 0, 'label': '216000.d.2.g1', 'maximal': True, 'maximal_normal': True, 'metabelian': False, 'metacyclic': False, 'minimal': False, 'minimal_normal': False, 'nilpotent': False, 'normal': True, 'old_label': '2.g1', 'outer_equivalence': True, 'perfect': False, 'proper': True, 'quotient': '2.1', 'quotient_Agroup': True, 'quotient_abelian': True, 'quotient_cyclic': True, 'quotient_hash': 1, 'quotient_metabelian': True, 'quotient_nilpotent': True, 'quotient_order': 2, 'quotient_simple': True, 'quotient_solvable': True, 'quotient_supersolvable': True, 'quotient_tex': 'C_2', 'simple': False, 'solvable': True, 'special_labels': [], 'split': True, 'standard_generators': False, 'stem': False, 'subgroup': '108000.p', 'subgroup_hash': 4258704782512683748, 'subgroup_order': 108000, 'subgroup_tex': 'D_5^3.C_3^2:D_6', 'supersolvable': False, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '216000.d', 'aut_centralizer_order': None, 'aut_label': '2.g1', 'aut_quo_index': None, 'aut_stab_index': None, 'aut_weyl_group': None, 'aut_weyl_index': None, 'centralizer': '216000.a1', 'complements': ['108000.E', '108000.F', '108000.G', '108000.H', '108000.C', '108000.D'], 'conjugacy_class_count': 1, 'contained_in': ['1.a1'], 'contains': ['4.e1', '4.g1', '4.i1', '6.t1', '6.v1', '8.n1'], 'core': '2.g1', 'coset_action_label': None, 'count': 1, 'diagramx': [1467, 3293, 1241, 3665], 'generators': [87264, 43200, 32400, 42, 2880, 80068, 189288, 113005, 72, 1488, 8640], 'label': '216000.d.2.g1', 'mobius_quo': 0, 'mobius_sub': None, 'normal_closure': '2.g1', 'normal_contained_in': ['1.a1'], 'normal_contains': ['4.e1', '4.g1', '4.i1'], 'normalizer': '1.a1', 'old_label': '2.g1', 'projective_image': '216000.d', 'quotient_action_image': '2.1', 'quotient_action_kernel': '1.1', 'quotient_action_kernel_order': 1, 'quotient_fusion': None, 'short_label': '2.g1', 'subgroup_fusion': None, 'weyl_group': '216000.d'}
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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': 4, 'aut_exponent': 60, 'aut_gen_orders': [12, 6, 12, 30, 10, 5], 'aut_gens': [[1, 2, 24, 144, 4320, 21600], [90577, 25114, 32424, 84528, 90720, 95040], [88665, 48170, 26472, 26328, 73440, 74304], [100945, 34442, 57912, 35568, 12960, 47520], [14257, 2186, 42360, 34704, 17280, 34560], [53569, 52778, 24, 15696, 4320, 21600], [2593, 59618, 2616, 144, 4320, 21600]], 'aut_group': '216000.d', 'aut_hash': 4125830444204071041, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 216000, 'aut_permdeg': 540, 'aut_perms': 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'aut_phi_ratio': 7.5, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 15, 1, 1], [2, 75, 1, 1], [2, 90, 1, 1], [2, 125, 1, 1], [2, 450, 1, 1], [3, 2, 1, 1], [3, 6, 1, 1], [3, 600, 1, 1], [3, 1200, 1, 1], [4, 450, 1, 3], [4, 1125, 1, 2], [4, 2250, 1, 3], [4, 3375, 1, 2], [5, 12, 1, 1], [5, 16, 1, 1], [5, 24, 1, 2], [5, 48, 1, 1], [6, 30, 1, 2], [6, 60, 1, 1], [6, 150, 1, 2], [6, 180, 1, 1], [6, 250, 1, 1], [6, 300, 1, 1], [6, 750, 1, 1], [6, 900, 1, 1], [6, 3000, 1, 1], [6, 6000, 1, 1], [10, 120, 1, 3], [10, 300, 1, 1], [10, 360, 1, 2], [10, 720, 1, 2], [10, 1800, 1, 1], [12, 900, 1, 3], [12, 4500, 1, 3], [12, 9000, 1, 2], [15, 12, 2, 1], [15, 16, 2, 1], [15, 24, 1, 1], [15, 24, 2, 3], [15, 48, 1, 2], [15, 48, 2, 3], [15, 96, 1, 2], [15, 96, 2, 1], [15, 2400, 1, 1], [15, 2400, 2, 1], [20, 1800, 1, 3], [30, 120, 2, 6], [30, 240, 1, 2], [30, 240, 2, 2], [30, 300, 2, 1], [30, 360, 2, 2], [30, 600, 1, 1], [30, 600, 2, 1], [30, 720, 2, 2], [30, 1800, 2, 1], [60, 1800, 2, 3]], 'aut_supersolvable': False, 'aut_tex': 'D_5^3:\\He_3.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': 60, 'autcentquo_group': '216000.d', 'autcentquo_hash': 4125830444204071041, 'autcentquo_nilpotent': False, 'autcentquo_order': 216000, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'D_5^3:\\He_3.C_2^3', 'cc_stats': [[1, 1, 1], [2, 15, 1], [2, 75, 1], [2, 90, 1], [2, 125, 1], [2, 450, 1], [3, 2, 1], [3, 6, 1], [3, 600, 1], [3, 1200, 1], [4, 450, 3], [4, 1125, 2], [4, 2250, 3], [4, 3375, 2], [5, 12, 1], [5, 16, 1], [5, 24, 2], [5, 48, 1], [6, 30, 2], [6, 60, 1], [6, 150, 2], [6, 180, 1], [6, 250, 1], [6, 300, 1], [6, 750, 1], [6, 900, 1], [6, 3000, 1], [6, 6000, 1], [10, 120, 3], [10, 300, 1], [10, 360, 2], [10, 720, 2], [10, 1800, 1], [12, 900, 3], [12, 4500, 3], [12, 9000, 2], [15, 12, 2], [15, 16, 2], [15, 24, 7], [15, 48, 8], [15, 96, 4], [15, 2400, 3], [20, 1800, 3], [30, 120, 12], [30, 240, 6], [30, 300, 2], [30, 360, 4], [30, 600, 3], [30, 720, 4], [30, 1800, 2], [60, 1800, 6]], 'center_label': '1.1', 'center_order': 1, 'central_product': False, 'central_quotient': '108000.p', 'commutator_count': 1, 'commutator_label': '13500.j', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '2.1', '3.1', '3.1', '3.1', '5.1', '5.1', '5.1'], 'composition_length': 11, 'conjugacy_classes_known': True, 'counter': 16, 'cyclic': False, 'derived_length': 4, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 15, 1, 1], [2, 75, 1, 1], [2, 90, 1, 1], [2, 125, 1, 1], [2, 450, 1, 1], [3, 2, 1, 1], [3, 6, 1, 1], [3, 600, 1, 1], [3, 1200, 1, 1], [4, 450, 1, 1], [4, 450, 2, 1], [4, 1125, 2, 1], [4, 2250, 1, 1], [4, 2250, 2, 1], [4, 3375, 2, 1], [5, 12, 1, 1], [5, 16, 1, 1], [5, 24, 1, 2], [5, 48, 1, 1], [6, 30, 1, 2], [6, 60, 1, 1], [6, 150, 1, 2], [6, 180, 1, 1], [6, 250, 1, 1], [6, 300, 1, 1], [6, 750, 1, 1], [6, 900, 1, 1], [6, 3000, 1, 1], [6, 6000, 1, 1], [10, 120, 1, 3], [10, 300, 1, 1], [10, 360, 1, 2], [10, 720, 1, 2], [10, 1800, 1, 1], [12, 900, 1, 1], [12, 900, 2, 1], [12, 4500, 1, 1], [12, 4500, 2, 1], [12, 9000, 2, 1], [15, 12, 2, 1], [15, 16, 2, 1], [15, 24, 1, 1], [15, 24, 2, 3], [15, 48, 1, 2], [15, 48, 2, 3], [15, 96, 1, 2], [15, 96, 2, 1], [15, 2400, 1, 1], [15, 2400, 2, 1], [20, 1800, 1, 1], [20, 1800, 2, 1], [30, 120, 2, 6], [30, 240, 1, 2], [30, 240, 2, 2], [30, 300, 2, 1], [30, 360, 2, 2], [30, 600, 1, 1], [30, 600, 2, 1], [30, 720, 2, 2], [30, 1800, 2, 1], [60, 1800, 2, 1], [60, 1800, 4, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 6696, 'exponent': 60, 'exponents_of_order': [5, 3, 3], 'factors_of_aut_order': [2, 3, 5], 'factors_of_order': [2, 3, 5], 'faithful_reps': [[12, 0, 8], [24, 0, 18], [48, -1, 1], [48, 0, 16], [48, 1, 1], [96, 0, 2]], 'familial': False, 'frattini_label': '3.1', 'frattini_quotient': '36000.n', 'hash': 4258704782512683748, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 60, 'inner_gen_orders': [4, 12, 6, 30, 5, 5], 'inner_gens': [[1, 79330, 35016, 86256, 56160, 43200], [95849, 2, 75648, 69864, 3456, 7776], [38449, 93002, 24, 58896, 4320, 95040], [47809, 25466, 75192, 144, 17280, 86400], [77761, 5186, 24, 8784, 4320, 21600], [86401, 39746, 56184, 43344, 4320, 21600]], 'inner_hash': 4258704782512683748, 'inner_nilpotent': False, 'inner_order': 108000, 'inner_split': True, 'inner_tex': 'D_5^3.C_3^2:D_6', 'inner_used': [1, 2], 'irrC_degree': 12, 'irrQ_degree': 24, 'irrQ_dim': 24, 'irrR_degree': 24, 'irrep_stats': [[1, 8], [2, 8], [3, 8], [4, 2], [6, 12], [12, 14], [16, 2], [24, 39], [32, 4], [48, 22], [96, 3]], 'label': '108000.p', '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': 'D5^3.C3^2:D6', 'ngens': 11, '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, 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': 94, 'number_characteristic_subgroups': 28, 'number_conjugacy_classes': 122, 'number_divisions': 85, 'number_normal_subgroups': 28, 'number_subgroup_autclasses': 1470, 'number_subgroup_classes': 1470, 'number_subgroups': 380384, 'old_label': None, 'order': 108000, 'order_factorization_type': 321, 'order_stats': [[1, 1], [2, 755], [3, 1808], [4, 17100], [5, 124], [6, 11800], [10, 4620], [12, 34200], [15, 8192], [20, 5400], [30, 13200], [60, 10800]], 'outer_abelian': True, 'outer_cyclic': True, 'outer_equivalence': True, 'outer_exponent': 2, 'outer_gen_orders': [2], 'outer_gen_pows': [81228], 'outer_gens': [[83377, 66962, 38904, 91728, 8640, 82080]], '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': 6, 'perfect': False, 'permutation_degree': 24, 'pgroup': 0, 'primary_abelian_invariants': [2, 4], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 6], [3, 4], [4, 4], [6, 12], [12, 7], [16, 2], [24, 13], [32, 2], [48, 19], [64, 1], [96, 10], [192, 1]], 'representations': {'PC': {'code': '9338506972852959753507913429740177953148245690545072543705763588401886144047912519833141814887387124558571325229456809341016903555590080143194254580513995495376039739281548828676512221938744999818413949196590119534167344255', 'gens': [1, 2, 5, 7, 10, 11], 'pres': [11, 2, 2, 2, 3, 2, 3, 2, 3, 5, 5, 5, 403920, 1745261, 56, 672278, 90, 77795, 1925884, 2080335, 386456, 698647, 158, 3184, 6641718, 2689781, 876442, 518172, 189008, 226, 354823, 4057170, 444605, 650536, 4275, 348, 342152, 2779939, 213870, 481181, 14308, 6177609, 190100, 2019631, 23802, 13275, 5227210, 470469, 2326136, 222199, 479214, 72676]}, 'Perm': {'d': 24, 'gens': [1129258880240916907469, 28210018194475458612661]}}, 'schur_multiplier': [2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 4], 'solvability_type': 11, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': 'D_5^3.C_3^2: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': '16.10', 'all_subgroups_known': False, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 4, 'aut_exponent': 60, 'aut_gen_orders': [12, 6, 12, 30, 10, 5], 'aut_gens': [[1, 2, 12, 144, 4320, 43200], [115537, 47458, 82788, 92664, 23328, 86400], [36177, 195098, 76812, 129024, 213912, 174528], [120409, 71306, 89004, 114192, 73440, 86400], [21673, 192026, 184044, 80496, 177120, 43200], [98497, 74738, 130476, 172944, 90720, 172800], [43201, 181442, 43212, 144, 133920, 43200]], 'aut_group': '216000.d', 'aut_hash': 4125830444204071041, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 216000, 'aut_permdeg': 224, 'aut_perms': [2536800249734081947349429693084845523971701346488732180273744419411012370631361347326554809586015725203813805714392371282277472413276103534041168619242545252618139665441710993258855810521148423052080862570069785698178863419678771350925371599051089226603884139002716047367414965261287467426321382616737278496530681335074217019745523367657978944110951389687873167404450448715251025248651767417260088782854109937684750620196567157606, 43675535342268978664089423895771121832907942680665041410162164882670625544397534159313930068908711924374631803310462303103375680028745553685772320284476893994298011337995967334900059575208547452326961761368807168964550038667204843614458461656358279380451687198829298463494598473488918875449226330082385224028825977920487454291431043230768309261890082199325562412239909430218068316746662497289325854315703278013694705009041666397335, 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38748285640019426263284475631115854741901219936959188781648286746931741045610633996250780755400865146595799990451444374693914951840823199524137067720659450047986952762005077037786600683280483730145276872785262842254357612396536387636506335916550919604864373090443587991813998450560860368580582807670735223833636384906122279287180775902085279694414111364107976130244111503206796437837291571783814497085339176109346338013034393882064, 32503100915024701308708298832234333341001964400143737477181440749000483856335537798954099254867371093598975665949556118754037831184169276212378989246843776411565314190367018519450366138390156698749253671224662844745999413684173787384369671017114156235116920374289794729464054333711564106303494227730403779477238938282731385364101460950966902658459196775681797220703144245154423443459797538131067402492233531789994913868067403941], 'aut_phi_ratio': 3.75, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 9, 1, 1], [2, 15, 1, 1], [2, 75, 1, 1], [2, 90, 1, 2], [2, 125, 1, 1], [2, 135, 1, 1], [2, 450, 1, 2], [2, 675, 1, 1], [2, 1125, 1, 1], [3, 2, 1, 1], [3, 6, 1, 1], [3, 600, 1, 1], [3, 1200, 1, 1], [4, 125, 1, 2], [4, 375, 1, 2], [4, 450, 1, 6], [4, 1125, 1, 2], [4, 2250, 1, 6], [4, 3375, 1, 2], [5, 12, 1, 1], [5, 16, 1, 1], [5, 24, 1, 2], [5, 48, 1, 1], [6, 30, 1, 2], [6, 60, 1, 1], [6, 150, 1, 2], [6, 180, 1, 2], [6, 250, 1, 1], [6, 300, 1, 1], [6, 750, 1, 1], [6, 900, 1, 2], [6, 1800, 1, 1], [6, 3000, 1, 1], [6, 6000, 1, 1], [6, 9000, 1, 1], [10, 108, 1, 1], [10, 120, 1, 3], [10, 144, 1, 1], [10, 216, 1, 2], [10, 300, 1, 1], [10, 360, 1, 4], [10, 432, 1, 1], [10, 720, 1, 4], [10, 1080, 1, 3], [10, 1800, 1, 2], [10, 2700, 1, 1], [12, 250, 1, 2], [12, 750, 1, 6], [12, 900, 1, 6], [12, 1500, 1, 2], [12, 3000, 1, 2], [12, 4500, 1, 6], [12, 6000, 1, 2], [12, 9000, 1, 2], [15, 24, 1, 2], [15, 32, 1, 1], [15, 48, 1, 5], [15, 96, 1, 5], [15, 192, 1, 1], [15, 2400, 1, 1], [15, 4800, 1, 1], [20, 1800, 1, 6], [30, 240, 1, 8], [30, 480, 1, 2], [30, 600, 1, 2], [30, 720, 1, 4], [30, 1200, 1, 1], [30, 1440, 1, 4], [30, 3600, 1, 2], [30, 7200, 1, 1], [60, 3600, 1, 6]], 'aut_supersolvable': False, 'aut_tex': 'D_5^3:\\He_3.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': 60, 'autcentquo_group': '216000.d', 'autcentquo_hash': 4125830444204071041, 'autcentquo_nilpotent': False, 'autcentquo_order': 216000, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'D_5^3:\\He_3.C_2^3', 'cc_stats': [[1, 1, 1], [2, 9, 1], [2, 15, 1], [2, 75, 1], [2, 90, 2], [2, 125, 1], [2, 135, 1], [2, 450, 2], [2, 675, 1], [2, 1125, 1], [3, 2, 1], [3, 6, 1], [3, 600, 1], [3, 1200, 1], [4, 125, 2], [4, 375, 2], [4, 450, 6], [4, 1125, 2], [4, 2250, 6], [4, 3375, 2], [5, 12, 1], [5, 16, 1], [5, 24, 2], [5, 48, 1], [6, 30, 2], [6, 60, 1], [6, 150, 2], [6, 180, 2], [6, 250, 1], [6, 300, 1], [6, 750, 1], [6, 900, 2], [6, 1800, 1], [6, 3000, 1], [6, 6000, 1], [6, 9000, 1], [10, 108, 1], [10, 120, 3], [10, 144, 1], [10, 216, 2], [10, 300, 1], [10, 360, 4], [10, 432, 1], [10, 720, 4], [10, 1080, 3], [10, 1800, 2], [10, 2700, 1], [12, 250, 2], [12, 750, 6], [12, 900, 6], [12, 1500, 2], [12, 3000, 2], [12, 4500, 6], [12, 6000, 2], [12, 9000, 2], [15, 24, 2], [15, 32, 1], [15, 48, 5], [15, 96, 5], [15, 192, 1], [15, 2400, 1], [15, 4800, 1], [20, 1800, 6], [30, 240, 8], [30, 480, 2], [30, 600, 2], [30, 720, 4], [30, 1200, 1], [30, 1440, 4], [30, 3600, 2], [30, 7200, 1], [60, 3600, 6]], 'center_label': '1.1', 'center_order': 1, 'central_product': False, 'central_quotient': '216000.d', 'commutator_count': 1, 'commutator_label': '13500.j', 'complements_known': True, 'complete': True, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '2.1', '2.1', '3.1', '3.1', '3.1', '5.1', '5.1', '5.1'], 'composition_length': 12, 'conjugacy_classes_known': True, 'counter': 4, 'cyclic': False, 'derived_length': 4, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 9, 1, 1], [2, 15, 1, 1], [2, 75, 1, 1], [2, 90, 1, 2], [2, 125, 1, 1], [2, 135, 1, 1], [2, 450, 1, 2], [2, 675, 1, 1], [2, 1125, 1, 1], [3, 2, 1, 1], [3, 6, 1, 1], [3, 600, 1, 1], [3, 1200, 1, 1], [4, 125, 2, 1], [4, 375, 2, 1], [4, 450, 1, 2], [4, 450, 2, 2], [4, 1125, 2, 1], [4, 2250, 1, 2], [4, 2250, 2, 2], [4, 3375, 2, 1], [5, 12, 1, 1], [5, 16, 1, 1], [5, 24, 1, 2], [5, 48, 1, 1], [6, 30, 1, 2], [6, 60, 1, 1], [6, 150, 1, 2], [6, 180, 1, 2], [6, 250, 1, 1], [6, 300, 1, 1], [6, 750, 1, 1], [6, 900, 1, 2], [6, 1800, 1, 1], [6, 3000, 1, 1], [6, 6000, 1, 1], [6, 9000, 1, 1], [10, 108, 1, 1], [10, 120, 1, 3], [10, 144, 1, 1], [10, 216, 1, 2], [10, 300, 1, 1], [10, 360, 1, 4], [10, 432, 1, 1], [10, 720, 1, 4], [10, 1080, 1, 3], [10, 1800, 1, 2], [10, 2700, 1, 1], [12, 250, 2, 1], [12, 750, 2, 3], [12, 900, 1, 2], [12, 900, 2, 2], [12, 1500, 2, 1], [12, 3000, 2, 1], [12, 4500, 1, 2], [12, 4500, 2, 2], [12, 6000, 2, 1], [12, 9000, 2, 1], [15, 24, 1, 2], [15, 32, 1, 1], [15, 48, 1, 5], [15, 96, 1, 5], [15, 192, 1, 1], [15, 2400, 1, 1], [15, 4800, 1, 1], [20, 1800, 1, 2], [20, 1800, 2, 2], [30, 240, 1, 8], [30, 480, 1, 2], [30, 600, 1, 2], [30, 720, 1, 4], [30, 1200, 1, 1], [30, 1440, 1, 4], [30, 3600, 1, 2], [30, 7200, 1, 1], [60, 3600, 1, 2], [60, 3600, 2, 2]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': None, 'exponent': 60, 'exponents_of_order': [6, 3, 3], 'factors_of_aut_order': [2, 3, 5], 'factors_of_order': [2, 3, 5], 'faithful_reps': [[24, 1, 4], [48, 0, 2], [48, 1, 11], [96, 1, 8], [192, 1, 1]], 'familial': False, 'frattini_label': '3.1', 'frattini_quotient': '72000.e', 'hash': 4125830444204071041, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 60, 'inner_gen_orders': [12, 6, 12, 30, 10, 5], 'inner_gens': [[1, 164338, 43044, 7128, 66528, 86400], [58013, 2, 196836, 154584, 130896, 8640], [126457, 68570, 12, 9648, 142560, 86400], [185833, 32330, 38028, 144, 82080, 43200], [71713, 136946, 120972, 181584, 4320, 172800], [172801, 77762, 172812, 144, 90720, 43200]], 'inner_hash': 4125830444204071041, 'inner_nilpotent': False, 'inner_order': 216000, 'inner_split': True, 'inner_tex': 'D_5^3:\\He_3.C_2^3', 'inner_used': [1, 2, 3], 'irrC_degree': 24, 'irrQ_degree': 24, 'irrQ_dim': 24, 'irrR_degree': 24, 'irrep_stats': [[1, 16], [2, 16], [3, 16], [4, 4], [6, 24], [12, 12], [16, 4], [24, 26], [32, 4], [48, 26], [64, 1], [96, 10], [192, 1]], 'label': '216000.d', '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': 'D5^3:He3.C2^3', 'ngens': 12, '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, 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': 160, 'number_characteristic_subgroups': 67, 'number_conjugacy_classes': 160, 'number_divisions': 136, 'number_normal_subgroups': 67, 'number_subgroup_autclasses': 5954, 'number_subgroup_classes': 5954, 'number_subgroups': 2002906, 'old_label': None, 'order': 216000, 'order_factorization_type': 321, 'order_stats': [[1, 1], [2, 3239], [3, 1808], [4, 26200], [5, 124], [6, 23680], [10, 15636], [12, 76400], [15, 8192], [20, 10800], [30, 28320], [60, 21600]], 'outer_abelian': True, 'outer_cyclic': True, 'outer_equivalence': True, 'outer_exponent': 1, 'outer_gen_orders': [], 'outer_gen_pows': [], 'outer_gens': [], 'outer_group': '1.1', 'outer_hash': 1, 'outer_nilpotent': True, 'outer_order': 1, 'outer_permdeg': 1, 'outer_perms': [], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_1', 'pc_rank': 6, 'perfect': False, 'permutation_degree': 24, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 4], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 8], [2, 12], [3, 8], [4, 6], [6, 16], [8, 1], [12, 16], [16, 4], [24, 23], [32, 4], [48, 24], [64, 1], [96, 12], [192, 1]], 'representations': {'PC': {'code': '187390058527099517228828497246061402247270299563914292561930924404806882088059101417156748621376375767390833450978923820149157895036193475439398836275089291316414802536032749852209063254781720113519314627114991047222745095608219416351890120427229377634325729885451767726120912716362730257609813284480163779023', 'gens': [1, 2, 4, 7, 10, 12], 'pres': [12, 2, 2, 3, 2, 2, 3, 2, 3, 5, 2, 5, 5, 2537568, 3944113, 61, 4919906, 2066115, 4724079, 903915, 135, 3134884, 64096, 2233828, 172, 110597, 55313, 26813, 598758, 6492546, 2643006, 67578, 130590, 246, 10229767, 8778259, 1893919, 16171, 4663, 379, 5598728, 6065300, 4059104, 15596, 15608, 7983369, 7853781, 6365553, 1425645, 1058457, 68481, 37893, 6705, 357, 17563402, 3022294, 1425634, 285166, 190138, 31762, 12441611, 622103, 1648547, 1036847, 1036859, 5879]}, 'Perm': {'d': 24, 'gens': [56524471011685912521276, 29481926098243258939281, 83539582165841850817661]}}, 'schur_multiplier': [2, 2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 4], 'solvability_type': 17, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 135, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': 'D_5^3:\\He_3.C_2^3', 'transitive_degree': 45, 'wreath_data': None, 'wreath_product': False}
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{'Agroup': True, 'Zgroup': True, 'abelian': True, 'abelian_quotient': '2.1', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': True, 'aut_cyclic': True, 'aut_derived_length': 0, 'aut_exponent': 1, 'aut_gen_orders': [], 'aut_gens': [[1]], 'aut_group': '1.1', 'aut_hash': 1, 'aut_nilpotency_class': 0, 'aut_nilpotent': True, 'aut_order': 1, 'aut_permdeg': 1, 'aut_perms': [], 'aut_phi_ratio': 1.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_1', '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': 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, 1]], 'center_label': '2.1', 'center_order': 2, 'central_product': False, 'central_quotient': '1.1', 'commutator_count': 0, 'commutator_label': '1.1', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1'], 'composition_length': 1, 'conjugacy_classes_known': True, 'counter': 1, 'cyclic': True, 'derived_length': 1, 'dihedral': True, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1]], 'element_repr_type': 'PC', 'elementary': 2, 'eulerian_function': 1, 'exponent': 2, 'exponents_of_order': [1], 'factors_of_aut_order': [], 'factors_of_order': [2], 'faithful_reps': [[1, 1, 1]], 'familial': True, 'frattini_label': '1.1', 'frattini_quotient': '2.1', 'hash': 1, 'hyperelementary': 2, 'inner_abelian': True, 'inner_cyclic': True, 'inner_exponent': 1, 'inner_gen_orders': [1], 'inner_gens': [[1]], '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, 2]], 'label': '2.1', 'linC_count': 1, 'linC_degree': 1, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 1, 'linQ_degree_count': 1, 'linQ_dim': 1, 'linQ_dim_count': 1, 'linR_count': 1, 'linR_degree': 1, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C2', 'ngens': 1, 'nilpotency_class': 1, 'nilpotent': True, 'normal_counts': [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': 2, 'number_divisions': 2, 'number_normal_subgroups': 2, 'number_subgroup_autclasses': 2, 'number_subgroup_classes': 2, 'number_subgroups': 2, 'old_label': None, 'order': 2, 'order_factorization_type': 1, 'order_stats': [[1, 1], [2, 1]], 'outer_abelian': True, 'outer_cyclic': True, 'outer_equivalence': False, 'outer_exponent': 1, 'outer_gen_orders': [], 'outer_gen_pows': [], 'outer_gens': [], 'outer_group': '1.1', 'outer_hash': 1, 'outer_nilpotent': True, 'outer_order': 1, 'outer_permdeg': 1, 'outer_perms': [], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_1', 'pc_rank': 1, 'perfect': False, 'permutation_degree': 2, 'pgroup': 2, 'primary_abelian_invariants': [2], 'quasisimple': False, 'rank': 1, 'rational': True, 'rational_characters_known': True, 'ratrep_stats': [[1, 2]], 'representations': {'PC': {'code': 0, 'gens': [1], 'pres': [1, -2]}, 'GLZ': {'b': 3, 'd': 2, 'gens': [12]}, 'Lie': [{'d': 2, 'q': 2, 'gens': [6], 'family': 'CSOPlus'}, {'d': 2, 'q': 2, 'gens': [6], 'family': 'COPlus'}, {'d': 1, 'q': 2, 'gens': [1], 'family': 'AGL'}, {'d': 1, 'q': 2, 'gens': [1], 'family': 'AGammaL'}, {'d': 1, 'q': 2, 'gens': [1], 'family': 'ASigmaL'}], 'GLFp': {'d': 2, 'p': 2, 'gens': [11]}, 'Perm': {'d': 2, 'gens': [1]}}, 'schur_multiplier': [], 'semidirect_product': False, 'simple': True, 'smith_abelian_invariants': [2], 'solvability_type': 0, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_2', 'transitive_degree': 2, 'wreath_data': None, 'wreath_product': False}
