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gps_subgroup_search • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'ambient': '768.1088764', 'ambient_counter': 1088764, 'ambient_order': 768, 'ambient_tex': 'D_4\\times \\GL(2,\\mathbb{Z}/4)', 'central': False, 'central_factor': False, 'centralizer_order': 32, 'characteristic': True, 'core_order': 128, 'counter': 41, 'cyclic': False, 'direct': False, 'hall': 0, 'label': '768.1088764.6.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': False, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': True, 'old_label': '6.a1', 'outer_equivalence': True, 'perfect': False, 'proper': True, 'quotient': '6.1', 'quotient_Agroup': True, 'quotient_abelian': False, 'quotient_cyclic': False, 'quotient_hash': 1, 'quotient_metabelian': True, 'quotient_nilpotent': False, 'quotient_order': 6, 'quotient_simple': False, 'quotient_solvable': True, 'quotient_supersolvable': True, 'quotient_tex': 'S_3', 'simple': False, 'solvable': True, 'special_labels': ['F'], 'split': True, 'standard_generators': False, 'stem': False, 'subgroup': '128.2320', 'subgroup_hash': 2320, 'subgroup_order': 128, 'subgroup_tex': 'D_4\\times C_2^4', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '768.1088764', 'aut_centralizer_order': None, 'aut_label': '6.a1', 'aut_quo_index': None, 'aut_stab_index': None, 'aut_weyl_group': None, 'aut_weyl_index': None, 'centralizer': '24.b1', 'complements': ['128.g1', '128.h1'], 'conjugacy_class_count': 1, 'contained_in': ['2.c1', '3.a1'], 'contains': ['12.a1', '12.b1', '12.c1', '12.d1', '12.e1', '12.j1', '12.l1', '12.r1', '12.bd1', '12.bz1', '12.cr1', '12.eb1', '12.ec1'], 'core': '6.a1', 'coset_action_label': None, 'count': 1, 'diagramx': None, 'generators': [1733, 2671, 12103, 12967, 6387, 12175], 'label': '768.1088764.6.a1', 'mobius_quo': None, 'mobius_sub': None, 'normal_closure': '6.a1', 'normal_contained_in': ['2.c1'], 'normal_contains': ['12.a1', '12.b1', '12.c1', '12.d1', '12.e1', '24.a1'], 'normalizer': '1.a1', 'old_label': '6.a1', 'projective_image': None, 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '6.a1', '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': '64.267', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 1, 'aut_exponent': 840, 'aut_gen_orders': [2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 4], 'aut_gens': [[1539, 1831, 2853, 2563, 517, 2577], [1539, 1831, 2853, 519, 517, 533], [1539, 1831, 2853, 2563, 517, 2609], [1539, 1831, 2853, 2563, 517, 1559], [1539, 1831, 2853, 2563, 517, 533], [1539, 1831, 2853, 1573, 517, 1559], [1539, 1831, 2853, 2563, 517, 3891], [1539, 1831, 2853, 2563, 517, 1811], [1539, 1831, 2853, 3841, 517, 3891], [1539, 1831, 2853, 1825, 517, 1811], [1539, 1831, 2853, 2595, 517, 2609], [2597, 2817, 2853, 2563, 3619, 2577], [1571, 1799, 2853, 2563, 549, 2577], [1539, 1831, 2853, 3857, 517, 2609], [1539, 1799, 2853, 2595, 549, 2609], [1539, 1831, 2821, 2595, 549, 2577], [1571, 1831, 2821, 2595, 517, 2577], [1799, 2593, 1571, 3877, 1575, 561]], 'aut_group': '660602880.c', 'aut_hash': 6486987916624719700, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 660602880, 'aut_permdeg': 64, 'aut_perms': [27925373857298700101304443730712399710262081586329416046897069550762755313462184528089, 12211491856806946221884584924742940091737652582463309599148861423245862258992091673178470, 102404794996782123462392660192805471798582839199543952216592397525412066182226691700754710, 96362309938133357311461207538011287094603394938956275173322526261380249628026555413317910, 29448114501063370286063432694762817170683277238760256181020153428336774521316010224477, 62123140004809474746001711732833183807195398201829491541034708759122265134456240876332310, 76819371923021977847853819281379189324319427512244019706727174515959797855698528318272870, 22849571707861473315251383310589309121758260113003601459229598462345508071516053941525, 25387472790756570442698189779798564609648887559509569832826795298431699380180623139288, 6607004854547298072657525309406596612894742363932387397586956458205005918839160562084, 101173954383185224791492223681705840951599146567975083463482658404353953612398184219527490, 11933737276424915160149104204098197388646906549707301386471973224774724653802135506499490, 104877534840351084087550402399443664552512767736392311681488215981366723819081653778874718, 321315232457699723798967278277648383805718663332350621628971324202704667895649551627083, 12217073546765387389355587546134055106884155128529197767968161204628173815138273534651296, 12216973584443448617451400429699823955311216099426642678105797861788471888490166908458810, 97148432559324839714280638310998106285933752039703756760231409630368716028141516107495780], 'aut_phi_ratio': 10321920.0, 'aut_solvable': False, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 1, 30, 1], [2, 2, 32, 1], [4, 2, 16, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_2^{10}.C_2^5.A_8', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 840, 'autcent_group': None, 'autcent_hash': 3030887972575308301, 'autcent_nilpotent': False, 'autcent_order': 330301440, 'autcent_solvable': False, 'autcent_split': True, 'autcent_supersolvable': False, 'autcent_tex': 'C_2^{10}.C_2^4.A_8', '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, 31], [2, 2, 32], [4, 2, 16]], 'center_label': '32.51', 'center_order': 32, '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', '2.1', '2.1'], 'composition_length': 7, 'conjugacy_classes_known': True, 'counter': 2320, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['2.1', 4], ['8.3', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 31], [2, 2, 1, 32], [4, 2, 1, 16]], 'element_repr_type': 'GLZq', 'elementary': 2, 'eulerian_function': 1953, 'exponent': 4, 'exponents_of_order': [7], 'factors_of_aut_order': [2, 3, 5, 7], 'factors_of_order': [2], 'faithful_reps': [], 'familial': False, 'frattini_label': '2.1', 'frattini_quotient': '64.267', 'hash': 2320, 'hyperelementary': 2, 'inner_abelian': True, 'inner_cyclic': False, 'inner_exponent': 2, 'inner_gen_orders': [1, 1, 1, 2, 1, 2], 'inner_gens': [[1539, 1831, 2853, 2563, 517, 2577], [1539, 1831, 2853, 2563, 517, 2577], [1539, 1831, 2853, 2563, 517, 2577], [1539, 1831, 2853, 2563, 517, 2609], [1539, 1831, 2853, 2563, 517, 2577], [1539, 1831, 2853, 2595, 517, 2577]], 'inner_hash': 2, 'inner_nilpotent': True, 'inner_order': 4, 'inner_split': False, 'inner_tex': 'C_2^2', 'inner_used': [4, 6], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 64], [2, 16]], 'label': '128.2320', 'linC_count': None, 'linC_degree': None, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 6, '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': 'D4*C2^4', 'ngens': 6, 'nilpotency_class': 2, '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': 5, 'number_characteristic_subgroups': 5, 'number_conjugacy_classes': 80, 'number_divisions': 80, 'number_normal_subgroups': 3132, 'number_subgroup_autclasses': 30, 'number_subgroup_classes': 5276, 'number_subgroups': 7420, 'old_label': None, 'order': 128, 'order_factorization_type': 7, 'order_stats': [[1, 1], [2, 95], [4, 32]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': True, 'outer_exponent': 840, 'outer_gen_orders': [5, 30], 'outer_gen_pows': [2577, 2577], 'outer_gens': [[801, 3587, 3847, 1573, 2853, 2577], [3619, 1795, 3847, 1557, 3843, 3863]], 'outer_group': None, 'outer_hash': 2127844866957932739, 'outer_nilpotent': False, 'outer_order': 165150720, 'outer_permdeg': 48, 'outer_perms': [1662063881944957846162354643155278210461523079996684890365802, 7638772802546096626716563913093369123194033338372308414709578], 'outer_solvable': False, 'outer_supersolvable': False, 'outer_tex': 'C_2^{12}.C_2.A_8', 'pc_rank': 6, 'perfect': False, 'permutation_degree': 12, 'pgroup': 2, 'primary_abelian_invariants': [2, 2, 2, 2, 2, 2], 'quasisimple': False, 'rank': 6, 'rational': True, 'rational_characters_known': True, 'ratrep_stats': [[1, 64], [2, 16]], 'representations': {'PC': {'code': 17318281248, 'gens': [1, 2, 3, 4, 5, 6], 'pres': [7, -2, 2, 2, 2, 2, 2, -2, 530, 124]}, 'GLZ': {'b': 3, 'd': 6, 'gens': [125101750005091132, 125101729084386182, 125101749995526650, 25038659807093174, 125101750005088214, 24992906222183252]}, 'GLZN': {'d': 2, 'p': 12, 'gens': [5601, 19949, 8645, 2665, 1733, 1801, 9587]}, 'GLZq': {'d': 2, 'q': 8, 'gens': [1831, 2853, 529, 1827, 545, 515, 1539]}, 'Perm': {'d': 12, 'gens': [7620607, 40284847, 87096366, 5328, 5167, 87096241, 87091200]}}, 'schur_multiplier': [2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 2, 2, 2, 2], 'solvability_type': 4, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'D_4\\times C_2^4', 'transitive_degree': 64, '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': '16.14', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 3, 'aut_exponent': 12, 'aut_gen_orders': [6, 4, 4, 6, 4, 6, 6, 2, 12], 'aut_gens': [[1733, 2671, 2707, 5922, 12967, 12175], [12107, 2671, 12601, 5394, 2665, 12967], [5811, 13033, 9155, 807, 12175, 12967], [6435, 13033, 17830, 1605, 12967, 2665], [19015, 13033, 2239, 6252, 2665, 12967], [16809, 13033, 9155, 1605, 12175, 12967], [5811, 2671, 19517, 16620, 2665, 12967], [1733, 19949, 13465, 6780, 2665, 12967], [1733, 2671, 2239, 10575, 12175, 12967], [6435, 9587, 12601, 5922, 2665, 12967]], 'aut_group': None, 'aut_hash': 1694722777932669743, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 12288, 'aut_permdeg': 36, 'aut_perms': [319779766416827653616641420083485802644640, 212980271589184922084851903942590410342543, 307145046823634039583972889993908647342283, 178810751301552218424035525181345488291387, 193434404152888823857033753329939122280285, 350519056774966131361470265833715782418932, 350684138360035321658046452568112556817000, 243295797354733915153967555162375513418097, 108127920886564277994955475515505991121905], 'aut_phi_ratio': 48.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [2, 2, 2, 1], [2, 2, 4, 1], [2, 3, 1, 4], [2, 4, 2, 1], [2, 6, 2, 1], [2, 6, 4, 1], [2, 12, 2, 2], [2, 24, 2, 1], [3, 8, 1, 1], [4, 2, 2, 1], [4, 4, 1, 1], [4, 6, 2, 1], [4, 12, 1, 1], [4, 12, 2, 3], [4, 24, 1, 4], [4, 24, 2, 3], [6, 8, 1, 3], [6, 8, 4, 1], [6, 16, 4, 2], [12, 16, 2, 2]], 'aut_supersolvable': False, 'aut_tex': 'A_4.C_2^6.C_2^4', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 2, 'autcent_group': '256.56092', 'autcent_hash': 56092, 'autcent_nilpotent': True, 'autcent_order': 256, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^8', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 12, 'autcentquo_group': '48.48', 'autcentquo_hash': 48, 'autcentquo_nilpotent': False, 'autcentquo_order': 48, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'C_2\\times S_4', 'cc_stats': [[1, 1, 1], [2, 1, 3], [2, 2, 6], [2, 3, 4], [2, 4, 2], [2, 6, 6], [2, 12, 4], [2, 24, 2], [3, 8, 1], [4, 2, 2], [4, 4, 1], [4, 6, 2], [4, 12, 7], [4, 24, 10], [6, 8, 7], [6, 16, 8], [12, 16, 4]], 'center_label': '4.2', 'center_order': 4, 'central_product': True, 'central_quotient': '192.1537', 'commutator_count': 1, 'commutator_label': '48.49', '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': 1088764, 'cyclic': False, 'derived_length': 3, 'dihedral': False, 'direct_factorization': [['8.3', 1], ['96.195', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [2, 2, 1, 6], [2, 3, 1, 4], [2, 4, 1, 2], [2, 6, 1, 6], [2, 12, 1, 4], [2, 24, 1, 2], [3, 8, 1, 1], [4, 2, 1, 2], [4, 4, 1, 1], [4, 6, 1, 2], [4, 12, 1, 7], [4, 24, 1, 10], [6, 8, 1, 3], [6, 8, 2, 2], [6, 16, 1, 4], [6, 16, 2, 2], [12, 16, 1, 2], [12, 16, 2, 1]], 'element_repr_type': 'GLZN', 'elementary': 1, 'eulerian_function': 8255520, 'exponent': 12, 'exponents_of_order': [8, 1], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [], 'familial': False, 'frattini_label': '4.2', 'frattini_quotient': '192.1537', 'hash': 1088764, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 12, 'inner_gen_orders': [2, 2, 2, 6, 2, 2], 'inner_gens': [[1733, 2671, 2707, 5394, 12967, 12175], [1733, 2671, 12997, 5922, 12967, 12175], [1733, 13033, 2707, 12039, 2665, 12175], [8641, 2671, 7456, 5922, 12175, 2665], [1733, 2671, 13069, 17148, 12967, 12175], [1733, 2671, 2707, 16356, 12967, 12175]], 'inner_hash': 1537, 'inner_nilpotent': False, 'inner_order': 192, 'inner_split': True, 'inner_tex': 'C_2^3\\times S_4', 'inner_used': [1, 2, 3, 4, 5], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 16], [2, 24], [3, 16], [4, 5], [6, 8], [12, 1]], 'label': '768.1088764', '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': 'D4*GL(2,Z/4)', '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': 39, 'number_characteristic_subgroups': 55, 'number_conjugacy_classes': 70, 'number_divisions': 65, 'number_normal_subgroups': 155, 'number_subgroup_autclasses': 1173, 'number_subgroup_classes': 3057, 'number_subgroups': 19814, 'old_label': None, 'order': 768, 'order_factorization_type': 51, 'order_stats': [[1, 1], [2, 167], [3, 8], [4, 344], [6, 184], [12, 64]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': True, 'outer_exponent': 4, 'outer_gen_orders': [2, 2, 2, 2, 4], 'outer_gen_pows': [1729, 1729, 1729, 1729, 1729], 'outer_gens': [[12107, 9587, 2707, 5922, 12967, 12175], [1733, 2671, 9623, 12039, 2665, 12175], [12107, 9587, 9623, 5922, 12967, 12175], [12107, 9587, 9623, 17226, 12967, 12175], [16185, 9587, 9623, 17226, 12967, 12175]], 'outer_group': '64.261', 'outer_hash': 261, 'outer_nilpotent': True, 'outer_order': 64, 'outer_permdeg': 10, 'outer_perms': [806400, 1174320, 127, 1174446, 368062], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'D_4\\times C_2^3', 'pc_rank': 6, 'perfect': False, 'permutation_degree': 12, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 2, 2], 'quasisimple': False, 'rank': 4, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 16], [2, 16], [3, 16], [4, 7], [6, 8], [8, 1], [12, 1]], 'representations': {'PC': {'code': '448255217457503097348806986207539605152256557068617541636', 'gens': [1, 2, 3, 5, 8, 9], 'pres': [9, -2, -2, -2, -2, -2, -2, -3, -2, 2, 173, 74, 5044, 922, 130, 2183, 158, 2040, 10393, 1771, 1348, 493, 2960, 539, 791]}, 'GLZN': {'d': 2, 'p': 12, 'gens': [5601, 19949, 1765, 8645, 18685, 2665, 1733, 1801, 9587]}, 'Perm': {'d': 12, 'gens': [1565, 5160, 40279680, 7632000, 87091200, 11520, 3, 7, 16]}}, 'schur_multiplier': [2, 2, 2, 2, 2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 2, 2], 'solvability_type': 11, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': 'D_4\\times \\GL(2,\\mathbb{Z}/4)', 'transitive_degree': 48, 'wreath_data': None, 'wreath_product': False}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': False, 'abelian_quotient': '2.1', '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, 4], [1, 3], [2, 4]], '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, 3, 1, 1], [3, 2, 1, 1]], 'aut_supersolvable': True, 'aut_tex': 'S_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': 6, 'autcentquo_group': '6.1', 'autcentquo_hash': 1, 'autcentquo_nilpotent': False, 'autcentquo_order': 6, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'S_3', 'cc_stats': [[1, 1, 1], [2, 3, 1], [3, 2, 1]], 'center_label': '1.1', 'center_order': 1, 'central_product': False, 'central_quotient': '6.1', 'commutator_count': 1, 'commutator_label': '3.1', 'complements_known': True, 'complete': True, 'complex_characters_known': True, 'composition_factors': ['2.1', '3.1'], 'composition_length': 2, 'conjugacy_classes_known': True, 'counter': 1, 'cyclic': False, 'derived_length': 2, 'dihedral': True, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 3, 1, 1], [3, 2, 1, 1]], 'element_repr_type': 'Perm', 'elementary': 1, 'eulerian_function': 3, 'exponent': 6, 'exponents_of_order': [1, 1], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [[2, 1, 1]], 'familial': True, 'frattini_label': '1.1', 'frattini_quotient': '6.1', 'hash': 1, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 6, 'inner_gen_orders': [2, 3], 'inner_gens': [[1, 3], [2, 4]], 'inner_hash': 1, 'inner_nilpotent': False, 'inner_order': 6, 'inner_split': True, 'inner_tex': 'S_3', 'inner_used': [1, 2], 'irrC_degree': 2, 'irrQ_degree': 2, 'irrQ_dim': 2, 'irrR_degree': 2, 'irrep_stats': [[1, 2], [2, 1]], 'label': '6.1', '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': 'S3', 'ngens': 2, 'nilpotency_class': -1, 'nilpotent': False, 'normal_counts': [0, 0, 0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 3, 'number_characteristic_subgroups': 3, 'number_conjugacy_classes': 3, 'number_divisions': 3, 'number_normal_subgroups': 3, 'number_subgroup_autclasses': 4, 'number_subgroup_classes': 4, 'number_subgroups': 6, 'old_label': None, 'order': 6, 'order_factorization_type': 11, 'order_stats': [[1, 1], [2, 3], [3, 2]], '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': 2, 'perfect': False, 'permutation_degree': 3, 'pgroup': 0, 'primary_abelian_invariants': [2], 'quasisimple': False, 'rank': 2, 'rational': True, 'rational_characters_known': True, 'ratrep_stats': [[1, 2], [2, 1]], 'representations': {'PC': {'code': 25, 'gens': [1, 2], 'pres': [2, -2, -3, 17]}, 'GLZ': {'b': 3, 'd': 2, 'gens': [28, 45]}, 'Lie': [{'d': 2, 'q': 2, 'gens': [6, 11], 'family': 'GL'}, {'d': 2, 'q': 2, 'gens': [6, 11], 'family': 'SL'}, {'d': 2, 'q': 2, 'gens': [6, 11], 'family': 'PSL'}, {'d': 2, 'q': 2, 'gens': [6, 11], 'family': 'PGL'}, {'d': 3, 'q': 2, 'gens': [84, 279], 'family': 'SO'}, {'d': 2, 'q': 2, 'gens': [20, 69], 'family': 'SU'}, {'d': 3, 'q': 2, 'gens': [84, 279], 'family': 'PSO'}, {'d': 2, 'q': 2, 'gens': [20, 69], 'family': 'PSU'}, {'d': 3, 'q': 2, 'gens': [84, 279], 'family': 'GO'}, {'d': 3, 'q': 2, 'gens': [84, 279], 'family': 'Omega'}, {'d': 3, 'q': 2, 'gens': [84, 279], 'family': 'PGO'}, {'d': 2, 'q': 2, 'gens': [20, 138], 'family': 'PGU'}, {'d': 3, 'q': 2, 'gens': [84, 279], 'family': 'POmega'}, {'d': 3, 'q': 2, 'gens': [84, 279], 'family': 'CSO'}, {'d': 2, 'q': 4, 'gens': [20, 194], 'family': 'CSOPlus'}, {'d': 2, 'q': 2, 'gens': [13, 14], 'family': 'CSOMinus'}, {'d': 2, 'q': 2, 'gens': [20, 69], 'family': 'CSU'}, {'d': 3, 'q': 2, 'gens': [84, 279], 'family': 'CO'}, {'d': 2, 'q': 2, 'gens': [13, 14], 'family': 'COMinus'}, {'d': 2, 'q': 2, 'gens': [6, 11], 'family': 'PGammaL'}, {'d': 2, 'q': 2, 'gens': [6, 11], 'family': 'PSigmaL'}, {'d': 2, 'q': 2, 'gens': [20, 138], 'family': 'PGammaU'}, {'d': 1, 'q': 3, 'gens': [1, 3], 'family': 'AGL'}, {'d': 1, 'q': 3, 'gens': [1, 3], 'family': 'AGammaL'}], 'GLFp': {'d': 2, 'p': 2, 'gens': [11, 6]}, 'Perm': {'d': 3, 'gens': [1, 4]}}, 'schur_multiplier': [], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2], 'solvability_type': 6, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'S_3', 'transitive_degree': 3, 'wreath_data': None, 'wreath_product': False}
