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gps_subgroup_search • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'ambient': '1344.2752', 'ambient_counter': 2752, 'ambient_order': 1344, 'ambient_tex': 'C_{168}.D_4', 'central': False, 'central_factor': False, 'centralizer_order': 4, 'characteristic': False, 'core_order': 24, 'counter': 98, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '1344.2752.28.d1.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': False, 'minimal': False, 'minimal_normal': False, 'nilpotent': False, 'normal': False, 'old_label': '28.d1.a1', 'outer_equivalence': False, 'perfect': False, 'proper': True, 'quotient': None, 'quotient_Agroup': None, 'quotient_abelian': None, 'quotient_cyclic': None, 'quotient_hash': None, 'quotient_metabelian': None, 'quotient_nilpotent': None, 'quotient_order': 28, 'quotient_simple': None, 'quotient_solvable': None, 'quotient_supersolvable': None, 'quotient_tex': None, 'simple': False, 'solvable': True, 'special_labels': [], 'split': None, 'standard_generators': False, 'stem': False, 'subgroup': '48.34', 'subgroup_hash': 34, 'subgroup_order': 48, 'subgroup_tex': 'C_6:Q_8', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '1344.2752', 'aut_centralizer_order': 24, 'aut_label': '28.d1', 'aut_quo_index': None, 'aut_stab_index': 14, 'aut_weyl_group': '192.1524', 'aut_weyl_index': 336, 'centralizer': '336.a1.a1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['4.b1.a1', '14.b1.a1', '14.c1.a1', '14.d1.a1'], 'contains': ['56.a1.a1', '56.g1.a1', '56.i1.a1', '56.k1.a1', '84.c1.a1'], 'core': '56.a1.a1', 'coset_action_label': None, 'count': 7, 'diagramx': [6595, -1, 4930, -1, 6677, -1, 5227, -1], 'generators': [699, 680, 684, 1008, 672], 'label': '1344.2752.28.d1.a1', 'mobius_quo': None, 'mobius_sub': -2, 'normal_closure': '4.b1.a1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '7.a1.a1', 'old_label': '28.d1.a1', 'projective_image': '672.604', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '28.d1.a1', 'subgroup_fusion': None, 'weyl_group': '48.41'}
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gps_groups • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '8.5', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 12, 'aut_gen_orders': [2, 2, 2, 2, 2, 2, 2, 3], 'aut_gens': [[1, 2, 4], [25, 2, 6], [1, 26, 4], [13, 2, 28], [1, 2, 20], [1, 2, 28], [3, 2, 4], [25, 2, 4], [17, 2, 4]], 'aut_group': '384.12882', 'aut_hash': 12882, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 384, 'aut_permdeg': 11, 'aut_perms': [1680, 40344, 19818024, 3669865, 744, 7985040, 3669864, 3], 'aut_phi_ratio': 24.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 1, 2, 1], [3, 2, 1, 1], [4, 2, 2, 1], [4, 6, 4, 1], [6, 2, 1, 1], [6, 2, 2, 1], [12, 2, 4, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_2\\wr C_2^2\\times S_3', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 4, 'autcent_group': '32.27', 'autcent_hash': 27, 'autcent_nilpotent': True, 'autcent_order': 32, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^2\\wr C_2', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 6, 'autcentquo_group': '12.4', 'autcentquo_hash': 4, 'autcentquo_nilpotent': False, 'autcentquo_order': 12, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'D_6', 'cc_stats': [[1, 1, 1], [2, 1, 3], [3, 2, 1], [4, 2, 2], [4, 6, 4], [6, 2, 3], [12, 2, 4]], 'center_label': '4.2', 'center_order': 4, 'central_product': True, 'central_quotient': '12.4', 'commutator_count': 1, 'commutator_label': '6.2', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '3.1'], 'composition_length': 5, 'conjugacy_classes_known': True, 'counter': 34, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['2.1', 1], ['24.4', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [3, 2, 1, 1], [4, 2, 1, 2], [4, 6, 1, 4], [6, 2, 1, 3], [12, 2, 2, 2]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 84, 'exponent': 12, 'exponents_of_order': [4, 1], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [], 'familial': False, 'frattini_label': '2.1', 'frattini_quotient': '24.14', 'hash': 34, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 6, 'inner_gen_orders': [2, 1, 6], 'inner_gens': [[1, 2, 44], [1, 2, 4], [9, 2, 4]], 'inner_hash': 4, 'inner_nilpotent': False, 'inner_order': 12, 'inner_split': False, 'inner_tex': 'D_6', 'inner_used': [1, 3], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 8], [2, 10]], 'label': '48.34', 'linC_count': 16, 'linC_degree': 3, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 4, 'linQ_degree_count': 4, 'linQ_dim': 6, 'linQ_dim_count': 4, 'linR_count': 16, 'linR_degree': 5, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'C6:Q8', 'ngens': 5, 'nilpotency_class': -1, 'nilpotent': False, 'normal_counts': [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': 9, 'number_characteristic_subgroups': 9, 'number_conjugacy_classes': 18, 'number_divisions': 16, 'number_normal_subgroups': 27, 'number_subgroup_autclasses': 20, 'number_subgroup_classes': 38, 'number_subgroups': 60, 'old_label': None, 'order': 48, 'order_factorization_type': 31, 'order_stats': [[1, 1], [2, 3], [3, 2], [4, 28], [6, 6], [12, 8]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 4, 'outer_gen_orders': [2, 2, 2, 4], 'outer_gen_pows': [0, 0, 0, 0], 'outer_gens': [[13, 26, 28], [1, 26, 4], [3, 2, 28], [27, 26, 30]], 'outer_group': '32.27', 'outer_hash': 27, 'outer_nilpotent': True, 'outer_order': 32, 'outer_permdeg': 8, 'outer_perms': [119, 25, 26875, 22924], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^2\\wr C_2', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 13, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 2], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 8], [2, 6], [4, 2]], 'representations': {'PC': {'code': 139448136244641, 'gens': [1, 2, 3], 'pres': [5, -2, -2, -2, -2, -3, 120, 662, 42, 803, 58, 804]}, 'GLZ': {'b': 3, 'd': 6, 'gens': [91793127399924017, 24992906222183252, 58415889162259849]}, 'GLFp': {'d': 3, 'p': 11, 'gens': [296010408, 1233225909, 186803620, 297800445, 2143735230]}, 'Perm': {'d': 13, 'gens': [40286719, 362880, 375473, 18619, 518918400]}}, 'schur_multiplier': [2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 2], 'solvability_type': 7, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_6:Q_8', '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': '8.5', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 168, 'aut_gen_orders': [24, 24, 12, 6, 6, 12, 12, 24], 'aut_gens': [[1, 2, 24], [1, 950, 1272], [681, 1054, 408], [13, 242, 1272], [21, 694, 552], [689, 926, 936], [13, 578, 744], [689, 250, 216], [677, 806, 408]], 'aut_group': None, 'aut_hash': 3795850347135509252, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 64512, 'aut_permdeg': 124, 'aut_perms': [11126230978578739812291992936190279809747674664347914128020836976564374581401618632104694681133092901007810624350408774044464231324196070211509273953745667171244336643186213028558700587093259764840294012670, 242802996244900404791719914821455150292658273535930239193724899108827416453051137762475776048158495719509690543055367285749571939050862653057392419522542571218580801627153915665774277638449697739592243366722, 967592075940559340091087351060751630406788368717735847379750795367323237660709350133629043004825043712065604870203461512518684267847303287672414408705374138733105883093889353782123019546417521828374987394048, 82055907346461475864405758496932748651849708403169463111227435463469956568466394107993962194687252023820039523584586198957376769419311884401189053674107590997824361732712828368339330702241420960499528819070, 829610671051694314906031726242255059174087642569554284523779801663359742019388637966972129243764092920096824864705438233411763165137757236546833782845252785249199899358841404772867432272294314362262496678156, 963633468077468678112976873526195821917374638824618630154223378163583189718766658522729701461900913477368117779495613174437590620657932915407984438922252974673391446038206708753775074606610339143733162337991, 833494079050670791618366903855510660686774904229216941930125983509066233333973239877966834166513877902488305718989089207633342835823417407656598298908130959700715925867627437570212594893803712003837101133254, 1472087686612061519065831636551900760944152721602015306547960989425641325853644492311384637020228720992051259617847920339573165872292008189180560992555693847532814585633174014817400000535635499090762811439278], 'aut_phi_ratio': 168.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 2, 1, 1], [2, 12, 1, 1], [3, 2, 1, 1], [4, 2, 1, 2], [4, 12, 1, 1], [4, 168, 2, 1], [6, 2, 1, 1], [6, 4, 1, 1], [7, 2, 3, 1], [8, 2, 2, 1], [8, 4, 1, 1], [8, 12, 2, 1], [8, 56, 2, 1], [12, 2, 2, 1], [12, 4, 1, 1], [14, 2, 3, 1], [14, 4, 3, 1], [14, 12, 6, 1], [21, 4, 3, 1], [24, 4, 2, 2], [24, 56, 4, 1], [28, 2, 6, 1], [28, 4, 3, 1], [28, 12, 6, 1], [42, 4, 3, 1], [42, 4, 6, 1], [56, 2, 12, 1], [56, 4, 6, 1], [56, 12, 12, 1], [84, 4, 6, 2], [168, 4, 12, 2]], 'aut_supersolvable': True, 'aut_tex': '(C_2\\times C_{84}).C_6.C_2^6', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 2, 'autcent_group': '8.5', 'autcent_hash': 5, 'autcent_nilpotent': True, 'autcent_order': 8, 'autcent_solvable': True, 'autcent_split': False, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^3', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 84, 'autcentquo_group': None, 'autcentquo_hash': 6707042392732806715, 'autcentquo_nilpotent': False, 'autcentquo_order': 8064, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': '(C_6\\times F_7).C_2^5', 'cc_stats': [[1, 1, 1], [2, 1, 1], [2, 2, 1], [2, 12, 1], [3, 2, 1], [4, 2, 2], [4, 12, 1], [4, 168, 2], [6, 2, 1], [6, 4, 1], [7, 2, 3], [8, 2, 2], [8, 4, 1], [8, 12, 2], [8, 56, 2], [12, 2, 2], [12, 4, 1], [14, 2, 3], [14, 4, 3], [14, 12, 6], [21, 4, 3], [24, 4, 4], [24, 56, 4], [28, 2, 6], [28, 4, 3], [28, 12, 6], [42, 4, 9], [56, 2, 12], [56, 4, 6], [56, 12, 12], [84, 4, 12], [168, 4, 24]], 'center_label': '2.1', 'center_order': 2, 'central_product': False, 'central_quotient': '672.604', 'commutator_count': 1, 'commutator_label': '168.39', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '2.1', '2.1', '3.1', '7.1'], 'composition_length': 8, 'conjugacy_classes_known': True, 'counter': 2752, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 2, 1, 1], [2, 12, 1, 1], [3, 2, 1, 1], [4, 2, 1, 2], [4, 12, 1, 1], [4, 168, 1, 2], [6, 2, 1, 1], [6, 4, 1, 1], [7, 2, 3, 1], [8, 2, 2, 1], [8, 4, 1, 1], [8, 12, 2, 1], [8, 56, 1, 2], [12, 2, 2, 1], [12, 4, 1, 1], [14, 2, 3, 1], [14, 4, 3, 1], [14, 12, 6, 1], [21, 4, 3, 1], [24, 4, 2, 2], [24, 56, 2, 2], [28, 2, 6, 1], [28, 4, 3, 1], [28, 12, 6, 1], [42, 4, 3, 1], [42, 4, 6, 1], [56, 2, 12, 1], [56, 4, 6, 1], [56, 12, 12, 1], [84, 4, 6, 2], [168, 4, 12, 2]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 10752, 'exponent': 168, 'exponents_of_order': [6, 1, 1], 'factors_of_aut_order': [2, 3, 7], 'factors_of_order': [2, 3, 7], 'faithful_reps': [[4, -1, 24]], 'familial': False, 'frattini_label': '8.2', 'frattini_quotient': '168.50', 'hash': 2752, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 84, 'inner_gen_orders': [2, 12, 28], 'inner_gens': [[1, 694, 24], [5, 2, 1320], [1, 50, 24]], 'inner_hash': 604, 'inner_nilpotent': False, 'inner_order': 672, 'inner_split': False, 'inner_tex': 'C_{28}:D_{12}', 'inner_used': [1, 2, 3], 'irrC_degree': 4, 'irrQ_degree': 96, 'irrQ_dim': 192, 'irrR_degree': 8, 'irrep_stats': [[1, 8], [2, 62], [4, 68]], 'label': '1344.2752', 'linC_count': 24, 'linC_degree': 4, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 16, 'linQ_degree_count': 16, 'linQ_dim': 24, 'linQ_dim_count': 16, 'linR_count': 24, 'linR_degree': 8, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'C168.D4', 'ngens': 8, '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], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 37, 'number_characteristic_subgroups': 50, 'number_conjugacy_classes': 138, 'number_divisions': 40, 'number_normal_subgroups': 60, 'number_subgroup_autclasses': 156, 'number_subgroup_classes': 200, 'number_subgroups': 1620, 'old_label': None, 'order': 1344, 'order_factorization_type': 311, 'order_stats': [[1, 1], [2, 15], [3, 2], [4, 352], [6, 6], [7, 6], [8, 144], [12, 8], [14, 90], [21, 12], [24, 240], [28, 96], [42, 36], [56, 192], [84, 48], [168, 96]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 6, 'outer_gen_orders': [2, 2, 2, 2, 6], 'outer_gen_pows': [0, 0, 12, 12, 1176], 'outer_gens': [[1, 682, 24], [1, 2, 696], [685, 2, 24], [685, 2, 360], [1, 170, 600]], 'outer_group': '96.231', 'outer_hash': 231, 'outer_nilpotent': True, 'outer_order': 96, 'outer_permdeg': 13, 'outer_perms': [3628800, 479001600, 744, 24, 40323], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^4\\times C_6', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 42, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 2], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 8], [2, 8], [4, 5], [6, 4], [8, 1], [12, 8], [16, 1], [24, 3], [48, 1], [96, 1]], 'representations': {'PC': {'code': 13483683952317226948964690732086380503702928655077694137365, 'gens': [1, 2, 5], 'pres': [8, -2, -2, -2, -3, -2, -2, -2, -7, 5376, 11105, 41, 16610, 66, 22019, 2715, 26412, 116, 31117, 141, 34958, 166, 36879]}, 'Perm': {'d': 42, 'gens': [816438677858700492620354429020016568359799691599, 8701403101138834385269859322574415483, 26614765275401854260263766328016220, 35114959815011836546591208481889644, 43338050520122600696845150113054948, 51835343089356623243872483634299795, 303916116658416027343136804044800000000, 35085403596887862568986022075080631753113600000000]}}, 'schur_multiplier': [2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 2], 'solvability_type': 7, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_{168}.D_4', 'transitive_degree': 672, 'wreath_data': None, 'wreath_product': False}
