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gps_subgroup_search • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'ambient': '11664.kv', 'ambient_counter': 282, 'ambient_order': 11664, 'ambient_tex': '\\He_3^2.(C_2\\times D_4)', 'central': False, 'central_factor': False, 'centralizer_order': 3, 'characteristic': False, 'core_order': 81, 'counter': 39, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '11664.kv.24.A', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': False, 'minimal': False, 'minimal_normal': False, 'nilpotent': False, 'normal': False, 'old_label': '24.a1', 'outer_equivalence': True, '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': 24, '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': '486.102', 'subgroup_hash': 102, 'subgroup_order': 486, 'subgroup_tex': 'C_3^4:C_6', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '11664.kv', 'aut_centralizer_order': None, 'aut_label': '24.A', 'aut_quo_index': None, 'aut_stab_index': None, 'aut_weyl_group': None, 'aut_weyl_index': None, 'centralizer': '3888.A', 'complements': None, 'conjugacy_class_count': 4, 'contained_in': ['8.A', '12.A', '12.B', '12.E'], 'contains': ['48.C', '72.B', '72.C', '72.D', '72.E'], 'core': '144.A', 'coset_action_label': None, 'count': 24, 'diagramx': [1446, -1, 3259, -1], 'generators': [76, 1296, 1496, 3600, 4320, 3888], 'label': '11664.kv.24.A', 'mobius_quo': None, 'mobius_sub': 0, 'normal_closure': '8.A', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '6.a1', 'old_label': '24.a1', 'projective_image': '11664.kv', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '24.A', 'subgroup_fusion': None, 'weyl_group': '648.555'}
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gps_groups • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '18.5', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 3, 'aut_exponent': 6, 'aut_gen_orders': [2, 3, 2, 3, 3, 3, 3, 2], 'aut_gens': [[24577758, 15477644, 24507118, 2939606, 36731395, 7046328], [43014227, 32319787, 20254942, 32864349, 36731395, 42803789], [7138662, 12564602, 32479219, 20201582, 36731395, 32653863], [24577758, 15477644, 20254942, 2939606, 11225629, 32552094], [24577758, 15477644, 32479219, 2939606, 36731395, 7046328], [24577758, 15477644, 24507118, 2939606, 36731395, 2794152], [24577758, 15477644, 24507118, 2939606, 36731395, 24577806], [24577758, 38113169, 32479219, 2939606, 36731395, 7046328], [20201582, 2414724, 24507118, 32864349, 36731395, 2670152]], 'aut_group': '5832.od', 'aut_hash': 2043999697674719391, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 5832, 'aut_permdeg': 63, 'aut_perms': [1868880048577573253314068272151853182452191819807674465114960121188869188293532433643029, 31978944192023857116454479538487640153356887775245970131484795750034251440984736799852, 1694470570403332968601306223428507482854551830536239576054726019415555918954788835799827, 31978944192023857116454479538487640153356887775245970131484795750034251440256174665560, 1678492865535159756931625760224650854376864553799595116663929306379146631867445700807067, 31981889445290642012176689817345551184253255237730176807230348127277263168749528578033, 31978944192023857139751674672499187132115226432799404575924433487463904405635910821174, 918069643619753828947295739762000027413710569692868877695744145438054318757218250573263], 'aut_phi_ratio': 36.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 9, 1, 1], [3, 1, 2, 1], [3, 2, 1, 1], [3, 2, 2, 1], [3, 3, 2, 1], [3, 6, 1, 1], [3, 6, 2, 2], [3, 6, 6, 1], [3, 9, 6, 1], [3, 18, 6, 1], [6, 9, 2, 1], [6, 27, 2, 1], [6, 27, 6, 1]], 'aut_supersolvable': True, 'aut_tex': '\\He_3^2:C_2^3', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 3, 'autcent_group': '9.2', 'autcent_hash': 2, 'autcent_nilpotent': True, 'autcent_order': 9, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_3^2', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 6, 'autcentquo_group': '648.555', 'autcentquo_hash': 555, 'autcentquo_nilpotent': False, 'autcentquo_order': 648, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'C_3.S_3^3', 'cc_stats': [[1, 1, 1], [2, 9, 1], [3, 1, 2], [3, 2, 3], [3, 3, 2], [3, 6, 11], [3, 9, 6], [3, 18, 6], [6, 9, 2], [6, 27, 8]], 'center_label': '3.1', 'center_order': 3, 'central_product': False, 'central_quotient': '162.34', 'commutator_count': 1, 'commutator_label': '27.5', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '3.1', '3.1', '3.1', '3.1', '3.1'], 'composition_length': 6, 'conjugacy_classes_known': True, 'counter': 102, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 9, 1, 1], [3, 1, 2, 1], [3, 2, 1, 1], [3, 2, 2, 1], [3, 3, 2, 1], [3, 6, 1, 1], [3, 6, 2, 5], [3, 9, 2, 3], [3, 18, 2, 3], [6, 9, 2, 1], [6, 27, 2, 4]], 'element_repr_type': 'GLFp', 'elementary': 1, 'eulerian_function': 12, 'exponent': 6, 'exponents_of_order': [5, 1], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [[6, 0, 6]], 'familial': False, 'frattini_label': '9.2', 'frattini_quotient': '54.12', 'hash': 102, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 6, 'inner_gen_orders': [3, 2, 3, 3, 1, 3], 'inner_gens': [[24577758, 24033196, 24507118, 2939606, 36731395, 24898803], [20201582, 15477644, 24507118, 32864349, 36731395, 20699747], [24577758, 15477644, 24507118, 2939606, 36731395, 24577806], [24577758, 2414724, 24507118, 2939606, 36731395, 32653863], [24577758, 15477644, 24507118, 2939606, 36731395, 7046328], [7138662, 32319787, 6973453, 20201582, 36731395, 7046328]], 'inner_hash': 34, 'inner_nilpotent': False, 'inner_order': 162, 'inner_split': True, 'inner_tex': 'C_3^3:C_6', 'inner_used': [1, 2, 3, 4, 6], 'irrC_degree': 6, 'irrQ_degree': 12, 'irrQ_dim': 12, 'irrR_degree': 12, 'irrep_stats': [[1, 18], [2, 9], [3, 4], [6, 11]], 'label': '486.102', 'linC_count': 6, 'linC_degree': 6, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 12, 'linQ_degree_count': 5, 'linQ_dim': 12, 'linQ_dim_count': 5, 'linR_count': 5, 'linR_degree': 12, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'C3^4:C6', 'ngens': 6, 'nilpotency_class': -1, 'nilpotent': False, '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': 15, 'number_characteristic_subgroups': 15, 'number_conjugacy_classes': 42, 'number_divisions': 23, 'number_normal_subgroups': 24, 'number_subgroup_autclasses': 83, 'number_subgroup_classes': 153, 'number_subgroups': 832, 'old_label': None, 'order': 486, 'order_factorization_type': 51, 'order_stats': [[1, 1], [2, 9], [3, 242], [6, 234]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 6, 'outer_gen_orders': [2, 2, 3, 3], 'outer_gen_pows': [14408200, 14408200, 14408200, 14408200], 'outer_gens': [[24577758, 15477644, 20254942, 2939606, 11225629, 7046328], [7138662, 15477644, 20254942, 2939606, 36731395, 24678894], [24577758, 15477644, 32479219, 2939606, 36731395, 7046328], [24577758, 15477644, 24507118, 2939606, 36731395, 20327817]], 'outer_group': '36.10', 'outer_hash': 10, 'outer_nilpotent': False, 'outer_order': 36, 'outer_permdeg': 6, 'outer_perms': [659, 719, 181, 373], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'S_3^2', 'pc_rank': 5, 'perfect': False, 'permutation_degree': 18, 'pgroup': 0, 'primary_abelian_invariants': [2, 3, 3], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 2], [2, 9], [4, 4], [6, 3], [12, 5]], 'representations': {'PC': {'code': 3642860151096234488119, 'gens': [1, 2, 3, 5, 6], 'pres': [6, 3, 3, 2, 3, 3, 3, 272, 50, 507, 1456, 652, 1313]}, 'GLFp': {'d': 4, 'p': 3, 'gens': [24577758, 15477644, 24507118, 2939606, 36731395, 7046328]}, 'Perm': {'d': 18, 'gens': [3402106385166720, 93485145748, 1134035330135490, 376703666887107, 2267507354250690, 711549292653507]}}, 'schur_multiplier': [3, 3, 3], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [3, 6], 'solvability_type': 7, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_3^4:C_6', 'transitive_degree': 18, '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': 4, 'aut_exponent': 24, 'aut_gen_orders': [6, 12, 12, 12, 12], 'aut_gens': [[1, 2, 8, 24, 144, 432, 1296, 3888], [10329, 10334, 5728, 9672, 2448, 1728, 1296, 7776], [2827, 298, 7248, 2908, 6576, 5584, 1296, 3888], [9835, 7222, 7592, 10496, 3024, 2880, 1296, 3888], [9805, 7302, 160, 10316, 10520, 2840, 2592, 3888], [11337, 8694, 8896, 3828, 2792, 96, 1296, 3888]], 'aut_group': None, 'aut_hash': 2302517250360555534, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 46656, 'aut_permdeg': 162, 'aut_perms': [4737916610715155664706401302694766945012519924953301976824808089393804293648194579990954855708401038921529071894620255454018529348958821742187275270695945806764351215513489066481636158403019556584192837638862119121133385707112792036951057084806190098427908414588436500510831271111715221668, 7974862195986620764570782997193519899216126092955316737965175675006575162213083837625994692455517747352360410253457681896986818390495762818061786351150192440709605742330913587291023397475729610782719227281892814675137859198102266080399923801657049270871296852140271291839578098418287100290, 5687185783820374802948162372881313873288657029100817877521171592239608525129547531421634988555609410451264108314162733724110875599238783186663663976977501492522999528748597520234911224346052528205006924135794191453459064143148642266678265761894686759015734690537447031245761010229366260312, 6586474084541327125109866304448262014215488733785111117583594827861397168026512246470539610687452602850772565875601767806545280352032774343041733839438622251684704378243091088784497279658999140002073882130055376186822302119601904539951498786240339296298690928769824906441783382117357307430, 2627297330428913050694626691194921378603014867027762817117397868331299490249951100562254549052047074189940586554463690006989256567298722129437919785090383471449797948618513294843022291565809830814791932930737254356274830645592657855370054688147965002285878690089072330711995391990521310230], 'aut_phi_ratio': 12.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 27, 2, 1], [2, 54, 2, 1], [2, 81, 1, 1], [2, 162, 2, 1], [3, 2, 1, 2], [3, 4, 1, 1], [3, 24, 2, 1], [3, 36, 4, 1], [3, 48, 2, 1], [3, 72, 2, 1], [3, 144, 2, 1], [4, 162, 1, 1], [4, 486, 1, 1], [6, 54, 2, 1], [6, 108, 2, 1], [6, 162, 1, 2], [6, 324, 1, 1], [6, 324, 4, 3], [6, 648, 2, 2], [12, 162, 2, 1], [12, 324, 1, 1], [12, 324, 2, 1], [12, 486, 2, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_3^2.C_3^4.D_4.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': 24, 'autcentquo_group': None, 'autcentquo_hash': 2302517250360555534, 'autcentquo_nilpotent': False, 'autcentquo_order': 46656, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'C_3^2.C_3^4.D_4.C_2^3', 'cc_stats': [[1, 1, 1], [2, 27, 2], [2, 54, 2], [2, 81, 1], [2, 162, 2], [3, 2, 2], [3, 4, 1], [3, 24, 2], [3, 36, 4], [3, 48, 2], [3, 72, 2], [3, 144, 2], [4, 162, 1], [4, 486, 1], [6, 54, 2], [6, 108, 2], [6, 162, 2], [6, 324, 13], [6, 648, 4], [12, 162, 2], [12, 324, 3], [12, 486, 2]], 'center_label': '1.1', 'center_order': 1, 'central_product': False, 'central_quotient': '11664.kv', 'commutator_count': 1, 'commutator_label': '1458.1576', 'complements_known': False, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '3.1', '3.1', '3.1', '3.1', '3.1', '3.1'], 'composition_length': 10, 'conjugacy_classes_known': True, 'counter': 282, 'cyclic': False, 'derived_length': 4, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 27, 1, 2], [2, 54, 1, 2], [2, 81, 1, 1], [2, 162, 1, 2], [3, 2, 1, 2], [3, 4, 1, 1], [3, 24, 1, 2], [3, 36, 1, 4], [3, 48, 1, 2], [3, 72, 1, 2], [3, 144, 1, 2], [4, 162, 1, 1], [4, 486, 1, 1], [6, 54, 1, 2], [6, 108, 1, 2], [6, 162, 1, 2], [6, 324, 1, 13], [6, 648, 1, 4], [12, 162, 2, 1], [12, 324, 1, 1], [12, 324, 2, 1], [12, 486, 2, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 10886400, 'exponent': 12, 'exponents_of_order': [6, 4], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [[12, 1, 4], [48, 1, 2]], 'familial': False, 'frattini_label': '9.2', 'frattini_quotient': '1296.3531', 'hash': 2955081780199608270, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 12, 'inner_gen_orders': [6, 4, 3, 6, 3, 3, 3, 3], 'inner_gens': [[1, 6110, 5728, 11256, 8928, 10800, 1296, 7776], [4813, 2, 3136, 5504, 11520, 720, 2592, 3888], [10001, 930, 8, 6648, 144, 5616, 1296, 3888], [5617, 1602, 9368, 24, 9360, 6048, 2592, 3888], [8497, 9794, 8, 6792, 144, 8208, 1296, 3888], [5185, 4034, 10376, 11256, 4032, 432, 1296, 3888], [1, 2594, 8, 2616, 144, 432, 1296, 3888], [7777, 2, 8, 24, 144, 432, 1296, 3888]], 'inner_hash': 2955081780199608270, 'inner_nilpotent': False, 'inner_order': 11664, 'inner_split': False, 'inner_tex': '\\He_3^2.(C_2\\times D_4)', 'inner_used': [1, 2, 3, 4], 'irrC_degree': 12, 'irrQ_degree': 12, 'irrQ_dim': 12, 'irrR_degree': 12, 'irrep_stats': [[1, 8], [2, 2], [4, 16], [8, 8], [12, 4], [16, 2], [18, 12], [36, 1], [48, 2]], 'label': '11664.kv', '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': 'He3^2.(C2*D4)', 'ngens': 10, '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], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 29, 'number_characteristic_subgroups': 13, 'number_conjugacy_classes': 55, 'number_divisions': 52, 'number_normal_subgroups': 27, 'number_subgroup_autclasses': 526, 'number_subgroup_classes': 1295, 'number_subgroups': 102579, 'old_label': None, 'order': 11664, 'order_factorization_type': 33, 'order_stats': [[1, 1], [2, 567], [3, 728], [4, 648], [6, 7452], [12, 2268]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': True, 'outer_exponent': 2, 'outer_gen_orders': [2, 2], 'outer_gen_pows': [2670, 4], 'outer_gens': [[3051, 2594, 8880, 9488, 2016, 144, 2592, 7776], [2689, 2, 9232, 5692, 2744, 6528, 1296, 7776]], 'outer_group': '4.2', 'outer_hash': 2, 'outer_nilpotent': True, 'outer_order': 4, 'outer_permdeg': 4, 'outer_perms': [1, 6], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^2', 'pc_rank': 8, 'perfect': False, 'permutation_degree': 36, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 2], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 8], [2, 2], [4, 16], [8, 8], [12, 2], [16, 2], [18, 8], [24, 1], [36, 3], [48, 2]], 'representations': {'PC': {'code': '3736206736914572732371176538338573298327729471332021339675670395103035095875381889405422227048755497135150582706638795111865356981490165501023', 'gens': [1, 2, 4, 5, 7, 8, 9, 10], 'pres': [10, 2, 2, 2, 3, 2, 3, 3, 3, 3, 3, 112800, 122201, 51, 260762, 229123, 62733, 103863, 562804, 137614, 1524, 41584, 144, 158405, 49455, 1465, 39275, 624966, 403216, 186506, 27346, 2156, 864007, 28817, 120987, 56197, 20207, 2937, 4627, 116658, 9768, 777609]}, 'Perm': {'d': 36, 'gens': [106363587271560335870164627814774715242610, 329447778369213946313011098570331825082185, 276255404203435341815859230992429384525196]}}, 'schur_multiplier': [2, 2, 2], 'semidirect_product': None, 'simple': False, 'smith_abelian_invariants': [2, 2, 2], 'solvability_type': 17, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': '\\He_3^2.(C_2\\times D_4)', 'transitive_degree': 36, 'wreath_data': None, 'wreath_product': False}
