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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': True, 'ambient': '6912.hn', 'ambient_counter': 196, 'ambient_order': 6912, 'ambient_tex': 'D_6^2:C_2^2:A_4', 'central': False, 'central_factor': False, 'centralizer_order': 48, 'characteristic': False, 'core_order': 2, 'counter': 1250, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '6912.hn.864.k1.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': False, 'old_label': '864.k1.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': 864, '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': '8.2', 'subgroup_hash': 2, 'subgroup_order': 8, 'subgroup_tex': 'C_2\\times C_4', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '6912.hn', 'aut_centralizer_order': None, 'aut_label': '864.k1', 'aut_quo_index': None, 'aut_stab_index': None, 'aut_weyl_group': None, 'aut_weyl_index': None, 'centralizer': '144.e1.a1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['288.k1.a1', '288.l1.a1', '432.j1.a1', '432.n1.a1', '432.s1.a1', '432.t1.a1', '432.u1.a1', '432.x1.a1', '432.bd1.a1'], 'contains': ['1728.a1.a1', '1728.o1.a1'], 'core': '3456.a1.a1', 'coset_action_label': None, 'count': 36, 'diagramx': [6820, -1, 9525, -1, 7296, -1, 7557, -1], 'generators': [7655590, 414, 40319], 'label': '6912.hn.864.k1.a1', 'mobius_quo': None, 'mobius_sub': 0, 'normal_closure': '6.a1.a1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '36.c1.a1', 'old_label': '864.k1.a1', 'projective_image': '3456.ou', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '864.k1.a1', 'subgroup_fusion': None, 'weyl_group': '4.2'}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': True, 'abelian_quotient': '8.2', '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, 3], [5, 3]], 'aut_group': '8.3', 'aut_hash': 3, 'aut_nilpotency_class': 2, 'aut_nilpotent': True, 'aut_order': 8, 'aut_permdeg': 4, 'aut_perms': [1, 22], 'aut_phi_ratio': 2.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 1, 2, 1], [4, 1, 4, 1]], 'aut_supersolvable': True, 'aut_tex': 'D_4', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 4, 'autcent_group': '8.3', 'autcent_hash': 3, 'autcent_nilpotent': True, 'autcent_order': 8, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'D_4', '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, 3], [4, 1, 4]], 'center_label': '8.2', 'center_order': 8, 'central_product': True, 'central_quotient': '1.1', 'commutator_count': 0, 'commutator_label': '1.1', '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': 2, 'cyclic': False, 'derived_length': 1, 'dihedral': False, 'direct_factorization': [['2.1', 1], ['4.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [4, 1, 2, 2]], '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': [], 'familial': True, 'frattini_label': '2.1', 'frattini_quotient': '4.2', 'hash': 2, 'hyperelementary': 2, 'inner_abelian': True, 'inner_cyclic': True, 'inner_exponent': 1, 'inner_gen_orders': [1, 1], 'inner_gens': [[1, 2], [1, 2]], '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, 8]], 'label': '8.2', 'linC_count': 12, 'linC_degree': 2, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 3, 'linQ_degree_count': 4, 'linQ_dim': 3, 'linQ_dim_count': 4, 'linR_count': 4, 'linR_degree': 3, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C2*C4', 'ngens': 2, 'nilpotency_class': 1, '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': 8, 'number_divisions': 6, 'number_normal_subgroups': 8, 'number_subgroup_autclasses': 6, 'number_subgroup_classes': 8, 'number_subgroups': 8, 'old_label': None, 'order': 8, 'order_factorization_type': 3, 'order_stats': [[1, 1], [2, 3], [4, 4]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 4, 'outer_gen_orders': [2, 4], 'outer_gen_pows': [0, 0], 'outer_gens': [[1, 3], [5, 3]], 'outer_group': '8.3', 'outer_hash': 3, 'outer_nilpotent': True, 'outer_order': 8, 'outer_permdeg': 4, 'outer_perms': [1, 22], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'D_4', 'pc_rank': 2, 'perfect': False, 'permutation_degree': 6, 'pgroup': 2, 'primary_abelian_invariants': [2, 4], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 2]], 'representations': {'PC': {'code': 34, 'gens': [1, 2], 'pres': [3, -2, 2, -2, 16]}, 'GLZ': {'b': 3, 'd': 3, 'gens': [3362, 16426]}, 'GLFp': {'d': 2, 'p': 5, 'gens': [251, 504]}, 'Perm': {'d': 6, 'gens': [22, 120, 7]}}, 'schur_multiplier': [2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 4], 'solvability_type': 1, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_2\\times C_4', 'transitive_degree': 8, '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': '12.5', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 3, 'aut_exponent': 24, 'aut_gen_orders': [6, 24, 8, 4, 12, 12], 'aut_gens': [[4142, 32140668018, 18498, 28003, 25434265614, 414, 40319, 14493436958, 997930107, 3254, 26824105451], [414, 6354422199, 35152, 16703, 26428579582, 3776, 40319, 13499163437, 7644906, 1032, 26352741221], [39905, 7655590, 34990, 21821, 87097614, 3776, 40319, 1956298302, 33178507045, 32741, 25390716794], [4142, 12933061776, 21821, 5329, 12940703147, 36543, 40319, 25390741037, 32140645970, 5764, 1005566739], [36543, 25430641996, 11536, 16703, 25434278586, 39905, 40319, 13016515358, 33178510540, 5764, 1005565246], [3776, 1959943037, 12316, 5329, 2043397813, 4142, 40319, 1956308179, 6267333459, 1032, 13016520662], [39905, 19679380390, 5167, 11536, 2043410181, 4142, 40319, 26824111699, 7640775, 5764, 14493429876]], 'aut_group': None, 'aut_hash': 2798513968372987562, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 110592, 'aut_permdeg': 128, 'aut_perms': [368409429390874840290324095706716312645058434846468251889945879089038996583271224866063644492226436085126076116977111583703699249951673691658484417302155189332260157019504624248678517052683633604673886148573542378260, 123386256315504207396731833749332496755818117727777263009479643607079968025879998816481405934796203673418760436121740760700568132553202797891210457799193375981773318807716746090732668480386366833866476135926863992324, 272136700984135931993133389305203962266147933491111409003233300867201182093219399210955144246311252209713405893245527153057922577758490007976821030742163286783355251924113003345920335433470420946314937146653276858523, 281866342886265949584185225492914553205281337733613266520251613080444546323855842771451020567974375556138031890188244883359419611343815020297734089475736161076926757123359825711809295691914162519328530270104140531551, 377564630288986285043385263702099697113871832251361876199442669787616625837719001528087425318554781498935984671594747887766812169117708943557062220578079824858228952588019099548262180083532944550440132101831701362557, 206335568408133332533776057973526062204675615912042804039125677760576880551435219069806922591088593251301767216116213250953850903240970954701540526356273281282474774521896999785754077288492289381057962013457805627096], 'aut_phi_ratio': 48.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 6, 1, 1], [2, 9, 1, 2], [2, 12, 1, 1], [2, 24, 4, 1], [2, 54, 1, 1], [2, 108, 1, 1], [3, 4, 2, 1], [3, 16, 2, 1], [3, 64, 4, 1], [4, 12, 1, 1], [4, 18, 2, 1], [4, 108, 1, 2], [4, 108, 2, 2], [4, 144, 2, 1], [6, 4, 2, 1], [6, 16, 2, 1], [6, 24, 2, 1], [6, 48, 2, 1], [6, 48, 4, 1], [6, 64, 4, 1], [6, 96, 8, 1], [6, 144, 2, 2], [6, 192, 8, 1], [12, 48, 2, 1], [12, 288, 2, 1], [12, 288, 4, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_6^2.(C_4\\times A_4).C_2^5.C_2', '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': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 24, 'autcentquo_group': None, 'autcentquo_hash': 8140822533548123779, 'autcentquo_nilpotent': False, 'autcentquo_order': 27648, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'C_6^2.(D_4\\times S_4).C_2^2', 'cc_stats': [[1, 1, 1], [2, 1, 1], [2, 6, 1], [2, 9, 2], [2, 12, 1], [2, 24, 4], [2, 54, 1], [2, 108, 1], [3, 4, 2], [3, 16, 2], [3, 64, 4], [4, 12, 1], [4, 18, 2], [4, 108, 6], [4, 144, 2], [6, 4, 2], [6, 16, 2], [6, 24, 2], [6, 48, 6], [6, 64, 4], [6, 96, 8], [6, 144, 4], [6, 192, 8], [12, 48, 2], [12, 288, 6]], 'center_label': '2.1', 'center_order': 2, 'central_product': False, 'central_quotient': '3456.ou', 'commutator_count': 1, 'commutator_label': '576.8639', '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', '3.1', '3.1'], 'composition_length': 11, 'conjugacy_classes_known': True, 'counter': 196, 'cyclic': False, 'derived_length': 3, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 6, 1, 1], [2, 9, 1, 2], [2, 12, 1, 1], [2, 24, 1, 4], [2, 54, 1, 1], [2, 108, 1, 1], [3, 4, 1, 2], [3, 16, 2, 1], [3, 64, 2, 2], [4, 12, 1, 1], [4, 18, 1, 2], [4, 108, 1, 2], [4, 108, 2, 2], [4, 144, 1, 2], [6, 4, 1, 2], [6, 16, 2, 1], [6, 24, 1, 2], [6, 48, 1, 6], [6, 64, 2, 2], [6, 96, 2, 4], [6, 144, 2, 2], [6, 192, 2, 4], [12, 48, 1, 2], [12, 288, 1, 2], [12, 288, 2, 2]], 'element_repr_type': 'Perm', 'elementary': 1, 'eulerian_function': 96, 'exponent': 12, 'exponents_of_order': [8, 3], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [[16, 0, 8], [16, 1, 4]], 'familial': False, 'frattini_label': '8.5', 'frattini_quotient': '864.4671', 'hash': 2001946764947785628, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 12, 'inner_gen_orders': [2, 4, 2, 2, 6, 2, 1, 6, 6, 3, 3], 'inner_gens': [[4142, 32140668018, 18498, 12316, 25434290939, 414, 40319, 14493457761, 997949728, 3194, 26824106119], [4142, 32140668018, 23616, 28783, 25434297521, 39905, 40319, 26352739119, 83499405, 25741, 26824122341], [4142, 32140663056, 18498, 28003, 25434296843, 39905, 40319, 14493457761, 997957185, 25741, 26824122983], [36177, 32140662622, 18498, 28003, 25434288580, 39905, 40319, 14493436958, 997943842, 6620, 26824104662], [36543, 32140656076, 12316, 5329, 25434265614, 36177, 40319, 26352730579, 1959944608, 20933, 13499169383], [4142, 32140661261, 21821, 12316, 25434267178, 414, 40319, 14493436958, 997945765, 1032, 26824111117], [4142, 32140668018, 18498, 28003, 25434265614, 414, 40319, 14493436958, 997930107, 3254, 26824105451], [36177, 6267323740, 21821, 28003, 13412064764, 414, 40319, 14493436958, 1959948501, 21517, 26824090632], [39905, 33178516312, 35152, 23616, 26911188778, 3776, 40319, 13499163437, 997930107, 32741, 26352709512], [414, 32140675161, 5329, 18498, 25434288382, 3776, 40319, 14493455297, 997944426, 3254, 26824097018], [39905, 32140679512, 34990, 21821, 1037857333, 36543, 40319, 14493455297, 1959921738, 29704, 26824105451]], 'inner_hash': 1574280891548733311, 'inner_nilpotent': False, 'inner_order': 3456, 'inner_split': False, 'inner_tex': 'C_6^2.(D_4\\times A_4)', 'inner_used': [1, 2, 3, 4, 5, 8, 9], 'irrC_degree': 16, 'irrQ_degree': 16, 'irrQ_dim': 16, 'irrR_degree': 16, 'irrep_stats': [[1, 12], [2, 3], [3, 4], [4, 24], [6, 9], [8, 3], [12, 4], [16, 12], [24, 4]], 'label': '6912.hn', 'linC_count': 144, 'linC_degree': 8, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 8, 'linQ_degree_count': 16, 'linQ_dim': 8, 'linQ_dim_count': 16, 'linR_count': 16, 'linR_degree': 8, 'maximal_subgroups_known': True, 'metabelian': False, 'metacyclic': False, 'monomial': True, 'name': 'D6^2:C2^2:A4', '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], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 32, 'number_characteristic_subgroups': 23, 'number_conjugacy_classes': 75, 'number_divisions': 55, 'number_normal_subgroups': 31, 'number_subgroup_autclasses': 788, 'number_subgroup_classes': 1430, 'number_subgroups': 50654, 'old_label': None, 'order': 6912, 'order_factorization_type': 33, 'order_stats': [[1, 1], [2, 295], [3, 296], [4, 984], [6, 3512], [12, 1824]], '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': [[4142, 32140668018, 18498, 28003, 25434265614, 414, 40319, 26352718238, 83472507, 3254, 26824105451], [3776, 32140672637, 12316, 21821, 25434266136, 414, 40319, 14493436880, 997935493, 2210, 26824100539], [4142, 32140668018, 18498, 28003, 25434265614, 414, 40319, 14493436958, 997950212, 3254, 26824105451], [4142, 7637261, 18498, 28003, 25434265614, 414, 40319, 1005550238, 6267310587, 3254, 25390729451]], 'outer_group': '32.46', 'outer_hash': 46, 'outer_nilpotent': True, 'outer_order': 32, 'outer_permdeg': 8, 'outer_perms': [5176, 7, 127, 11647], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^2\\times D_4', 'pc_rank': 6, 'perfect': False, 'permutation_degree': 14, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 3], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 5], [3, 4], [4, 9], [6, 5], [8, 9], [12, 6], [16, 5], [24, 4], [32, 4]], 'representations': {'PC': {'code': '73864501470118747308358512871772801427904812240883775199203329937877835752809411149446012282107965529111468045529066004', 'gens': [1, 3, 5, 6, 8, 10], 'pres': [11, 2, 3, 2, 2, 2, 2, 3, 2, 3, 2, 2, 22, 204932, 59908, 90, 47524, 24435, 346901, 205936, 51243, 1358, 9685, 192, 8652, 1271, 145735, 51762, 74653, 10600, 260, 123560, 20622, 9545, 18489, 341900, 2066, 328]}, 'Perm': {'d': 14, 'gens': [4142, 32140668018, 18498, 28003, 25434265614, 414, 40319, 14493436958, 997930107, 3254, 26824105451]}}, 'schur_multiplier': [2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 6], 'solvability_type': 11, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': 'D_6^2:C_2^2:A_4', 'transitive_degree': 48, 'wreath_data': None, 'wreath_product': False}
