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gps_subgroup_search • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'ambient': '1344.5659', 'ambient_counter': 5659, 'ambient_order': 1344, 'ambient_tex': 'D_{84}.D_4', 'central': False, 'central_factor': False, 'centralizer_order': 4, 'characteristic': False, 'core_order': 24, 'counter': 141, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '1344.5659.28.i1.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': False, 'minimal': False, 'minimal_normal': False, 'nilpotent': False, 'normal': False, 'old_label': '28.i1.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': '1344.5659', 'aut_centralizer_order': 96, 'aut_label': '28.i1', 'aut_quo_index': None, 'aut_stab_index': 7, 'aut_weyl_group': '96.209', 'aut_weyl_index': 672, 'centralizer': '336.a1.a1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['4.d1.a1', '14.c1.a1', '14.d1.a1', '14.f1.a1'], 'contains': ['56.b1.a1', '56.f1.a1', '56.o1.a1', '56.r1.a1', '84.i1.a1'], 'core': '56.b1.a1', 'coset_action_label': None, 'count': 7, 'diagramx': [1713, -1, 7065, -1, 9191, -1, 8078, -1], 'generators': [3, 448, 4, 16, 8], 'label': '1344.5659.28.i1.a1', 'mobius_quo': None, 'mobius_sub': -2, 'normal_closure': '4.d1.a1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '7.a1.a1', 'old_label': '28.i1.a1', 'projective_image': '336.196', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '28.i1.a1', 'subgroup_fusion': None, 'weyl_group': '48.36'}
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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': 84, 'aut_gen_orders': [6, 6, 2, 6, 6, 12, 6, 12, 12, 6], 'aut_gens': [[1, 2, 16, 32], [977, 10, 16, 824], [73, 26, 16, 440], [905, 22, 16, 1320], [449, 26, 16, 744], [409, 22, 16, 624], [1245, 26, 16, 176], [721, 22, 16, 168], [717, 26, 16, 1008], [93, 2, 16, 360], [665, 10, 16, 1200]], 'aut_group': None, 'aut_hash': 5255589997166976132, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 64512, 'aut_permdeg': 180, 'aut_perms': [113672824983227793230469008019902425327539566621473720030011731750812358727261314541342539957575113191587303250342093729489092893647827704902417346176588457318975121252971554982314941891029975051945041619499659416127558672668509421473800382379816447453457182250294081150085515915461575696510259391055531289737125850868859221016993, 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'aut_phi_ratio': 168.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [2, 4, 1, 1], [2, 84, 2, 1], [3, 2, 1, 1], [4, 2, 1, 2], [4, 2, 2, 1], [4, 84, 2, 1], [4, 168, 1, 2], [6, 2, 1, 3], [6, 4, 2, 1], [7, 2, 3, 1], [8, 4, 2, 2], [12, 2, 2, 2], [12, 4, 2, 1], [14, 2, 3, 3], [14, 4, 6, 1], [21, 2, 6, 1], [24, 4, 4, 2], [28, 2, 6, 2], [28, 4, 6, 1], [42, 2, 6, 3], [42, 4, 12, 1], [56, 4, 12, 2], [84, 2, 12, 2], [84, 4, 12, 1], [168, 4, 24, 2]], 'aut_supersolvable': True, 'aut_tex': 'C_{42}.(C_2^5\\times C_6).C_2^3', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 2, 'autcent_group': '64.267', 'autcent_hash': 267, 'autcent_nilpotent': True, 'autcent_order': 64, 'autcent_solvable': True, 'autcent_split': False, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^6', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 42, 'autcentquo_group': '1008.906', 'autcentquo_hash': 906, 'autcentquo_nilpotent': False, 'autcentquo_order': 1008, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'C_2\\times D_6\\times F_7', 'cc_stats': [[1, 1, 1], [2, 1, 3], [2, 4, 1], [2, 84, 2], [3, 2, 1], [4, 2, 4], [4, 84, 2], [4, 168, 2], [6, 2, 3], [6, 4, 2], [7, 2, 3], [8, 4, 4], [12, 2, 4], [12, 4, 2], [14, 2, 9], [14, 4, 6], [21, 2, 6], [24, 4, 8], [28, 2, 12], [28, 4, 6], [42, 2, 18], [42, 4, 12], [56, 4, 24], [84, 2, 24], [84, 4, 12], [168, 4, 48]], 'center_label': '4.2', 'center_order': 4, 'central_product': False, 'central_quotient': '336.196', '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': 5659, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [2, 4, 1, 1], [2, 84, 1, 2], [3, 2, 1, 1], [4, 2, 1, 2], [4, 2, 2, 1], [4, 84, 1, 2], [4, 168, 1, 2], [6, 2, 1, 3], [6, 4, 2, 1], [7, 2, 3, 1], [8, 4, 2, 2], [12, 2, 2, 2], [12, 4, 2, 1], [14, 2, 3, 3], [14, 4, 6, 1], [21, 2, 6, 1], [24, 4, 4, 2], [28, 2, 6, 2], [28, 4, 6, 1], [42, 2, 6, 3], [42, 4, 12, 1], [56, 4, 12, 2], [84, 2, 12, 2], [84, 4, 12, 1], [168, 4, 24, 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': [], 'familial': False, 'frattini_label': '8.2', 'frattini_quotient': '168.56', 'hash': 5659, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 84, 'inner_gen_orders': [2, 4, 1, 42], 'inner_gens': [[1, 6, 16, 1320], [13, 2, 16, 48], [1, 2, 16, 32], [73, 18, 16, 32]], 'inner_hash': 196, 'inner_nilpotent': False, 'inner_order': 336, 'inner_split': None, 'inner_tex': 'C_2\\times D_{84}', 'inner_used': [1, 2, 4], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 8], [2, 170], [4, 41]], 'label': '1344.5659', 'linC_count': 192, 'linC_degree': 4, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 14, 'linQ_degree_count': 72, 'linQ_dim': 18, 'linQ_dim_count': 144, 'linR_count': 96, 'linR_degree': 6, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'D84.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': 44, 'number_characteristic_subgroups': 67, 'number_conjugacy_classes': 219, 'number_divisions': 46, 'number_normal_subgroups': 71, 'number_subgroup_autclasses': 244, 'number_subgroup_classes': 304, 'number_subgroups': 3432, 'old_label': None, 'order': 1344, 'order_factorization_type': 311, 'order_stats': [[1, 1], [2, 175], [3, 2], [4, 512], [6, 14], [7, 6], [8, 16], [12, 16], [14, 42], [21, 12], [24, 32], [28, 48], [42, 84], [56, 96], [84, 96], [168, 192]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 6, 'outer_gen_orders': [2, 2, 2, 2, 2, 6], 'outer_gen_pows': [0, 448, 896, 576, 0, 4], 'outer_gens': [[17, 10, 16, 56], [73, 26, 16, 440], [321, 30, 16, 416], [601, 18, 16, 952], [1105, 22, 16, 1320], [717, 26, 16, 1008]], 'outer_group': '192.1543', 'outer_hash': 1543, 'outer_nilpotent': True, 'outer_order': 192, 'outer_permdeg': 192, 'outer_perms': 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True, 'ratrep_stats': [[1, 8], [2, 10], [4, 5], [6, 4], [8, 2], [12, 8], [16, 1], [24, 5], [48, 2], [96, 1]], 'representations': {'PC': {'code': 34684952322691369526056445042832616069810595339, 'gens': [1, 2, 5, 6], 'pres': [8, -2, -2, -2, -2, 2, 2, -3, -7, 97, 41, 290, 66, 63365, 1165, 141, 71686, 222, 73735]}, 'Perm': {'d': 30, 'gens': [8888283220636017750421426274665, 19253942259196942134867124972800, 28432161090949839894603242956800, 6266937600, 36894763266095368932801414912000, 46390244693197320526866038784000, 3, 444984]}}, 'schur_multiplier': [2, 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': 'D_{84}.D_4', 'transitive_degree': 672, 'wreath_data': None, 'wreath_product': False}
