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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': True, 'ambient': '256.53563', 'ambient_counter': 53563, 'ambient_order': 256, 'ambient_tex': 'C_4^2.C_2^4', 'central': False, 'central_factor': False, 'centralizer_order': 32, 'characteristic': False, 'core_order': 4, 'counter': 834, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '256.53563.32.bb1.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': False, 'old_label': '32.bb1.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': 32, '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': '256.53563', 'aut_centralizer_order': None, 'aut_label': '32.bb1', 'aut_quo_index': None, 'aut_stab_index': None, 'aut_weyl_group': None, 'aut_weyl_index': None, 'centralizer': '8.gd1.a1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['16.be1.a1', '16.bx1.a1', '16.cy1.a1', '16.cy1.b1', '16.dc1.a1', '16.dd1.a1', '16.dk1.a1', '16.do1.a1', '16.dr1.a1', '16.ee1.a1', '16.ee1.b1', '16.eg1.a1', '16.eg1.b1', '16.ej1.a1', '16.ej1.b1'], 'contains': ['64.a1.a1', '64.y1.a1'], 'core': '64.a1.a1', 'coset_action_label': None, 'count': 2, 'diagramx': [3760, -1, 8816, -1, 3757, -1, 8801, -1], 'generators': [46, 128], 'label': '256.53563.32.bb1.a1', 'mobius_quo': None, 'mobius_sub': 0, 'normal_closure': '16.be1.a1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '2.l1.a1', 'old_label': '32.bb1.a1', 'projective_image': None, 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '32.bb1.a1', 'subgroup_fusion': None, 'weyl_group': None}
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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': '32.51', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 4, 'aut_gen_orders': [4, 4, 2, 4, 4, 4, 4, 2, 4, 4, 4], 'aut_gens': [[1, 2, 4, 8, 32], [69, 146, 148, 152, 96], [21, 210, 20, 8, 176], [17, 146, 20, 24, 32], [21, 66, 132, 152, 32], [197, 18, 4, 152, 160], [209, 82, 148, 24, 48], [193, 210, 4, 152, 160], [129, 18, 20, 24, 176], [133, 82, 20, 8, 240], [149, 82, 132, 24, 48], [193, 194, 148, 24, 32]], 'aut_group': None, 'aut_hash': 8629282911301114921, 'aut_nilpotency_class': 2, 'aut_nilpotent': True, 'aut_order': 8192, 'aut_permdeg': 28, 'aut_perms': [107077900298823049332231288483, 245153381026689305381191891211, 207470592193846155904946107645, 240173802677333567541019031481, 213970744620876774852267294801, 249838041843115351998860503628, 279168123845224661836841620220, 168803420436965589278204330047, 1598918940645192738655810901, 250008674145597799573364937826, 65755796486105479361393386885], 'aut_phi_ratio': 64.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [2, 2, 2, 1], [2, 4, 2, 2], [2, 4, 4, 1], [2, 8, 1, 1], [4, 2, 1, 2], [4, 2, 2, 5], [4, 4, 2, 6], [4, 4, 4, 1], [4, 8, 1, 3], [4, 8, 2, 2], [8, 8, 1, 4], [8, 8, 2, 2]], 'aut_supersolvable': True, 'aut_tex': 'C_2^9.C_2^4', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 2, 'autcent_group': '1024.djt', 'autcent_hash': 1957374279625110839, 'autcent_nilpotent': True, 'autcent_order': 1024, 'autcent_solvable': True, 'autcent_split': False, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^{10}', 'autcentquo_abelian': True, 'autcentquo_cyclic': False, 'autcentquo_exponent': 2, 'autcentquo_group': '8.5', 'autcentquo_hash': 5, 'autcentquo_nilpotent': True, 'autcentquo_order': 8, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'C_2^3', 'cc_stats': [[1, 1, 1], [2, 1, 3], [2, 2, 2], [2, 4, 8], [2, 8, 1], [4, 2, 12], [4, 4, 16], [4, 8, 7], [8, 8, 8]], 'center_label': '4.2', 'center_order': 4, 'central_product': False, 'central_quotient': '64.261', 'commutator_count': 1, 'commutator_label': '8.2', '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'], 'composition_length': 8, 'conjugacy_classes_known': True, 'counter': 53563, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [2, 2, 1, 2], [2, 4, 1, 8], [2, 8, 1, 1], [4, 2, 1, 4], [4, 2, 2, 4], [4, 4, 1, 8], [4, 4, 2, 4], [4, 8, 1, 7], [8, 8, 1, 8]], 'element_repr_type': 'PC', 'elementary': 2, 'eulerian_function': 39997440, 'exponent': 8, 'exponents_of_order': [8], 'factors_of_aut_order': [2], 'factors_of_order': [2], 'faithful_reps': [], 'familial': False, 'frattini_label': '8.2', 'frattini_quotient': '32.51', 'hash': 53563, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 4, 'inner_gen_orders': [2, 2, 2, 2, 4], 'inner_gens': [[1, 2, 4, 8, 240], [1, 2, 132, 8, 224], [1, 130, 4, 8, 32], [1, 2, 4, 8, 160], [81, 66, 4, 136, 32]], 'inner_hash': 261, 'inner_nilpotent': True, 'inner_order': 64, 'inner_split': None, 'inner_tex': 'D_4\\times C_2^3', 'inner_used': [1, 2, 3, 4, 5], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 32], [2, 24], [8, 2]], 'label': '256.53563', 'linC_count': 32, 'linC_degree': 10, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 12, 'linQ_degree_count': 16, 'linQ_dim': 12, 'linQ_dim_count': 8, 'linR_count': 8, 'linR_degree': 12, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'C4^2.C2^4', 'ngens': 5, 'nilpotency_class': 3, 'nilpotent': True, 'normal_counts': [0, 0, 0, 0, 0, 0, 0, 0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 34, 'number_characteristic_subgroups': 143, 'number_conjugacy_classes': 58, 'number_divisions': 50, 'number_normal_subgroups': 455, 'number_subgroup_autclasses': 592, 'number_subgroup_classes': 1040, 'number_subgroups': 1983, 'old_label': None, 'order': 256, 'order_factorization_type': 7, 'order_stats': [[1, 1], [2, 47], [4, 144], [8, 64]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 4, 'outer_gen_orders': [4, 4, 2, 2, 2, 2, 2, 2, 4, 2, 2], 'outer_gen_pows': [0, 0, 0, 192, 196, 192, 192, 0, 0, 196, 192], 'outer_gens': [[69, 146, 148, 152, 96], [21, 210, 20, 8, 176], [17, 146, 20, 24, 32], [21, 66, 132, 152, 32], [197, 18, 4, 152, 160], [209, 82, 148, 24, 48], [193, 210, 4, 152, 160], [129, 18, 20, 24, 176], [133, 82, 20, 8, 240], [197, 18, 4, 152, 160], [193, 194, 148, 24, 32]], 'outer_group': '128.2320', 'outer_hash': 2320, 'outer_nilpotent': True, 'outer_order': 128, 'outer_permdeg': 64, 'outer_perms': [2270593173496970456544013998671653392619652730544112298438779855376992108314772773153713, 4284679856563135271478408937217278372255413320053611805650993968564612175235550589164420, 354436061395690320680056180853551317276392807756217244815793844943197502771472220917569, 6327495781516968087196122707857682334241371396491049672821453589376278221002161007764836, 8096470311358869273525103073254845194790208555305239207192228332601686523786017394594060, 10331423583991866657948611541304517612037518096210577653788562542589984646353082158382476, 12031373727147530462952462388213046397440158114934954782243779243145030488163471199395929, 14329201563188950229033342073107338130493793110584538913525295135508193219990159685491221, 16343795894130308808158237492323398317882745320535281636984199144214949186421247162281722, 8096470311358869273525103073254845194790208555305239207192228332601686523786017394594060, 18358388378932299605546426331483902233216860823210245000949006719953520943719855024628900], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'D_4\\times C_2^4', 'pc_rank': 5, 'perfect': False, 'permutation_degree': 24, 'pgroup': 2, 'primary_abelian_invariants': [2, 2, 2, 2, 2], 'quasisimple': False, 'rank': 5, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 32], [2, 8], [4, 8], [8, 2]], 'representations': {'PC': {'code': 636330232345205365887463741544, 'gens': [1, 2, 3, 4, 6], 'pres': [8, 2, 2, 2, 2, 2, 2, 2, 2, 1594, 91, 11525, 5389, 989, 141, 10758, 5390, 166]}, 'Perm': {'d': 24, 'gens': [29441336855979612979589, 57332193832825718693455, 84505521347099318194255, 105016267535968244346094, 85936113038586329164800, 17764, 138425821237256267249764, 114008274095960303498880]}}, 'schur_multiplier': [2, 2, 2, 2, 2, 2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [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': 'C_4^2.C_2^4', 'transitive_degree': 64, 'wreath_data': None, 'wreath_product': False}
