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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': True, 'ambient': '1024.dii', 'ambient_counter': 2245, 'ambient_order': 1024, 'ambient_tex': 'C_4:D_8.D_8', 'central': False, 'central_factor': False, 'centralizer_order': 16, 'characteristic': False, 'core_order': 8, 'counter': 169, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '1024.dii.64.g1.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': False, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': False, 'old_label': '64.g1.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': 64, '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': '16.10', 'subgroup_hash': 10, 'subgroup_order': 16, 'subgroup_tex': 'C_2^2\\times C_4', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '1024.dii', 'aut_centralizer_order': 64, 'aut_label': '64.g1', 'aut_quo_index': None, 'aut_stab_index': 8, 'aut_weyl_group': '16.11', 'aut_weyl_index': 512, 'centralizer': '64.g1.a1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['32.h1.a1', '32.i1.a1', '32.l1.a1'], 'contains': ['128.a1.a1', '128.p1.a1', '128.bi1.a1'], 'core': '128.a1.a1', 'coset_action_label': None, 'count': 8, 'diagramx': [6384, -1, 7598, -1, 6375, -1, 7607, -1], 'generators': [940, 832, 512], 'label': '1024.dii.64.g1.a1', 'mobius_quo': None, 'mobius_sub': 0, 'normal_closure': '8.b1.a1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '8.i1.a1', 'old_label': '64.g1.a1', 'projective_image': '512.1866', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '64.g1.a1', 'subgroup_fusion': None, 'weyl_group': '8.3'}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': True, 'abelian_quotient': '16.10', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 4, 'aut_exponent': 12, 'aut_gen_orders': [2, 3, 2, 2, 2, 2, 2], 'aut_gens': [[1, 2, 4], [10, 9, 7], [10, 3, 15], [1, 2, 5], [1, 2, 14], [9, 2, 14], [1, 10, 13], [1, 2, 12]], 'aut_group': '192.1493', 'aut_hash': 1493, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 192, 'aut_permdeg': 8, 'aut_perms': [17764, 31287, 5329, 40319, 37965, 21769, 12316], 'aut_phi_ratio': 24.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 1, 6, 1], [4, 1, 8, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_2^3:S_4', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 12, 'autcent_group': '192.1493', 'autcent_hash': 1493, 'autcent_nilpotent': False, 'autcent_order': 192, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': False, 'autcent_tex': 'C_2^3:S_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, 7], [4, 1, 8]], 'center_label': '16.10', 'center_order': 16, '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', '2.1'], 'composition_length': 4, 'conjugacy_classes_known': True, 'counter': 10, 'cyclic': False, 'derived_length': 1, 'dihedral': False, 'direct_factorization': [['2.1', 2], ['4.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 7], [4, 1, 2, 4]], 'element_repr_type': 'PC', 'elementary': 2, 'eulerian_function': 7, 'exponent': 4, 'exponents_of_order': [4], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2], 'faithful_reps': [], 'familial': False, 'frattini_label': '2.1', 'frattini_quotient': '8.5', 'hash': 10, 'hyperelementary': 2, 'inner_abelian': True, 'inner_cyclic': True, 'inner_exponent': 1, 'inner_gen_orders': [1, 1, 1], 'inner_gens': [[1, 2, 4], [1, 2, 4], [1, 2, 4]], '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, 16]], 'label': '16.10', 'linC_count': 224, 'linC_degree': 3, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 4, 'linQ_degree_count': 48, 'linQ_dim': 4, 'linQ_dim_count': 48, 'linR_count': 48, 'linR_degree': 4, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'C2^2*C4', 'ngens': 3, 'nilpotency_class': 1, 'nilpotent': True, 'normal_counts': [0, 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': 16, 'number_divisions': 12, 'number_normal_subgroups': 27, 'number_subgroup_autclasses': 9, 'number_subgroup_classes': 27, 'number_subgroups': 27, 'old_label': None, 'order': 16, 'order_factorization_type': 3, 'order_stats': [[1, 1], [2, 7], [4, 8]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 12, 'outer_gen_orders': [2, 3, 2, 2, 2, 2, 2], 'outer_gen_pows': [0, 0, 0, 0, 0, 0, 0], 'outer_gens': [[10, 9, 7], [10, 3, 15], [1, 2, 5], [1, 2, 14], [9, 2, 14], [1, 10, 13], [1, 2, 12]], 'outer_group': '192.1493', 'outer_hash': 1493, 'outer_nilpotent': False, 'outer_order': 192, 'outer_permdeg': 8, 'outer_perms': [17764, 31287, 5329, 40319, 37965, 21769, 12316], 'outer_solvable': True, 'outer_supersolvable': False, 'outer_tex': 'C_2^3:S_4', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 8, 'pgroup': 2, 'primary_abelian_invariants': [2, 2, 4], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 8], [2, 4]], 'representations': {'PC': {'code': 516, 'gens': [1, 2, 3], 'pres': [4, -2, 2, 2, -2, 34]}, 'GLZ': {'b': 3, 'd': 4, 'gens': [7233746, 7115326, 7115160]}, 'GLFp': {'d': 3, 'p': 5, 'gens': [1173753, 1832002, 438254, 1565004]}, 'Perm': {'d': 8, 'gens': [22, 5040, 120, 7]}}, 'schur_multiplier': [2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 4], 'solvability_type': 2, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_2^2\\times C_4', 'transitive_degree': 16, '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': '8.2', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 3, 'aut_exponent': 16, 'aut_gen_orders': [2, 4, 4, 4, 4, 8], 'aut_gens': [[1, 4, 16, 128], [3, 716, 600, 896], [769, 4, 856, 640], [673, 580, 16, 128], [769, 516, 848, 640], [1, 556, 28, 40], [933, 932, 16, 384]], 'aut_group': None, 'aut_hash': 79916486570037805, 'aut_nilpotency_class': 6, 'aut_nilpotent': True, 'aut_order': 8192, 'aut_permdeg': 288, 'aut_perms': 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213043553545280607970306202270065685155411456843944840529076376752285371036315694876483655551070974488488122606568037191853173555957568011886051642287876236800968450177651341378417236430747504423139385208632724933662728765217008105091302850878235378785234574139585997696802646845711501136888135892676960244706643889631900728302425016923058375927700786385469541708095106948008120359789369147438722617680558165663283869536267988327607299369801183210454577111272990615229058985164917661508540216231931264222383885329028272353412118396612104631229979793039877350195062898529173933860356920], 'aut_phi_ratio': 16.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 2, 1, 3], [2, 32, 1, 3], [4, 4, 1, 4], [4, 8, 1, 1], [4, 32, 1, 1], [4, 64, 1, 1], [4, 64, 4, 1], [8, 8, 1, 4], [8, 8, 4, 1], [8, 16, 2, 1], [8, 32, 2, 1], [16, 32, 4, 1], [16, 32, 8, 1]], 'aut_supersolvable': True, 'aut_tex': '(C_2\\times C_4:D_4).C_2^6.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': False, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^2', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 8, 'autcentquo_group': '2048.cpu', 'autcentquo_hash': 3144402051166287857, 'autcentquo_nilpotent': True, 'autcentquo_order': 2048, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'D_4^2.C_2^5', 'cc_stats': [[1, 1, 1], [2, 1, 1], [2, 2, 3], [2, 32, 3], [4, 4, 4], [4, 8, 1], [4, 32, 1], [4, 64, 5], [8, 8, 8], [8, 16, 2], [8, 32, 2], [16, 32, 12]], 'center_label': '2.1', 'center_order': 2, 'central_product': False, 'central_quotient': '512.1866', 'commutator_count': 1, 'commutator_label': '128.576', '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', '2.1', '2.1'], 'composition_length': 10, 'conjugacy_classes_known': True, 'counter': 2245, 'cyclic': False, 'derived_length': 3, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 2, 1, 3], [2, 32, 1, 3], [4, 4, 1, 4], [4, 8, 1, 1], [4, 32, 1, 1], [4, 64, 1, 1], [4, 64, 2, 2], [8, 8, 1, 4], [8, 8, 4, 1], [8, 16, 2, 1], [8, 32, 2, 1], [16, 32, 2, 2], [16, 32, 4, 2]], 'element_repr_type': 'PC', 'elementary': 2, 'eulerian_function': 48, 'exponent': 16, 'exponents_of_order': [10], 'factors_of_aut_order': [2], 'factors_of_order': [2], 'faithful_reps': [[16, 1, 2]], 'familial': False, 'frattini_label': '256.2825', 'frattini_quotient': '4.2', 'hash': 2101610243620390082, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 16, 'inner_gen_orders': [4, 4, 8, 4], 'inner_gens': [[1, 556, 28, 40], [105, 4, 752, 384], [933, 932, 16, 384], [745, 772, 784, 128]], 'inner_hash': 8985888352205780980, 'inner_nilpotent': True, 'inner_order': 512, 'inner_split': False, 'inner_tex': 'C_2^3.C_2\\wr C_4', 'inner_used': [1, 2, 3], 'irrC_degree': 16, 'irrQ_degree': 16, 'irrQ_dim': 16, 'irrR_degree': 16, 'irrep_stats': [[1, 8], [2, 10], [4, 21], [8, 2], [16, 2]], 'label': '1024.dii', 'linC_count': 2, 'linC_degree': 16, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 16, 'linQ_degree_count': 2, 'linQ_dim': 16, 'linQ_dim_count': 2, 'linR_count': 2, 'linR_degree': 16, 'maximal_subgroups_known': True, 'metabelian': False, 'metacyclic': False, 'monomial': True, 'name': 'C4:D8.D8', 'ngens': 2, 'nilpotency_class': 7, 'nilpotent': True, 'normal_counts': [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': 25, 'number_characteristic_subgroups': 27, 'number_conjugacy_classes': 43, 'number_divisions': 28, 'number_normal_subgroups': 27, 'number_subgroup_autclasses': 279, 'number_subgroup_classes': 313, 'number_subgroups': 2901, 'old_label': None, 'order': 1024, 'order_factorization_type': 7, 'order_stats': [[1, 1], [2, 103], [4, 376], [8, 160], [16, 384]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 2, 'outer_gen_orders': [2, 2, 2, 2], 'outer_gen_pows': [0, 8, 64, 8], 'outer_gens': [[3, 716, 600, 896], [769, 4, 856, 640], [673, 580, 16, 128], [769, 516, 848, 640]], 'outer_group': '16.14', 'outer_hash': 14, 'outer_nilpotent': True, 'outer_order': 16, 'outer_permdeg': 8, 'outer_perms': [5040, 120, 6, 1], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^4', 'pc_rank': 4, 'perfect': False, 'permutation_degree': 32, 'pgroup': 2, 'primary_abelian_invariants': [2, 4], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 4], [4, 9], [8, 8], [16, 3]], 'representations': {'PC': {'code': '33440014478390050552912017046848909401624210676892595164840054554654072041162920565732080483829', 'gens': [1, 3, 5, 8], 'pres': [10, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 20, 16682, 14112, 82, 20803, 1404, 2814, 9424, 144, 23525, 18255, 1465, 175, 53766, 20176, 10106, 626, 3207, 35857, 7707, 1967, 237, 5768, 34578, 17308, 4368, 268]}, 'Perm': {'d': 32, 'gens': [195152660219665819875502796339963392, 118943267166205684098373210499907247]}}, 'schur_multiplier': [2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 4], 'solvability_type': 5, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_4:D_8.D_8', 'transitive_degree': 32, 'wreath_data': None, 'wreath_product': False}
