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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': True, 'ambient': '1600.1933', 'ambient_counter': 1933, 'ambient_order': 1600, 'ambient_tex': 'C_{20}^2.C_2^2', 'central': True, 'central_factor': False, 'centralizer_order': 1600, 'characteristic': True, 'core_order': 4, 'counter': 77, 'cyclic': False, 'direct': False, 'hall': 0, 'label': '1600.1933.400.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': True, 'old_label': '400.a1', 'outer_equivalence': True, 'perfect': False, 'proper': True, 'quotient': '400.107', 'quotient_Agroup': False, 'quotient_abelian': False, 'quotient_cyclic': False, 'quotient_hash': 107, 'quotient_metabelian': True, 'quotient_nilpotent': False, 'quotient_order': 400, 'quotient_simple': False, 'quotient_solvable': True, 'quotient_supersolvable': True, 'quotient_tex': 'C_{10}^2:C_4', 'simple': False, 'solvable': True, 'special_labels': ['Z', 'U2'], 'split': False, 'standard_generators': False, 'stem': True, 'subgroup': '4.2', 'subgroup_hash': 2, 'subgroup_order': 4, 'subgroup_tex': 'C_2^2', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '1600.1933', 'aut_centralizer_order': 3072000, 'aut_label': '400.a1', 'aut_quo_index': 2, 'aut_stab_index': 1, 'aut_weyl_group': '1.1', 'aut_weyl_index': 3072000, 'centralizer': '1.a1', 'complements': [], 'conjugacy_class_count': 1, 'contained_in': ['80.a1', '200.a1', '200.b1', '200.c1', '200.d1'], 'contains': ['800.a1', '800.b1', '800.c1'], 'core': '400.a1', 'coset_action_label': None, 'count': 1, 'diagramx': [4526, 5624, 4281, 6308], 'generators': [40, 800], 'label': '1600.1933.400.a1', 'mobius_quo': 2, 'mobius_sub': 0, 'normal_closure': '400.a1', 'normal_contained_in': ['80.a1', '200.a1', '200.b1', '200.c1'], 'normal_contains': ['800.a1', '800.b1', '800.c1'], 'normalizer': '1.a1', 'old_label': '400.a1', 'projective_image': '400.107', 'quotient_action_image': '1.1', 'quotient_action_kernel': '400.107', 'quotient_action_kernel_order': 400, 'quotient_fusion': None, 'short_label': '400.a1', 'subgroup_fusion': None, 'weyl_group': '1.1'}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': True, 'abelian_quotient': '4.2', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 6, 'aut_gen_orders': [2, 3], 'aut_gens': [[1, 2], [3, 2], [2, 3]], 'aut_group': '6.1', 'aut_hash': 1, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 6, 'aut_permdeg': 3, 'aut_perms': [1, 4], 'aut_phi_ratio': 3.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 3, 1]], 'aut_supersolvable': True, 'aut_tex': 'S_3', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 6, 'autcent_group': '6.1', 'autcent_hash': 1, 'autcent_nilpotent': False, 'autcent_order': 6, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'S_3', '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]], 'center_label': '4.2', 'center_order': 4, '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'], 'composition_length': 2, 'conjugacy_classes_known': True, 'counter': 2, 'cyclic': False, 'derived_length': 1, 'dihedral': True, 'direct_factorization': [['2.1', 2]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3]], 'element_repr_type': 'PC', 'elementary': 2, 'eulerian_function': 1, 'exponent': 2, 'exponents_of_order': [2], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2], 'faithful_reps': [], 'familial': True, 'frattini_label': '1.1', '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, 4]], 'label': '4.2', 'linC_count': 3, 'linC_degree': 2, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 2, 'linQ_degree_count': 3, 'linQ_dim': 2, 'linQ_dim_count': 3, 'linR_count': 3, 'linR_degree': 2, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C2^2', 'ngens': 2, 'nilpotency_class': 1, 'nilpotent': True, 'normal_counts': [0, 0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 2, 'number_characteristic_subgroups': 2, 'number_conjugacy_classes': 4, 'number_divisions': 4, 'number_normal_subgroups': 5, 'number_subgroup_autclasses': 3, 'number_subgroup_classes': 5, 'number_subgroups': 5, 'old_label': None, 'order': 4, 'order_factorization_type': 2, 'order_stats': [[1, 1], [2, 3]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 6, 'outer_gen_orders': [2, 3], 'outer_gen_pows': [0, 0], 'outer_gens': [[3, 2], [2, 3]], 'outer_group': '6.1', 'outer_hash': 1, 'outer_nilpotent': False, 'outer_order': 6, 'outer_permdeg': 3, 'outer_perms': [1, 4], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'S_3', 'pc_rank': 2, 'perfect': False, 'permutation_degree': 4, 'pgroup': 2, 'primary_abelian_invariants': [2, 2], 'quasisimple': False, 'rank': 2, 'rational': True, 'rational_characters_known': True, 'ratrep_stats': [[1, 4]], 'representations': {'PC': {'code': 0, 'gens': [1, 2], 'pres': [2, -2, 2]}, 'GLZ': {'b': 3, 'd': 2, 'gens': [14, 12]}, 'GLFp': {'d': 2, 'p': 3, 'gens': [55, 56]}, 'Perm': {'d': 4, 'gens': [6, 1]}}, 'schur_multiplier': [2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2], 'solvability_type': 1, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_2^2', 'transitive_degree': 4, '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': 2, 'aut_exponent': 120, 'aut_gen_orders': [10, 24, 24, 20, 4, 20, 12, 10], 'aut_gens': [[1, 4, 80], [769, 510, 288], [867, 334, 1112], [851, 1540, 1568], [339, 526, 1080], [801, 598, 280], [51, 958, 1264], [963, 172, 472], [745, 1486, 1520]], 'aut_group': None, 'aut_hash': 2834573797362515646, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 3072000, 'aut_permdeg': 416, 'aut_perms': 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3], [4, 2, 1, 4], [4, 4, 1, 1], [4, 8, 2, 1], [5, 2, 12, 1], [8, 100, 4, 2], [10, 2, 12, 3], [20, 4, 12, 4], [20, 4, 24, 1], [20, 8, 48, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_{10}^2.C_2^4.C_2^4.S_5', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 2, 'autcent_group': '16.14', 'autcent_hash': 14, 'autcent_nilpotent': True, 'autcent_order': 16, 'autcent_solvable': True, 'autcent_split': False, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^4', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 120, 'autcentquo_group': None, 'autcentquo_hash': 8686510852381798618, 'autcentquo_nilpotent': False, 'autcentquo_order': 192000, 'autcentquo_solvable': False, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'C_2^4\\times C_5^2:C_4.S_5', 'cc_stats': [[1, 1, 1], [2, 1, 3], [4, 2, 4], [4, 4, 1], [4, 8, 2], [5, 2, 12], [8, 100, 8], [10, 2, 36], [20, 4, 72], [20, 8, 48]], 'center_label': '4.2', 'center_order': 4, 'central_product': False, 'central_quotient': '400.107', 'commutator_count': 2, 'commutator_label': '200.37', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '2.1', '2.1', '5.1', '5.1'], 'composition_length': 8, 'conjugacy_classes_known': True, 'counter': 1933, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [4, 2, 1, 4], [4, 4, 1, 1], [4, 8, 1, 2], [5, 2, 2, 6], [8, 100, 4, 2], [10, 2, 2, 18], [20, 4, 2, 24], [20, 4, 4, 6], [20, 8, 4, 12]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 672, 'exponent': 40, 'exponents_of_order': [6, 2], 'factors_of_aut_order': [2, 3, 5], 'factors_of_order': [2, 5], 'faithful_reps': [], 'familial': False, 'frattini_label': '16.2', 'frattini_quotient': '100.15', 'hash': 1933, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 20, 'inner_gen_orders': [4, 10, 10], 'inner_gens': [[1, 1276, 1560], [409, 4, 880], [201, 804, 80]], 'inner_hash': 107, 'inner_nilpotent': False, 'inner_order': 400, 'inner_split': True, 'inner_tex': 'C_{10}^2:C_4', 'inner_used': [1, 2, 3], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 8], [2, 106], [4, 73]], 'label': '1600.1933', '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': True, 'metacyclic': False, 'monomial': True, 'name': 'C20^2.C2^2', '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], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 22, 'number_characteristic_subgroups': 31, 'number_conjugacy_classes': 187, 'number_divisions': 79, 'number_normal_subgroups': 131, 'number_subgroup_autclasses': 87, 'number_subgroup_classes': 256, 'number_subgroups': 1176, 'old_label': None, 'order': 1600, 'order_factorization_type': 32, 'order_stats': [[1, 1], [2, 3], [4, 28], [5, 24], [8, 800], [10, 72], [20, 672]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': True, 'outer_exponent': 120, 'outer_gen_orders': [2, 2, 8, 2, 5, 4, 4, 5], 'outer_gen_pows': [0, 0, 0, 0, 0, 0, 0, 0], 'outer_gens': [[1, 340, 416], [801, 4, 80], [3, 934, 592], [3, 1284, 760], [1, 1300, 1376], [1, 908, 584], [43, 30, 1400], [1, 1348, 416]], 'outer_group': '7680.l', 'outer_hash': 230622914464470435, 'outer_nilpotent': False, 'outer_order': 7680, 'outer_permdeg': 32, 'outer_perms': [36959649061688220820691354112000, 7, 100970087257375343903351508398438423, 194796596688315575831314068338352016, 25903528126242395981075250910387200, 174000775000659543679594939735096, 135589052664116702891571820889041927, 146482890515752047114163796582400], 'outer_solvable': False, 'outer_supersolvable': False, 'outer_tex': 'C_2^4\\times \\GL(2,5)', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 26, 'pgroup': 0, 'primary_abelian_invariants': [2, 4], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 4], [4, 29], [8, 36], [16, 6]], 'representations': {'PC': {'code': 7187580092729912892798483186053695849409856016351633539267, 'gens': [1, 3, 6], 'pres': [8, 2, 2, 2, 2, 5, 2, 2, 5, 16, 329, 30626, 538, 66, 2307, 91, 2564, 74885, 10581, 141, 80646, 166, 81927]}, 'Perm': {'d': 26, 'gens': [17427786990278895653109847, 31103501533603057354118400, 49769633213511844527417600, 65896948806048336936960000, 1178234244633600, 78825098250903121108992000, 444240, 444277]}}, 'schur_multiplier': [10], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 4], 'solvability_type': 7, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_{20}^2.C_2^2', 'transitive_degree': 1600, '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': 2, 'aut_exponent': 120, 'aut_gen_orders': [12, 12, 4, 20, 24], 'aut_gens': [[1, 4, 40], [151, 124, 304], [245, 196, 392], [373, 126, 384], [213, 148, 280], [29, 148, 72]], 'aut_group': None, 'aut_hash': 2802418470926142505, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 384000, 'aut_permdeg': 200, 'aut_perms': [455773741080477467034735534115940404183908025401908906714364286690116024567853504747615849593109247914221039876856974528537921060722172699931726708440611088007935276839904891040169348946796258524884590040204806764544810281293558742419139958332378002037633159740383421163375321862004502300590554968681043962076251660128311382291177195037693834503449467160655415964798978487618, 478302318665108962854312932147227403138904263850937681530554049897495693585400002091169886317093319559684247710881253395003308540088350761634795011520337592303291077242710830759422940104155772375018449383592095010830225299663586016570854999475296475920033809611516413538465675945179570540729304201647979587404698305976697408508159763085773479420687609208623737830039643435828, 414683307297065809647763320442971723164725483320152959836872356253788835179346379690540058528933121110777392232812361011692156308899833700510014107831009759524194550367475108728725360835280547101533803100454091684891108823530183016427149162041172039653245501883039832812044764908097594574570768251493688291391742052009078169224675622919071365015545361123001944688370212219248, 637940182778480774892275767958345851429868952381931806813877922203910728857479515775608939928268052926761340867699951272101888430765375671596340490052980509266382979724066204071934838968237696728939903970564737588488534343209481362599174309445958698936249146563964578378891006714454603162283838539210805426400749824129474141694540231923747968151011131813613004220003569430522, 562477266638267891065291400098660221750072157011985511539522746807397326480225361678297774364222458123564481770236522509998740518758830862139465490328068496490211550349377684405621352273296987748106263483099217763172311375814047864882300821501120308730955040621141275461543931354082604749314662755970803583398399866386545487272977020215887115127408175385700233886651445276224], 'aut_phi_ratio': 2400.0, 'aut_solvable': False, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 1, 2, 1], [2, 2, 2, 1], [4, 50, 4, 1], [5, 2, 12, 1], [10, 2, 12, 1], [10, 2, 24, 1], [10, 2, 48, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_5^2.C_2^5.C_2^2.S_5', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 2, 'autcent_group': '16.14', 'autcent_hash': 14, 'autcent_nilpotent': True, 'autcent_order': 16, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^4', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 120, 'autcentquo_group': '24000.bv', 'autcentquo_hash': 6497402662917918296, 'autcentquo_nilpotent': False, 'autcentquo_order': 24000, 'autcentquo_solvable': False, 'autcentquo_supersolvable': False, 'autcentquo_tex': '(C_5\\times C_{10}):\\GL(2,5)', 'cc_stats': [[1, 1, 1], [2, 1, 3], [2, 2, 2], [4, 50, 4], [5, 2, 12], [10, 2, 84]], 'center_label': '4.2', 'center_order': 4, 'central_product': False, 'central_quotient': '100.15', 'commutator_count': 1, 'commutator_label': '50.5', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '5.1', '5.1'], 'composition_length': 6, 'conjugacy_classes_known': True, 'counter': 107, '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], [4, 50, 2, 2], [5, 2, 2, 6], [10, 2, 2, 18], [10, 2, 4, 12]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 84, 'exponent': 20, 'exponents_of_order': [4, 2], 'factors_of_aut_order': [2, 3, 5], 'factors_of_order': [2, 5], 'faithful_reps': [], 'familial': False, 'frattini_label': '4.2', 'frattini_quotient': '100.15', 'hash': 107, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 10, 'inner_gen_orders': [2, 10, 5], 'inner_gens': [[1, 236, 360], [209, 4, 40], [81, 4, 40]], 'inner_hash': 15, 'inner_nilpotent': False, 'inner_order': 100, 'inner_split': True, 'inner_tex': 'C_5:D_{10}', 'inner_used': [1, 2, 3], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 8], [2, 98]], 'label': '400.107', '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': True, 'metacyclic': False, 'monomial': True, 'name': 'C10^2:C4', 'ngens': 6, 'nilpotency_class': -1, 'nilpotent': False, 'normal_counts': [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': 9, 'number_characteristic_subgroups': 9, 'number_conjugacy_classes': 106, 'number_divisions': 44, 'number_normal_subgroups': 67, 'number_subgroup_autclasses': 33, 'number_subgroup_classes': 136, 'number_subgroups': 520, 'old_label': None, 'order': 400, 'order_factorization_type': 32, 'order_stats': [[1, 1], [2, 7], [4, 200], [5, 24], [10, 168]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': True, 'outer_exponent': 120, 'outer_gen_orders': [2, 4, 3, 2, 2], 'outer_gen_pows': [0, 0, 0, 0, 0], 'outer_gens': [[1, 6, 40], [1, 92, 216], [1, 340, 392], [21, 4, 40], [223, 204, 40]], 'outer_group': '3840.g', 'outer_hash': 5839622094200158414, 'outer_nilpotent': False, 'outer_order': 3840, 'outer_permdeg': 28, 'outer_perms': [16, 179264660911629365283380472967, 170894953514334592753142291280, 1, 6], 'outer_solvable': False, 'outer_supersolvable': False, 'outer_tex': 'D_4\\times \\GL(2,5)', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 18, 'pgroup': 0, 'primary_abelian_invariants': [2, 4], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 4], [4, 24], [8, 12]], 'representations': {'PC': {'code': 52955791448541918505291827, 'gens': [1, 3, 5], 'pres': [6, -2, -2, -2, -5, -2, -5, 12, 4250, 50, 771, 10804, 88, 11525]}, 'GLZN': {'d': 2, 'p': 55, 'gens': [7669476, 166981, 3493896, 6899511, 5355593, 5656784]}, 'Perm': {'d': 18, 'gens': [21010055276599, 12341, 13075, 19342, 397620792172800, 1041828480]}}, 'schur_multiplier': [2, 10], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 4], 'solvability_type': 7, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_{10}^2:C_4', 'transitive_degree': 200, 'wreath_data': None, 'wreath_product': False}
