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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': False, 'ambient': '384.17956', 'ambient_counter': 17956, 'ambient_order': 384, 'ambient_tex': '\\GL(2,3):D_4', 'central': False, 'central_factor': False, 'centralizer_order': 8, 'characteristic': False, 'core_order': 1, 'counter': 267, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '384.17956.64.f1.a2', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': False, 'minimal_normal': False, 'nilpotent': False, 'normal': False, 'old_label': '64.f1.a2', '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': '6.1', 'subgroup_hash': 1, 'subgroup_order': 6, 'subgroup_tex': 'S_3', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '384.17956', 'aut_centralizer_order': 16, 'aut_label': '64.f1', 'aut_quo_index': None, 'aut_stab_index': 16, 'aut_weyl_group': '6.1', 'aut_weyl_index': 256, 'centralizer': '48.a1.a1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['32.d1.a2', '32.g1.a1', '32.l1.a1', '32.o1.a2', '32.o1.b2', '32.o1.c1', '32.o1.d1'], 'contains': ['128.a1.a1', '192.g1.a1'], 'core': '384.a1.a1', 'coset_action_label': None, 'count': 8, 'diagramx': [432, -1, 1016, -1, 781, -1, 9310, -1], 'generators': [5734448, 3333742], 'label': '384.17956.64.f1.a2', 'mobius_quo': None, 'mobius_sub': 0, 'normal_closure': '4.d1.a1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '8.c1.a1', 'old_label': '64.f1.a2', 'projective_image': '384.17956', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '64.f1.a2', 'subgroup_fusion': None, 'weyl_group': '6.1'}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': False, 'abelian_quotient': '2.1', '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, 4], [1, 3], [2, 4]], '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, 3, 1, 1], [3, 2, 1, 1]], 'aut_supersolvable': True, 'aut_tex': 'S_3', 'autcent_abelian': True, 'autcent_cyclic': True, 'autcent_exponent': 1, 'autcent_group': '1.1', 'autcent_hash': 1, 'autcent_nilpotent': True, 'autcent_order': 1, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_1', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 6, 'autcentquo_group': '6.1', 'autcentquo_hash': 1, 'autcentquo_nilpotent': False, 'autcentquo_order': 6, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'S_3', 'cc_stats': [[1, 1, 1], [2, 3, 1], [3, 2, 1]], 'center_label': '1.1', 'center_order': 1, 'central_product': False, 'central_quotient': '6.1', 'commutator_count': 1, 'commutator_label': '3.1', 'complements_known': True, 'complete': True, 'complex_characters_known': True, 'composition_factors': ['2.1', '3.1'], 'composition_length': 2, 'conjugacy_classes_known': True, 'counter': 1, 'cyclic': False, 'derived_length': 2, 'dihedral': True, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 3, 1, 1], [3, 2, 1, 1]], 'element_repr_type': 'Perm', 'elementary': 1, 'eulerian_function': 3, 'exponent': 6, 'exponents_of_order': [1, 1], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [[2, 1, 1]], 'familial': True, 'frattini_label': '1.1', 'frattini_quotient': '6.1', 'hash': 1, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 6, 'inner_gen_orders': [2, 3], 'inner_gens': [[1, 3], [2, 4]], 'inner_hash': 1, 'inner_nilpotent': False, 'inner_order': 6, 'inner_split': True, 'inner_tex': 'S_3', 'inner_used': [1, 2], 'irrC_degree': 2, 'irrQ_degree': 2, 'irrQ_dim': 2, 'irrR_degree': 2, 'irrep_stats': [[1, 2], [2, 1]], 'label': '6.1', 'linC_count': 1, 'linC_degree': 2, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 2, 'linQ_degree_count': 1, 'linQ_dim': 2, 'linQ_dim_count': 1, 'linR_count': 1, 'linR_degree': 2, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'S3', 'ngens': 2, 'nilpotency_class': -1, 'nilpotent': False, 'normal_counts': [0, 0, 0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 3, 'number_characteristic_subgroups': 3, 'number_conjugacy_classes': 3, 'number_divisions': 3, 'number_normal_subgroups': 3, 'number_subgroup_autclasses': 4, 'number_subgroup_classes': 4, 'number_subgroups': 6, 'old_label': None, 'order': 6, 'order_factorization_type': 11, 'order_stats': [[1, 1], [2, 3], [3, 2]], 'outer_abelian': True, 'outer_cyclic': True, 'outer_equivalence': False, 'outer_exponent': 1, 'outer_gen_orders': [], 'outer_gen_pows': [], 'outer_gens': [], 'outer_group': '1.1', 'outer_hash': 1, 'outer_nilpotent': True, 'outer_order': 1, 'outer_permdeg': 1, 'outer_perms': [], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_1', 'pc_rank': 2, 'perfect': False, 'permutation_degree': 3, 'pgroup': 0, 'primary_abelian_invariants': [2], 'quasisimple': False, 'rank': 2, 'rational': True, 'rational_characters_known': True, 'ratrep_stats': [[1, 2], [2, 1]], 'representations': {'PC': {'code': 25, 'gens': [1, 2], 'pres': [2, -2, -3, 17]}, 'GLZ': {'b': 3, 'd': 2, 'gens': [28, 45]}, 'Lie': [{'d': 2, 'q': 2, 'gens': [6, 11], 'family': 'GL'}, {'d': 2, 'q': 2, 'gens': [6, 11], 'family': 'SL'}, {'d': 2, 'q': 2, 'gens': [6, 11], 'family': 'PSL'}, {'d': 2, 'q': 2, 'gens': [6, 11], 'family': 'PGL'}, {'d': 3, 'q': 2, 'gens': [84, 279], 'family': 'SO'}, {'d': 2, 'q': 2, 'gens': [20, 69], 'family': 'SU'}, {'d': 3, 'q': 2, 'gens': [84, 279], 'family': 'PSO'}, {'d': 2, 'q': 2, 'gens': [20, 69], 'family': 'PSU'}, {'d': 3, 'q': 2, 'gens': [84, 279], 'family': 'GO'}, {'d': 3, 'q': 2, 'gens': [84, 279], 'family': 'Omega'}, {'d': 3, 'q': 2, 'gens': [84, 279], 'family': 'PGO'}, {'d': 2, 'q': 2, 'gens': [20, 138], 'family': 'PGU'}, {'d': 3, 'q': 2, 'gens': [84, 279], 'family': 'POmega'}, {'d': 3, 'q': 2, 'gens': [84, 279], 'family': 'CSO'}, {'d': 2, 'q': 4, 'gens': [20, 194], 'family': 'CSOPlus'}, {'d': 2, 'q': 2, 'gens': [13, 14], 'family': 'CSOMinus'}, {'d': 2, 'q': 2, 'gens': [20, 69], 'family': 'CSU'}, {'d': 3, 'q': 2, 'gens': [84, 279], 'family': 'CO'}, {'d': 2, 'q': 2, 'gens': [13, 14], 'family': 'COMinus'}, {'d': 2, 'q': 2, 'gens': [6, 11], 'family': 'PGammaL'}, {'d': 2, 'q': 2, 'gens': [6, 11], 'family': 'PSigmaL'}, {'d': 2, 'q': 2, 'gens': [20, 138], 'family': 'PGammaU'}, {'d': 1, 'q': 3, 'gens': [1, 3], 'family': 'AGL'}, {'d': 1, 'q': 3, 'gens': [1, 3], 'family': 'AGammaL'}], 'GLFp': {'d': 2, 'p': 2, 'gens': [11, 6]}, 'Perm': {'d': 3, 'gens': [1, 4]}}, 'schur_multiplier': [], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2], 'solvability_type': 6, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'S_3', 'transitive_degree': 3, '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': 3, 'aut_exponent': 12, 'aut_gen_orders': [6, 2, 2, 2, 6, 2, 4, 2], 'aut_gens': [[132667, 5042476, 293524, 6164698, 4605946], [5814484, 531509, 697135, 4605946, 2981098], [2296241, 531509, 3042404, 6164698, 4549642], [3559400, 6236344, 6643792, 2990890, 6174490], [2122417, 5042476, 316168, 2981098, 4549642], [5720389, 6236344, 6606980, 2981098, 2990890], [5720389, 531509, 4273084, 4605946, 6164698], [5728804, 531509, 4374370, 4605946, 2990890], [6632585, 1725377, 2636141, 6164698, 4605946]], 'aut_group': '1536.408641059', 'aut_hash': 1418641417957792246, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 1536, 'aut_permdeg': 32, 'aut_perms': [103071517019489506059338745333332770, 207617191100129693064065287926675448, 259575283414634561869365509135851361, 222319126012780672796347464770620937, 13519617270249274355879087240875112, 71032795751090259932839295571284748, 212545512234755561918632447964533413, 188280573621665042303723697751914314], 'aut_phi_ratio': 12.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [2, 2, 2, 1], [2, 12, 1, 1], [2, 12, 2, 1], [2, 24, 1, 1], [3, 8, 1, 1], [4, 2, 2, 2], [4, 6, 1, 2], [4, 12, 1, 2], [4, 24, 1, 2], [6, 8, 1, 3], [6, 16, 2, 1], [8, 6, 4, 1], [8, 12, 2, 3], [12, 16, 2, 2]], 'aut_supersolvable': False, 'aut_tex': 'C_2^6\\times S_4', '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': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^6', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 12, 'autcentquo_group': '24.12', 'autcentquo_hash': 12, 'autcentquo_nilpotent': False, 'autcentquo_order': 24, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'S_4', 'cc_stats': [[1, 1, 1], [2, 1, 3], [2, 2, 2], [2, 12, 3], [2, 24, 1], [3, 8, 1], [4, 2, 4], [4, 6, 2], [4, 12, 2], [4, 24, 2], [6, 8, 3], [6, 16, 2], [8, 6, 4], [8, 12, 6], [12, 16, 4]], 'center_label': '4.2', 'center_order': 4, 'central_product': True, 'central_quotient': '96.226', 'commutator_count': 1, 'commutator_label': '48.32', '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', '3.1'], 'composition_length': 8, 'conjugacy_classes_known': True, 'counter': 17956, 'cyclic': False, 'derived_length': 4, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [2, 2, 1, 2], [2, 12, 1, 3], [2, 24, 1, 1], [3, 8, 1, 1], [4, 2, 2, 2], [4, 6, 1, 2], [4, 12, 1, 2], [4, 24, 1, 2], [6, 8, 1, 3], [6, 16, 1, 2], [8, 6, 2, 2], [8, 12, 2, 3], [12, 16, 2, 2]], 'element_repr_type': 'GLZN', 'elementary': 1, 'eulerian_function': 10080, 'exponent': 24, 'exponents_of_order': [7, 1], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [], 'familial': False, 'frattini_label': '4.2', 'frattini_quotient': '96.226', 'hash': 17956, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 12, 'inner_gen_orders': [2, 2, 6, 2, 2], 'inner_gens': [[132667, 5042476, 284803, 6174490, 4549642], [132667, 5042476, 2636141, 6164698, 4605946], [3435199, 531509, 293524, 2981098, 2990890], [4604263, 5042476, 697135, 6164698, 4549642], [2122417, 5042476, 394810, 2990890, 4605946]], 'inner_hash': 226, 'inner_nilpotent': False, 'inner_order': 96, 'inner_split': True, 'inner_tex': 'C_2^2\\times S_4', 'inner_used': [1, 2, 3, 4], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 8], [2, 14], [3, 8], [4, 7], [6, 2], [8, 1]], 'label': '384.17956', 'linC_count': 16, 'linC_degree': 4, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 10, 'linQ_degree_count': 12, 'linQ_dim': 10, 'linQ_dim_count': 12, 'linR_count': 8, 'linR_degree': 6, 'maximal_subgroups_known': True, 'metabelian': False, 'metacyclic': False, 'monomial': False, 'name': 'GL(2,3):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], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 27, 'number_characteristic_subgroups': 32, 'number_conjugacy_classes': 40, 'number_divisions': 31, 'number_normal_subgroups': 36, 'number_subgroup_autclasses': 249, 'number_subgroup_classes': 305, 'number_subgroups': 1617, 'old_label': None, 'order': 384, 'order_factorization_type': 51, 'order_stats': [[1, 1], [2, 67], [3, 8], [4, 92], [6, 56], [8, 96], [12, 64]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 2, 'outer_gen_orders': [2, 2, 2, 2], 'outer_gen_pows': [132652, 132652, 132652, 132652], 'outer_gens': [[132667, 5042476, 394810, 2990890, 4605946], [4642835, 5042476, 293524, 6164698, 4605946], [132667, 6236344, 293524, 6164698, 4605946], [132667, 5042476, 4264363, 6164698, 4605946]], 'outer_group': '16.14', 'outer_hash': 14, 'outer_nilpotent': True, 'outer_order': 16, 'outer_permdeg': 16, 'outer_perms': [1327912203241, 2966658509284, 4396538203938, 5800737348096], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^4', 'pc_rank': 5, 'perfect': False, 'permutation_degree': 20, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 2], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 8], [2, 6], [3, 8], [4, 1], [6, 2], [8, 6]], 'representations': {'PC': {'code': 768726718584351456422895533306329201463632754539650575880, 'gens': [1, 2, 3, 6, 7], 'pres': [8, -2, -2, -2, -2, -3, -2, 2, -2, 1545, 482, 2650, 66, 1283, 91, 1284, 16133, 1749, 1757, 469, 237, 16134, 3382, 2382, 878, 382, 166]}, 'GLZN': {'d': 2, 'p': 51, 'gens': [6721000, 2122432, 6324778, 132667, 1772389, 6632600, 3919406, 3909614]}, 'Perm': {'d': 20, 'gens': [2283861441336487, 1, 141986429951392096, 7, 524608910715000, 270143702759846880, 399507013713750600, 527966508282077520]}}, 'schur_multiplier': [2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 2], 'solvability_type': 12, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': '\\GL(2,3):D_4', 'transitive_degree': 32, 'wreath_data': None, 'wreath_product': False}
