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gps_subgroup_search • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'ambient': '128.734', 'ambient_counter': 734, 'ambient_order': 128, 'ambient_tex': 'C_4^2:D_4', 'central': False, 'central_factor': False, 'centralizer_order': 8, 'characteristic': False, 'core_order': 8, 'counter': 78, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '128.734.8.bb1.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': False, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': False, 'old_label': '8.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': 8, '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': '16.13', 'subgroup_hash': 13, 'subgroup_order': 16, 'subgroup_tex': 'D_4:C_2', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '128.734', 'aut_centralizer_order': 64, 'aut_label': '8.bb1', 'aut_quo_index': None, 'aut_stab_index': 2, 'aut_weyl_group': '16.11', 'aut_weyl_index': 128, 'centralizer': '16.c1.a1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['4.c1.a1', '4.q1.a1', '4.q1.b1'], 'contains': ['16.e1.a1', '16.o1.a1', '16.bc1.a1', '16.be1.a1', '16.bg1.a1'], 'core': '16.e1.a1', 'coset_action_label': None, 'count': 2, 'diagramx': [6987, -1, 7255, -1, 6987, -1, 7244, -1], 'generators': [49029, 48259, 21517], 'label': '128.734.8.bb1.a1', 'mobius_quo': None, 'mobius_sub': 0, 'normal_closure': '4.c1.a1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '2.d1.a1', 'old_label': '8.bb1.a1', 'projective_image': '64.73', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '8.bb1.a1', 'subgroup_fusion': None, 'weyl_group': '8.3'}
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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': [2, 2, 3, 2, 2], 'aut_gens': [[30, 377, 501], [120, 105, 65], [120, 377, 129], [65, 90, 120], [120, 253, 129], [120, 377, 501]], 'aut_group': '48.48', 'aut_hash': 48, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 48, 'aut_permdeg': 6, 'aut_perms': [415, 450, 233, 403, 444], 'aut_phi_ratio': 6.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 2, 3, 1], [4, 1, 2, 1], [4, 2, 3, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_2\\times S_4', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 2, 'autcent_group': '8.5', 'autcent_hash': 5, 'autcent_nilpotent': True, 'autcent_order': 8, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^3', '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, 1, 1], [2, 2, 3], [4, 1, 2], [4, 2, 3]], 'center_label': '4.1', 'center_order': 4, 'central_product': True, 'central_quotient': '4.2', 'commutator_count': 1, 'commutator_label': '2.1', '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': 13, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 2, 1, 3], [4, 1, 2, 1], [4, 2, 1, 3]], 'element_repr_type': 'GLFp', 'elementary': 2, 'eulerian_function': 28, 'exponent': 4, 'exponents_of_order': [4], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2], 'faithful_reps': [[2, 0, 2]], 'familial': False, 'frattini_label': '2.1', 'frattini_quotient': '8.5', 'hash': 13, 'hyperelementary': 2, 'inner_abelian': True, 'inner_cyclic': False, 'inner_exponent': 2, 'inner_gen_orders': [2, 2, 2], 'inner_gens': [[30, 253, 129], [120, 377, 501], [120, 377, 501]], 'inner_hash': 2, 'inner_nilpotent': True, 'inner_order': 4, 'inner_split': True, 'inner_tex': 'C_2^2', 'inner_used': [1, 2], 'irrC_degree': 2, 'irrQ_degree': 4, 'irrQ_dim': 4, 'irrR_degree': 4, 'irrep_stats': [[1, 8], [2, 2]], 'label': '16.13', 'linC_count': 2, 'linC_degree': 2, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 4, 'linQ_degree_count': 1, 'linQ_dim': 4, 'linQ_dim_count': 1, 'linR_count': 1, 'linR_degree': 4, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'D4:C2', 'ngens': 3, 'nilpotency_class': 2, 'nilpotent': True, 'normal_counts': [0, 0, 0, 0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 5, 'number_characteristic_subgroups': 5, 'number_conjugacy_classes': 10, 'number_divisions': 9, 'number_normal_subgroups': 17, 'number_subgroup_autclasses': 10, 'number_subgroup_classes': 20, 'number_subgroups': 23, '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': 6, 'outer_gen_orders': [2, 6], 'outer_gen_pows': [377, 126], 'outer_gens': [[65, 377, 501], [65, 90, 30]], 'outer_group': '12.4', 'outer_hash': 4, 'outer_nilpotent': False, 'outer_order': 12, 'outer_permdeg': 5, 'outer_perms': [6, 31], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'D_6', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 8, 'pgroup': 2, 'primary_abelian_invariants': [2, 2, 2], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 8], [4, 1]], 'representations': {'PC': {'code': 8716, 'gens': [1, 2, 3], 'pres': [4, -2, 2, 2, -2, 81, 34]}, 'GLZ': {'b': 3, 'd': 4, 'gens': [25793872, 22766861, 817574]}, 'GLFp': {'d': 2, 'p': 5, 'gens': [30, 377, 501]}, 'Perm': {'d': 8, 'gens': [35278, 23616, 16582, 5167]}}, 'schur_multiplier': [2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 2], 'solvability_type': 4, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'D_4:C_2', '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': '8.5', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 4, 'aut_gen_orders': [2, 2, 4, 2, 4, 2, 2, 2, 4], 'aut_gens': [[13485, 4455, 48323, 56453], [13485, 4455, 56389, 56453], [13485, 4455, 13387, 21645], [13485, 20843, 48323, 56453], [13485, 4455, 30919, 56453], [61473, 20963, 48323, 56453], [13485, 4455, 13515, 56453], [13485, 32673, 13387, 21645], [21611, 4455, 30919, 56453], [48165, 53739, 48323, 56453]], 'aut_group': None, 'aut_hash': 1969822349860123513, 'aut_nilpotency_class': 2, 'aut_nilpotent': True, 'aut_order': 2048, 'aut_permdeg': 40, 'aut_perms': [670133381802392074356705057145363591391970435664, 732486157433238425487085569061992167711096513524, 418446495558843215148740322471238700710688602920, 104689544639787197571408603032417378878992530070, 2631130187942707189740971091974242579789169931, 418700856322987460676327928399762803632210768404, 732486457400032849919621180284034830752172737000, 121965968971197275514009201971499993035604165561, 7865355689016556955349915239823725256074607915], 'aut_phi_ratio': 32.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 1, 2, 1], [2, 2, 1, 2], [2, 8, 1, 1], [2, 8, 2, 2], [4, 2, 1, 4], [4, 4, 4, 1], [4, 8, 1, 1], [4, 8, 2, 1], [8, 8, 2, 2]], 'aut_supersolvable': True, 'aut_tex': 'C_2^3.C_2^6.C_2^2', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 2, 'autcent_group': None, 'autcent_hash': 267, 'autcent_nilpotent': True, 'autcent_order': 64, 'autcent_solvable': True, 'autcent_split': None, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^6', 'autcentquo_abelian': None, 'autcentquo_cyclic': None, 'autcentquo_exponent': None, 'autcentquo_group': None, 'autcentquo_hash': None, 'autcentquo_nilpotent': None, 'autcentquo_order': 32, 'autcentquo_solvable': None, 'autcentquo_supersolvable': None, 'autcentquo_tex': None, 'cc_stats': [[1, 1, 1], [2, 1, 3], [2, 2, 2], [2, 8, 5], [4, 2, 4], [4, 4, 4], [4, 8, 3], [8, 8, 4]], 'center_label': '4.2', 'center_order': 4, 'central_product': False, 'central_quotient': '32.27', 'commutator_count': 1, 'commutator_label': '16.10', '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'], 'composition_length': 7, 'conjugacy_classes_known': True, 'counter': 734, '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, 8, 1, 5], [4, 2, 1, 4], [4, 4, 1, 4], [4, 8, 1, 3], [8, 8, 1, 2], [8, 8, 2, 1]], 'element_repr_type': 'GLZq', 'elementary': 2, 'eulerian_function': 336, 'exponent': 8, 'exponents_of_order': [7], 'factors_of_aut_order': [2], 'factors_of_order': [2], 'faithful_reps': [], 'familial': False, 'frattini_label': '16.10', 'frattini_quotient': '8.5', 'hash': 734, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 4, 'inner_gen_orders': [2, 2, 4, 2], 'inner_gens': [[13485, 32673, 13387, 21645], [21611, 4455, 30919, 56453], [48165, 53739, 48323, 56453], [48293, 4455, 48323, 56453]], 'inner_hash': 27, 'inner_nilpotent': True, 'inner_order': 32, 'inner_split': None, 'inner_tex': 'C_2^2\\wr C_2', 'inner_used': [1, 2, 3], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 8], [2, 14], [4, 4]], 'label': '128.734', 'linC_count': 32, 'linC_degree': 6, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 6, 'linQ_degree_count': 24, 'linQ_dim': 6, 'linQ_dim_count': 24, 'linR_count': 24, 'linR_degree': 6, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'C4^2:D4', 'ngens': 3, 'nilpotency_class': 3, 'nilpotent': True, 'normal_counts': [0, 0, 0, 0, 0, 0, 0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 17, 'number_characteristic_subgroups': 34, 'number_conjugacy_classes': 26, 'number_divisions': 25, 'number_normal_subgroups': 46, 'number_subgroup_autclasses': 139, 'number_subgroup_classes': 239, 'number_subgroups': 656, 'old_label': None, 'order': 128, 'order_factorization_type': 7, 'order_stats': [[1, 1], [2, 47], [4, 48], [8, 32]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 4, 'outer_gen_orders': [2, 2, 2, 4, 2, 4], 'outer_gen_pows': [56453, 4097, 39049, 4097, 4097, 4097], 'outer_gens': [[61473, 37351, 13515, 56453], [13485, 4455, 21709, 21645], [13485, 20843, 61647, 56453], [61473, 37351, 21709, 21645], [61473, 20963, 13387, 21645], [61473, 37351, 4289, 21645]], 'outer_group': '64.261', 'outer_hash': 261, 'outer_nilpotent': True, 'outer_order': 64, 'outer_permdeg': 32, 'outer_perms': [196162835435764981665662870785606822, 8348204302821459038416821492019701, 187660892854463646960044104980798094, 228196900270941822509039869105384465, 66990084564814550180466325318835161, 263041787167960847292089916291039577], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'D_4\\times C_2^3', 'pc_rank': 4, 'perfect': False, 'permutation_degree': 12, 'pgroup': 2, 'primary_abelian_invariants': [2, 2, 2], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 8], [2, 12], [4, 5]], 'representations': {'PC': {'code': 79104398895142578185578, 'gens': [1, 2, 4, 6], 'pres': [7, 2, 2, 2, 2, 2, 2, 2, 85, 36, 675, 794, 80, 1411, 4037, 124]}, 'GLZ': {'b': 3, 'd': 6, 'gens': [125101729081201910, 58301495106334124, 25038645869518594]}, 'GLZN': {'d': 2, 'p': 48, 'gens': [111169, 111487, 829447, 111745, 113483, 2821273, 4176589]}, 'GLZq': {'d': 2, 'q': 16, 'gens': [4225, 21709, 48259, 4419, 4161, 37163, 39049]}, 'Perm': {'d': 12, 'gens': [40764967, 12783016, 1865642, 95242440, 23, 18558023, 139725360]}}, 'schur_multiplier': [2, 2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [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:D_4', 'transitive_degree': 16, 'wreath_data': None, 'wreath_product': False}
