-
gps_subgroup_search • Show schema
Hide schema
{'Agroup': False, 'Zgroup': False, 'abelian': False, 'ambient': '256.11936', 'ambient_counter': 11936, 'ambient_order': 256, 'ambient_tex': 'C_4^2.C_2^4', 'central': False, 'central_factor': False, 'centralizer_order': 4, 'characteristic': False, 'core_order': 32, 'counter': 60, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '256.11936.4.bb1.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': False, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': False, 'old_label': '4.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': 4, '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': '64.133', 'subgroup_hash': 133, 'subgroup_order': 64, 'subgroup_tex': 'D_4.D_4', 'supersolvable': True, 'sylow': 0}
-
gps_subgroup_data • Show schema
Hide schema
{'ambient': '256.11936', 'aut_centralizer_order': None, 'aut_label': '4.bb1', 'aut_quo_index': None, 'aut_stab_index': None, 'aut_weyl_group': None, 'aut_weyl_index': None, 'centralizer': '64.a1.a1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['2.d1.a1'], 'contains': ['8.c1.a1', '8.x1.a1', '8.bd1.a1', '8.ca1.a1', '8.cb1.b1', '8.cn1.a1', '8.co1.b1'], 'core': '8.c1.a1', 'coset_action_label': None, 'count': 2, 'diagramx': [4405, -1, 9543, -1, 4405, -1, 9503, -1], 'generators': [5, 2, 32], 'label': '256.11936.4.bb1.a1', 'mobius_quo': None, 'mobius_sub': 0, 'normal_closure': '2.d1.a1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '2.d1.a1', 'old_label': '4.bb1.a1', 'projective_image': None, 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '4.bb1.a1', 'subgroup_fusion': None, 'weyl_group': None}
-
gps_groups • Show schema
Hide schema
{'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, 2, 2, 2, 2, 4], 'aut_gens': [[1, 2, 16, 32], [1, 34, 16, 32], [1, 2, 48, 32], [33, 2, 16, 32], [1, 2, 24, 32], [1, 10, 16, 32], [1, 6, 24, 32], [13, 2, 48, 32]], 'aut_group': '256.56083', 'aut_hash': 56083, 'aut_nilpotency_class': 2, 'aut_nilpotent': True, 'aut_order': 256, 'aut_permdeg': 20, 'aut_perms': [3768968421792296, 61391729505859592, 1344892980246134581, 40866443034990152, 2450807383475816, 356977849161323, 2379437827792543260], 'aut_phi_ratio': 8.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [2, 4, 1, 1], [2, 4, 2, 1], [4, 2, 1, 2], [4, 2, 2, 1], [4, 4, 2, 1], [4, 8, 1, 2], [8, 4, 2, 2]], 'aut_supersolvable': True, 'aut_tex': 'D_4\\times C_2^5', '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': False, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^6', 'autcentquo_abelian': True, 'autcentquo_cyclic': False, 'autcentquo_exponent': 2, 'autcentquo_group': '4.2', 'autcentquo_hash': 2, 'autcentquo_nilpotent': True, 'autcentquo_order': 4, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'C_2^2', 'cc_stats': [[1, 1, 1], [2, 1, 3], [2, 4, 3], [4, 2, 4], [4, 4, 2], [4, 8, 2], [8, 4, 4]], 'center_label': '4.2', 'center_order': 4, 'central_product': False, 'central_quotient': '16.11', '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'], 'composition_length': 6, 'conjugacy_classes_known': True, 'counter': 133, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [2, 4, 1, 3], [4, 2, 1, 2], [4, 2, 2, 1], [4, 4, 1, 2], [4, 8, 1, 2], [8, 4, 2, 2]], 'element_repr_type': 'PC', 'elementary': 2, 'eulerian_function': 336, 'exponent': 8, 'exponents_of_order': [6], 'factors_of_aut_order': [2], 'factors_of_order': [2], 'faithful_reps': [], 'familial': False, 'frattini_label': '8.2', 'frattini_quotient': '8.5', 'hash': 133, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 4, 'inner_gen_orders': [2, 4, 2, 1], 'inner_gens': [[1, 6, 24, 32], [13, 2, 48, 32], [9, 34, 16, 32], [1, 2, 16, 32]], 'inner_hash': 11, 'inner_nilpotent': True, 'inner_order': 16, 'inner_split': True, 'inner_tex': 'C_2\\times D_4', 'inner_used': [1, 2, 3], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 8], [2, 10], [4, 1]], 'label': '64.133', 'linC_count': 16, 'linC_degree': 4, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 6, 'linQ_degree_count': 4, 'linQ_dim': 10, 'linQ_dim_count': 8, 'linR_count': 8, 'linR_degree': 6, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'D4.D4', 'ngens': 3, 'nilpotency_class': 3, 'nilpotent': True, 'normal_counts': [0, 0, 0, 0, 0, 0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 14, 'number_characteristic_subgroups': 25, 'number_conjugacy_classes': 19, 'number_divisions': 16, 'number_normal_subgroups': 29, 'number_subgroup_autclasses': 61, 'number_subgroup_classes': 76, 'number_subgroups': 137, 'old_label': None, 'order': 64, 'order_factorization_type': 3, 'order_stats': [[1, 1], [2, 15], [4, 32], [8, 16]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 2, 'outer_gen_orders': [2, 2, 2, 2], 'outer_gen_pows': [0, 0, 0, 0], 'outer_gens': [[1, 2, 48, 32], [33, 2, 16, 32], [1, 2, 24, 32], [1, 10, 16, 32]], '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': 20, 'pgroup': 2, 'primary_abelian_invariants': [2, 2, 2], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 8], [2, 6], [4, 1], [8, 1]], 'representations': {'PC': {'code': 1197402799214950, 'gens': [1, 2, 5, 6], 'pres': [6, 2, 2, 2, 2, 2, 2, 73, 31, 218, 50, 724, 730]}, 'GLZN': {'d': 2, 'p': 48, 'gens': [111169, 1734447, 1880081, 111745, 1071039, 1034049]}, 'Perm': {'d': 20, 'gens': [150232650287264041, 275481921337343407, 406634628889772776, 534602333071628760, 663089032007500327, 663089032007500320]}}, 'schur_multiplier': [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': 'D_4.D_4', 'transitive_degree': 32, 'wreath_data': None, 'wreath_product': False}
-
gps_groups • Show schema
Hide schema
{'Agroup': False, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '16.14', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 4, 'aut_gen_orders': [4, 2, 4, 4, 4, 2, 4, 4, 4], 'aut_gens': [[1, 2, 4, 32, 64], [17, 74, 148, 32, 64], [1, 130, 76, 176, 64], [121, 226, 28, 32, 64], [193, 218, 20, 32, 64], [249, 98, 76, 176, 192], [1, 2, 28, 160, 192], [41, 186, 156, 48, 64], [121, 186, 212, 48, 64], [121, 58, 84, 32, 64]], 'aut_group': None, 'aut_hash': 335283721508395749, 'aut_nilpotency_class': 2, 'aut_nilpotent': True, 'aut_order': 8192, 'aut_permdeg': 48, 'aut_perms': [6640946682292402294710229971418177233331700319098181692262784, 6471259266873542938914365723471612299561398238272996075244173, 10126666582739159462025187884551614301453918000695027825132649, 6737672350716508656177301954169122886247204883127165402407328, 10533982395544150432648000846995971987015070021508134399000698, 1560717256927836226161057024557776380883251508892968630991556, 1683356412311516255268662274464929859355216367120264058720422, 34701646834614846401416528601924880787758804509459926086865, 10126703271304787244812574488006400313518680575228990961218577], 'aut_phi_ratio': 64.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [2, 4, 1, 1], [2, 8, 2, 1], [2, 16, 1, 1], [4, 2, 1, 2], [4, 2, 2, 1], [4, 4, 1, 4], [4, 8, 2, 5], [4, 16, 1, 1], [8, 8, 2, 2], [8, 8, 4, 1], [8, 16, 2, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_2^9.C_2^4', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 2, 'autcent_group': '256.56092', 'autcent_hash': 56092, 'autcent_nilpotent': True, 'autcent_order': 256, 'autcent_solvable': True, 'autcent_split': False, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^8', 'autcentquo_abelian': True, 'autcentquo_cyclic': False, 'autcentquo_exponent': 2, 'autcentquo_group': '32.51', 'autcentquo_hash': 51, 'autcentquo_nilpotent': True, 'autcentquo_order': 32, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'C_2^5', 'cc_stats': [[1, 1, 1], [2, 1, 3], [2, 4, 1], [2, 8, 2], [2, 16, 1], [4, 2, 4], [4, 4, 4], [4, 8, 10], [4, 16, 1], [8, 8, 8], [8, 16, 2]], 'center_label': '4.2', 'center_order': 4, 'central_product': False, 'central_quotient': '64.202', 'commutator_count': 1, 'commutator_label': '16.2', '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': 11936, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [2, 4, 1, 1], [2, 8, 1, 2], [2, 16, 1, 1], [4, 2, 1, 4], [4, 4, 1, 4], [4, 8, 1, 10], [4, 16, 1, 1], [8, 8, 2, 4], [8, 16, 1, 2]], 'element_repr_type': 'PC', 'elementary': 2, 'eulerian_function': 161280, 'exponent': 8, 'exponents_of_order': [8], 'factors_of_aut_order': [2], 'factors_of_order': [2], 'faithful_reps': [], 'familial': False, 'frattini_label': '16.2', 'frattini_quotient': '16.14', 'hash': 11936, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 4, 'inner_gen_orders': [2, 2, 4, 2, 2], 'inner_gens': [[1, 2, 68, 176, 192], [1, 2, 76, 160, 64], [193, 218, 4, 160, 64], [145, 130, 132, 32, 64], [129, 2, 4, 32, 64]], 'inner_hash': 202, 'inner_nilpotent': True, 'inner_order': 64, 'inner_split': True, 'inner_tex': 'C_2^3:D_4', 'inner_used': [1, 2, 3, 4], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 16], [2, 12], [4, 8], [8, 1]], 'label': '256.11936', 'linC_count': 16, 'linC_degree': 8, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 16, 'linQ_degree_count': 8, 'linQ_dim': 16, 'linQ_dim_count': 2, 'linR_count': 12, 'linR_degree': 12, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'C4^2.C2^4', 'ngens': 4, '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': 24, 'number_characteristic_subgroups': 51, 'number_conjugacy_classes': 37, 'number_divisions': 33, 'number_normal_subgroups': 113, 'number_subgroup_autclasses': 269, 'number_subgroup_classes': 440, 'number_subgroups': 1103, 'old_label': None, 'order': 256, 'order_factorization_type': 7, 'order_stats': [[1, 1], [2, 39], [4, 120], [8, 96]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 4, 'outer_gen_orders': [2, 2, 2, 2, 2, 4], 'outer_gen_pows': [72, 72, 72, 72, 0, 0], 'outer_gens': [[17, 90, 4, 160, 64], [145, 74, 148, 160, 64], [145, 74, 4, 32, 64], [17, 90, 4, 176, 64], [1, 130, 4, 32, 64], [185, 58, 148, 32, 192]], 'outer_group': '128.2163', 'outer_hash': 2163, 'outer_nilpotent': True, 'outer_order': 128, 'outer_permdeg': 12, 'outer_perms': [101693647, 138989543, 124104960, 135359400, 138989520, 212088976], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^4:D_4', 'pc_rank': 5, 'perfect': False, 'permutation_degree': 32, 'pgroup': 2, 'primary_abelian_invariants': [2, 2, 2, 2], 'quasisimple': False, 'rank': 4, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 16], [2, 12], [8, 5]], 'representations': {'PC': {'code': 10676834086057431390851236309732875086, 'gens': [1, 2, 3, 6, 7], 'pres': [8, 2, 2, 2, 2, 2, 2, 2, 2, 1161, 1634, 922, 66, 4355, 2443, 91, 8453, 3853, 1941, 10758, 166]}, 'Perm': {'d': 32, 'gens': [8286344635498329334875575644845436, 17809543778754126056629296828236856, 8260994866546642485758158138091114, 26331987703400895738173053070813855, 34553639408742526768007116164045099, 4503899085099, 43051791510835987321438619650596101, 5895092772101]}}, 'schur_multiplier': [2, 2, 2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [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': 128, 'wreath_data': None, 'wreath_product': False}
