-
gps_subgroup_search • Show schema
Hide schema
{'Agroup': True, 'Zgroup': True, 'abelian': True, 'ambient': '256.4554', 'ambient_counter': 4554, 'ambient_order': 256, 'ambient_tex': '(C_2^3\\times C_4).D_4', 'central': False, 'central_factor': False, 'centralizer_order': 64, 'characteristic': False, 'core_order': 2, 'counter': 212, 'cyclic': True, 'direct': None, 'hall': 0, 'label': '256.4554.64.k1.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': False, 'old_label': '64.k1.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': '4.1', 'subgroup_hash': 1, 'subgroup_order': 4, 'subgroup_tex': 'C_4', 'supersolvable': True, 'sylow': 0}
-
gps_subgroup_data • Show schema
Hide schema
{'ambient': '256.4554', 'aut_centralizer_order': None, 'aut_label': '64.k1', 'aut_quo_index': None, 'aut_stab_index': None, 'aut_weyl_group': None, 'aut_weyl_index': None, 'centralizer': '4.p1.a1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['32.k1.a1', '32.s1.a1', '32.x1.a1', '32.y1.a1', '32.y1.a2', '32.ba1.a1', '32.ba1.b1'], 'contains': ['128.a1.a1'], 'core': '128.a1.a1', 'coset_action_label': None, 'count': 4, 'diagramx': None, 'generators': [154], 'label': '256.4554.64.k1.a1', 'mobius_quo': None, 'mobius_sub': None, 'normal_closure': '16.f1.a1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '4.p1.a1', 'old_label': '64.k1.a1', 'projective_image': None, 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '64.k1.a1', 'subgroup_fusion': None, 'weyl_group': None}
-
gps_groups • Show schema
Hide schema
{'Agroup': True, 'Zgroup': True, 'abelian': True, 'abelian_quotient': '4.1', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': True, 'aut_cyclic': True, 'aut_derived_length': 1, 'aut_exponent': 2, 'aut_gen_orders': [2], 'aut_gens': [[1], [3]], 'aut_group': '2.1', 'aut_hash': 1, 'aut_nilpotency_class': 1, 'aut_nilpotent': True, 'aut_order': 2, 'aut_permdeg': 2, 'aut_perms': [1], 'aut_phi_ratio': 1.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [4, 1, 2, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_2', 'autcent_abelian': True, 'autcent_cyclic': True, 'autcent_exponent': 2, 'autcent_group': '2.1', 'autcent_hash': 1, 'autcent_nilpotent': True, 'autcent_order': 2, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2', '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, 1], [4, 1, 2]], 'center_label': '4.1', 'center_order': 4, 'central_product': False, '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': 1, 'cyclic': True, 'derived_length': 1, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [4, 1, 2, 1]], 'element_repr_type': 'PC', 'elementary': 2, 'eulerian_function': 1, 'exponent': 4, 'exponents_of_order': [2], 'factors_of_aut_order': [2], 'factors_of_order': [2], 'faithful_reps': [[1, 0, 2]], 'familial': True, 'frattini_label': '2.1', 'frattini_quotient': '2.1', 'hash': 1, 'hyperelementary': 2, 'inner_abelian': True, 'inner_cyclic': True, 'inner_exponent': 1, 'inner_gen_orders': [1], 'inner_gens': [[1]], 'inner_hash': 1, 'inner_nilpotent': True, 'inner_order': 1, 'inner_split': True, 'inner_tex': 'C_1', 'inner_used': [], 'irrC_degree': 1, 'irrQ_degree': 2, 'irrQ_dim': 2, 'irrR_degree': 2, 'irrep_stats': [[1, 4]], 'label': '4.1', 'linC_count': 2, 'linC_degree': 1, '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': 'C4', 'ngens': 1, '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': 3, 'number_characteristic_subgroups': 3, 'number_conjugacy_classes': 4, 'number_divisions': 3, 'number_normal_subgroups': 3, 'number_subgroup_autclasses': 3, 'number_subgroup_classes': 3, 'number_subgroups': 3, 'old_label': None, 'order': 4, 'order_factorization_type': 2, 'order_stats': [[1, 1], [2, 1], [4, 2]], 'outer_abelian': True, 'outer_cyclic': True, 'outer_equivalence': False, 'outer_exponent': 2, 'outer_gen_orders': [2], 'outer_gen_pows': [0], 'outer_gens': [[3]], 'outer_group': '2.1', 'outer_hash': 1, 'outer_nilpotent': True, 'outer_order': 2, 'outer_permdeg': 2, 'outer_perms': [1], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2', 'pc_rank': 1, 'perfect': False, 'permutation_degree': 4, 'pgroup': 2, 'primary_abelian_invariants': [4], 'quasisimple': False, 'rank': 1, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 2], [2, 1]], 'representations': {'PC': {'code': 5, 'gens': [1], 'pres': [2, -2, -2, 4]}, 'GLZ': {'b': 3, 'd': 2, 'gens': [46]}, 'Lie': [{'d': 2, 'q': 3, 'gens': [56, 15], 'family': 'CSOPlus'}], 'GLFp': {'d': 2, 'p': 3, 'gens': [21]}, 'Perm': {'d': 4, 'gens': [22, 7]}}, 'schur_multiplier': [], 'semidirect_product': False, 'simple': False, 'smith_abelian_invariants': [4], 'solvability_type': 0, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_4', 'transitive_degree': 4, 'wreath_data': None, 'wreath_product': False}
-
gps_groups • Show schema
Hide schema
{'Agroup': False, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '16.10', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 4, 'aut_gen_orders': [2, 4, 2, 4, 4, 4, 4, 4], 'aut_gens': [[1, 4, 16, 64], [43, 188, 16, 224], [145, 148, 184, 64], [137, 36, 16, 192], [195, 132, 24, 96], [115, 20, 144, 96], [203, 148, 144, 64], [17, 148, 184, 96], [195, 12, 56, 192]], 'aut_group': None, 'aut_hash': 5074040165636225497, 'aut_nilpotency_class': 3, 'aut_nilpotent': True, 'aut_order': 4096, 'aut_permdeg': 80, 'aut_perms': [20444710710774564985177209771245365628953936662367577509314946798973025673346537949293445977549995656213875089613662317, 15621448355732972785184071404033980872675412072013521620103143478506193553584039161566735130243308625816295428485968316, 15557964103119327606452876504751542284443498312555989439022259838243664529358983540469068476987786018860820034548041431, 53436211184657174653671928950334601873093234823996145185533715805890933955373174666337465653084886902267832618322034420, 58803801850952988408948710144338742566167051245040221064898603344581623747980283233147593752453659460621383262451372996, 18637806501072822349079874927140032144530705265965711569658382768323396031786593515827644639833564139373691649471848960, 68684004578791068557496212656390474754962614561508770779030837482771449504193841190500633244739949822476283526147732479, 58731609698623016812631376293460253501992425220329276416761473927027496154521329610046694865330409168179869033595056729], 'aut_phi_ratio': 32.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [2, 2, 1, 2], [2, 4, 2, 1], [2, 8, 1, 1], [2, 16, 1, 1], [4, 2, 2, 1], [4, 4, 1, 1], [4, 4, 2, 3], [4, 8, 1, 1], [4, 8, 2, 2], [4, 16, 1, 1], [8, 8, 8, 1], [8, 16, 4, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_2^8.C_2^4', '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': False, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^6', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 4, 'autcentquo_group': '64.202', 'autcentquo_hash': 202, 'autcentquo_nilpotent': True, 'autcentquo_order': 64, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'C_2^3:D_4', 'cc_stats': [[1, 1, 1], [2, 1, 3], [2, 2, 2], [2, 4, 2], [2, 8, 1], [2, 16, 1], [4, 2, 2], [4, 4, 7], [4, 8, 5], [4, 16, 1], [8, 8, 8], [8, 16, 4]], 'center_label': '4.2', 'center_order': 4, 'central_product': False, 'central_quotient': '64.90', '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', '2.1'], 'composition_length': 8, 'conjugacy_classes_known': True, 'counter': 4554, '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, 4, 1, 2], [2, 8, 1, 1], [2, 16, 1, 1], [4, 2, 2, 1], [4, 4, 1, 3], [4, 4, 2, 2], [4, 8, 1, 3], [4, 8, 2, 1], [4, 16, 1, 1], [8, 8, 4, 2], [8, 16, 2, 2]], 'element_repr_type': 'PC', 'elementary': 2, 'eulerian_function': 1344, 'exponent': 8, 'exponents_of_order': [8], 'factors_of_aut_order': [2], 'factors_of_order': [2], 'faithful_reps': [], 'familial': False, 'frattini_label': '32.23', 'frattini_quotient': '8.5', 'hash': 4554, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 4, 'inner_gen_orders': [4, 4, 2, 2], 'inner_gens': [[1, 156, 152, 64], [185, 4, 16, 224], [137, 4, 16, 64], [1, 164, 16, 64]], 'inner_hash': 90, 'inner_nilpotent': True, 'inner_order': 64, 'inner_split': None, 'inner_tex': 'C_2^4:C_4', 'inner_used': [1, 2, 4], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 16], [2, 12], [4, 8], [8, 1]], 'label': '256.4554', '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': '(C2^3*C4).D4', 'ngens': 3, 'nilpotency_class': 4, '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': 20, 'number_characteristic_subgroups': 39, 'number_conjugacy_classes': 37, 'number_divisions': 25, 'number_normal_subgroups': 51, 'number_subgroup_autclasses': 199, 'number_subgroup_classes': 247, 'number_subgroups': 879, 'old_label': None, 'order': 256, 'order_factorization_type': 7, 'order_stats': [[1, 1], [2, 39], [4, 88], [8, 128]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 4, 'outer_gen_orders': [2, 2, 2, 2, 4], 'outer_gen_pows': [0, 154, 154, 18, 0], 'outer_gens': [[43, 188, 16, 224], [25, 180, 184, 192], [185, 180, 184, 224], [17, 148, 184, 96], [219, 28, 176, 64]], 'outer_group': '64.261', 'outer_hash': 261, 'outer_nilpotent': True, 'outer_order': 64, 'outer_permdeg': 32, 'outer_perms': [8268628444230288865648587289804816, 18038196755204260493677924422300247, 112260373682779536584766187963204207, 52090320484070171814847344681975168, 146321174753995094309650109067370298], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'D_4\\times C_2^3', 'pc_rank': 4, 'perfect': False, 'permutation_degree': 24, 'pgroup': 2, 'primary_abelian_invariants': [2, 2, 4], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 8], [2, 8], [4, 4], [8, 5]], 'representations': {'PC': {'code': 208351728608403483687705687424583511, 'gens': [1, 3, 5, 7], 'pres': [8, 2, 2, 2, 2, 2, 2, 2, 2, 16, 1033, 3746, 1690, 66, 1283, 6084, 972, 116, 3158, 166]}, 'Perm': {'d': 24, 'gens': [30823692590920178845863, 58471055449100406970370, 85947181262698245651694, 111787520114829945659909, 138277539401031942925294, 10169658974225696175844, 165493671219134058908644, 165493671219134058890880]}}, 'schur_multiplier': [2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 4], 'solvability_type': 4, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': '(C_2^3\\times C_4).D_4', 'transitive_degree': 32, 'wreath_data': None, 'wreath_product': False}
