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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': True, 'ambient': '256.4541', 'ambient_counter': 4541, 'ambient_order': 256, 'ambient_tex': '(C_2^3\\times C_4).D_4', 'central': False, 'central_factor': False, 'centralizer_order': 32, 'characteristic': False, 'core_order': 2, 'counter': 160, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '256.4541.32.bj1.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': False, 'old_label': '32.bj1.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': 32, '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': '8.2', 'subgroup_hash': 2, 'subgroup_order': 8, 'subgroup_tex': 'C_2\\times C_4', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '256.4541', 'aut_centralizer_order': None, 'aut_label': '32.bj1', 'aut_quo_index': None, 'aut_stab_index': None, 'aut_weyl_group': None, 'aut_weyl_index': None, 'centralizer': '8.c1.a1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['16.s1.a1', '16.bj1.a1', '16.bp1.a1'], 'contains': ['64.l1.a1', '64.p1.a1', '64.q1.a1'], 'core': '128.b1.a1', 'coset_action_label': None, 'count': 4, 'diagramx': None, 'generators': [33, 4], 'label': '256.4541.32.bj1.a1', 'mobius_quo': None, 'mobius_sub': None, 'normal_closure': '8.c1.a1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '4.m1.a1', 'old_label': '32.bj1.a1', 'projective_image': None, 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '32.bj1.a1', 'subgroup_fusion': None, 'weyl_group': None}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': True, 'abelian_quotient': '8.2', '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], 'aut_gens': [[1, 2], [1, 3], [5, 3]], 'aut_group': '8.3', 'aut_hash': 3, 'aut_nilpotency_class': 2, 'aut_nilpotent': True, 'aut_order': 8, 'aut_permdeg': 4, 'aut_perms': [1, 22], 'aut_phi_ratio': 2.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 1, 2, 1], [4, 1, 4, 1]], 'aut_supersolvable': True, 'aut_tex': 'D_4', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 4, 'autcent_group': '8.3', 'autcent_hash': 3, 'autcent_nilpotent': True, 'autcent_order': 8, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'D_4', '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], [4, 1, 4]], 'center_label': '8.2', 'center_order': 8, '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', '2.1'], 'composition_length': 3, 'conjugacy_classes_known': True, 'counter': 2, 'cyclic': False, 'derived_length': 1, 'dihedral': False, 'direct_factorization': [['2.1', 1], ['4.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [4, 1, 2, 2]], 'element_repr_type': 'PC', 'elementary': 2, 'eulerian_function': 3, 'exponent': 4, 'exponents_of_order': [3], 'factors_of_aut_order': [2], 'factors_of_order': [2], 'faithful_reps': [], 'familial': True, 'frattini_label': '2.1', '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, 8]], 'label': '8.2', 'linC_count': 12, 'linC_degree': 2, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 3, 'linQ_degree_count': 4, 'linQ_dim': 3, 'linQ_dim_count': 4, 'linR_count': 4, 'linR_degree': 3, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C2*C4', 'ngens': 2, 'nilpotency_class': 1, 'nilpotent': True, 'normal_counts': [0, 0, 0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 4, 'number_characteristic_subgroups': 4, 'number_conjugacy_classes': 8, 'number_divisions': 6, 'number_normal_subgroups': 8, 'number_subgroup_autclasses': 6, 'number_subgroup_classes': 8, 'number_subgroups': 8, 'old_label': None, 'order': 8, 'order_factorization_type': 3, 'order_stats': [[1, 1], [2, 3], [4, 4]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 4, 'outer_gen_orders': [2, 4], 'outer_gen_pows': [0, 0], 'outer_gens': [[1, 3], [5, 3]], 'outer_group': '8.3', 'outer_hash': 3, 'outer_nilpotent': True, 'outer_order': 8, 'outer_permdeg': 4, 'outer_perms': [1, 22], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'D_4', 'pc_rank': 2, 'perfect': False, 'permutation_degree': 6, 'pgroup': 2, 'primary_abelian_invariants': [2, 4], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 2]], 'representations': {'PC': {'code': 34, 'gens': [1, 2], 'pres': [3, -2, 2, -2, 16]}, 'GLZ': {'b': 3, 'd': 3, 'gens': [3362, 16426]}, 'GLFp': {'d': 2, 'p': 5, 'gens': [251, 504]}, 'Perm': {'d': 6, 'gens': [22, 120, 7]}}, 'schur_multiplier': [2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 4], 'solvability_type': 1, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_2\\times C_4', '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': '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': [4, 2, 4, 2, 2, 4, 2, 4], 'aut_gens': [[1, 2, 16, 64], [129, 99, 152, 200], [137, 10, 56, 192], [9, 3, 176, 72], [129, 66, 184, 192], [9, 166, 248, 64], [129, 202, 240, 224], [129, 142, 240, 64], [9, 111, 240, 200]], 'aut_group': None, 'aut_hash': 5074040165636225497, 'aut_nilpotency_class': 3, 'aut_nilpotent': True, 'aut_order': 4096, 'aut_permdeg': 80, 'aut_perms': [30086677259283781355101789948615335627375594993962058949539742258358375005296896715666828356348200429821543241500720655, 18695119855788781322137882857354267241091770783454715468016703691816219548598411889878641475305040027408146433453661093, 1110196421012318252815997976958674758026673648434238862890742597251211983992812502894361795210709407618661280842336258, 9797156189394293458464775467821805951228182655297205196266662386087611355667236361463344500236154049271516358425736739, 40072038584173574037232485502509561911325721089492348757461767097418161763012365842241274252767808871148344771063829495, 11076737825249006856905964422483029827347291196193396667490478357099159413086455332725994597872016413201316818472605946, 6534823012636180300047436708192483399297418072686741851477131006962025697876834288385784728882097141443797851282849720, 52314815603585851928015140704915820423371383909400473330592286033706480608348494273044550640760097664951709634231046741], 'aut_phi_ratio': 32.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [2, 2, 1, 2], [2, 2, 2, 1], [2, 4, 1, 1], [2, 16, 1, 1], [4, 4, 2, 4], [4, 8, 1, 2], [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, 4], [2, 4, 1], [2, 16, 1], [4, 4, 8], [4, 8, 6], [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': 4541, '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, 4], [2, 4, 1, 1], [2, 16, 1, 1], [4, 4, 1, 2], [4, 4, 2, 3], [4, 8, 1, 2], [4, 8, 2, 2], [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': 4541, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 4, 'inner_gen_orders': [2, 4, 4, 2], 'inner_gens': [[1, 138, 152, 64], [137, 2, 240, 224], [137, 98, 16, 64], [1, 162, 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, 3], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 16], [2, 12], [4, 8], [8, 1]], 'label': '256.4541', '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': 24, 'number_normal_subgroups': 51, 'number_subgroup_autclasses': 187, 'number_subgroup_classes': 220, 'number_subgroups': 735, 'old_label': None, 'order': 256, 'order_factorization_type': 7, 'order_stats': [[1, 1], [2, 31], [4, 96], [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': [224, 64, 100, 0, 0], 'outer_gens': [[137, 230, 208, 192], [137, 198, 112, 64], [129, 202, 240, 224], [137, 10, 56, 192], [129, 99, 152, 200]], 'outer_group': '64.261', 'outer_hash': 261, 'outer_nilpotent': True, 'outer_order': 64, 'outer_permdeg': 32, 'outer_perms': [108515588221673033044774095420099247, 139037960750367351826143160640858839, 35662793851488007153826828450354880, 1865928164078413711469332196467488, 9924827213105952537994574152089616], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'D_4\\times C_2^3', 'pc_rank': 4, 'perfect': False, 'permutation_degree': 32, '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, 2], [8, 6]], 'representations': {'PC': {'code': 42352842923149934072359470842184918, 'gens': [1, 2, 5, 7], 'pres': [8, 2, 2, 2, 2, 2, 2, 2, 2, 2209, 41, 66, 6084, 4812, 1780, 116, 3853, 6286, 2710, 166]}, 'Perm': {'d': 32, 'gens': [8781134879511054717135913785715645, 1125375612389524035253853252874262, 2967137878828, 17801276690826926280512419258368000, 26331761458188395569896346562283474, 1865928384952036414407030657024000, 34812457740773315563841488910255778, 34812457740773315563835694403584000]}}, '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': 64, 'wreath_data': None, 'wreath_product': False}
