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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': True, 'ambient': '4896.bd', 'ambient_counter': 30, 'ambient_order': 4896, 'ambient_tex': 'C_2\\times \\PSL(2,17)', 'central': True, 'central_factor': False, 'centralizer_order': 4896, 'characteristic': True, 'core_order': 2, 'counter': 60, 'cyclic': True, 'direct': True, 'hall': 0, 'label': '4896.bd.2448.a1.a1', 'maximal': False, 'maximal_normal': True, 'metabelian': True, 'metacyclic': True, 'minimal': True, 'minimal_normal': True, 'nilpotent': True, 'normal': True, 'old_label': '2448.a1.a1', 'outer_equivalence': False, 'perfect': False, 'proper': True, 'quotient': '2448.a', 'quotient_Agroup': False, 'quotient_abelian': False, 'quotient_cyclic': False, 'quotient_hash': 7454658504626110115, 'quotient_metabelian': True, 'quotient_nilpotent': False, 'quotient_order': 2448, 'quotient_simple': True, 'quotient_solvable': False, 'quotient_supersolvable': False, 'quotient_tex': '\\PSL(2,17)', 'simple': True, 'solvable': True, 'special_labels': ['Z', 'F', 'R', 'U0', 'C1'], 'split': True, 'standard_generators': False, 'stem': False, 'subgroup': '2.1', 'subgroup_hash': 1, 'subgroup_order': 2, 'subgroup_tex': 'C_2', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '4896.bd', 'aut_centralizer_order': 4896, 'aut_label': '2448.a1', 'aut_quo_index': 1, 'aut_stab_index': 1, 'aut_weyl_group': '1.1', 'aut_weyl_index': 4896, 'centralizer': '1.a1.a1', 'complements': ['2.a1.a1'], 'conjugacy_class_count': 1, 'contained_in': ['144.a1.a1', '816.a1.a1', '1224.a1.a1'], 'contains': ['4896.a1.a1'], 'core': '2448.a1.a1', 'coset_action_label': None, 'count': 1, 'diagramx': [6397, 6592, 2842, 3390, 4742, 7153, 3879, 2895], 'generators': [1], 'label': '4896.bd.2448.a1.a1', 'mobius_quo': -1, 'mobius_sub': 0, 'normal_closure': '2448.a1.a1', 'normal_contained_in': ['1.a1.a1'], 'normal_contains': ['4896.a1.a1'], 'normalizer': '1.a1.a1', 'old_label': '2448.a1.a1', 'projective_image': '2448.a', 'quotient_action_image': '1.1', 'quotient_action_kernel': '2448.a', 'quotient_action_kernel_order': 2448, 'quotient_fusion': None, 'short_label': '2448.a1.a1', 'subgroup_fusion': None, 'weyl_group': '1.1'}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': True, 'abelian_quotient': '2.1', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': True, 'aut_cyclic': True, 'aut_derived_length': 0, 'aut_exponent': 1, 'aut_gen_orders': [], 'aut_gens': [[1]], 'aut_group': '1.1', 'aut_hash': 1, 'aut_nilpotency_class': 0, 'aut_nilpotent': True, 'aut_order': 1, 'aut_permdeg': 1, 'aut_perms': [], 'aut_phi_ratio': 1.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_1', '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': 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]], 'center_label': '2.1', 'center_order': 2, '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'], 'composition_length': 1, 'conjugacy_classes_known': True, 'counter': 1, 'cyclic': True, 'derived_length': 1, 'dihedral': True, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1]], 'element_repr_type': 'PC', 'elementary': 2, 'eulerian_function': 1, 'exponent': 2, 'exponents_of_order': [1], 'factors_of_aut_order': [], 'factors_of_order': [2], 'faithful_reps': [[1, 1, 1]], 'familial': True, 'frattini_label': '1.1', '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': 1, 'irrQ_dim': 1, 'irrR_degree': 1, 'irrep_stats': [[1, 2]], 'label': '2.1', 'linC_count': 1, 'linC_degree': 1, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 1, 'linQ_degree_count': 1, 'linQ_dim': 1, 'linQ_dim_count': 1, 'linR_count': 1, 'linR_degree': 1, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C2', 'ngens': 1, 'nilpotency_class': 1, 'nilpotent': True, 'normal_counts': [0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 2, 'number_characteristic_subgroups': 2, 'number_conjugacy_classes': 2, 'number_divisions': 2, 'number_normal_subgroups': 2, 'number_subgroup_autclasses': 2, 'number_subgroup_classes': 2, 'number_subgroups': 2, 'old_label': None, 'order': 2, 'order_factorization_type': 1, 'order_stats': [[1, 1], [2, 1]], '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': 1, 'perfect': False, 'permutation_degree': 2, 'pgroup': 2, 'primary_abelian_invariants': [2], 'quasisimple': False, 'rank': 1, 'rational': True, 'rational_characters_known': True, 'ratrep_stats': [[1, 2]], 'representations': {'PC': {'code': 0, 'gens': [1], 'pres': [1, -2]}, 'GLZ': {'b': 3, 'd': 2, 'gens': [12]}, 'Lie': [{'d': 2, 'q': 2, 'gens': [6], 'family': 'CSOPlus'}, {'d': 2, 'q': 2, 'gens': [6], 'family': 'COPlus'}, {'d': 1, 'q': 2, 'gens': [1], 'family': 'AGL'}, {'d': 1, 'q': 2, 'gens': [1], 'family': 'AGammaL'}, {'d': 1, 'q': 2, 'gens': [1], 'family': 'ASigmaL'}], 'GLFp': {'d': 2, 'p': 2, 'gens': [11]}, 'Perm': {'d': 2, 'gens': [1]}}, 'schur_multiplier': [], 'semidirect_product': False, 'simple': True, 'smith_abelian_invariants': [2], 'solvability_type': 0, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_2', 'transitive_degree': 2, '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': '2.1', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 1, 'aut_exponent': 2448, 'aut_gen_orders': [16, 3], 'aut_gens': [[135543539892112974, 264010865022884329], [2149254125133149160, 2172566552297122969], [219022444164965400, 1640688163172519455]], 'aut_group': '4896.b', 'aut_hash': 2876299048306215829, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 4896, 'aut_permdeg': 153, 'aut_perms': [54890881609024359888675753853316152083985303113277480739552057756328936732430398279921048494060907080311882510048026982887601202633078260925380064391939175656046913109932796825008884652479965776931291782722921952935372358484772713209795114613482147789155676156381151653, 170367645667895131090345731231047522453666398200409924060399138181249147572171548193166633531229625550109840756112758768046233213343694282805038562539885691358831594870683284637644237087170281298276535499683252037098538709847867433538144354183262327417344692889248925574], 'aut_phi_ratio': 3.1875, 'aut_solvable': False, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 153, 1, 2], [3, 272, 1, 1], [4, 306, 1, 2], [6, 272, 1, 1], [8, 306, 1, 4], [9, 272, 1, 3], [17, 144, 2, 1], [18, 272, 1, 3], [34, 144, 2, 1]], 'aut_supersolvable': False, 'aut_tex': '\\PGL(2,17)', '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': 2448, 'autcentquo_group': '4896.b', 'autcentquo_hash': 2876299048306215829, 'autcentquo_nilpotent': False, 'autcentquo_order': 4896, 'autcentquo_solvable': False, 'autcentquo_supersolvable': False, 'autcentquo_tex': '\\PGL(2,17)', 'cc_stats': [[1, 1, 1], [2, 1, 1], [2, 153, 2], [3, 272, 1], [4, 306, 2], [6, 272, 1], [8, 306, 4], [9, 272, 3], [17, 144, 2], [18, 272, 3], [34, 144, 2]], 'center_label': '2.1', 'center_order': 2, 'central_product': True, 'central_quotient': '2448.a', 'commutator_count': 1, 'commutator_label': '2448.a', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2448.a'], 'composition_length': 2, 'conjugacy_classes_known': True, 'counter': 30, 'cyclic': False, 'derived_length': 1, 'dihedral': False, 'direct_factorization': [['2.1', 1], ['2448.a', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 153, 1, 2], [3, 272, 1, 1], [4, 306, 1, 2], [6, 272, 1, 1], [8, 306, 2, 2], [9, 272, 3, 1], [17, 144, 2, 1], [18, 272, 3, 1], [34, 144, 2, 1]], 'element_repr_type': 'Perm', 'elementary': 1, 'eulerian_function': 3396, 'exponent': 1224, 'exponents_of_order': [5, 2, 1], 'factors_of_aut_order': [2, 3, 17], 'factors_of_order': [2, 3, 17], 'faithful_reps': [[9, 1, 2], [16, 1, 4], [17, 1, 1], [18, 1, 3]], 'familial': False, 'frattini_label': '1.1', 'frattini_quotient': '4896.bd', 'hash': 8320548675179894970, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 1224, 'inner_gen_orders': [9, 17], 'inner_gens': [[135543539892112974, 1861105183396077265], [1173274868321445744, 264010865022884329]], 'inner_hash': 7454658504626110115, 'inner_nilpotent': False, 'inner_order': 2448, 'inner_split': True, 'inner_tex': '\\PSL(2,17)', 'inner_used': [1, 2], 'irrC_degree': 9, 'irrQ_degree': 16, 'irrQ_dim': 16, 'irrR_degree': 9, 'irrep_stats': [[1, 2], [9, 4], [16, 8], [17, 2], [18, 6]], 'label': '4896.bd', 'linC_count': 2, 'linC_degree': 9, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 16, 'linQ_degree_count': 1, 'linQ_dim': 16, 'linQ_dim_count': 1, 'linR_count': 2, 'linR_degree': 9, 'maximal_subgroups_known': True, 'metabelian': False, 'metacyclic': False, 'monomial': False, 'name': 'C2*PSL(2,17)', 'ngens': 2, 'nilpotency_class': -1, 'nilpotent': False, 'normal_counts': [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 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': 20, 'number_characteristic_subgroups': 4, 'number_conjugacy_classes': 22, 'number_divisions': 14, 'number_normal_subgroups': 4, 'number_subgroup_autclasses': 50, 'number_subgroup_classes': 63, 'number_subgroups': 8090, 'old_label': None, 'order': 4896, 'order_factorization_type': 321, 'order_stats': [[1, 1], [2, 307], [3, 272], [4, 612], [6, 272], [8, 1224], [9, 816], [17, 288], [18, 816], [34, 288]], 'outer_abelian': True, 'outer_cyclic': True, 'outer_equivalence': False, 'outer_exponent': 2, 'outer_gen_orders': [2], 'outer_gen_pows': [0], 'outer_gens': [[1672206430552091454, 52361246387299687]], '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': None, 'perfect': False, 'permutation_degree': 20, 'pgroup': 0, 'primary_abelian_invariants': [2], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 2], [16, 2], [17, 2], [18, 4], [36, 2], [48, 2]], 'representations': {'Perm': {'d': 20, 'gens': [135543539892112974, 264010865022884329]}}, 'schur_multiplier': [2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2], 'solvability_type': 13, 'solvable': False, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': 'C_2\\times \\PSL(2,17)', 'transitive_degree': 36, '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': '1.1', 'all_subgroups_known': True, 'almost_simple': True, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 1, 'aut_exponent': 2448, 'aut_gen_orders': [16, 3], 'aut_gens': [[29481, 4657], [14881, 72507], [11838, 83096]], 'aut_group': '4896.b', 'aut_hash': 2876299048306215829, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 4896, 'aut_permdeg': 153, 'aut_perms': [126433037395781308191752831730948071705295179547638450660075145121114988819146559058062013198709398919298942547528431562454414304749622582453967352385456558299296110878841297028032236947606217110531781133136839923307307040823030235361462428802933982332744384181750676906, 33446647067438622332164923145896361945417459819576376143924702938622391143474787266927644711973487372571071995622854390347478961423261692014769907883109790853988427511696749612116366079470484640774865377208832718258695949276963520594469616604076893956763143556847659357], 'aut_phi_ratio': 6.375, 'aut_solvable': False, 'aut_stats': [[1, 1, 1, 1], [2, 153, 1, 1], [3, 272, 1, 1], [4, 306, 1, 1], [8, 306, 1, 2], [9, 272, 1, 3], [17, 144, 2, 1]], 'aut_supersolvable': False, 'aut_tex': '\\PGL(2,17)', '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': 2448, 'autcentquo_group': '4896.b', 'autcentquo_hash': 2876299048306215829, 'autcentquo_nilpotent': False, 'autcentquo_order': 4896, 'autcentquo_solvable': False, 'autcentquo_supersolvable': False, 'autcentquo_tex': '\\PGL(2,17)', 'cc_stats': [[1, 1, 1], [2, 153, 1], [3, 272, 1], [4, 306, 1], [8, 306, 2], [9, 272, 3], [17, 144, 2]], 'center_label': '1.1', 'center_order': 1, 'central_product': False, 'central_quotient': '2448.a', 'commutator_count': 1, 'commutator_label': '2448.a', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2448.a'], 'composition_length': 1, 'conjugacy_classes_known': True, 'counter': 1, 'cyclic': False, 'derived_length': 0, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 153, 1, 1], [3, 272, 1, 1], [4, 306, 1, 1], [8, 306, 2, 1], [9, 272, 3, 1], [17, 144, 2, 1]], 'element_repr_type': 'Lie', 'elementary': 1, 'eulerian_function': 1132, 'exponent': 1224, 'exponents_of_order': [4, 2, 1], 'factors_of_aut_order': [2, 3, 17], 'factors_of_order': [2, 3, 17], 'faithful_reps': [[9, 1, 2], [16, 1, 4], [17, 1, 1], [18, 1, 3]], 'familial': True, 'frattini_label': '1.1', 'frattini_quotient': '2448.a', 'hash': 7454658504626110115, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 1224, 'inner_gen_orders': [8, 3], 'inner_gens': [[29481, 2362], [15612, 4657]], 'inner_hash': 7454658504626110115, 'inner_nilpotent': False, 'inner_order': 2448, 'inner_split': True, 'inner_tex': '\\PSL(2,17)', 'inner_used': [1, 2], 'irrC_degree': 9, 'irrQ_degree': 16, 'irrQ_dim': 16, 'irrR_degree': 9, 'irrep_stats': [[1, 1], [9, 2], [16, 4], [17, 1], [18, 3]], 'label': '2448.a', 'linC_count': 2, 'linC_degree': 9, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 16, 'linQ_degree_count': 1, 'linQ_dim': 16, 'linQ_dim_count': 1, 'linR_count': 2, 'linR_degree': 9, 'maximal_subgroups_known': True, 'metabelian': False, 'metacyclic': False, 'monomial': False, 'name': 'PSL(2,17)', 'ngens': 2, 'nilpotency_class': -1, 'nilpotent': False, 'normal_counts': [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 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': 10, 'number_characteristic_subgroups': 2, 'number_conjugacy_classes': 11, 'number_divisions': 7, 'number_normal_subgroups': 2, 'number_subgroup_autclasses': 18, 'number_subgroup_classes': 22, 'number_subgroups': 2420, 'old_label': None, 'order': 2448, 'order_factorization_type': 321, 'order_stats': [[1, 1], [2, 153], [3, 272], [4, 306], [8, 612], [9, 816], [17, 288]], 'outer_abelian': True, 'outer_cyclic': True, 'outer_equivalence': False, 'outer_exponent': 2, 'outer_gen_orders': [2], 'outer_gen_pows': [12752], 'outer_gens': [[5549, 15109]], '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': None, 'perfect': True, 'permutation_degree': 18, 'pgroup': 0, 'primary_abelian_invariants': [], 'quasisimple': True, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 1], [16, 1], [17, 1], [18, 2], [36, 1], [48, 1]], 'representations': {'Lie': [{'d': 2, 'q': 17, 'gens': [4657, 29481], 'family': 'PSL'}, {'d': 2, 'q': 17, 'family': 'PSU'}, {'d': 3, 'q': 17, 'gens': [62781900492, 387620707], 'family': 'Omega'}, {'d': 3, 'q': 17, 'family': 'POmega'}, {'d': 2, 'q': 17, 'family': 'PSigmaL'}], 'Perm': {'d': 18, 'gens': [1524124145418846, 2784456841984956]}}, 'schur_multiplier': [2], 'semidirect_product': False, 'simple': True, 'smith_abelian_invariants': [], 'solvability_type': 13, 'solvable': False, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': '\\PSL(2,17)', 'transitive_degree': 18, 'wreath_data': None, 'wreath_product': False}
