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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': True, 'ambient': '1190.4', 'ambient_counter': 4, 'ambient_order': 1190, 'ambient_tex': 'D_5\\times C_{119}', 'central': False, 'central_factor': False, 'centralizer_order': 595, 'characteristic': True, 'core_order': 5, 'counter': 14, 'cyclic': True, 'direct': False, 'hall': 5, 'label': '1190.4.238.a1.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': True, 'minimal_normal': True, 'nilpotent': True, 'normal': True, 'old_label': '238.a1.a1', 'outer_equivalence': False, 'perfect': False, 'proper': True, 'quotient': '238.4', 'quotient_Agroup': True, 'quotient_abelian': True, 'quotient_cyclic': True, 'quotient_hash': 4, 'quotient_metabelian': True, 'quotient_nilpotent': True, 'quotient_order': 238, 'quotient_simple': False, 'quotient_solvable': True, 'quotient_supersolvable': True, 'quotient_tex': 'C_{238}', 'simple': True, 'solvable': True, 'special_labels': ['D', 'L1', 'D1'], 'split': True, 'standard_generators': False, 'stem': False, 'subgroup': '5.1', 'subgroup_hash': 1, 'subgroup_order': 5, 'subgroup_tex': 'C_5', 'supersolvable': True, 'sylow': 5}
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gps_subgroup_data • Show schema
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{'ambient': '1190.4', 'aut_centralizer_order': 480, 'aut_label': '238.a1', 'aut_quo_index': 1, 'aut_stab_index': 1, 'aut_weyl_group': '4.1', 'aut_weyl_index': 480, 'centralizer': '2.a1.a1', 'complements': ['5.a1.a1'], 'conjugacy_class_count': 1, 'contained_in': ['14.a1.a1', '34.a1.a1', '119.a1.a1'], 'contains': ['1190.a1.a1'], 'core': '238.a1.a1', 'coset_action_label': None, 'count': 1, 'diagramx': [3320, 4857, 6839, 5318, 6086, 5795, 4517, 5795], 'generators': [952], 'label': '1190.4.238.a1.a1', 'mobius_quo': -1, 'mobius_sub': -1, 'normal_closure': '238.a1.a1', 'normal_contained_in': ['14.a1.a1', '34.a1.a1', '119.a1.a1'], 'normal_contains': ['1190.a1.a1'], 'normalizer': '1.a1.a1', 'old_label': '238.a1.a1', 'projective_image': '1190.4', 'quotient_action_image': '2.1', 'quotient_action_kernel': '119.1', 'quotient_action_kernel_order': 119, 'quotient_fusion': None, 'short_label': '238.a1.a1', 'subgroup_fusion': None, 'weyl_group': '2.1'}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': True, 'abelian_quotient': '5.1', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': True, 'aut_cyclic': True, 'aut_derived_length': 1, 'aut_exponent': 4, 'aut_gen_orders': [4], 'aut_gens': [[1], [2]], 'aut_group': '4.1', 'aut_hash': 1, 'aut_nilpotency_class': 1, 'aut_nilpotent': True, 'aut_order': 4, 'aut_permdeg': 4, 'aut_perms': [9], 'aut_phi_ratio': 1.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [5, 1, 4, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_4', 'autcent_abelian': True, 'autcent_cyclic': True, 'autcent_exponent': 4, 'autcent_group': '4.1', 'autcent_hash': 1, 'autcent_nilpotent': True, 'autcent_order': 4, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_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], [5, 1, 4]], 'center_label': '5.1', 'center_order': 5, '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': ['5.1'], 'composition_length': 1, 'conjugacy_classes_known': True, 'counter': 1, 'cyclic': True, 'derived_length': 1, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [5, 1, 4, 1]], 'element_repr_type': 'PC', 'elementary': 5, 'eulerian_function': 1, 'exponent': 5, 'exponents_of_order': [1], 'factors_of_aut_order': [2], 'factors_of_order': [5], 'faithful_reps': [[1, 0, 4]], 'familial': True, 'frattini_label': '1.1', 'frattini_quotient': '5.1', 'hash': 1, 'hyperelementary': 5, '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': 4, 'irrQ_dim': 4, 'irrR_degree': 2, 'irrep_stats': [[1, 5]], 'label': '5.1', 'linC_count': 4, 'linC_degree': 1, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 4, 'linQ_degree_count': 1, 'linQ_dim': 4, 'linQ_dim_count': 1, 'linR_count': 2, 'linR_degree': 2, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C5', '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': 5, 'number_divisions': 2, 'number_normal_subgroups': 2, 'number_subgroup_autclasses': 2, 'number_subgroup_classes': 2, 'number_subgroups': 2, 'old_label': None, 'order': 5, 'order_factorization_type': 1, 'order_stats': [[1, 1], [5, 4]], 'outer_abelian': True, 'outer_cyclic': True, 'outer_equivalence': False, 'outer_exponent': 4, 'outer_gen_orders': [4], 'outer_gen_pows': [0], 'outer_gens': [[2]], 'outer_group': '4.1', 'outer_hash': 1, 'outer_nilpotent': True, 'outer_order': 4, 'outer_permdeg': 4, 'outer_perms': [9], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_4', 'pc_rank': 1, 'perfect': False, 'permutation_degree': 5, 'pgroup': 5, 'primary_abelian_invariants': [5], 'quasisimple': False, 'rank': 1, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 1], [4, 1]], 'representations': {'PC': {'code': 0, 'gens': [1], 'pres': [1, -5]}, 'GLZ': {'b': 3, 'd': 4, 'gens': [26329372]}, 'Lie': [{'d': 1, 'q': 5, 'gens': [33], 'family': 'ASigmaL'}], 'GLFp': {'d': 2, 'p': 5, 'gens': [131]}, 'Perm': {'d': 5, 'gens': [96]}}, 'schur_multiplier': [], 'semidirect_product': False, 'simple': True, 'smith_abelian_invariants': [5], 'solvability_type': 0, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_5', 'transitive_degree': 5, 'wreath_data': None, 'wreath_product': False}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': False, 'abelian_quotient': '238.4', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 240, 'aut_gen_orders': [16, 6, 4, 5], 'aut_gens': [[1, 2], [1, 282], [1, 542], [1, 716], [953, 2]], 'aut_group': '1920.38210', 'aut_hash': 5717208605233935285, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 1920, 'aut_permdeg': 27, 'aut_perms': [239495901217564176449108937, 288874431142094179561456867, 23669960485813123916312, 10098447527665271987895116312], 'aut_phi_ratio': 5.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 5, 1, 1], [5, 2, 2, 1], [7, 1, 6, 1], [14, 5, 6, 1], [17, 1, 16, 1], [34, 5, 16, 1], [35, 2, 12, 1], [85, 2, 32, 1], [119, 1, 96, 1], [238, 5, 96, 1], [595, 2, 192, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_2\\times C_{48}\\times F_5', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 48, 'autcent_group': '96.59', 'autcent_hash': 59, 'autcent_nilpotent': True, 'autcent_order': 96, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2\\times C_{48}', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 20, 'autcentquo_group': '20.3', 'autcentquo_hash': 3, 'autcentquo_nilpotent': False, 'autcentquo_order': 20, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'F_5', 'cc_stats': [[1, 1, 1], [2, 5, 1], [5, 2, 2], [7, 1, 6], [14, 5, 6], [17, 1, 16], [34, 5, 16], [35, 2, 12], [85, 2, 32], [119, 1, 96], [238, 5, 96], [595, 2, 192]], 'center_label': '119.1', 'center_order': 119, 'central_product': True, 'central_quotient': '10.1', 'commutator_count': 1, 'commutator_label': '5.1', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '5.1', '7.1', '17.1'], 'composition_length': 4, 'conjugacy_classes_known': True, 'counter': 4, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['10.1', 1], ['17.1', 1], ['7.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 5, 1, 1], [5, 2, 2, 1], [7, 1, 6, 1], [14, 5, 6, 1], [17, 1, 16, 1], [34, 5, 16, 1], [35, 2, 12, 1], [85, 2, 32, 1], [119, 1, 96, 1], [238, 5, 96, 1], [595, 2, 192, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 432, 'exponent': 1190, 'exponents_of_order': [1, 1, 1, 1], 'factors_of_aut_order': [2, 3, 5], 'factors_of_order': [2, 5, 7, 17], 'faithful_reps': [[2, 0, 192]], 'familial': False, 'frattini_label': '1.1', 'frattini_quotient': '1190.4', 'hash': 4, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 10, 'inner_gen_orders': [2, 5], 'inner_gens': [[1, 478], [715, 2]], 'inner_hash': 1, 'inner_nilpotent': False, 'inner_order': 10, 'inner_split': False, 'inner_tex': 'D_5', 'inner_used': [1, 2], 'irrC_degree': 2, 'irrQ_degree': 384, 'irrQ_dim': 384, 'irrR_degree': 4, 'irrep_stats': [[1, 238], [2, 238]], 'label': '1190.4', 'linC_count': 192, 'linC_degree': 2, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 26, 'linQ_degree_count': 4, 'linQ_dim': 26, 'linQ_dim_count': 4, 'linR_count': 288, 'linR_degree': 4, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'D5*C119', 'ngens': 4, 'nilpotency_class': -1, 'nilpotent': False, 'normal_counts': [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': 12, 'number_characteristic_subgroups': 12, 'number_conjugacy_classes': 476, 'number_divisions': 12, 'number_normal_subgroups': 12, 'number_subgroup_autclasses': 16, 'number_subgroup_classes': 16, 'number_subgroups': 32, 'old_label': None, 'order': 1190, 'order_factorization_type': 11, 'order_stats': [[1, 1], [2, 5], [5, 4], [7, 6], [14, 30], [17, 16], [34, 80], [35, 24], [85, 64], [119, 96], [238, 480], [595, 384]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 48, 'outer_gen_orders': [2, 2, 48], 'outer_gen_pows': [0, 1, 0], 'outer_gens': [[1, 712], [1, 954], [1, 932]], 'outer_group': '192.935', 'outer_hash': 935, 'outer_nilpotent': True, 'outer_order': 192, 'outer_permdeg': 192, 'outer_perms': [59477336681893871063998436910138657035849835005431922430814043397623172012773675919791929673355932439995826851830117381352700639534080304671478001824499524887025096467761843621535183826228520344284741251866951088882172131694792914611926543443598145273723117397436549661796722427788090004938157987925897648024605513860239586484390294548363490029156774332752, 1848992581474398139915385700582335214304909119859105759881437402168469054722279352408872365821917419936094621453172414440070496141911124131961188973087854785759030077376799898641313782825874938555077283844030499836063704344252112166662992552681107558165502312385582699072047467351771211962729587629262490089849187610131268709401146867497898583379392681647, 33456219558634517153591499725167966198921296397344956061896503630815732018298739789543549465444400214778262097846767716731086511810512088637602898593501003299672341135979488177574512216093053505803595170516324063729396196209306120202580702141099452098499298137959137465699389407016133151013964747146655761084313094657517383467413531076270747422652008767296], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^2\\times C_{48}', 'pc_rank': 2, 'perfect': False, 'permutation_degree': 29, 'pgroup': 0, 'primary_abelian_invariants': [2, 7, 17], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 2], [4, 1], [6, 2], [16, 2], [24, 1], [64, 1], [96, 2], [384, 1]], 'representations': {'PC': {'code': 21287401658647391, 'gens': [1, 2], 'pres': [4, -2, -5, -7, -17, 3825, 45, 94]}, 'GLFp': {'d': 2, 'p': 239, 'gens': [3249156723, 2493886790]}, 'Perm': {'d': 29, 'gens': [10904380660461683146752000000, 27029669358316584960000, 22324392524313, 326681594722593895809024000000]}}, 'schur_multiplier': [], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [238], 'solvability_type': 6, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'D_5\\times C_{119}', 'transitive_degree': 595, 'wreath_data': None, 'wreath_product': False}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': True, 'abelian_quotient': '238.4', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': True, 'aut_cyclic': False, 'aut_derived_length': 1, 'aut_exponent': 48, 'aut_gen_orders': [2, 48], 'aut_gens': [[1], [69], [3]], 'aut_group': '96.59', 'aut_hash': 59, 'aut_nilpotency_class': 1, 'aut_nilpotent': True, 'aut_order': 96, 'aut_permdeg': 21, 'aut_perms': [2439325392333218664, 19984155209275203], 'aut_phi_ratio': 1.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [7, 1, 6, 1], [14, 1, 6, 1], [17, 1, 16, 1], [34, 1, 16, 1], [119, 1, 96, 1], [238, 1, 96, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_2\\times C_{48}', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 48, 'autcent_group': '96.59', 'autcent_hash': 59, 'autcent_nilpotent': True, 'autcent_order': 96, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2\\times C_{48}', '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], [7, 1, 6], [14, 1, 6], [17, 1, 16], [34, 1, 16], [119, 1, 96], [238, 1, 96]], 'center_label': '238.4', 'center_order': 238, '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', '7.1', '17.1'], 'composition_length': 3, 'conjugacy_classes_known': True, 'counter': 4, 'cyclic': True, 'derived_length': 1, 'dihedral': False, 'direct_factorization': [['17.1', 1], ['2.1', 1], ['7.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [7, 1, 6, 1], [14, 1, 6, 1], [17, 1, 16, 1], [34, 1, 16, 1], [119, 1, 96, 1], [238, 1, 96, 1]], 'element_repr_type': 'PC', 'elementary': 238, 'eulerian_function': 1, 'exponent': 238, 'exponents_of_order': [1, 1, 1], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 7, 17], 'faithful_reps': [[1, 0, 96]], 'familial': True, 'frattini_label': '1.1', 'frattini_quotient': '238.4', 'hash': 4, 'hyperelementary': 238, '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': 96, 'irrQ_dim': 96, 'irrR_degree': 2, 'irrep_stats': [[1, 238]], 'label': '238.4', 'linC_count': 96, 'linC_degree': 1, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 22, 'linQ_degree_count': 3, 'linQ_dim': 22, 'linQ_dim_count': 3, 'linR_count': 48, 'linR_degree': 2, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C238', 'ngens': 3, 'nilpotency_class': 1, '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': 8, 'number_characteristic_subgroups': 8, 'number_conjugacy_classes': 238, 'number_divisions': 8, 'number_normal_subgroups': 8, 'number_subgroup_autclasses': 8, 'number_subgroup_classes': 8, 'number_subgroups': 8, 'old_label': None, 'order': 238, 'order_factorization_type': 11, 'order_stats': [[1, 1], [2, 1], [7, 6], [14, 6], [17, 16], [34, 16], [119, 96], [238, 96]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 48, 'outer_gen_orders': [2, 48], 'outer_gen_pows': [0, 0], 'outer_gens': [[69], [3]], 'outer_group': '96.59', 'outer_hash': 59, 'outer_nilpotent': True, 'outer_order': 96, 'outer_permdeg': 21, 'outer_perms': [2439325392333218664, 19984155209275203], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2\\times C_{48}', 'pc_rank': 1, 'perfect': False, 'permutation_degree': 26, 'pgroup': 0, 'primary_abelian_invariants': [2, 7, 17], 'quasisimple': False, 'rank': 1, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 2], [6, 2], [16, 2], [96, 2]], 'representations': {'PC': {'code': 33435743, 'gens': [1], 'pres': [3, -2, -7, -17, 6, 46]}, 'GLFp': {'d': 2, 'p': 239, 'gens': [1064849869]}, 'Perm': {'d': 26, 'gens': [15511210043330985984000000, 155112100433309859840000, 334764638208000]}}, 'schur_multiplier': [], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [238], 'solvability_type': 0, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_{238}', 'transitive_degree': 238, 'wreath_data': None, 'wreath_product': False}
