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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': True, 'ambient': '1600.10114', 'ambient_counter': 10114, 'ambient_order': 1600, 'ambient_tex': 'C_2^2\\times C_{20}:F_5', 'central': False, 'central_factor': False, 'centralizer_order': 64, 'characteristic': False, 'core_order': 2, 'counter': 139, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '1600.10114.400.h1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': False, 'old_label': '400.h1', 'outer_equivalence': True, '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': 400, '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': '4.2', 'subgroup_hash': 2, 'subgroup_order': 4, 'subgroup_tex': 'C_2^2', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '1600.10114', 'aut_centralizer_order': 16384, 'aut_label': '400.h1', 'aut_quo_index': None, 'aut_stab_index': 75, 'aut_weyl_group': '2.1', 'aut_weyl_index': 1228800, 'centralizer': '25.a1', 'complements': None, 'conjugacy_class_count': 3, 'contained_in': ['80.n1', '80.o1', '200.d1', '200.e1', '200.k1'], 'contains': ['800.b1', '800.d1'], 'core': '800.b1', 'coset_action_label': None, 'count': 75, 'diagramx': [5171, -1, 4838, -1], 'generators': [5, 800], 'label': '1600.10114.400.h1', 'mobius_quo': None, 'mobius_sub': 0, 'normal_closure': '16.g1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '25.a1', 'old_label': '400.h1', 'projective_image': '800.1206', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '400.h1', 'subgroup_fusion': None, 'weyl_group': '1.1'}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': True, 'abelian_quotient': '4.2', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 6, 'aut_gen_orders': [2, 3], 'aut_gens': [[1, 2], [3, 2], [2, 3]], 'aut_group': '6.1', 'aut_hash': 1, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 6, 'aut_permdeg': 3, 'aut_perms': [1, 4], 'aut_phi_ratio': 3.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 3, 1]], 'aut_supersolvable': True, 'aut_tex': 'S_3', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 6, 'autcent_group': '6.1', 'autcent_hash': 1, 'autcent_nilpotent': False, 'autcent_order': 6, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'S_3', '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]], 'center_label': '4.2', 'center_order': 4, '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'], 'composition_length': 2, 'conjugacy_classes_known': True, 'counter': 2, 'cyclic': False, 'derived_length': 1, 'dihedral': True, 'direct_factorization': [['2.1', 2]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3]], 'element_repr_type': 'PC', 'elementary': 2, 'eulerian_function': 1, 'exponent': 2, 'exponents_of_order': [2], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2], 'faithful_reps': [], 'familial': True, 'frattini_label': '1.1', '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, 4]], 'label': '4.2', 'linC_count': 3, 'linC_degree': 2, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 2, 'linQ_degree_count': 3, 'linQ_dim': 2, 'linQ_dim_count': 3, 'linR_count': 3, 'linR_degree': 2, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C2^2', 'ngens': 2, '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': 2, 'number_characteristic_subgroups': 2, 'number_conjugacy_classes': 4, 'number_divisions': 4, 'number_normal_subgroups': 5, 'number_subgroup_autclasses': 3, 'number_subgroup_classes': 5, 'number_subgroups': 5, 'old_label': None, 'order': 4, 'order_factorization_type': 2, 'order_stats': [[1, 1], [2, 3]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 6, 'outer_gen_orders': [2, 3], 'outer_gen_pows': [0, 0], 'outer_gens': [[3, 2], [2, 3]], 'outer_group': '6.1', 'outer_hash': 1, 'outer_nilpotent': False, 'outer_order': 6, 'outer_permdeg': 3, 'outer_perms': [1, 4], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'S_3', 'pc_rank': 2, 'perfect': False, 'permutation_degree': 4, 'pgroup': 2, 'primary_abelian_invariants': [2, 2], 'quasisimple': False, 'rank': 2, 'rational': True, 'rational_characters_known': True, 'ratrep_stats': [[1, 4]], 'representations': {'PC': {'code': 0, 'gens': [1, 2], 'pres': [2, -2, 2]}, 'GLZ': {'b': 3, 'd': 2, 'gens': [14, 12]}, 'GLFp': {'d': 2, 'p': 3, 'gens': [55, 56]}, 'Perm': {'d': 4, 'gens': [6, 1]}}, 'schur_multiplier': [2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2], 'solvability_type': 1, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_2^2', 'transitive_degree': 4, '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': '32.45', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 4, 'aut_exponent': 120, 'aut_gen_orders': [60, 8, 8, 4, 8], 'aut_gens': [[1, 2, 8, 80], [41, 546, 49, 560], [841, 487, 520, 1257], [41, 126, 360, 1217], [40, 715, 65, 721], [801, 594, 825, 1080]], 'aut_group': None, 'aut_hash': 903259633507885408, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 2457600, 'aut_permdeg': 800, 'aut_perms': 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'aut_phi_ratio': 3840.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 1, 6, 1], [2, 25, 1, 2], [2, 25, 6, 1], [4, 2, 4, 1], [4, 50, 4, 1], [4, 50, 16, 1], [5, 4, 2, 1], [5, 4, 4, 1], [10, 4, 2, 1], [10, 4, 4, 1], [10, 4, 12, 1], [10, 4, 24, 1], [20, 4, 16, 1], [20, 4, 32, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_{10}^2.C_2^6.C_{12}.C_2.C_2^4', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 12, 'autcent_group': None, 'autcent_hash': 4599629986281706675, 'autcent_nilpotent': False, 'autcent_order': 1536, 'autcent_solvable': True, 'autcent_split': None, 'autcent_supersolvable': False, 'autcent_tex': 'C_2^4.C_2^4.S_3', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 40, 'autcentquo_group': None, 'autcentquo_hash': 10254, 'autcentquo_nilpotent': False, 'autcentquo_order': 1600, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'F_5^2.C_2^2', 'cc_stats': [[1, 1, 1], [2, 1, 7], [2, 25, 8], [4, 2, 4], [4, 50, 20], [5, 4, 6], [10, 4, 42], [20, 4, 48]], 'center_label': '8.5', 'center_order': 8, 'central_product': True, 'central_quotient': '200.48', 'commutator_count': 1, 'commutator_label': '50.5', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '2.1', '2.1', '5.1', '5.1'], 'composition_length': 8, 'conjugacy_classes_known': True, 'counter': 10114, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['2.1', 2], ['400.159', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 7], [2, 25, 1, 8], [4, 2, 1, 4], [4, 50, 1, 4], [4, 50, 2, 8], [5, 4, 1, 2], [5, 4, 2, 2], [10, 4, 1, 14], [10, 4, 2, 14], [20, 4, 2, 8], [20, 4, 4, 8]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 807240, 'exponent': 20, 'exponents_of_order': [6, 2], 'factors_of_aut_order': [2, 3, 5], 'factors_of_order': [2, 5], 'faithful_reps': [], 'familial': False, 'frattini_label': '2.1', 'frattini_quotient': '800.1206', 'hash': 10114, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 20, 'inner_gen_orders': [1, 4, 5, 10], 'inner_gens': [[1, 2, 8, 80], [1, 2, 56, 240], [1, 34, 8, 80], [1, 1442, 8, 80]], 'inner_hash': 48, 'inner_nilpotent': False, 'inner_order': 200, 'inner_split': False, 'inner_tex': 'C_{10}:F_5', 'inner_used': [2, 3, 4], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 32], [2, 8], [4, 96]], 'label': '1600.10114', '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': 'C2^2*C20:F5', 'ngens': 8, '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], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 17, 'number_characteristic_subgroups': 15, 'number_conjugacy_classes': 136, 'number_divisions': 80, 'number_normal_subgroups': 250, 'number_subgroup_autclasses': 147, 'number_subgroup_classes': 1072, 'number_subgroups': 10028, 'old_label': None, 'order': 1600, 'order_factorization_type': 32, 'order_stats': [[1, 1], [2, 207], [4, 1008], [5, 24], [10, 168], [20, 192]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': True, 'outer_exponent': 12, 'outer_gen_orders': [12, 4, 4, 4, 4], 'outer_gen_pows': [32, 996, 1300, 0, 400], 'outer_gens': [[41, 546, 49, 560], [841, 487, 520, 1257], [41, 126, 360, 1217], [40, 715, 65, 721], [801, 594, 825, 1080]], 'outer_group': None, 'outer_hash': 2344437273852571667, 'outer_nilpotent': False, 'outer_order': 12288, 'outer_permdeg': 128, 'outer_perms': [3132040021508802927290269141570017422857330610671891354553566641039637464402080576138684591249920663426037382124505111153183243768833527067275224016116457536465203329156010655890757625195161321067596594665233227696, 6169364629371319899755151898881083788298654028346707871761536469177509373782261576737311687857680425328409426777232513766239155014632270354730962870411445244661396702959791208346656577568035874347299915799297171171, 9205936166363081301780858153393229230684411824316496454106022942306972704075666064999925615488354822023767062974601570917810771466024574249828326570758081965613897047819153727720975181907674495650051149248281063857, 12242507703354842703805881744463049061479108027061598797626361906958168685160122627291683937118911740542720732204103661908481996702160829786556225597264530992260629745067894582260618220991691119868027797596686483732, 15279079240331077769413962336919678282484802373724633899067864127023365610082439349663594010204575718831809897884624410978249109658620250724257654749479671937309217662894520353555404307364105683316548587558731889514], 'outer_solvable': True, 'outer_supersolvable': False, 'outer_tex': 'C_2^5.A_4.C_2^4.C_2', 'pc_rank': 4, 'perfect': False, 'permutation_degree': 18, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 2, 4], 'quasisimple': False, 'rank': 4, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 16], [2, 16], [4, 16], [8, 24], [16, 8]], 'representations': {'PC': {'code': 3044897548897492230344640933393952742542178913312530104515, 'gens': [1, 2, 4, 6], 'pres': [8, -2, -2, -2, -2, -5, -2, -2, -5, 41, 907, 595, 91, 652, 660, 5773, 8661, 141, 13454, 20182, 166, 30735, 20503]}, 'GLZN': {'d': 2, 'p': 100, 'gens': [1007249, 1006343, 49000049, 1005001, 81208021, 1002001, 1002501, 99000099]}, 'Perm': {'d': 18, 'gens': [359804574219007, 16, 5183, 11656, 5606234726400, 5160, 777140346124800, 1157666954129280]}}, 'schur_multiplier': [2, 2, 2, 2, 2, 10], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 2, 4], 'solvability_type': 7, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_2^2\\times C_{20}:F_5', 'transitive_degree': 160, 'wreath_data': None, 'wreath_product': False}
