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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': True, 'ambient': '1020.29', 'ambient_counter': 29, 'ambient_order': 1020, 'ambient_tex': 'A_4\\times C_{85}', 'central': True, 'central_factor': False, 'centralizer_order': 1020, 'characteristic': True, 'core_order': 17, 'counter': 12, 'cyclic': True, 'direct': True, 'hall': 17, 'label': '1020.29.60.a1.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': True, 'minimal_normal': True, 'nilpotent': True, 'normal': True, 'old_label': '60.a1.a1', 'outer_equivalence': False, 'perfect': False, 'proper': True, 'quotient': '60.9', 'quotient_Agroup': True, 'quotient_abelian': False, 'quotient_cyclic': False, 'quotient_hash': 9, 'quotient_metabelian': True, 'quotient_nilpotent': False, 'quotient_order': 60, 'quotient_simple': False, 'quotient_solvable': True, 'quotient_supersolvable': False, 'quotient_tex': 'C_5\\times A_4', 'simple': True, 'solvable': True, 'special_labels': [], 'split': True, 'standard_generators': False, 'stem': False, 'subgroup': '17.1', 'subgroup_hash': 1, 'subgroup_order': 17, 'subgroup_tex': 'C_{17}', 'supersolvable': True, 'sylow': 17}
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gps_subgroup_data • Show schema
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{'ambient': '1020.29', 'aut_centralizer_order': 96, 'aut_label': '60.a1', 'aut_quo_index': 1, 'aut_stab_index': 1, 'aut_weyl_group': '16.1', 'aut_weyl_index': 96, 'centralizer': '1.a1.a1', 'complements': ['17.a1.a1'], 'conjugacy_class_count': 1, 'contained_in': ['12.a1.a1', '20.a1.a1', '30.a1.a1'], 'contains': ['1020.a1.a1'], 'core': '60.a1.a1', 'coset_action_label': None, 'count': 1, 'diagramx': [2267, 687, 732, 8952, 5748, 3217, 4121, 4650], 'generators': [60], 'label': '1020.29.60.a1.a1', 'mobius_quo': -1, 'mobius_sub': -4, 'normal_closure': '60.a1.a1', 'normal_contained_in': ['12.a1.a1', '15.a1.a1'], 'normal_contains': ['1020.a1.a1'], 'normalizer': '1.a1.a1', 'old_label': '60.a1.a1', 'projective_image': '60.9', 'quotient_action_image': '1.1', 'quotient_action_kernel': '60.9', 'quotient_action_kernel_order': 60, 'quotient_fusion': None, 'short_label': '60.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': '17.1', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': True, 'aut_cyclic': True, 'aut_derived_length': 1, 'aut_exponent': 16, 'aut_gen_orders': [16], 'aut_gens': [[1], [3]], 'aut_group': '16.1', 'aut_hash': 1, 'aut_nilpotency_class': 1, 'aut_nilpotent': True, 'aut_order': 16, 'aut_permdeg': 16, 'aut_perms': [1401602636313], 'aut_phi_ratio': 1.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [17, 1, 16, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_{16}', 'autcent_abelian': True, 'autcent_cyclic': True, 'autcent_exponent': 16, 'autcent_group': '16.1', 'autcent_hash': 1, 'autcent_nilpotent': True, 'autcent_order': 16, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_{16}', '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], [17, 1, 16]], 'center_label': '17.1', 'center_order': 17, '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': ['17.1'], '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], [17, 1, 16, 1]], 'element_repr_type': 'PC', 'elementary': 17, 'eulerian_function': 1, 'exponent': 17, 'exponents_of_order': [1], 'factors_of_aut_order': [2], 'factors_of_order': [17], 'faithful_reps': [[1, 0, 16]], 'familial': True, 'frattini_label': '1.1', 'frattini_quotient': '17.1', 'hash': 1, 'hyperelementary': 17, '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': 16, 'irrQ_dim': 16, 'irrR_degree': 2, 'irrep_stats': [[1, 17]], 'label': '17.1', 'linC_count': 16, 'linC_degree': 1, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 16, 'linQ_degree_count': 1, 'linQ_dim': 16, 'linQ_dim_count': 1, 'linR_count': 8, 'linR_degree': 2, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C17', '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': 17, 'number_divisions': 2, 'number_normal_subgroups': 2, 'number_subgroup_autclasses': 2, 'number_subgroup_classes': 2, 'number_subgroups': 2, 'old_label': None, 'order': 17, 'order_factorization_type': 1, 'order_stats': [[1, 1], [17, 16]], 'outer_abelian': True, 'outer_cyclic': True, 'outer_equivalence': False, 'outer_exponent': 16, 'outer_gen_orders': [16], 'outer_gen_pows': [0], 'outer_gens': [[3]], 'outer_group': '16.1', 'outer_hash': 1, 'outer_nilpotent': True, 'outer_order': 16, 'outer_permdeg': 16, 'outer_perms': [1401602636313], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_{16}', 'pc_rank': 1, 'perfect': False, 'permutation_degree': 17, 'pgroup': 17, 'primary_abelian_invariants': [17], 'quasisimple': False, 'rank': 1, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 1], [16, 1]], 'representations': {'PC': {'code': 0, 'gens': [1], 'pres': [1, -17]}, 'Lie': [{'d': 1, 'q': 17, 'gens': [22324392524313], 'family': 'ASigmaL'}], 'GLFp': {'d': 2, 'p': 17, 'gens': [4931]}, 'Perm': {'d': 17, 'gens': [334764638208000]}}, 'schur_multiplier': [], 'semidirect_product': False, 'simple': True, 'smith_abelian_invariants': [17], 'solvability_type': 0, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_{17}', 'transitive_degree': 17, 'wreath_data': None, 'wreath_product': False}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '255.1', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 3, 'aut_exponent': 48, 'aut_gen_orders': [48, 16, 4, 8], 'aut_gens': [[1, 3, 30], [1, 531, 375], [17, 9, 825], [512, 519, 630], [512, 3, 465]], 'aut_group': '1536.408556214', 'aut_hash': 5231633552952528856, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 1536, 'aut_permdeg': 28, 'aut_perms': [219500756849448963859564756878, 157745315915127726930155614699, 59449552956461522004381782644, 262070669072392446759249889254], 'aut_phi_ratio': 6.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 3, 1, 1], [3, 4, 2, 1], [5, 1, 4, 1], [10, 3, 4, 1], [15, 4, 8, 1], [17, 1, 16, 1], [34, 3, 16, 1], [51, 4, 32, 1], [85, 1, 64, 1], [170, 3, 64, 1], [255, 4, 128, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_4\\times C_{16}\\times S_4', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 16, 'autcent_group': '64.26', 'autcent_hash': 26, 'autcent_nilpotent': True, 'autcent_order': 64, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_4\\times C_{16}', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 12, 'autcentquo_group': '24.12', 'autcentquo_hash': 12, 'autcentquo_nilpotent': False, 'autcentquo_order': 24, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'S_4', 'cc_stats': [[1, 1, 1], [2, 3, 1], [3, 4, 2], [5, 1, 4], [10, 3, 4], [15, 4, 8], [17, 1, 16], [34, 3, 16], [51, 4, 32], [85, 1, 64], [170, 3, 64], [255, 4, 128]], 'center_label': '85.1', 'center_order': 85, 'central_product': True, 'central_quotient': '12.3', 'commutator_count': 1, 'commutator_label': '4.2', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '3.1', '5.1', '17.1'], 'composition_length': 5, 'conjugacy_classes_known': True, 'counter': 29, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['12.3', 1], ['17.1', 1], ['5.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 3, 1, 1], [3, 4, 2, 1], [5, 1, 4, 1], [10, 3, 4, 1], [15, 4, 8, 1], [17, 1, 16, 1], [34, 3, 16, 1], [51, 4, 32, 1], [85, 1, 64, 1], [170, 3, 64, 1], [255, 4, 128, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 432, 'exponent': 510, 'exponents_of_order': [2, 1, 1, 1], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3, 5, 17], 'faithful_reps': [[3, 0, 64]], 'familial': False, 'frattini_label': '1.1', 'frattini_quotient': '1020.29', 'hash': 29, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 6, 'inner_gen_orders': [3, 2, 2], 'inner_gens': [[1, 528, 45], [526, 3, 30], [16, 3, 30]], 'inner_hash': 3, 'inner_nilpotent': False, 'inner_order': 12, 'inner_split': True, 'inner_tex': 'A_4', 'inner_used': [1, 2], 'irrC_degree': 3, 'irrQ_degree': 192, 'irrQ_dim': 192, 'irrR_degree': 6, 'irrep_stats': [[1, 255], [3, 85]], 'label': '1020.29', 'linC_count': 64, 'linC_degree': 3, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 23, 'linQ_degree_count': 1, 'linQ_dim': 23, 'linQ_dim_count': 1, 'linR_count': 96, 'linR_degree': 5, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'A4*C85', 'ngens': 5, '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], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 12, 'number_characteristic_subgroups': 12, 'number_conjugacy_classes': 340, 'number_divisions': 12, 'number_normal_subgroups': 12, 'number_subgroup_autclasses': 20, 'number_subgroup_classes': 20, 'number_subgroups': 40, 'old_label': None, 'order': 1020, 'order_factorization_type': 222, 'order_stats': [[1, 1], [2, 3], [3, 8], [5, 4], [10, 12], [15, 32], [17, 16], [34, 48], [51, 128], [85, 64], [170, 192], [255, 512]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 16, 'outer_gen_orders': [2, 4, 16], 'outer_gen_pows': [0, 0, 0], 'outer_gens': [[2, 3, 1005], [1, 21, 990], [2, 3, 705]], 'outer_group': '128.837', 'outer_hash': 837, 'outer_nilpotent': True, 'outer_order': 128, 'outer_permdeg': 128, 'outer_perms': [63743519918226595519532610091666961074471673434843162611275101793322393119916585961282157117653654118898770628539436178567478662255606483647465433246789677363837306338978660157020320344411320948234577376174443689647, 48585144591868183331893406816054554144395592813214240733865262280699902039330192777198079449216805902647305992902078476598794650190142130005049934184200458640493659470657506597616315752393840124137757693082875148096, 136621236806028869617879690549755517237077712606235813048662859750680473713298230292464057209607037828367833245427118494120050138867071616006624121138130828992115347739023819441915915067231719149985746361124621504703], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2\\times C_4\\times C_{16}', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 26, 'pgroup': 0, 'primary_abelian_invariants': [3, 5, 17], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 1], [2, 1], [3, 1], [4, 1], [8, 1], [12, 1], [16, 1], [32, 1], [48, 1], [64, 1], [128, 1], [192, 1]], 'representations': {'PC': {'code': 4343777542098519130895, 'gens': [1, 2, 4], 'pres': [5, 3, 2, 5, 2, 17, 5281, 26, 903, 58]}, 'GLFq': {'d': 2, 'q': 256, 'gens': [1571094549, 1602740575, 1442840662, 251658255, 1992390262]}, 'Perm': {'d': 26, 'gens': [1240896803466478878720000, 53651891654000640000, 22324392524313, 15537062060069870960640000, 32263316890128450846720000]}}, 'schur_multiplier': [2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [255], 'solvability_type': 8, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': 'A_4\\times C_{85}', 'transitive_degree': 340, 'wreath_data': None, 'wreath_product': False}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '15.1', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 3, 'aut_exponent': 12, 'aut_gen_orders': [2, 2, 4, 4, 2, 6], 'aut_gens': [[1, 3, 6], [1, 3, 54], [34, 3, 54], [4, 3, 18], [1, 3, 42], [35, 33, 54], [31, 30, 57]], 'aut_group': '96.186', 'aut_hash': 186, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 96, 'aut_permdeg': 8, 'aut_perms': [16, 5176, 16689, 18, 15960, 10936], 'aut_phi_ratio': 6.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 3, 1, 1], [3, 4, 2, 1], [5, 1, 4, 1], [10, 3, 4, 1], [15, 4, 8, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_4\\times S_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': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 12, 'autcentquo_group': '24.12', 'autcentquo_hash': 12, 'autcentquo_nilpotent': False, 'autcentquo_order': 24, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'S_4', 'cc_stats': [[1, 1, 1], [2, 3, 1], [3, 4, 2], [5, 1, 4], [10, 3, 4], [15, 4, 8]], 'center_label': '5.1', 'center_order': 5, 'central_product': True, 'central_quotient': '12.3', 'commutator_count': 1, 'commutator_label': '4.2', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '3.1', '5.1'], 'composition_length': 4, 'conjugacy_classes_known': True, 'counter': 9, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['12.3', 1], ['5.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 3, 1, 1], [3, 4, 2, 1], [5, 1, 4, 1], [10, 3, 4, 1], [15, 4, 8, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 24, 'exponent': 30, 'exponents_of_order': [2, 1, 1], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3, 5], 'faithful_reps': [[3, 0, 4]], 'familial': False, 'frattini_label': '1.1', 'frattini_quotient': '60.9', 'hash': 9, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 6, 'inner_gen_orders': [3, 2, 2], 'inner_gens': [[1, 30, 9], [34, 3, 6], [4, 3, 6]], 'inner_hash': 3, 'inner_nilpotent': False, 'inner_order': 12, 'inner_split': True, 'inner_tex': 'A_4', 'inner_used': [1, 2], 'irrC_degree': 3, 'irrQ_degree': 12, 'irrQ_dim': 12, 'irrR_degree': 6, 'irrep_stats': [[1, 15], [3, 5]], 'label': '60.9', 'linC_count': 4, 'linC_degree': 3, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 7, 'linQ_degree_count': 1, 'linQ_dim': 7, 'linQ_dim_count': 1, 'linR_count': 6, 'linR_degree': 5, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'C5*A4', 'ngens': 4, 'nilpotency_class': -1, 'nilpotent': False, 'normal_counts': [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': 6, 'number_characteristic_subgroups': 6, 'number_conjugacy_classes': 20, 'number_divisions': 6, 'number_normal_subgroups': 6, 'number_subgroup_autclasses': 10, 'number_subgroup_classes': 10, 'number_subgroups': 20, 'old_label': None, 'order': 60, 'order_factorization_type': 222, 'order_stats': [[1, 1], [2, 3], [3, 8], [5, 4], [10, 12], [15, 32]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 4, 'outer_gen_orders': [2, 4], 'outer_gen_pows': [0, 0], 'outer_gens': [[2, 30, 27], [1, 3, 18]], 'outer_group': '8.2', 'outer_hash': 2, 'outer_nilpotent': True, 'outer_order': 8, 'outer_permdeg': 6, 'outer_perms': [120, 9], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2\\times C_4', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 9, 'pgroup': 0, 'primary_abelian_invariants': [3, 5], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 1], [2, 1], [3, 1], [4, 1], [8, 1], [12, 1]], 'representations': {'PC': {'code': 4852423747, 'gens': [1, 2, 3], 'pres': [4, -3, -2, 2, -5, 241, 110, 34]}, 'GLFp': {'d': 3, 'p': 11, 'gens': [656967451, 1691722594, 857494092, 1907820797]}, 'Perm': {'d': 9, 'gens': [5760, 33, 41040, 90720]}}, 'schur_multiplier': [2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [15], 'solvability_type': 8, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': 'C_5\\times A_4', 'transitive_degree': 20, 'wreath_data': None, 'wreath_product': False}
