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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': True, 'ambient': '508032.a', 'ambient_counter': 1, 'ambient_order': 508032, 'ambient_tex': '\\SOPlus(4,8)', 'central': False, 'central_factor': False, 'centralizer_order': 63, 'characteristic': False, 'core_order': 1, 'counter': 67, 'cyclic': True, 'direct': None, 'hall': 0, 'label': '508032.a.8064.a1.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': False, 'old_label': '8064.a1.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': 8064, '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': '63.2', 'subgroup_hash': 2, 'subgroup_order': 63, 'subgroup_tex': 'C_{63}', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '508032.a', 'aut_centralizer_order': None, 'aut_label': '8064.a1', 'aut_quo_index': None, 'aut_stab_index': None, 'aut_weyl_group': None, 'aut_weyl_index': None, 'centralizer': '8064.a1.a1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['1008.a1.a1', '4032.a1.a1', '4032.b1.a1', '4032.c1.a1'], 'contains': ['24192.a1.a1', '56448.a1.a1'], 'core': '508032.a1.a1', 'coset_action_label': None, 'count': 2016, 'diagramx': [8202, -1, 323, -1, 7377, -1, 7769, -1], 'generators': [2517535212476564, 2517535212192000, 736239829387632], 'label': '508032.a.8064.a1.a1', 'mobius_quo': None, 'mobius_sub': 0, 'normal_closure': '2.a1.a1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '2016.a1.a1', 'old_label': '8064.a1.a1', 'projective_image': '508032.a', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '8064.a1.a1', 'subgroup_fusion': None, 'weyl_group': '4.2'}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': True, 'abelian_quotient': '63.2', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': True, 'aut_cyclic': False, 'aut_derived_length': 1, 'aut_exponent': 6, 'aut_gen_orders': [6, 6], 'aut_gens': [[1], [29], [10]], 'aut_group': '36.14', 'aut_hash': 14, 'aut_nilpotency_class': 1, 'aut_nilpotent': True, 'aut_order': 36, 'aut_permdeg': 10, 'aut_perms': [363024, 5043], 'aut_phi_ratio': 1.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [3, 1, 2, 1], [7, 1, 6, 1], [9, 1, 6, 1], [21, 1, 12, 1], [63, 1, 36, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_6^2', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 6, 'autcent_group': '36.14', 'autcent_hash': 14, 'autcent_nilpotent': True, 'autcent_order': 36, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_6^2', '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], [3, 1, 2], [7, 1, 6], [9, 1, 6], [21, 1, 12], [63, 1, 36]], 'center_label': '63.2', 'center_order': 63, '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': ['3.1', '3.1', '7.1'], 'composition_length': 3, 'conjugacy_classes_known': True, 'counter': 2, 'cyclic': True, 'derived_length': 1, 'dihedral': False, 'direct_factorization': [['7.1', 1], ['9.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [3, 1, 2, 1], [7, 1, 6, 1], [9, 1, 6, 1], [21, 1, 12, 1], [63, 1, 36, 1]], 'element_repr_type': 'PC', 'elementary': 21, 'eulerian_function': 1, 'exponent': 63, 'exponents_of_order': [2, 1], 'factors_of_aut_order': [2, 3], 'factors_of_order': [3, 7], 'faithful_reps': [[1, 0, 36]], 'familial': True, 'frattini_label': '3.1', 'frattini_quotient': '21.2', 'hash': 2, 'hyperelementary': 21, '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': 36, 'irrQ_dim': 36, 'irrR_degree': 2, 'irrep_stats': [[1, 63]], 'label': '63.2', 'linC_count': 36, 'linC_degree': 1, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 12, 'linQ_degree_count': 1, 'linQ_dim': 12, 'linQ_dim_count': 1, 'linR_count': 18, 'linR_degree': 2, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C63', 'ngens': 3, 'nilpotency_class': 1, 'nilpotent': True, 'normal_counts': [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': 63, 'number_divisions': 6, 'number_normal_subgroups': 6, 'number_subgroup_autclasses': 6, 'number_subgroup_classes': 6, 'number_subgroups': 6, 'old_label': None, 'order': 63, 'order_factorization_type': 22, 'order_stats': [[1, 1], [3, 2], [7, 6], [9, 6], [21, 12], [63, 36]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 6, 'outer_gen_orders': [6, 6], 'outer_gen_pows': [0, 0], 'outer_gens': [[29], [10]], 'outer_group': '36.14', 'outer_hash': 14, 'outer_nilpotent': True, 'outer_order': 36, 'outer_permdeg': 10, 'outer_perms': [363024, 5043], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_6^2', 'pc_rank': 1, 'perfect': False, 'permutation_degree': 16, 'pgroup': 0, 'primary_abelian_invariants': [9, 7], 'quasisimple': False, 'rank': 1, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 1], [2, 1], [6, 2], [12, 1], [36, 1]], 'representations': {'PC': {'code': 484535, 'gens': [1], 'pres': [3, -3, -3, -7, 9, 22]}, 'GLFp': {'d': 2, 'p': 71, 'gens': [22033491]}, 'Perm': {'d': 16, 'gens': [11021826816000, 4320, 2616307464960]}}, 'schur_multiplier': [], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [63], 'solvability_type': 0, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_{63}', 'transitive_degree': 63, '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': 252, 'aut_gen_orders': [6, 6], 'aut_gens': [[4184295005372912, 5936573788871207], [4888067627399207, 5942052209990490], [5183428073553753, 5286577798461023]], 'aut_group': '1524096.a', 'aut_hash': 1525363166326822, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 1524096, 'aut_permdeg': 504, 'aut_perms': [74014840397779115029613280111352007605730183792379434221809802184380227930877015414025993375999885572254707181323512597066111903349503164153187667770150946859295917533760992521506842998321421814375016936917171394663367268137537350054316748118128426684335070975287547413897675062982043478782413956460427902570309789772857628278148698725163824198580731081624383811031752441218602337614906924255249040889335767764046867488583537929238461817240539418909642396326693247302055519190842075605027027934954922416650567850252894329599620614618989931232073052350819160451227021955722099571883404522390533685485662015238140581190128098019556047579601331209679892356353558977988033809076145054796262634463837474590170189221471037384830584343444561719056827590115781443216271809537027452305849212615479006251692856348481483588940484646442152806854984662524084683454171555328986241712258117145885190271018684596315581970040119893029478525235265279097620510495364182493315813551528901119655533681660551318405973017585366008869544243712752739919174885494881791450162509178889414430797659473621656693335420313666267872386107578557479239095432308817339109640974943, 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'aut_phi_ratio': 10.5, 'aut_solvable': False, 'aut_stats': [[1, 1, 1, 1], [2, 126, 1, 1], [2, 504, 1, 1], [2, 3969, 1, 1], [3, 112, 1, 1], [3, 3136, 1, 1], [4, 31752, 1, 1], [6, 7056, 1, 1], [6, 28224, 1, 1], [7, 144, 3, 1], [7, 5184, 3, 1], [7, 10368, 3, 1], [9, 112, 3, 1], [9, 3136, 3, 1], [9, 6272, 3, 2], [14, 9072, 3, 1], [14, 36288, 3, 1], [18, 7056, 3, 1], [18, 28224, 3, 1], [21, 8064, 3, 1], [63, 8064, 3, 3]], 'aut_supersolvable': False, 'aut_tex': '\\SL(2,8)^2:C_6', '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': 252, 'autcentquo_group': '1524096.a', 'autcentquo_hash': 1525363166326822, 'autcentquo_nilpotent': False, 'autcentquo_order': 1524096, 'autcentquo_solvable': False, 'autcentquo_supersolvable': False, 'autcentquo_tex': '\\SL(2,8)^2:C_6', 'cc_stats': [[1, 1, 1], [2, 126, 1], [2, 504, 1], [2, 3969, 1], [3, 112, 1], [3, 3136, 1], [4, 31752, 1], [6, 7056, 1], [6, 28224, 1], [7, 144, 3], [7, 5184, 3], [7, 10368, 3], [9, 112, 3], [9, 3136, 3], [9, 6272, 6], [14, 9072, 3], [14, 36288, 3], [18, 7056, 3], [18, 28224, 3], [21, 8064, 3], [63, 8064, 9]], 'center_label': '1.1', 'center_order': 1, 'central_product': False, 'central_quotient': '508032.a', 'commutator_count': 1, 'commutator_label': '254016.a', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '504.156', '504.156'], 'composition_length': 3, 'conjugacy_classes_known': True, 'counter': 1, 'cyclic': False, 'derived_length': 1, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 126, 1, 1], [2, 504, 1, 1], [2, 3969, 1, 1], [3, 112, 1, 1], [3, 3136, 1, 1], [4, 31752, 1, 1], [6, 7056, 1, 1], [6, 28224, 1, 1], [7, 144, 3, 1], [7, 5184, 3, 1], [7, 10368, 3, 1], [9, 112, 3, 1], [9, 3136, 3, 1], [9, 6272, 3, 2], [14, 9072, 3, 1], [14, 36288, 3, 1], [18, 7056, 3, 1], [18, 28224, 3, 1], [21, 8064, 3, 1], [63, 8064, 9, 1]], 'element_repr_type': 'Perm', 'elementary': 1, 'eulerian_function': 124995, 'exponent': 252, 'exponents_of_order': [7, 4, 2], 'factors_of_aut_order': [2, 3, 7], 'factors_of_order': [2, 3, 7], 'faithful_reps': [[14, 1, 4], [16, 1, 1], [18, 1, 3], [49, 1, 8], [64, 1, 2], [81, 1, 6], [98, 1, 6], [112, 1, 4], [126, 1, 12], [144, 1, 3], [162, 1, 3]], 'familial': True, 'frattini_label': '1.1', 'frattini_quotient': '508032.a', 'hash': 8458028148835551398, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 252, 'inner_gen_orders': [18, 14], 'inner_gens': [[4184295005372912, 5542857462215742], [4909861202321684, 5936573788871207]], 'inner_hash': 8458028148835551398, 'inner_nilpotent': False, 'inner_order': 508032, 'inner_split': True, 'inner_tex': '\\SOPlus(4,8)', 'inner_used': [1, 2], 'irrC_degree': 14, 'irrQ_degree': 14, 'irrQ_dim': 14, 'irrR_degree': 14, 'irrep_stats': [[1, 2], [14, 4], [16, 1], [18, 3], [49, 8], [64, 2], [81, 6], [98, 6], [112, 4], [126, 12], [144, 3], [162, 3]], 'label': '508032.a', 'linC_count': 4, 'linC_degree': 14, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 14, 'linQ_degree_count': 1, 'linQ_dim': 14, 'linQ_dim_count': 1, 'linR_count': 4, 'linR_degree': 14, 'maximal_subgroups_known': True, 'metabelian': False, 'metacyclic': False, 'monomial': False, 'name': 'SO+(4,8)', '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, 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, 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, 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': 24, 'number_characteristic_subgroups': 3, 'number_conjugacy_classes': 54, 'number_divisions': 22, 'number_normal_subgroups': 3, 'number_subgroup_autclasses': 188, 'number_subgroup_classes': 226, 'number_subgroups': 1148786, 'old_label': None, 'order': 508032, 'order_factorization_type': 321, 'order_stats': [[1, 1], [2, 4599], [3, 3248], [4, 31752], [6, 35280], [7, 47088], [9, 47376], [14, 136080], [18, 105840], [21, 24192], [63, 72576]], 'outer_abelian': True, 'outer_cyclic': True, 'outer_equivalence': False, 'outer_exponent': 3, 'outer_gen_orders': [3], 'outer_gen_pows': [6357632078612804], 'outer_gens': [[5582193113276414, 4833138514291730]], 'outer_group': '3.1', 'outer_hash': 1, 'outer_nilpotent': True, 'outer_order': 3, 'outer_permdeg': 3, 'outer_perms': [3], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_3', 'pc_rank': None, 'perfect': False, 'permutation_degree': 18, 'pgroup': 0, 'primary_abelian_invariants': [2], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 2], [14, 1], [16, 1], [42, 1], [49, 2], [54, 1], [64, 2], [112, 1], [147, 2], [243, 2], [294, 2], [336, 1], [378, 1], [432, 1], [486, 1], [1134, 1]], 'representations': {'Lie': [{'d': 4, 'q': 8, 'gens': [141326051967552, 69793251840, 145144133587464, 35188667285505], 'family': 'SOPlus'}, {'d': 4, 'q': 8, 'family': 'GOPlus'}, {'d': 4, 'q': 8, 'family': 'PSOPlus'}, {'d': 4, 'q': 8, 'family': 'PGOPlus'}, {'d': 4, 'q': 8, 'family': 'CSOPlus'}], 'Perm': {'d': 18, 'gens': [5936573788871207, 4184295005372912]}}, 'schur_multiplier': [], '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': '\\SOPlus(4,8)', 'transitive_degree': 18, 'wreath_data': ['1008.b1.a1', '254016.b1.a1', '508032.a1.a1', '2T1'], 'wreath_product': True}
