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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': False, 'ambient': '9000.o', 'ambient_counter': 15, 'ambient_order': 9000, 'ambient_tex': 'C_5^3:(S_3\\times C_{12})', 'central': False, 'central_factor': False, 'centralizer_order': 15, 'characteristic': False, 'core_order': 375, 'counter': 28, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '9000.o.12.f1.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': False, 'minimal': False, 'minimal_normal': False, 'nilpotent': False, 'normal': False, 'old_label': '12.f1.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': 12, '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': '750.33', 'subgroup_hash': 33, 'subgroup_order': 750, 'subgroup_tex': 'C_5^2:C_{30}', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '9000.o', 'aut_centralizer_order': None, 'aut_label': '12.f1', 'aut_quo_index': None, 'aut_stab_index': None, 'aut_weyl_group': None, 'aut_weyl_index': None, 'centralizer': '600.d1.a1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['4.b1.a1', '6.e1.a1'], 'contains': ['24.a1.a1', '36.c1.a1', '60.f1.a1', '60.g1.a1', '60.g1.b1', '60.g1.c1', '60.g1.d1', '60.h1.a1', '60.j1.a1'], 'core': '24.a1.a1', 'coset_action_label': None, 'count': 3, 'diagramx': [3442, -1, 8485, -1, 3705, -1, 3511, -1], 'generators': [114, 1800, 24, 4, 360], 'label': '9000.o.12.f1.a1', 'mobius_quo': None, 'mobius_sub': 0, 'normal_closure': '4.b1.a1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '3.b1.a1', 'old_label': '12.f1.a1', 'projective_image': '3000.bf', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '12.f1.a1', 'subgroup_fusion': None, 'weyl_group': '200.47'}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '30.4', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 1, 'aut_exponent': 120, 'aut_gen_orders': [4, 6, 12, 12], 'aut_gens': [[1, 10, 50], [49, 160, 540], [649, 460, 510], [177, 640, 210], [169, 180, 710]], 'aut_group': None, 'aut_hash': 1645357508925994488, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 96000, 'aut_permdeg': 31, 'aut_perms': [5863191199773667989983329144885827, 7399730451033377056969931398314993, 6369430250586138591754640441908450, 1107826715597689337426119589158384], 'aut_phi_ratio': 480.0, 'aut_solvable': False, 'aut_stats': [[1, 1, 1, 1], [2, 25, 1, 1], [3, 1, 2, 1], [5, 1, 4, 1], [5, 2, 12, 1], [5, 2, 48, 1], [6, 25, 2, 1], [10, 25, 4, 1], [15, 1, 8, 1], [15, 2, 24, 1], [15, 2, 96, 1], [30, 25, 8, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_2\\times C_4\\times C_5^2:C_4.S_5', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 4, 'autcent_group': '8.2', 'autcent_hash': 2, 'autcent_nilpotent': True, 'autcent_order': 8, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2\\times C_4', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 120, 'autcentquo_group': '12000.a', 'autcentquo_hash': 5912173750379521722, 'autcentquo_nilpotent': False, 'autcentquo_order': 12000, 'autcentquo_solvable': False, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'C_5^2.\\GL(2,5)', 'cc_stats': [[1, 1, 1], [2, 25, 1], [3, 1, 2], [5, 1, 4], [5, 2, 60], [6, 25, 2], [10, 25, 4], [15, 1, 8], [15, 2, 120], [30, 25, 8]], 'center_label': '15.1', 'center_order': 15, 'central_product': True, 'central_quotient': '50.4', 'commutator_count': 1, 'commutator_label': '25.2', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '3.1', '5.1', '5.1', '5.1'], 'composition_length': 5, 'conjugacy_classes_known': True, 'counter': 33, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['3.1', 1], ['5.1', 1], ['50.4', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 25, 1, 1], [3, 1, 2, 1], [5, 1, 4, 1], [5, 2, 2, 6], [5, 2, 4, 12], [6, 25, 2, 1], [10, 25, 4, 1], [15, 1, 8, 1], [15, 2, 4, 6], [15, 2, 8, 12], [30, 25, 8, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 2821, 'exponent': 30, 'exponents_of_order': [3, 1, 1], 'factors_of_aut_order': [2, 3, 5], 'factors_of_order': [2, 3, 5], 'faithful_reps': [], 'familial': False, 'frattini_label': '1.1', 'frattini_quotient': '750.33', 'hash': 33, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 10, 'inner_gen_orders': [2, 5, 5], 'inner_gens': [[1, 40, 200], [21, 10, 50], [601, 10, 50]], 'inner_hash': 4, 'inner_nilpotent': False, 'inner_order': 50, 'inner_split': False, 'inner_tex': 'C_5:D_5', 'inner_used': [1, 2, 3], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 30], [2, 180]], 'label': '750.33', 'linC_count': 11520, 'linC_degree': 4, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 14, 'linQ_degree_count': 180, 'linQ_dim': 14, 'linQ_dim_count': 180, 'linR_count': 960, 'linR_degree': 6, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'C5^2:C30', '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], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 12, 'number_characteristic_subgroups': 12, 'number_conjugacy_classes': 210, 'number_divisions': 44, 'number_normal_subgroups': 36, 'number_subgroup_autclasses': 28, 'number_subgroup_classes': 112, 'number_subgroups': 352, 'old_label': None, 'order': 750, 'order_factorization_type': 311, 'order_stats': [[1, 1], [2, 25], [3, 2], [5, 124], [6, 50], [10, 100], [15, 248], [30, 200]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 60, 'outer_gen_orders': [12, 12, 4], 'outer_gen_pows': [0, 1, 1], 'outer_gens': [[7, 640, 110], [7, 480, 90], [3, 460, 740]], 'outer_group': '1920.240479', 'outer_hash': 229582038051242566, 'outer_nilpotent': False, 'outer_order': 1920, 'outer_permdeg': 15, 'outer_perms': [6706390338, 93446040378, 41086098], 'outer_solvable': False, 'outer_supersolvable': False, 'outer_tex': 'C_2\\times A_5:C_4^2', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 18, 'pgroup': 0, 'primary_abelian_invariants': [2, 3, 5], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 2], [2, 2], [4, 8], [8, 20], [16, 12]], 'representations': {'PC': {'code': 15932750575121442039, 'gens': [1, 3, 4], 'pres': [5, -2, -5, -5, -3, -5, 10, 602, 4003, 78, 15004]}, 'GLZN': {'d': 2, 'p': 50, 'gens': [125031, 125049, 125501, 188776, 202031]}, 'Perm': {'d': 18, 'gens': [359977885707624, 3, 799438429440000, 1179698193273600, 1552236302540904]}}, 'schur_multiplier': [5], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [30], 'solvability_type': 7, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_5^2:C_{30}', 'transitive_degree': 150, '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': '24.9', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 3, 'aut_exponent': 60, 'aut_gen_orders': [20, 4, 12, 20, 12], 'aut_gens': [[1, 12, 120, 360, 1800], [197, 5316, 8160, 3240, 1800], [7273, 8364, 5880, 1440, 7200], [3365, 7428, 8760, 5400, 6120], [3553, 8316, 6000, 360, 7560], [5545, 7428, 4440, 5400, 6120]], 'aut_group': None, 'aut_hash': 9056236928001170395, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 24000, 'aut_permdeg': 142, 'aut_perms': [6305272414242887059426578678321553369260281456352809910310134029639890750484460027579361036705636040274264226769503914134012639170362023088058552550153282420007526082598100551174841379621234022025091624219390900159714403067362450158698833842950, 159702007891857697698146808975708009605794611996087052873616980783079576628870865442821548532431404437196244857528852921544626036027593649764379423122910623302261583500753774385365151652373894253807022889031811193621693037027971708193900344148174, 268447636188760560829050854936783894908457827881992175053391314630480119338608984928299369839373927708754644901503570169360288319714613164488956170170915763442161204740394494010250181445913498858670401680260293864305928017236388783835378735508802, 204956542203982195499446150503782617163252684625366533042748246673644349231698829744605712256026123965743992971336823335374576464728144061846690419183948987231699293695662903386405761978019798316048061504762752916629849297429423242291964871931623, 138478588004196894520796397144826298903298267101126154278018141372278284908305935306462654001441200892367883558788432298964245705385082790614618638385544279211208946583185609685131257620654905146288038860835244404748801284361801401157341300198761], 'aut_phi_ratio': 10.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 15, 1, 1], [2, 75, 1, 1], [2, 125, 1, 1], [3, 1, 2, 1], [3, 50, 1, 1], [3, 50, 2, 1], [4, 125, 1, 2], [4, 375, 1, 2], [5, 4, 1, 1], [5, 12, 1, 2], [5, 12, 4, 1], [5, 24, 2, 1], [6, 15, 2, 1], [6, 75, 2, 1], [6, 125, 2, 1], [6, 250, 1, 1], [6, 250, 2, 1], [10, 60, 1, 2], [10, 60, 4, 1], [10, 300, 1, 1], [12, 125, 2, 2], [12, 250, 1, 2], [12, 250, 2, 2], [12, 375, 2, 2], [15, 4, 2, 1], [15, 12, 2, 2], [15, 12, 8, 1], [15, 24, 4, 1], [15, 200, 1, 1], [15, 200, 2, 1], [30, 60, 2, 2], [30, 60, 8, 1], [30, 300, 2, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_2\\times C_5^3.(C_4^2\\times S_3)', 'autcent_abelian': True, 'autcent_cyclic': True, 'autcent_exponent': 2, 'autcent_group': '2.1', 'autcent_hash': 1, 'autcent_nilpotent': True, 'autcent_order': 2, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 60, 'autcentquo_group': '12000.cm', 'autcentquo_hash': 5340715652065179242, 'autcentquo_nilpotent': False, 'autcentquo_order': 12000, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'C_5^3:(S_3\\times C_4^2)', 'cc_stats': [[1, 1, 1], [2, 15, 1], [2, 75, 1], [2, 125, 1], [3, 1, 2], [3, 50, 3], [4, 125, 2], [4, 375, 2], [5, 4, 1], [5, 12, 6], [5, 24, 2], [6, 15, 2], [6, 75, 2], [6, 125, 2], [6, 250, 3], [10, 60, 6], [10, 300, 1], [12, 125, 4], [12, 250, 6], [12, 375, 4], [15, 4, 2], [15, 12, 12], [15, 24, 4], [15, 200, 3], [30, 60, 12], [30, 300, 2]], 'center_label': '3.1', 'center_order': 3, 'central_product': True, 'central_quotient': '3000.bf', 'commutator_count': 1, 'commutator_label': '375.6', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '3.1', '3.1', '5.1', '5.1', '5.1'], 'composition_length': 8, 'conjugacy_classes_known': True, 'counter': 15, 'cyclic': False, 'derived_length': 3, 'dihedral': False, 'direct_factorization': [['3.1', 1], ['3000.bf', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 15, 1, 1], [2, 75, 1, 1], [2, 125, 1, 1], [3, 1, 2, 1], [3, 50, 1, 1], [3, 50, 2, 1], [4, 125, 2, 1], [4, 375, 2, 1], [5, 4, 1, 1], [5, 12, 1, 6], [5, 24, 1, 2], [6, 15, 2, 1], [6, 75, 2, 1], [6, 125, 2, 1], [6, 250, 1, 1], [6, 250, 2, 1], [10, 60, 1, 6], [10, 300, 1, 1], [12, 125, 4, 1], [12, 250, 2, 1], [12, 250, 4, 1], [12, 375, 4, 1], [15, 4, 2, 1], [15, 12, 2, 6], [15, 24, 2, 2], [15, 200, 1, 1], [15, 200, 2, 1], [30, 60, 2, 6], [30, 300, 2, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 576, 'exponent': 60, 'exponents_of_order': [3, 3, 2], 'factors_of_aut_order': [2, 3, 5], 'factors_of_order': [2, 3, 5], 'faithful_reps': [[12, 0, 16], [24, 0, 4]], 'familial': False, 'frattini_label': '1.1', 'frattini_quotient': '9000.o', 'hash': 1735341798720924668, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 60, 'inner_gen_orders': [4, 10, 3, 5, 5], 'inner_gens': [[1, 1236, 3840, 3600, 720], [5737, 12, 6360, 360, 7560], [4921, 972, 120, 3240, 1440], [5761, 12, 8040, 360, 1800], [2881, 5052, 2280, 360, 1800]], 'inner_hash': 3629711919683738508, 'inner_nilpotent': False, 'inner_order': 3000, 'inner_split': True, 'inner_tex': 'C_5^3:(C_4\\times S_3)', 'inner_used': [1, 2, 3], 'irrC_degree': 12, 'irrQ_degree': 24, 'irrQ_dim': 24, 'irrR_degree': 24, 'irrep_stats': [[1, 24], [2, 12], [4, 6], [8, 3], [12, 36], [24, 6]], 'label': '9000.o', 'linC_count': 16, 'linC_degree': 12, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 14, 'linQ_degree_count': 32, 'linQ_dim': 14, 'linQ_dim_count': 32, 'linR_count': 64, 'linR_degree': 14, 'maximal_subgroups_known': True, 'metabelian': False, 'metacyclic': False, 'monomial': True, 'name': 'C5^3:(S3*C12)', '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, 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': 44, 'number_characteristic_subgroups': 32, 'number_conjugacy_classes': 87, 'number_divisions': 52, 'number_normal_subgroups': 32, 'number_subgroup_autclasses': 248, 'number_subgroup_classes': 356, 'number_subgroups': 14168, 'old_label': None, 'order': 9000, 'order_factorization_type': 321, 'order_stats': [[1, 1], [2, 215], [3, 152], [4, 1000], [5, 124], [6, 1180], [10, 660], [12, 3500], [15, 848], [30, 1320]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 4, 'outer_gen_orders': [2, 4], 'outer_gen_pows': [1208, 24], 'outer_gens': [[1205, 1212, 4800, 1800, 3240], [49, 12, 840, 720, 3600]], 'outer_group': '8.2', 'outer_hash': 2, 'outer_nilpotent': True, 'outer_order': 8, 'outer_permdeg': 6, 'outer_perms': [120, 10], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2\\times C_4', 'pc_rank': 5, 'perfect': False, 'permutation_degree': 18, 'pgroup': 0, 'primary_abelian_invariants': [2, 4, 3], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 8], [4, 7], [8, 4], [12, 12], [16, 1], [24, 14], [48, 2]], 'representations': {'PC': {'code': '100150550806413291036784627529661301996636702880962974953856348810262400591398111', 'gens': [1, 4, 6, 7, 8], 'pres': [8, -2, -2, -3, -2, -5, -3, -5, 5, 16, 41, 39555, 1739, 91, 2884, 1932, 184325, 11533, 25469, 201606, 40334, 1558, 46087, 230415, 40351, 815]}, 'Perm': {'d': 18, 'gens': [401818317466347, 779836733359444]}}, 'schur_multiplier': [2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 12], 'solvability_type': 11, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': 'C_5^3:(S_3\\times C_{12})', 'transitive_degree': 45, 'wreath_data': None, 'wreath_product': False}
