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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': False, 'ambient': '105468750.d', 'ambient_counter': 4, 'ambient_order': 105468750, 'ambient_tex': 'C_5^9.C_9.C_6', 'central': False, 'central_factor': False, 'centralizer_order': None, 'characteristic': True, 'core_order': 234375, 'counter': 848, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '105468750.d.450._.A', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': False, 'minimal': False, 'minimal_normal': False, 'nilpotent': False, 'normal': True, 'old_label': '450.A', 'outer_equivalence': False, 'perfect': False, 'proper': True, 'quotient': '450.20', 'quotient_Agroup': True, 'quotient_abelian': False, 'quotient_cyclic': False, 'quotient_hash': None, 'quotient_metabelian': False, 'quotient_nilpotent': False, 'quotient_order': 450, 'quotient_simple': False, 'quotient_solvable': True, 'quotient_supersolvable': False, 'quotient_tex': 'C_3\\times C_5^2:S_3', 'simple': False, 'solvable': True, 'special_labels': [], 'split': None, 'standard_generators': False, 'stem': False, 'subgroup': None, 'subgroup_hash': None, 'subgroup_order': 234375, 'subgroup_tex': 'C_5^7.C_3', 'supersolvable': False, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '105468750.d', 'aut_centralizer_order': None, 'aut_label': None, 'aut_quo_index': None, 'aut_stab_index': None, 'aut_weyl_group': None, 'aut_weyl_index': None, 'centralizer': None, 'complements': None, 'conjugacy_class_count': 1, 'contained_in': None, 'contains': None, 'core': None, 'coset_action_label': None, 'count': 1, 'diagramx': None, 'generators': [8156060555424805018740973109226208799535102728263288416, 8156060595146997316763947789275702448268839885745030433, 62762715007740501701527364525087661677945836890, 2720125338365526731828405412110470925566690407648950913, 5440183729808487870669617242225455533817162135842079360, 8156060534490508864483525976566401201104258796969984033, 43463007039748412956337036474721226175577717196687543015, 2720125317431229092468728339464123217499168490013872480], 'label': '105468750.d.450._.A', 'mobius_quo': None, 'mobius_sub': None, 'normal_closure': None, 'normal_contained_in': None, 'normal_contains': None, 'normalizer': None, 'old_label': '450.A', 'projective_image': None, 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '450._.A', 'subgroup_fusion': None, 'weyl_group': None}
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label None does not appear in gps_groups
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gps_groups • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '6.2', 'all_subgroups_known': False, 'almost_simple': False, 'aut_abelian': None, 'aut_cyclic': None, 'aut_derived_length': None, 'aut_exponent': None, 'aut_gen_orders': None, 'aut_gens': [[59842705998931307835842054050696123748996585351739091319, 73443336875014425639649660978906763726375055200996947290], [62558582824547624375538460059482615895925124946851948279, 95204414540966286638576930002482409284996685817744591400], [62558582824547624679725948708564930295586824832162896599, 116961105700319328588877479703323859233155217936815347109], [81787872897900368211845240086975940153340535617005587176, 111525170441432376381871984003630246598952645430949116255], [19040814815242119227109133824104981079423935673445072583, 68003153187075274672490612807530496721290747102277817056], [54402522248229443380251733794280441370075687282094470429, 73443336875014425639649660978906763726375055200996947290], [65035608610145952817601199580739985254930535429234174806, 68003153187075274672490612807530496721290747009831550720], [65035608547383238452426237038340739249451032372043262839, 68003153145247601629939061312797972739643625246953868864], [59842705957103634792492842682323263111902069421849680599, 73443336854121049628099273284113816961338590124691436224], [65035608610145952245028506340120366407591888170342892279, 78636239423466356829604195850774216136558451029390539616], [65035608610145952548418361324666524897273903540777350519, 78636239423466355648672999692355170952477079497552057024], [65035608610145951940857509123374004193384669918007293399, 73443336875014425639649660978907306618956345770132989313], [59842705998931308712585779875106306549998061872525944439, 68003153187075274672490612807528557820291048906097458016], [59842705998931307835842054050694289430899794467715587799, 73443336875014425639649660978907950671143588074491907713], [54402522329820971705041507228840920854177134635561592349, 78636239423466355648672999692357005270087289979941849690], [62558582803654248094805209129550441469788234555931116215, 70723278420808749832243642009828474374846199022662617633], [54402522310992157745426731703727557751077046738974419319, 76159213679737365291051971783568541809237595682459833050]], 'aut_group': None, 'aut_hash': None, 'aut_nilpotency_class': None, 'aut_nilpotent': None, 'aut_order': 6750000000, 'aut_permdeg': None, 'aut_perms': None, 'aut_phi_ratio': None, 'aut_solvable': None, 'aut_stats': None, 'aut_supersolvable': None, 'aut_tex': None, 'autcent_abelian': None, 'autcent_cyclic': None, 'autcent_exponent': None, 'autcent_group': None, 'autcent_hash': None, 'autcent_nilpotent': None, 'autcent_order': None, 'autcent_solvable': None, 'autcent_split': None, 'autcent_supersolvable': None, 'autcent_tex': None, 'autcentquo_abelian': None, 'autcentquo_cyclic': None, 'autcentquo_exponent': None, 'autcentquo_group': None, 'autcentquo_hash': None, 'autcentquo_nilpotent': None, 'autcentquo_order': None, 'autcentquo_solvable': None, 'autcentquo_supersolvable': None, 'autcentquo_tex': None, 'cc_stats': None, 'center_label': '1.1', 'center_order': 1, 'central_product': None, 'central_quotient': '105468750.d', 'commutator_count': None, 'commutator_label': '17578125.a', 'complements_known': False, 'complete': False, 'complex_characters_known': False, 'composition_factors': ['2.1', '3.1', '3.1', '3.1', '5.1', '5.1', '5.1', '5.1', '5.1', '5.1', '5.1', '5.1', '5.1'], 'composition_length': 13, 'conjugacy_classes_known': False, 'counter': 4, 'cyclic': False, 'derived_length': 3, 'dihedral': False, 'direct_factorization': None, 'direct_product': None, 'div_stats': None, 'element_repr_type': 'Perm', 'elementary': 1, 'eulerian_function': None, 'exponent': 90, 'exponents_of_order': [9, 3, 1], 'factors_of_aut_order': [2, 3, 5], 'factors_of_order': [2, 3, 5], 'faithful_reps': None, 'familial': False, 'frattini_label': '1.1', 'frattini_quotient': '105468750.d', 'hash': 987084575953134364, 'hyperelementary': 1, 'inner_abelian': None, 'inner_cyclic': None, 'inner_exponent': None, 'inner_gen_orders': None, 'inner_gens': None, 'inner_hash': None, 'inner_nilpotent': None, 'inner_order': 105468750, 'inner_split': None, 'inner_tex': 'C_5^9.C_9.C_6', 'inner_used': None, 'irrC_degree': None, 'irrQ_degree': None, 'irrQ_dim': None, 'irrR_degree': None, 'irrep_stats': None, 'label': '105468750.d', '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': False, 'metacyclic': False, 'monomial': None, 'name': 'C5^9.C9.C6', 'ngens': 2, 'nilpotency_class': -1, 'nilpotent': False, 'normal_counts': [1, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 2, 0, 0, 1, 0, 1, 0, 2, 0, 1, 1, 1], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': None, 'number_characteristic_subgroups': 22, 'number_conjugacy_classes': None, 'number_divisions': None, 'number_normal_subgroups': 22, 'number_subgroup_autclasses': None, 'number_subgroup_classes': None, 'number_subgroups': None, 'old_label': None, 'order': 105468750, 'order_factorization_type': 321, 'order_stats': None, 'outer_abelian': None, 'outer_cyclic': None, 'outer_equivalence': False, 'outer_exponent': None, 'outer_gen_orders': None, 'outer_gen_pows': None, 'outer_gens': None, 'outer_group': None, 'outer_hash': None, 'outer_nilpotent': None, 'outer_order': 64, 'outer_permdeg': None, 'outer_perms': None, 'outer_solvable': None, 'outer_supersolvable': None, 'outer_tex': None, 'pc_rank': None, 'perfect': False, 'permutation_degree': 45, 'pgroup': 0, 'primary_abelian_invariants': [2, 3], 'quasisimple': False, 'rank': 2, 'rational': None, 'rational_characters_known': False, 'ratrep_stats': None, 'representations': {'PC': {'code': '22165402742326926404910806111710546172171626678469499143343295142937313560521300394129971778170439329960048271950010632871607583663762266245472731930100787488893084438927670579076699737415335999226857322245595973119192485143897101150521363718313917579921942958886466687159350699720874756975491324279501828900214785427946065906735752508364958905711122610340233215', 'gens': [1, 3, 6, 7, 8, 9, 10, 11, 12, 13], 'pres': [13, -2, -3, -3, -3, -5, -5, 5, 5, 5, 5, 5, 5, 5, 26, 263973074, 3158827778, 469999662, 145, 1804869147, 2542567264, 198, 14044, 3243245, 312804198, 77251, 231704, 124692756, 661510414, 655232, 18785175, 142534087, 758103860, 25332873, 43435126, 881645318, 1038173781, 74168089, 124534847, 982519209, 1159230172, 253445435, 6516948, 6872590, 304208213, 470591586, 275510284, 1752276251, 944541024, 756229537, 274191890, 17778178092, 3048928180, 2434003103, 281141691]}, 'Perm': {'d': 45, 'gens': [73443336875014425639649660978906763726375055200996947290, 59842705998931307835842054050696123748996585351739091319]}}, 'schur_multiplier': None, 'semidirect_product': None, 'simple': False, 'smith_abelian_invariants': [6], 'solvability_type': 17, 'solvable': True, 'subgroup_inclusions_known': False, 'subgroup_index_bound': 250, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': 'C_5^9.C_9.C_6', 'transitive_degree': 45, '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': '6.2', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 3, 'aut_exponent': 60, 'aut_gen_orders': [4, 2, 2, 3, 5, 5, 2], 'aut_gens': [[1, 2, 6, 90], [235, 254, 78, 270], [307, 290, 84, 360], [57, 202, 132, 342], [419, 416, 438, 108], [307, 290, 6, 90], [379, 434, 6, 90], [1, 2, 66, 90]], 'aut_group': '1200.951', 'aut_hash': 951, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 1200, 'aut_permdeg': 17, 'aut_perms': [161962148252563, 45849626322441, 92122999329435, 107621682549173, 44281709476155, 235938134016283, 31112], 'aut_phi_ratio': 10.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 15, 1, 1], [3, 1, 2, 1], [3, 50, 1, 1], [3, 50, 2, 1], [5, 3, 4, 1], [5, 6, 2, 1], [6, 15, 2, 1], [10, 15, 4, 1], [15, 3, 8, 1], [15, 6, 4, 1], [30, 15, 8, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_5^2:(C_4\\times D_6)', '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': '600.151', 'autcentquo_hash': 151, 'autcentquo_nilpotent': False, 'autcentquo_order': 600, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'C_5^2:(C_4\\times S_3)', 'cc_stats': [[1, 1, 1], [2, 15, 1], [3, 1, 2], [3, 50, 3], [5, 3, 4], [5, 6, 2], [6, 15, 2], [10, 15, 4], [15, 3, 8], [15, 6, 4], [30, 15, 8]], 'center_label': '3.1', 'center_order': 3, 'central_product': True, 'central_quotient': '150.5', 'commutator_count': 1, 'commutator_label': '75.2', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '3.1', '3.1', '5.1', '5.1'], 'composition_length': 5, 'conjugacy_classes_known': True, 'counter': 20, 'cyclic': False, 'derived_length': 3, 'dihedral': False, 'direct_factorization': [['150.5', 1], ['3.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 15, 1, 1], [3, 1, 2, 1], [3, 50, 1, 1], [3, 50, 2, 1], [5, 3, 4, 1], [5, 6, 2, 1], [6, 15, 2, 1], [10, 15, 4, 1], [15, 3, 8, 1], [15, 6, 4, 1], [30, 15, 8, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 72, 'exponent': 30, 'exponents_of_order': [2, 2, 1], 'factors_of_aut_order': [2, 3, 5], 'factors_of_order': [2, 3, 5], 'faithful_reps': [[3, 0, 16], [6, 0, 4]], 'familial': False, 'frattini_label': '1.1', 'frattini_quotient': '450.20', 'hash': 20, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 30, 'inner_gen_orders': [2, 3, 5, 5], 'inner_gens': [[1, 4, 384, 90], [5, 2, 438, 108], [163, 110, 6, 90], [1, 74, 6, 90]], 'inner_hash': 5, 'inner_nilpotent': False, 'inner_order': 150, 'inner_split': True, 'inner_tex': 'C_5^2:S_3', 'inner_used': [1, 2, 3], 'irrC_degree': 3, 'irrQ_degree': 24, 'irrQ_dim': 24, 'irrR_degree': 6, 'irrep_stats': [[1, 6], [2, 3], [3, 24], [6, 6]], 'label': '450.20', 'linC_count': 16, 'linC_degree': 3, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 14, 'linQ_degree_count': 6, 'linQ_dim': 14, 'linQ_dim_count': 6, 'linR_count': 8, 'linR_degree': 6, 'maximal_subgroups_known': True, 'metabelian': False, 'metacyclic': False, 'monomial': True, 'name': 'C3*C5^2:S3', '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], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 12, 'number_characteristic_subgroups': 8, 'number_conjugacy_classes': 39, 'number_divisions': 12, 'number_normal_subgroups': 8, 'number_subgroup_autclasses': 26, 'number_subgroup_classes': 26, 'number_subgroups': 244, 'old_label': None, 'order': 450, 'order_factorization_type': 222, 'order_stats': [[1, 1], [2, 15], [3, 152], [5, 24], [6, 30], [10, 60], [15, 48], [30, 120]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 4, 'outer_gen_orders': [2, 4], 'outer_gen_pows': [0, 0], 'outer_gens': [[1, 2, 66, 90], [1, 2, 42, 180]], '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': 4, 'perfect': False, 'permutation_degree': 18, 'pgroup': 0, 'primary_abelian_invariants': [2, 3], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 2], [2, 3], [4, 1], [12, 3], [24, 3]], 'representations': {'PC': {'code': 700301926691713929039967, 'gens': [1, 2, 3, 5], 'pres': [5, -2, -3, -3, -5, 5, 41, 5762, 3292, 57, 5043, 2348, 1359]}, 'GLZN': {'d': 2, 'p': 50, 'gens': [125049, 188776, 2702031, 4671337, 3976021]}, 'Perm': {'d': 18, 'gens': [64165879816560, 3, 425258361649560, 798157540354320, 1180064644017840]}}, 'schur_multiplier': [], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [6], 'solvability_type': 11, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': 'C_3\\times C_5^2:S_3', 'transitive_degree': 45, 'wreath_data': None, 'wreath_product': False}
