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gps_subgroup_search • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'ambient': '279936.de', 'ambient_counter': 83, 'ambient_order': 279936, 'ambient_tex': 'C_6^4.C_6:S_3^2', 'central': False, 'central_factor': False, 'centralizer_order': None, 'characteristic': False, 'core_order': 1296, 'counter': 142, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '279936.de.18.CC', 'maximal': False, 'maximal_normal': False, 'metabelian': False, 'metacyclic': False, 'minimal': False, 'minimal_normal': False, 'nilpotent': False, 'normal': False, 'old_label': '18.cc1', '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': 18, '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': '15552.fe', 'subgroup_hash': 1428624750080802414, 'subgroup_order': 15552, 'subgroup_tex': 'C_6^4:D_6', 'supersolvable': False, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '279936.de', 'aut_centralizer_order': None, 'aut_label': '18.CC', 'aut_quo_index': None, 'aut_stab_index': None, 'aut_weyl_group': None, 'aut_weyl_index': None, 'centralizer': None, 'complements': None, 'conjugacy_class_count': 4, 'contained_in': None, 'contains': None, 'core': '216.B', 'coset_action_label': None, 'count': 36, 'diagramx': [146, -1, 2399, -1], 'generators': [2044440, 362904, 2859081884082664608685363200, 2027760, 177665441247705770014089960, 143746366066846067735619132967, 190881685080524713002089026488, 20514203349831944463478734, 13490570060918805254400, 139629562531758669847276800, 43994938594516804808065152000], 'label': '279936.de.18.CC', 'mobius_quo': None, 'mobius_sub': None, 'normal_closure': '2.D', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '9.A', 'old_label': '18.cc1', 'projective_image': '279936.de', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '18.CC', 'subgroup_fusion': None, 'weyl_group': '5324.n'}
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gps_groups • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '4.2', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 3, 'aut_exponent': 36, 'aut_gen_orders': [12, 6, 6, 12, 6], 'aut_gens': [[122357957335944037, 122003577585805156, 256850958414048127], [256136968169351928, 281771227930288598, 769792599540835190], [750942641659041865, 769397849974908782, 134452469439603450], [769732614573219389, 757343707686748277, 244064350350504813], [730584686146817448, 256096262210364702, 633837792879524155], [282835668642245589, 281771227893771159, 243647370320445967]], 'aut_group': None, 'aut_hash': 4745204657315933787, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 186624, 'aut_permdeg': 120, 'aut_perms': [1488039246943298570475460723363265773325791278224007680882044093427855659019315769393726969178288025728060558733903991284295400956952928491020213467176667016119809479258199127433975293466178418615010, 6166316221841771608246484735805431145461703225453108724877241810177714163941782338000168196679439374654601108788256913850845979815572756235853933760192040794972051255802313893988483020952175060285109, 285863382673252068239513620980240152182512342834003850971392739821419918245900134047073362963781081728660827610343338186730602414230877185827245736557747820038073229589683356240026155087196607691079, 5004320497870857733097276451677992637160687137095227801394744982866305580324371112991219446215686484949417783603873810377906739254100521509741123887666366438787206853690077930721654101666366053141358, 580213626596158989026165738929924506213749963822085963328307572573130273163500842711020274104983388097376827891225428312495625912366664688067721588279532758106336316886363465784270691862452753499582], 'aut_phi_ratio': 36.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 3, 1, 1], [2, 6, 1, 2], [2, 12, 1, 1], [2, 108, 1, 1], [2, 324, 1, 1], [3, 2, 1, 2], [3, 3, 2, 1], [3, 4, 1, 1], [3, 6, 2, 1], [3, 6, 3, 1], [3, 12, 3, 1], [3, 288, 1, 1], [3, 576, 1, 1], [4, 36, 1, 1], [4, 108, 1, 1], [4, 216, 1, 1], [4, 324, 1, 1], [4, 648, 1, 1], [6, 3, 2, 1], [6, 6, 1, 2], [6, 6, 2, 5], [6, 6, 3, 1], [6, 6, 4, 1], [6, 6, 6, 1], [6, 12, 1, 4], [6, 12, 2, 6], [6, 12, 3, 3], [6, 12, 4, 2], [6, 12, 6, 5], [6, 12, 12, 2], [6, 24, 1, 1], [6, 36, 2, 1], [6, 72, 3, 1], [6, 108, 2, 1], [6, 216, 1, 1], [6, 216, 2, 1], [6, 324, 2, 1], [6, 864, 1, 1], [9, 288, 2, 1], [9, 576, 2, 1], [12, 36, 2, 1], [12, 72, 1, 1], [12, 72, 2, 1], [12, 72, 3, 1], [12, 72, 6, 1], [12, 108, 2, 1], [12, 216, 1, 1], [12, 216, 2, 3], [12, 216, 4, 1], [12, 324, 2, 1], [12, 648, 2, 1], [18, 864, 2, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_6^4.C_6^2.C_2^2', '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': 36, 'autcentquo_group': None, 'autcentquo_hash': 4745204657315933787, 'autcentquo_nilpotent': False, 'autcentquo_order': 186624, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'C_6^4.C_6^2.C_2^2', 'cc_stats': [[1, 1, 1], [2, 3, 1], [2, 6, 2], [2, 12, 1], [2, 108, 1], [2, 324, 1], [3, 2, 2], [3, 3, 2], [3, 4, 1], [3, 6, 5], [3, 12, 3], [3, 288, 1], [3, 576, 1], [4, 36, 1], [4, 108, 1], [4, 216, 1], [4, 324, 1], [4, 648, 1], [6, 3, 2], [6, 6, 25], [6, 12, 87], [6, 24, 1], [6, 36, 2], [6, 72, 3], [6, 108, 2], [6, 216, 3], [6, 324, 2], [6, 864, 1], [9, 288, 2], [9, 576, 2], [12, 36, 2], [12, 72, 12], [12, 108, 2], [12, 216, 11], [12, 324, 2], [12, 648, 2], [18, 864, 2]], 'center_label': '1.1', 'center_order': 1, 'central_product': False, 'central_quotient': '15552.fe', 'commutator_count': 2, 'commutator_label': None, 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '2.1', '2.1', '3.1', '3.1', '3.1', '3.1', '3.1'], 'composition_length': 11, 'conjugacy_classes_known': True, 'counter': 135, 'cyclic': False, 'derived_length': 3, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 3, 1, 1], [2, 6, 1, 2], [2, 12, 1, 1], [2, 108, 1, 1], [2, 324, 1, 1], [3, 2, 1, 2], [3, 3, 2, 1], [3, 4, 1, 1], [3, 6, 1, 3], [3, 6, 2, 1], [3, 12, 1, 3], [3, 288, 1, 1], [3, 576, 1, 1], [4, 36, 1, 1], [4, 108, 1, 1], [4, 216, 1, 1], [4, 324, 1, 1], [4, 648, 1, 1], [6, 3, 2, 1], [6, 6, 1, 5], [6, 6, 2, 10], [6, 12, 1, 21], [6, 12, 2, 33], [6, 24, 1, 1], [6, 36, 2, 1], [6, 72, 1, 3], [6, 108, 2, 1], [6, 216, 1, 1], [6, 216, 2, 1], [6, 324, 2, 1], [6, 864, 1, 1], [9, 288, 1, 2], [9, 576, 1, 2], [12, 36, 2, 1], [12, 72, 1, 4], [12, 72, 2, 4], [12, 108, 2, 1], [12, 216, 1, 1], [12, 216, 2, 5], [12, 324, 2, 1], [12, 648, 2, 1], [18, 864, 1, 2]], 'element_repr_type': 'Perm', 'elementary': 1, 'eulerian_function': 6531840, 'exponent': 36, 'exponents_of_order': [6, 5], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [[12, 0, 30], [12, 1, 6]], 'familial': False, 'frattini_label': '36.14', 'frattini_quotient': '432.747', 'hash': 1428624750080802414, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 36, 'inner_gen_orders': [18, 6, 12], 'inner_gens': [[122357957335944037, 122398327151906117, 264299384472260645], [122063562437381589, 122003577585805156, 257958384247317595], [608960508448120746, 769397849974814389, 256850958414048127]], 'inner_hash': 1428624750080802414, 'inner_nilpotent': False, 'inner_order': 15552, 'inner_split': True, 'inner_tex': 'C_6^4:D_6', 'inner_used': [1, 2, 3], 'irrC_degree': 12, 'irrQ_degree': 12, 'irrQ_dim': 12, 'irrR_degree': 12, 'irrep_stats': [[1, 4], [2, 10], [3, 20], [4, 4], [6, 64], [12, 90]], 'label': '15552.fe', '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': True, 'name': 'C6^4:D6', 'ngens': 3, '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], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 78, 'number_characteristic_subgroups': 42, 'number_conjugacy_classes': 192, 'number_divisions': 129, 'number_normal_subgroups': 48, 'number_subgroup_autclasses': 2304, 'number_subgroup_classes': 4264, 'number_subgroups': 127860, 'old_label': None, 'order': 15552, 'order_factorization_type': 33, 'order_stats': [[1, 1], [2, 459], [3, 944], [4, 1332], [6, 3888], [9, 1728], [12, 5472], [18, 1728]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': True, 'outer_exponent': 6, 'outer_gen_orders': [2, 6], 'outer_gen_pows': [0, 0], 'outer_gens': [[122357957335999058, 122003577585813690, 256850958413996386], [122357957335995460, 122003577585805156, 646660667112057727]], 'outer_group': '12.5', 'outer_hash': 5, 'outer_nilpotent': True, 'outer_order': 12, 'outer_permdeg': 7, 'outer_perms': [720, 28], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2\\times C_6', 'pc_rank': None, 'perfect': False, 'permutation_degree': 20, 'pgroup': 0, 'primary_abelian_invariants': [2, 2], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 10], [3, 4], [4, 4], [6, 28], [12, 46], [24, 33]], 'representations': {'PC': {'code': '3929627720187871043739034835840074615441314560391053974831614911390847430443817602179300109244969573133945496121280417539325872872062900650', 'gens': [1, 2, 4, 5, 7, 9, 11], 'pres': [11, 2, 2, 3, 3, 2, 3, 2, 3, 2, 3, 2, 264, 124301, 56, 199850, 542, 57181, 19044, 13864, 82185, 21982, 158, 33269, 190096, 2810, 124746, 207917, 85278, 226, 266119, 133074, 22216, 908828, 737767, 43700, 294, 285140, 1032382, 46870]}, 'Perm': {'d': 20, 'gens': [122357957335944037, 122003577585805156, 256850958414048127]}}, 'schur_multiplier': [2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2], 'solvability_type': 11, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': 'C_6^4:D_6', 'transitive_degree': 36, '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': '8.5', 'all_subgroups_known': False, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 3, 'aut_exponent': 36, 'aut_gen_orders': [18, 6, 12, 6, 18], 'aut_gens': [[809848354096830107461617030, 12162760293420376556061214254, 23472373873229817363580853815], [190929475401762225338272910280, 24244213911353091963441167785, 163569103832272813474012876950], [5274622029229935955643718030, 14754426097447319124979087974, 157289546631726707422129259761], [71576298153704127118753775088, 175915142195271320282707209144, 168330393152316928090152531751], [81904967780289520689554269128, 165927402307444059674670156895, 36367027171408509408042923694], [55755694181113540848356540647, 154994508008652612006637985521, 91609519056767846642986349808]], 'aut_group': None, 'aut_hash': 8684730641443794311, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 5038848, 'aut_permdeg': 864, 'aut_perms': 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'aut_supersolvable': False, 'aut_tex': 'C_2^2\\times C_6^4.C_3^5.C_2^2', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 2, 'autcent_group': '4.2', 'autcent_hash': 2, 'autcent_nilpotent': True, 'autcent_order': 4, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^2', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 36, 'autcentquo_group': None, 'autcentquo_hash': 8627725503505202077, 'autcentquo_nilpotent': False, 'autcentquo_order': 1259712, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'C_6^4.C_3^5.C_2^2', 'cc_stats': [[1, 1, 1], [2, 1, 1], [2, 6, 2], [2, 9, 2], [2, 108, 2], [2, 324, 2], [2, 972, 2], [3, 2, 2], [3, 4, 1], [3, 6, 2], [3, 9, 2], [3, 12, 2], [3, 18, 4], [3, 36, 3], [3, 432, 3], [3, 2592, 1], [4, 324, 2], [4, 972, 4], [4, 1944, 2], [6, 2, 2], [6, 4, 1], [6, 6, 6], [6, 9, 6], [6, 12, 26], [6, 18, 52], [6, 36, 171], [6, 216, 2], [6, 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None, 'exponent': 36, 'exponents_of_order': [7, 7], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [[12, 0, 18], [12, 1, 9], [36, 0, 36], [36, 1, 6]], 'familial': False, 'frattini_label': '81.15', 'frattini_quotient': '3456.gl', 'hash': 445500119779896023, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 36, 'inner_gen_orders': [12, 12, 18], 'inner_gens': [[809848354096830107461617030, 35090736695131498364998392030, 12616354363359272616928738801], [65930199507481865750344897344, 12162760293420376556061214254, 23875666510062741846614171575], [5667390751377177467876420160, 10944226556978134553619113574, 23472373873229817363580853815]], 'inner_hash': 17262214244556444, 'inner_nilpotent': False, 'inner_order': 139968, 'inner_split': None, 'inner_tex': 'C_3^5.\\POPlus(4,3)', 'inner_used': [1, 2, 3], 'irrC_degree': 12, 'irrQ_degree': 12, 'irrQ_dim': None, 'irrR_degree': None, 'irrep_stats': [[1, 8], [2, 20], [3, 16], [4, 8], [6, 32], [9, 24], [12, 102], [18, 112], [36, 174]], 'label': '279936.de', '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': 'C6^4.C6:S3^2', 'ngens': 3, '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], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 277, 'number_characteristic_subgroups': 67, 'number_conjugacy_classes': 496, 'number_divisions': 336, 'number_normal_subgroups': 91, 'number_subgroup_autclasses': None, 'number_subgroup_classes': None, 'number_subgroups': None, 'old_label': None, 'order': 279936, 'order_factorization_type': 33, 'order_stats': [[1, 1], [2, 2839], [3, 4130], [4, 8424], [6, 61070], [9, 15552], [12, 79056], [18, 108864]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': True, 'outer_exponent': 6, 'outer_gen_orders': [6, 6], 'outer_gen_pows': [46519545854591391905840563200, 0], 'outer_gens': [[190929475401762225338272910280, 24244213911353091963441167785, 163569103832272813474012876950], [60718662072902996557525071769, 154558336800651636133634432095, 23051711032189683200432023854]], 'outer_group': '36.14', 'outer_hash': 14, 'outer_nilpotent': True, 'outer_order': 36, 'outer_permdeg': 36, 'outer_perms': [10664175605322357673051203243021989000289, 297531412133913132430890600099471149909405], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_6^2', 'pc_rank': None, 'perfect': False, 'permutation_degree': 28, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 2], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': False, 'ratrep_stats': [[1, 8], [2, 20], [4, 8], [6, 16], [9, 8], [12, 58], [18, 48], [24, 28], [36, 74], [72, 68]], 'representations': {'PC': {'code': '23107825341088972967270859062998801312184095923853983504505747607901120010604447735444915225775881568815383839177975100200050134123849879986273907408782323116954026249874464362492925663891787013064490097338577675411702288320072182863860693997699907650072756448192607105447536934932423296566213320023114051527209098192733753008342429080564483353683116781711041650817035876320383367891924', 'gens': [1, 2, 4, 6, 7, 9, 11, 13, 14], 'pres': [14, 2, 2, 3, 2, 3, 3, 3, 3, 2, 3, 2, 3, 2, 2, 1759968, 3284345, 71, 11512706, 88482, 6697827, 3113729, 3790951, 157, 14022964, 433458, 4845572, 11400, 7850309, 3437299, 4355097, 1787735, 497005, 4611, 22586262, 3185804, 6562114, 2157420, 649214, 167950, 384, 145159, 12145, 14451704, 7103398, 3368016, 2592374, 592012, 197394, 62462, 372, 1088649, 2494810, 14968824, 5987558, 889864, 1654950, 551708, 110974, 458, 5443211, 2286169, 1306407, 771173, 167899, 56025, 24287, 22820642, 8226076, 1680642, 1371074, 457084, 152430, 34292173, 17146107, 1714649, 3524527, 619233, 206471, 31849]}, 'Perm': {'d': 28, 'gens': [809848354096830107461617030, 12162760293420376556061214254, 23472373873229817363580853815]}}, 'schur_multiplier': [2, 2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 2], 'solvability_type': 17, 'solvable': True, 'subgroup_inclusions_known': False, 'subgroup_index_bound': 48, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': 'C_6^4.C_6:S_3^2', 'transitive_degree': 36, 'wreath_data': None, 'wreath_product': False}
