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gps_subgroup_search • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'ambient': '1344.2716', 'ambient_counter': 2716, 'ambient_order': 1344, 'ambient_tex': 'C_6.(C_4\\times D_{28})', 'central': False, 'central_factor': False, 'centralizer_order': 4, 'characteristic': False, 'core_order': 112, 'counter': 30, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '1344.2716.6.f1.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': None, 'minimal': False, 'minimal_normal': False, 'nilpotent': False, 'normal': False, 'old_label': '6.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': 6, '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': '224.23', 'subgroup_hash': 23, 'subgroup_order': 224, 'subgroup_tex': 'C_{56}:C_4', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '1344.2716', 'aut_centralizer_order': 8, 'aut_label': '6.f1', 'aut_quo_index': None, 'aut_stab_index': 3, 'aut_weyl_group': '2688.ek', 'aut_weyl_index': 24, 'centralizer': '336.a1.a1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['2.f1.a1', '3.a1.a1'], 'contains': ['12.b1.a1', '12.f1.a1', '12.g1.a1', '42.f1.a1'], 'core': '12.b1.a1', 'coset_action_label': None, 'count': 3, 'diagramx': [3907, -1, 4619, -1, 3880, -1, 4626, -1], 'generators': [701, 192, 1012, 336, 2, 672], 'label': '1344.2716.6.f1.a1', 'mobius_quo': None, 'mobius_sub': 1, 'normal_closure': '2.f1.a1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '3.a1.a1', 'old_label': '6.f1.a1', 'projective_image': '336.149', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '6.f1.a1', 'subgroup_fusion': None, 'weyl_group': '112.29'}
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gps_groups • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '8.2', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 168, 'aut_gen_orders': [6, 14, 8, 2, 4, 2], 'aut_gens': [[1, 4], [1, 68], [97, 118], [29, 4], [1, 174], [171, 116], [1, 116]], 'aut_group': '5376.cy', 'aut_hash': 7842614414034835782, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 5376, 'aut_permdeg': 19, 'aut_perms': [6446835113241761, 6445526959914211, 20716451813796480, 6407704514534400, 13896668441779200, 2703006028800], 'aut_phi_ratio': 56.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 1, 2, 1], [4, 2, 1, 2], [4, 28, 4, 1], [7, 2, 3, 1], [8, 2, 4, 1], [14, 2, 3, 1], [14, 2, 6, 1], [28, 2, 6, 2], [56, 2, 24, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_2^4.D_4\\times F_7', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 2, 'autcent_group': '16.14', 'autcent_hash': 14, 'autcent_nilpotent': True, 'autcent_order': 16, 'autcent_solvable': True, 'autcent_split': False, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^4', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 84, 'autcentquo_group': '336.125', 'autcentquo_hash': 125, 'autcentquo_nilpotent': False, 'autcentquo_order': 336, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'D_4\\times F_7', 'cc_stats': [[1, 1, 1], [2, 1, 3], [4, 2, 2], [4, 28, 4], [7, 2, 3], [8, 2, 4], [14, 2, 9], [28, 2, 12], [56, 2, 24]], 'center_label': '4.2', 'center_order': 4, 'central_product': False, 'central_quotient': '56.5', 'commutator_count': 1, 'commutator_label': '28.2', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '2.1', '7.1'], 'composition_length': 6, 'conjugacy_classes_known': True, 'counter': 23, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [4, 2, 1, 2], [4, 28, 2, 2], [7, 2, 3, 1], [8, 2, 2, 2], [14, 2, 3, 3], [28, 2, 6, 2], [56, 2, 12, 2]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 3, 'exponent': 56, 'exponents_of_order': [5, 1], 'factors_of_aut_order': [2, 3, 7], 'factors_of_order': [2, 7], 'faithful_reps': [], 'familial': False, 'frattini_label': '8.2', 'frattini_quotient': '28.3', 'hash': 23, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 28, 'inner_gen_orders': [2, 28], 'inner_gens': [[1, 108], [121, 4]], 'inner_hash': 5, 'inner_nilpotent': False, 'inner_order': 56, 'inner_split': False, 'inner_tex': 'D_{28}', 'inner_used': [1, 2], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 8], [2, 54]], 'label': '224.23', 'linC_count': 96, 'linC_degree': 3, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 10, 'linQ_degree_count': 4, 'linQ_dim': 12, 'linQ_dim_count': 8, 'linR_count': 24, 'linR_degree': 6, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C56:C4', 'ngens': 6, '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': 13, 'number_characteristic_subgroups': 15, 'number_conjugacy_classes': 62, 'number_divisions': 18, 'number_normal_subgroups': 25, 'number_subgroup_autclasses': 26, 'number_subgroup_classes': 36, 'number_subgroups': 142, 'old_label': None, 'order': 224, 'order_factorization_type': 51, 'order_stats': [[1, 1], [2, 3], [4, 116], [7, 6], [8, 8], [14, 18], [28, 24], [56, 48]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 12, 'outer_gen_orders': [2, 2, 2, 12], 'outer_gen_pows': [0, 0, 196, 0], 'outer_gens': [[1, 116], [1, 172], [29, 4], [29, 190]], 'outer_group': '96.221', 'outer_hash': 221, 'outer_nilpotent': True, 'outer_order': 96, 'outer_permdeg': 11, 'outer_perms': [744, 1680, 3669864, 4033683], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_{12}:C_2^3', 'pc_rank': 2, 'perfect': False, 'permutation_degree': 19, 'pgroup': 0, 'primary_abelian_invariants': [2, 4], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 4], [4, 2], [6, 4], [12, 2], [24, 2]], 'representations': {'PC': {'code': 6959153731676131594298463557, 'gens': [1, 3], 'pres': [6, -2, -2, -2, -2, -2, -7, 12, 1946, 50, 5187, 69, 6244, 88, 6917]}, 'GLZN': {'d': 2, 'p': 39, 'gens': [1483000, 2294707, 2204419, 1642187, 830480, 450560]}, 'Perm': {'d': 19, 'gens': [357001373395972, 87097257, 87091200, 12568, 18498, 7116370137676800]}}, 'schur_multiplier': [2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 4], 'solvability_type': 6, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_{56}:C_4', 'transitive_degree': 224, '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': '16.10', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 84, 'aut_gen_orders': [6, 6, 6, 6, 6, 6, 6, 12, 28, 6], 'aut_gens': [[1, 2, 8], [629, 1014, 828], [1205, 230, 1048], [581, 6, 1144], [917, 114, 1208], [385, 1014, 1308], [1, 338, 524], [917, 1234, 108], [145, 790, 1096], [241, 674, 572], [1057, 902, 1196]], 'aut_group': None, 'aut_hash': 5255589997166976132, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 64512, 'aut_permdeg': 76, 'aut_perms': [1371925242141516793580746341247666218601114641544876713515065913252134099616094560176708300149092481648803540343, 1250827666161109583713345474658474920781807033404276890642731396547534786834830730088743506928820191214880729565, 979984247360773397490926949572660171156257820950419976840585445570919572797730928010522783618933195735692089792, 640309126107309908489693027629109075071624938220251395061180255771034481639069686550924815361335229226060139514, 1809749822949201633207597956347086391564374251975337772242162997243006423705962540101016545279820346696326260053, 10219290358168062990929984351471666207673509968729263567065261686947221286102503446241592283231044382161129, 640327245864878701465646666831006698478180872847449674940128002097910037332513036212123635531279249769327999449, 1715876364497148816910294916839595282463476373052291678159124235359939833314699028208149654683281893553170473372, 415396613665191011767705238029720906292226017376634438093715856706654005600392114840185333880720962791978779356, 1784936015238607448198461359934347681453009526665815268426811926593902771128202278925950757156742567809897444276], 'aut_phi_ratio': 168.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [3, 2, 1, 1], [4, 2, 1, 2], [4, 6, 2, 2], [4, 28, 2, 2], [4, 84, 2, 2], [6, 2, 1, 3], [7, 2, 3, 1], [8, 4, 2, 1], [8, 12, 2, 1], [12, 4, 1, 2], [12, 56, 2, 2], [14, 2, 3, 3], [21, 4, 3, 1], [24, 4, 4, 1], [28, 2, 6, 2], [28, 12, 6, 2], [42, 4, 3, 3], [56, 4, 12, 1], [56, 12, 12, 1], [84, 4, 6, 2], [168, 4, 24, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_{42}.(C_2^5\\times C_6).C_2^3', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 2, 'autcent_group': '64.267', 'autcent_hash': 267, 'autcent_nilpotent': True, 'autcent_order': 64, 'autcent_solvable': True, 'autcent_split': False, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^6', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 42, 'autcentquo_group': '1008.906', 'autcentquo_hash': 906, 'autcentquo_nilpotent': False, 'autcentquo_order': 1008, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'C_2\\times D_6\\times F_7', 'cc_stats': [[1, 1, 1], [2, 1, 3], [3, 2, 1], [4, 2, 2], [4, 6, 4], [4, 28, 4], [4, 84, 4], [6, 2, 3], [7, 2, 3], [8, 4, 2], [8, 12, 2], [12, 4, 2], [12, 56, 4], [14, 2, 9], [21, 4, 3], [24, 4, 4], [28, 2, 12], [28, 12, 12], [42, 4, 9], [56, 4, 12], [56, 12, 12], [84, 4, 12], [168, 4, 24]], 'center_label': '4.2', 'center_order': 4, 'central_product': False, 'central_quotient': '336.149', 'commutator_count': 1, 'commutator_label': '84.6', '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', '7.1'], 'composition_length': 8, 'conjugacy_classes_known': True, 'counter': 2716, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [3, 2, 1, 1], [4, 2, 1, 2], [4, 6, 2, 2], [4, 28, 1, 2], [4, 28, 2, 1], [4, 84, 1, 2], [4, 84, 2, 1], [6, 2, 1, 3], [7, 2, 3, 1], [8, 4, 2, 1], [8, 12, 1, 2], [12, 4, 1, 2], [12, 56, 1, 2], [12, 56, 2, 1], [14, 2, 3, 3], [21, 4, 3, 1], [24, 4, 4, 1], [28, 2, 6, 2], [28, 12, 6, 2], [42, 4, 3, 3], [56, 4, 12, 1], [56, 12, 6, 2], [84, 4, 6, 2], [168, 4, 24, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 10752, 'exponent': 168, 'exponents_of_order': [6, 1, 1], 'factors_of_aut_order': [2, 3, 7], 'factors_of_order': [2, 3, 7], 'faithful_reps': [], 'familial': False, 'frattini_label': '8.2', 'frattini_quotient': '168.50', 'hash': 2716, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 84, 'inner_gen_orders': [2, 4, 84], 'inner_gens': [[1, 678, 780], [5, 2, 232], [1253, 1122, 8]], 'inner_hash': 149, 'inner_nilpotent': False, 'inner_order': 336, 'inner_split': None, 'inner_tex': 'S_3\\times D_{28}', 'inner_used': [1, 2, 3], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 16], [2, 60], [4, 68]], 'label': '1344.2716', 'linC_count': 192, 'linC_degree': 5, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 14, 'linQ_degree_count': 152, 'linQ_dim': 16, 'linQ_dim_count': 16, 'linR_count': 72, 'linR_degree': 10, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'C6.(C4*D28)', '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], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 40, 'number_characteristic_subgroups': 60, 'number_conjugacy_classes': 144, 'number_divisions': 45, 'number_normal_subgroups': 74, 'number_subgroup_autclasses': 184, 'number_subgroup_classes': 216, 'number_subgroups': 1588, 'old_label': None, 'order': 1344, 'order_factorization_type': 311, 'order_stats': [[1, 1], [2, 3], [3, 2], [4, 476], [6, 6], [7, 6], [8, 32], [12, 232], [14, 18], [21, 12], [24, 16], [28, 168], [42, 36], [56, 192], [84, 48], [168, 96]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 6, 'outer_gen_orders': [2, 2, 2, 2, 2, 6], 'outer_gen_pows': [0, 912, 0, 512, 1152, 0], 'outer_gens': [[1205, 1014, 444], [241, 674, 572], [389, 6, 1336], [1301, 562, 1016], [1153, 1014, 348], [1, 338, 524]], 'outer_group': '192.1543', 'outer_hash': 1543, 'outer_nilpotent': True, 'outer_order': 192, 'outer_permdeg': 192, 'outer_perms': [170442631972567820409474121584110168361480324103518494115779734776646134295950167065124572080972484180306961106520738641787277862018663995645181393144037252596629213711176385520636101522901178919606954863299864362048884453238303062997950082682952889598290466549788795298395308927680583895317370329642177669650057855688002819521221002244635331109196414245900, 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263940028353310144980870683021382314437904982986844318855370086421971383403631108341143018976686019894394187483992620271564348749958213418726823019756777037488327456786413575697891259723195443390667462371673199125045611286543252532619055963250035885957443295186056285752901675345954047624059694259313842537891615728891270186780715977858676394247712921041939, 176998097373790704008945462156071573133113620924617921648424393349487432881101916071414008889030539541008320530083417741551260669856200775822455804094728230537494021143701832683047026834357508251018563063130638005233791383054047000520092323539602557268364870247467131365024684713669020543020229608745839736784782718544499776124728031408619126388858106361700, 11279115237680986864334149239244643242595541564496334927963150102778168667679424873159160903575299285309258988603281124640573035369132872628658065164443711538477763786423704514037196171111480496565674721892621175127956248872977533278677558853910566526654349592083604594775313006162956385542754883580662035882419247224112092522587377720457579762922599820989], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^5\\times C_6', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 30, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 4], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 8], [2, 10], [4, 7], [6, 4], [8, 2], [12, 6], [24, 6], [48, 2]], 'representations': {'PC': {'code': 79238447993863849739566662197694298823625540237545469567371374584291083, 'gens': [1, 2, 4], 'pres': [8, -2, -2, -2, 2, -2, -2, -3, -7, 5376, 10849, 41, 16226, 4050, 24963, 3723, 91, 35204, 9292, 116, 19973, 22285, 141, 46598, 14350, 222, 73735]}, 'Perm': {'d': 30, 'gens': [305292256521398230336080117967, 51373800639369815040, 409805665629298577, 539080928542276567, 669445860923048280, 798023019859233840, 1175091669949317120000, 9452345886333784620057231360000]}}, 'schur_multiplier': [2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 4], 'solvability_type': 7, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_6.(C_4\\times D_{28})', 'transitive_degree': 1344, 'wreath_data': None, 'wreath_product': False}
