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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': True, 'ambient': '928.31', 'ambient_counter': 31, 'ambient_order': 928, 'ambient_tex': 'C_4.D_{116}', 'central': False, 'central_factor': False, 'centralizer_order': 464, 'characteristic': True, 'core_order': 29, 'counter': 20, 'cyclic': True, 'direct': False, 'hall': 29, 'label': '928.31.32.a1.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': True, 'minimal_normal': True, 'nilpotent': True, 'normal': True, 'old_label': '32.a1.a1', 'outer_equivalence': False, 'perfect': False, 'proper': True, 'quotient': '32.8', 'quotient_Agroup': False, 'quotient_abelian': False, 'quotient_cyclic': False, 'quotient_hash': 8, 'quotient_metabelian': True, 'quotient_nilpotent': True, 'quotient_order': 32, 'quotient_simple': False, 'quotient_solvable': True, 'quotient_supersolvable': True, 'quotient_tex': 'C_4.D_4', 'simple': True, 'solvable': True, 'special_labels': ['L3', 'C5'], 'split': True, 'standard_generators': False, 'stem': False, 'subgroup': '29.1', 'subgroup_hash': 1, 'subgroup_order': 29, 'subgroup_tex': 'C_{29}', 'supersolvable': True, 'sylow': 29}
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gps_subgroup_data • Show schema
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{'ambient': '928.31', 'aut_centralizer_order': 1856, 'aut_label': '32.a1', 'aut_quo_index': 2, 'aut_stab_index': 1, 'aut_weyl_group': '28.2', 'aut_weyl_index': 1856, 'centralizer': '2.b1.a1', 'complements': [], 'conjugacy_class_count': 1, 'contained_in': ['16.a1.a1', '16.b1.a1'], 'contains': ['928.a1.a1'], 'core': '32.a1.a1', 'coset_action_label': None, 'count': 1, 'diagramx': [1190, 2022, 9640, 3975, 2309, 3686, 6571, 5299], 'generators': [32], 'label': '928.31.32.a1.a1', 'mobius_quo': -1, 'mobius_sub': 0, 'normal_closure': '32.a1.a1', 'normal_contained_in': ['16.a1.a1'], 'normal_contains': ['928.a1.a1'], 'normalizer': '1.a1.a1', 'old_label': '32.a1.a1', 'projective_image': '928.31', 'quotient_action_image': '2.1', 'quotient_action_kernel': '16.6', 'quotient_action_kernel_order': 16, 'quotient_fusion': None, 'short_label': '32.a1.a1', 'subgroup_fusion': None, 'weyl_group': '2.1'}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': True, 'abelian_quotient': '29.1', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': True, 'aut_cyclic': True, 'aut_derived_length': 1, 'aut_exponent': 28, 'aut_gen_orders': [28], 'aut_gens': [[1], [2]], 'aut_group': '28.2', 'aut_hash': 2, 'aut_nilpotency_class': 1, 'aut_nilpotent': True, 'aut_order': 28, 'aut_permdeg': 11, 'aut_perms': [4032873], 'aut_phi_ratio': 1.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [29, 1, 28, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_{28}', 'autcent_abelian': True, 'autcent_cyclic': True, 'autcent_exponent': 28, 'autcent_group': '28.2', 'autcent_hash': 2, 'autcent_nilpotent': True, 'autcent_order': 28, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_{28}', '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], [29, 1, 28]], 'center_label': '29.1', 'center_order': 29, 'central_product': False, 'central_quotient': '1.1', 'commutator_count': 0, 'commutator_label': '1.1', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['29.1'], 'composition_length': 1, 'conjugacy_classes_known': True, 'counter': 1, 'cyclic': True, 'derived_length': 1, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [29, 1, 28, 1]], 'element_repr_type': 'PC', 'elementary': 29, 'eulerian_function': 1, 'exponent': 29, 'exponents_of_order': [1], 'factors_of_aut_order': [2, 7], 'factors_of_order': [29], 'faithful_reps': [[1, 0, 28]], 'familial': True, 'frattini_label': '1.1', 'frattini_quotient': '29.1', 'hash': 1, 'hyperelementary': 29, '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': 28, 'irrQ_dim': 28, 'irrR_degree': 2, 'irrep_stats': [[1, 29]], 'label': '29.1', 'linC_count': 28, 'linC_degree': 1, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 28, 'linQ_degree_count': 1, 'linQ_dim': 28, 'linQ_dim_count': 1, 'linR_count': 14, 'linR_degree': 2, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C29', 'ngens': 1, 'nilpotency_class': 1, 'nilpotent': True, 'normal_counts': [0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 2, 'number_characteristic_subgroups': 2, 'number_conjugacy_classes': 29, 'number_divisions': 2, 'number_normal_subgroups': 2, 'number_subgroup_autclasses': 2, 'number_subgroup_classes': 2, 'number_subgroups': 2, 'old_label': None, 'order': 29, 'order_factorization_type': 1, 'order_stats': [[1, 1], [29, 28]], 'outer_abelian': True, 'outer_cyclic': True, 'outer_equivalence': False, 'outer_exponent': 28, 'outer_gen_orders': [28], 'outer_gen_pows': [0], 'outer_gens': [[2]], 'outer_group': '28.2', 'outer_hash': 2, 'outer_nilpotent': True, 'outer_order': 28, 'outer_permdeg': 11, 'outer_perms': [4032873], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_{28}', 'pc_rank': 1, 'perfect': False, 'permutation_degree': 29, 'pgroup': 29, 'primary_abelian_invariants': [29], 'quasisimple': False, 'rank': 1, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 1], [28, 1]], 'representations': {'PC': {'code': 0, 'gens': [1], 'pres': [1, -29]}, 'Lie': [{'d': 1, 'q': 29, 'gens': [316196664211373618851684940313], 'family': 'ASigmaL'}], 'GLFp': {'d': 2, 'p': 29, 'gens': [24419]}, 'Perm': {'d': 29, 'gens': [8536873649127988094042112000000]}}, 'schur_multiplier': [], 'semidirect_product': False, 'simple': True, 'smith_abelian_invariants': [29], 'solvability_type': 0, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_{29}', 'transitive_degree': 29, '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': '8.2', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 812, 'aut_gen_orders': [28, 14, 28, 28, 28, 28], 'aut_gens': [[1, 2, 4], [881, 466, 780], [115, 2, 492], [281, 466, 446], [155, 466, 342], [761, 466, 382], [827, 2, 476]], 'aut_group': None, 'aut_hash': 6488921755636882477, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 51968, 'aut_permdeg': 240, 'aut_perms': [1580746536284944163951949583332633842276102265021413383657626455402600994014619882641703588903263792912357841984599472104571054269978860016712519264268532322166054702321887631676240853447848608963980863694786715783049627103905361839302565025358075257547806222902534159769681565834752128317720162430594275732497610733826730105852864388084330105969520486580081486146795240471298929557825153219964318232719532342922307689532998417866124700763936481875129861423010935453356, 2304877192926077160045627800980930264627160486446362536670707152677186250726450105736405972217740424658207320793331357549500882475428509461593923458229778783940432810134886060922017218761147936793709147191295185317838884543086363154494144499136339946223419438702316134215159541888037288159213179846815460555749796972977643635822630852866409891059586435798310283563162051965611077464374298802714057964455924031581084319976714859997159521195116332738939060717555124938037, 2681421535677846637792911218598803588872951558740556883537977861812652430037852775840554025943010090599506660166222990230724327004988350152128060343155950001495985891657315454955666469309902449102038806715928227886538050224733523760866808173114413066778587823459004958393733816323097704682271439165866518637555982554674854620368524200524843452505537216280489147477137870778606013978142555105407203643489003191040923431424097507492248731624056614240305402523938234575362, 450066341157136133186625243516731212927675962433152594320712568709595474854125218985599747594395456019498652794254841637205400328121414420683707066367691893273983248655324258224065813273802996261436207463948069226969721671115739189752096159118320266450330921379035578081752560625502109490969525211578754327986930596854777910569239121019646315497537367728037643169767254835558177554971338029073599857245450729616243897626975119812607086089423727997105327125544565031665, 3105136833319743486493698633497364752177064379362332613433068884885748922993762565382052384054609489634070591416471754022956545099828406679914270395083357411139479798634117479661991337054167845381123599159126211580303404199310365094440547633833829193382565217861797781046519925722709002723887640838943173386512817785941239328398326187844146502046595686457750820263435759558272094555860081747491435407718658206935665842218578073141617091460474378991680700512661914988003, 3876959510860266790140466158325783557722741289603385340397932895442043872991289576669692969844122593129736543552163737389593887653138756593449140337704087657182384817381896897987036541938960896080719891037717876748653853733343112003863077598826103539163686731223027108094006739309449483085199083290896231316785885972125644593623094759034097762652253544066927592255619503779219033896683904376368314999876560342964306377693182769040711299301246247893497899782341278571495], 'aut_phi_ratio': 116.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 2, 1, 1], [4, 2, 1, 2], [4, 116, 2, 1], [8, 4, 2, 1], [8, 116, 2, 1], [29, 2, 14, 1], [58, 2, 14, 1], [58, 4, 14, 1], [116, 2, 28, 1], [116, 4, 14, 1], [232, 4, 56, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_{58}.C_{14}.C_4.C_2^4', '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': False, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^2', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 812, 'autcentquo_group': None, 'autcentquo_hash': 865129260256801227, 'autcentquo_nilpotent': False, 'autcentquo_order': 12992, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'D_{29}:C_7.C_2^5', 'cc_stats': [[1, 1, 1], [2, 1, 1], [2, 2, 1], [4, 2, 2], [4, 116, 2], [8, 4, 2], [8, 116, 2], [29, 2, 14], [58, 2, 14], [58, 4, 14], [116, 2, 28], [116, 4, 14], [232, 4, 56]], 'center_label': '2.1', 'center_order': 2, 'central_product': False, 'central_quotient': '464.14', 'commutator_count': 1, 'commutator_label': '116.5', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '2.1', '29.1'], 'composition_length': 6, 'conjugacy_classes_known': True, 'counter': 31, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 2, 1, 1], [4, 2, 1, 2], [4, 116, 1, 2], [8, 4, 2, 1], [8, 116, 2, 1], [29, 2, 14, 1], [58, 2, 14, 1], [58, 4, 14, 1], [116, 2, 28, 1], [116, 4, 14, 1], [232, 4, 56, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 6, 'exponent': 232, 'exponents_of_order': [5, 1], 'factors_of_aut_order': [2, 7, 29], 'factors_of_order': [2, 29], 'faithful_reps': [[4, -1, 28]], 'familial': False, 'frattini_label': '8.2', 'frattini_quotient': '116.4', 'hash': 31, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 116, 'inner_gen_orders': [2, 2, 116], 'inner_gens': [[1, 2, 230], [1, 2, 468], [243, 466, 4]], 'inner_hash': 14, 'inner_nilpotent': False, 'inner_order': 464, 'inner_split': True, 'inner_tex': 'D_{58}:C_4', 'inner_used': [1, 2, 3], 'irrC_degree': 4, 'irrQ_degree': 112, 'irrQ_dim': 224, 'irrR_degree': 8, 'irrep_stats': [[1, 8], [2, 114], [4, 29]], 'label': '928.31', 'linC_count': 28, 'linC_degree': 4, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 32, 'linQ_degree_count': 2, 'linQ_dim': 36, 'linQ_dim_count': 2, 'linR_count': 28, 'linR_degree': 8, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'C4.D116', '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': 14, 'number_characteristic_subgroups': 17, 'number_conjugacy_classes': 151, 'number_divisions': 15, 'number_normal_subgroups': 19, 'number_subgroup_autclasses': 34, 'number_subgroup_classes': 38, 'number_subgroups': 472, 'old_label': None, 'order': 928, 'order_factorization_type': 51, 'order_stats': [[1, 1], [2, 3], [4, 236], [8, 240], [29, 28], [58, 84], [116, 112], [232, 224]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 28, 'outer_gen_orders': [2, 2, 28], 'outer_gen_pows': [232, 0, 0], 'outer_gens': [[233, 2, 228], [1, 466, 460], [465, 2, 452]], 'outer_group': '112.37', 'outer_hash': 37, 'outer_nilpotent': True, 'outer_order': 112, 'outer_permdeg': 15, 'outer_perms': [87178291200, 479001600, 11613033], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^2\\times C_{28}', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 45, 'pgroup': 0, 'primary_abelian_invariants': [2, 4], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 4], [4, 1], [28, 2], [56, 3], [112, 1]], 'representations': {'PC': {'code': 11958988553440742549594200029616600698907, 'gens': [1, 2, 3], 'pres': [6, -2, -2, 2, -2, -2, -29, 2784, 4142, 4220, 50, 22083, 69, 27364, 88, 32261]}, 'Perm': {'d': 45, 'gens': [60448736001643175572648702094559546123986556057980932, 2979643062695, 4404290803109, 50350415795, 5807914735709, 2779169046791735119188923215203734982567252731473920000]}}, 'schur_multiplier': [2], 'semidirect_product': False, 'simple': False, 'smith_abelian_invariants': [2, 4], 'solvability_type': 7, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_4.D_{116}', 'transitive_degree': 464, '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': '8.2', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 3, 'aut_exponent': 8, 'aut_gen_orders': [2, 2, 2, 2, 2, 2, 2], 'aut_gens': [[1, 4, 8], [9, 4, 24], [23, 20, 24], [3, 20, 10], [5, 4, 8], [19, 4, 12], [1, 4, 24], [17, 4, 8]], 'aut_group': '128.928', 'aut_hash': 928, 'aut_nilpotency_class': 4, 'aut_nilpotent': True, 'aut_order': 128, 'aut_permdeg': 8, 'aut_perms': [5612, 40031, 13792, 18619, 111, 3758, 13798], 'aut_phi_ratio': 8.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 2, 1, 1], [4, 2, 2, 1], [4, 4, 2, 1], [8, 4, 4, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_2\\wr D_4', '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': False, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^2', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 4, 'autcentquo_group': '32.27', 'autcentquo_hash': 27, 'autcentquo_nilpotent': True, 'autcentquo_order': 32, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'C_2^2\\wr C_2', 'cc_stats': [[1, 1, 1], [2, 1, 1], [2, 2, 1], [4, 2, 2], [4, 4, 2], [8, 4, 4]], 'center_label': '2.1', 'center_order': 2, 'central_product': False, 'central_quotient': '16.3', 'commutator_count': 1, 'commutator_label': '4.2', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '2.1'], 'composition_length': 5, 'conjugacy_classes_known': True, 'counter': 8, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 2, 1, 1], [4, 2, 1, 2], [4, 4, 1, 2], [8, 4, 2, 2]], 'element_repr_type': 'PC', 'elementary': 2, 'eulerian_function': 3, 'exponent': 8, 'exponents_of_order': [5], 'factors_of_aut_order': [2], 'factors_of_order': [2], 'faithful_reps': [[4, -1, 1]], 'familial': False, 'frattini_label': '8.2', 'frattini_quotient': '4.2', 'hash': 8, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 4, 'inner_gen_orders': [4, 2, 2], 'inner_gens': [[1, 20, 12], [17, 4, 8], [5, 4, 8]], 'inner_hash': 3, 'inner_nilpotent': True, 'inner_order': 16, 'inner_split': True, 'inner_tex': 'C_2^2:C_4', 'inner_used': [1, 2, 3], 'irrC_degree': 4, 'irrQ_degree': 4, 'irrQ_dim': 8, 'irrR_degree': 8, 'irrep_stats': [[1, 8], [2, 2], [4, 1]], 'label': '32.8', 'linC_count': 1, 'linC_degree': 4, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 4, 'linQ_degree_count': 1, 'linQ_dim': 8, 'linQ_dim_count': 1, 'linR_count': 1, 'linR_degree': 8, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'C4.D4', 'ngens': 2, 'nilpotency_class': 3, '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': 11, 'number_divisions': 9, 'number_normal_subgroups': 12, 'number_subgroup_autclasses': 13, 'number_subgroup_classes': 19, 'number_subgroups': 26, 'old_label': None, 'order': 32, 'order_factorization_type': 3, 'order_stats': [[1, 1], [2, 3], [4, 12], [8, 16]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 4, 'outer_gen_orders': [2, 4], 'outer_gen_pows': [4, 4], 'outer_gens': [[9, 4, 8], [9, 4, 14]], 'outer_group': '8.3', 'outer_hash': 3, 'outer_nilpotent': True, 'outer_order': 8, 'outer_permdeg': 4, 'outer_perms': [5, 18], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'D_4', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 16, 'pgroup': 2, 'primary_abelian_invariants': [2, 4], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 4], [4, 1]], 'representations': {'PC': {'code': 619029594731, 'gens': [1, 3, 4], 'pres': [5, 2, 2, 2, 2, 2, 10, 86, 302, 243, 248, 58]}, 'GLFp': {'d': 4, 'p': 3, 'gens': [8256119, 34247624, 37630912, 28816400, 28233362]}, 'Perm': {'d': 16, 'gens': [15310288183680, 6920136143976, 5167, 4097467157057, 1313941673647]}}, 'schur_multiplier': [2], 'semidirect_product': False, 'simple': False, 'smith_abelian_invariants': [2, 4], 'solvability_type': 4, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_4.D_4', 'transitive_degree': 16, 'wreath_data': None, 'wreath_product': False}
