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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': False, 'ambient': '4116.ba', 'ambient_counter': 27, 'ambient_order': 4116, 'ambient_tex': 'C_{21}.D_7^2', 'central': False, 'central_factor': False, 'centralizer_order': 21, 'characteristic': False, 'core_order': 21, 'counter': 20, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '4116.ba.14.d1.b1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': False, 'minimal_normal': False, 'nilpotent': False, 'normal': False, 'old_label': '14.d1.b1', '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': 14, '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': '294.18', 'subgroup_hash': 18, 'subgroup_order': 294, 'subgroup_tex': 'D_7\\times C_{21}', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '4116.ba', 'aut_centralizer_order': 7, 'aut_label': '14.d1', 'aut_quo_index': None, 'aut_stab_index': 42, 'aut_weyl_group': '168.47', 'aut_weyl_index': 294, 'centralizer': '196.a1.a1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['2.b1.a1'], 'contains': ['28.b1.b1', '42.d1.b1', '98.d1.a1', '98.f1.b1'], 'core': '196.a1.a1', 'coset_action_label': None, 'count': 14, 'diagramx': [2109, -1, 1028, -1, 1723, -1, 2001, -1], 'generators': [1, 588, 684, 28], 'label': '4116.ba.14.d1.b1', 'mobius_quo': None, 'mobius_sub': 0, 'normal_closure': '2.b1.a1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '14.d1.b1', 'old_label': '14.d1.b1', 'projective_image': '1372.16', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '14.d1.b1', 'subgroup_fusion': None, 'weyl_group': '14.1'}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '42.6', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 42, 'aut_gen_orders': [2, 6, 42], 'aut_gens': [[1, 14], [13, 14], [1, 140], [95, 112]], 'aut_group': '504.178', 'aut_hash': 178, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 504, 'aut_permdeg': 15, 'aut_perms': [1053144299670, 56087101641, 695538237481], 'aut_phi_ratio': 6.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 7, 1, 1], [3, 1, 2, 1], [6, 7, 2, 1], [7, 1, 6, 1], [7, 2, 3, 1], [7, 2, 18, 1], [14, 7, 6, 1], [21, 1, 12, 1], [21, 2, 6, 1], [21, 2, 36, 1], [42, 7, 12, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_{14}:C_6^2', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 6, 'autcent_group': '12.5', 'autcent_hash': 5, 'autcent_nilpotent': True, 'autcent_order': 12, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2\\times C_6', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 42, 'autcentquo_group': '42.1', 'autcentquo_hash': 1, 'autcentquo_nilpotent': False, 'autcentquo_order': 42, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'F_7', 'cc_stats': [[1, 1, 1], [2, 7, 1], [3, 1, 2], [6, 7, 2], [7, 1, 6], [7, 2, 21], [14, 7, 6], [21, 1, 12], [21, 2, 42], [42, 7, 12]], 'center_label': '21.2', 'center_order': 21, 'central_product': True, 'central_quotient': '14.1', 'commutator_count': 1, 'commutator_label': '7.1', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '3.1', '7.1', '7.1'], 'composition_length': 4, 'conjugacy_classes_known': True, 'counter': 18, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['14.1', 1], ['3.1', 1], ['7.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 7, 1, 1], [3, 1, 2, 1], [6, 7, 2, 1], [7, 1, 6, 1], [7, 2, 3, 1], [7, 2, 6, 3], [14, 7, 6, 1], [21, 1, 12, 1], [21, 2, 6, 1], [21, 2, 12, 3], [42, 7, 12, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 96, 'exponent': 42, 'exponents_of_order': [2, 1, 1], 'factors_of_aut_order': [2, 3, 7], 'factors_of_order': [2, 3, 7], 'faithful_reps': [[2, 0, 36]], 'familial': False, 'frattini_label': '1.1', 'frattini_quotient': '294.18', 'hash': 18, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 14, 'inner_gen_orders': [2, 7], 'inner_gens': [[1, 182], [127, 14]], 'inner_hash': 1, 'inner_nilpotent': False, 'inner_order': 14, 'inner_split': True, 'inner_tex': 'D_7', 'inner_used': [1, 2], 'irrC_degree': 2, 'irrQ_degree': 24, 'irrQ_dim': 24, 'irrR_degree': 4, 'irrep_stats': [[1, 42], [2, 63]], 'label': '294.18', 'linC_count': 36, 'linC_degree': 2, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 14, 'linQ_degree_count': 10, 'linQ_dim': 14, 'linQ_dim_count': 10, 'linR_count': 54, 'linR_degree': 4, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'D7*C21', 'ngens': 4, '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': 12, 'number_characteristic_subgroups': 12, 'number_conjugacy_classes': 105, 'number_divisions': 16, 'number_normal_subgroups': 12, 'number_subgroup_autclasses': 18, 'number_subgroup_classes': 22, 'number_subgroups': 52, 'old_label': None, 'order': 294, 'order_factorization_type': 222, 'order_stats': [[1, 1], [2, 7], [3, 2], [6, 14], [7, 48], [14, 42], [21, 96], [42, 84]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 6, 'outer_gen_orders': [6, 6], 'outer_gen_pows': [0, 0], 'outer_gens': [[3, 14], [1, 238]], 'outer_group': '36.14', 'outer_hash': 14, 'outer_nilpotent': True, 'outer_order': 36, 'outer_permdeg': 10, 'outer_perms': [5043, 363024], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_6^2', 'pc_rank': 2, 'perfect': False, 'permutation_degree': 17, 'pgroup': 0, 'primary_abelian_invariants': [2, 3, 7], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 2], [2, 2], [6, 3], [12, 6], [24, 3]], 'representations': {'PC': {'code': 84541155798287, 'gens': [1, 3], 'pres': [4, -2, -7, -3, -7, 8, 2186, 46, 4035]}, 'GLFp': {'d': 2, 'p': 43, 'gens': [1892, 1669688, 1908206]}, 'Perm': {'d': 17, 'gens': [127, 403200, 22324388486400, 973]}}, 'schur_multiplier': [], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [42], 'solvability_type': 6, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'D_7\\times C_{21}', 'transitive_degree': 42, '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': '12.5', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 4, 'aut_exponent': 84, 'aut_gen_orders': [6, 2, 7, 42, 14, 14, 6, 7, 14], 'aut_gens': [[1, 2, 84, 588], [85, 62, 252, 1176], [1, 58, 84, 588], [2245, 2186, 1848, 588], [169, 1558, 336, 588], [2665, 1511, 12, 3528], [373, 2182, 3612, 3528], [337, 46, 168, 2352], [1, 3530, 84, 588], [169, 814, 84, 588]], 'aut_group': '49392.bb', 'aut_hash': 2546147806097575046, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 49392, 'aut_permdeg': 100, 'aut_perms': [39886716433554786861147705782310725141475047354673966892028177187731843174987122473938402855981931768154376797833269726339902141573895020208313972673768067990, 4968829558669574315901366089774607151601309468731257825252349565120650249417866453441784108315939393476283967989447976890625163264000000000000000000000, 79372305083324720254885468194430400227144281942377529629751945413504723940014534052622479723486203270579673375424349747883789459085272605303635184272208354613, 92810245609406383804541843152439771765670218351316333545882750215076314437138865096201387556272502018101432314366866275990966969751552116102059349436862003048, 65116831137900535716334245244604348712325533481243246541899993869225462917513699937880995651690670839536840314634101780766259948259879263903206697917318023374, 17541624061438210713937913701470845456726373877349001575035524504127914759059580579588587590552331921073069563877840211209077501628710774429224414115764378384, 84629917523447951167792536336580414106175815273956419543440049509533462460693551140163770172630860431021699828222202885963907030800324933617570156651444233345, 44310909635164709489252015010530322820270091040017078534072989859918023242554817548046259891161801426904580952431257550985867626331566550566934951621100694880, 74792379734320000475441707095824666889200384957467316058389597393083866337981042412306141326622249240868331416995047726606183103676981956804250099704139912338], 'aut_phi_ratio': 42.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 49, 1, 1], [2, 49, 2, 1], [3, 1, 2, 1], [6, 49, 2, 1], [6, 49, 4, 1], [7, 2, 3, 1], [7, 14, 6, 1], [7, 28, 9, 1], [14, 98, 3, 1], [14, 98, 6, 1], [21, 2, 6, 1], [21, 14, 12, 1], [21, 28, 18, 1], [42, 98, 6, 1], [42, 98, 12, 1]], 'aut_supersolvable': False, 'aut_tex': '\\He_7:(C_6^2:C_2^2)', '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': 84, 'autcentquo_group': '24696.h', 'autcentquo_hash': 6460325027391172650, 'autcentquo_nilpotent': False, 'autcentquo_order': 24696, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': '\\He_7:C_6\\wr C_2', 'cc_stats': [[1, 1, 1], [2, 49, 3], [3, 1, 2], [6, 49, 6], [7, 2, 3], [7, 14, 6], [7, 28, 9], [14, 98, 9], [21, 2, 6], [21, 14, 12], [21, 28, 18], [42, 98, 18]], 'center_label': '3.1', 'center_order': 3, 'central_product': True, 'central_quotient': '1372.16', 'commutator_count': 1, 'commutator_label': '343.3', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '3.1', '7.1', '7.1', '7.1'], 'composition_length': 6, 'conjugacy_classes_known': True, 'counter': 27, 'cyclic': False, 'derived_length': 3, 'dihedral': False, 'direct_factorization': [['1372.16', 1], ['3.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 49, 1, 3], [3, 1, 2, 1], [6, 49, 2, 3], [7, 2, 3, 1], [7, 14, 3, 2], [7, 28, 3, 3], [14, 98, 3, 3], [21, 2, 6, 1], [21, 14, 6, 2], [21, 28, 6, 3], [42, 98, 6, 3]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 84, 'exponent': 42, 'exponents_of_order': [3, 2, 1], 'factors_of_aut_order': [2, 3, 7], 'factors_of_order': [2, 3, 7], 'faithful_reps': [[14, 0, 12]], 'familial': False, 'frattini_label': '7.1', 'frattini_quotient': '588.49', 'hash': 2041031394734020119, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 14, 'inner_gen_orders': [2, 14, 7, 7], 'inner_gens': [[1, 1286, 504, 588], [4093, 2, 2856, 3528], [169, 1934, 84, 588], [1, 1178, 84, 588]], 'inner_hash': 16, 'inner_nilpotent': False, 'inner_order': 1372, 'inner_split': True, 'inner_tex': 'C_7^2:D_{14}', 'inner_used': [1, 2], 'irrC_degree': 14, 'irrQ_degree': 84, 'irrQ_dim': 84, 'irrR_degree': 28, 'irrep_stats': [[1, 12], [2, 36], [4, 27], [14, 18]], 'label': '4116.ba', 'linC_count': 12, 'linC_degree': 14, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 44, 'linQ_degree_count': 8, 'linQ_dim': 44, 'linQ_dim_count': 8, 'linR_count': 24, 'linR_degree': 16, 'maximal_subgroups_known': True, 'metabelian': False, 'metacyclic': False, 'monomial': True, 'name': 'C21.D7^2', 'ngens': 6, '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], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 16, 'number_characteristic_subgroups': 10, 'number_conjugacy_classes': 93, 'number_divisions': 26, 'number_normal_subgroups': 22, 'number_subgroup_autclasses': 52, 'number_subgroup_classes': 94, 'number_subgroups': 2218, 'old_label': None, 'order': 4116, 'order_factorization_type': 321, 'order_stats': [[1, 1], [2, 147], [3, 2], [6, 294], [7, 342], [14, 882], [21, 684], [42, 1764]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 6, 'outer_gen_orders': [6, 6], 'outer_gen_pows': [253, 0], 'outer_gens': [[2041, 435, 48, 2352], [169, 22, 420, 3528]], 'outer_group': '36.12', 'outer_hash': 12, 'outer_nilpotent': False, 'outer_order': 36, 'outer_permdeg': 8, 'outer_perms': [750, 10081], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_6\\times S_3', 'pc_rank': 4, 'perfect': False, 'permutation_degree': 52, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 3], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 4], [6, 4], [12, 7], [24, 3], [42, 2], [84, 2]], 'representations': {'PC': {'code': '66374914638807909022467554599475681325088559', 'gens': [1, 2, 5, 6], 'pres': [6, -2, -2, -3, -7, -7, -7, 15433, 31, 32690, 68, 29955, 15124, 42850, 27106, 6112, 63515]}, 'GLFp': {'d': 4, 'p': 7, 'gens': [31392350697539, 611588865287, 9495688004000, 31584908075045, 7394473631462, 28186167033742]}, 'Perm': {'d': 52, 'gens': [122392157881291898313947223698298773040565732814893270413422844720, 1706346053456648766825121617584478599019749040109854405960246182760, 3289727285998847184752714187827877242951891636603417354089227828384, 4871879539460433276768188427234411992507894929765460423018638091667, 3, 6454033016996107993701221292664290746258824536151317386943091056960]}}, 'schur_multiplier': [2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 6], 'solvability_type': 10, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_{21}.D_7^2', 'transitive_degree': 147, 'wreath_data': None, 'wreath_product': False}
