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gps_subgroup_search • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'ambient': '4096.buh', 'ambient_counter': 1204, 'ambient_order': 4096, 'ambient_tex': 'C_4^2:C_2^2.D_4^2', 'central': False, 'central_factor': False, 'centralizer_order': None, 'characteristic': False, 'core_order': 256, 'counter': 210, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '4096.buh.8._.DY', 'maximal': False, 'maximal_normal': False, 'metabelian': False, 'metacyclic': False, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': False, 'old_label': '8.DY', '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': 8, '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': '512.60809', 'subgroup_hash': None, 'subgroup_order': 512, 'subgroup_tex': 'D_8\\wr C_2', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '4096.buh', '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': 2, 'diagramx': None, 'generators': [1, 2200, 512], 'label': '4096.buh.8._.DY', 'mobius_quo': None, 'mobius_sub': None, 'normal_closure': None, 'normal_contained_in': None, 'normal_contains': None, 'normalizer': None, 'old_label': '8.DY', 'projective_image': None, 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '8._.DY', 'subgroup_fusion': None, 'weyl_group': None}
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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': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 3, 'aut_exponent': 16, 'aut_gen_orders': [2, 2, 2, 2, 2, 2, 2], 'aut_gens': [[1, 2, 8, 64], [1, 34, 8, 64], [1, 34, 8, 96], [5, 6, 424, 224], [33, 34, 40, 352], [305, 182, 8, 456], [33, 230, 56, 120], [29, 286, 152, 352]], 'aut_group': '4096.brd', 'aut_hash': 5830821769979981028, 'aut_nilpotency_class': 6, 'aut_nilpotent': True, 'aut_order': 4096, 'aut_permdeg': 48, 'aut_perms': [1783695150402495069294310932636883913049412181984824819938371, 1783094320606454130075446691322640360335694161432706524328529, 5357992358992059921476714661111567210702372017934917776760, 3846642698600853553456182128479492607973304506144693712712, 5837149538456654733880306131955503407384486594606488779689364, 4063816081365005558226645640473410027397918129834380293866977, 8303704181275948731262704876027745672110166722044616431520343], 'aut_phi_ratio': 16.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 2, 1, 1], [2, 8, 4, 1], [2, 16, 1, 1], [2, 16, 2, 1], [2, 32, 1, 1], [4, 4, 1, 3], [4, 16, 1, 1], [4, 16, 2, 1], [4, 64, 2, 1], [8, 4, 2, 1], [8, 4, 4, 1], [8, 8, 1, 1], [8, 8, 2, 1], [8, 16, 4, 1], [8, 32, 1, 1], [16, 32, 2, 1]], 'aut_supersolvable': True, 'aut_tex': '(C_4\\times \\OD_{16}).D_4^2', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 2, 'autcent_group': '8.5', 'autcent_hash': 5, 'autcent_nilpotent': True, 'autcent_order': 8, 'autcent_solvable': True, 'autcent_split': False, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^3', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 8, 'autcentquo_group': '512.420058', 'autcentquo_hash': 4262671825627166463, 'autcentquo_nilpotent': True, 'autcentquo_order': 512, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'D_4^2.C_2^3', 'cc_stats': [[1, 1, 1], [2, 1, 1], [2, 2, 1], [2, 8, 4], [2, 16, 3], [2, 32, 1], [4, 4, 3], [4, 16, 3], [4, 64, 2], [8, 4, 6], [8, 8, 3], [8, 16, 4], [8, 32, 1], [16, 32, 2]], 'center_label': '2.1', 'center_order': 2, 'central_product': False, 'central_quotient': '256.6661', 'commutator_count': 1, 'commutator_label': '64.174', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '2.1', '2.1', '2.1', '2.1', '2.1'], 'composition_length': 9, 'conjugacy_classes_known': True, 'counter': 60809, 'cyclic': False, 'derived_length': 3, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 2, 1, 1], [2, 8, 1, 4], [2, 16, 1, 3], [2, 32, 1, 1], [4, 4, 1, 3], [4, 16, 1, 3], [4, 64, 1, 2], [8, 4, 2, 3], [8, 8, 1, 1], [8, 8, 2, 1], [8, 16, 2, 2], [8, 32, 1, 1], [16, 32, 2, 1]], 'element_repr_type': 'PC', 'elementary': 2, 'eulerian_function': 10752, 'exponent': 16, 'exponents_of_order': [9], 'factors_of_aut_order': [2], 'factors_of_order': [2], 'faithful_reps': [[4, 1, 8], [8, 1, 2]], 'familial': False, 'frattini_label': '64.174', 'frattini_quotient': '8.5', 'hash': 60809, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 8, 'inner_gen_orders': [2, 4, 4, 8], 'inner_gens': [[1, 6, 8, 456], [5, 2, 184, 120], [1, 402, 8, 64], [185, 10, 8, 64]], 'inner_hash': 6661, 'inner_nilpotent': True, 'inner_order': 256, 'inner_split': False, 'inner_tex': 'D_4^2:C_2^2', 'inner_used': [1, 2, 3, 4], 'irrC_degree': 4, 'irrQ_degree': 8, 'irrQ_dim': 8, 'irrR_degree': 4, 'irrep_stats': [[1, 8], [2, 6], [4, 18], [8, 3]], 'label': '512.60809', 'linC_count': 8, 'linC_degree': 4, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 8, 'linQ_degree_count': 4, 'linQ_dim': 8, 'linQ_dim_count': 4, 'linR_count': 8, 'linR_degree': 4, 'maximal_subgroups_known': True, 'metabelian': False, 'metacyclic': False, 'monomial': True, 'name': 'D8wrC2', 'ngens': 3, 'nilpotency_class': 6, 'nilpotent': True, 'normal_counts': [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': 20, 'number_characteristic_subgroups': 16, 'number_conjugacy_classes': 35, 'number_divisions': 28, 'number_normal_subgroups': 30, 'number_subgroup_autclasses': 201, 'number_subgroup_classes': 357, 'number_subgroups': 2598, 'old_label': None, 'order': 512, 'order_factorization_type': 7, 'order_stats': [[1, 1], [2, 115], [4, 188], [8, 144], [16, 64]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 4, 'outer_gen_orders': [2, 2, 2, 2], 'outer_gen_pows': [0, 0, 0, 0], 'outer_gens': [[1, 2, 8, 96], [153, 94, 8, 456], [33, 2, 40, 320], [33, 2, 8, 64]], 'outer_group': '16.11', 'outer_hash': 11, 'outer_nilpotent': True, 'outer_order': 16, 'outer_permdeg': 6, 'outer_perms': [126, 55, 289, 288], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2\\times D_4', 'pc_rank': 4, 'perfect': False, 'permutation_degree': 16, 'pgroup': 2, 'primary_abelian_invariants': [2, 2, 2], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 8], [2, 6], [4, 6], [8, 7], [16, 1]], 'representations': {'PC': {'code': '202050378031308156279437029341143090179702581064918285869530', 'gens': [1, 2, 4, 7], 'pres': [9, 2, 2, 2, 2, 2, 2, 2, 2, 2, 109, 46, 3324, 525, 102, 6853, 562, 130, 28734, 3795, 7080, 186, 28807, 6352, 6937, 214, 23336, 11681]}, 'Perm': {'d': 16, 'gens': [11212821043200, 1313941697263, 1307761464367, 5606234750016, 6354391823, 2796059347200, 4097467152142, 5167, 1313941673647]}}, 'schur_multiplier': [2, 2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 2], 'solvability_type': 5, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'D_8\\wr C_2', 'transitive_degree': 16, 'wreath_data': ['D_8', 'C_2', '2T1'], 'wreath_product': True}
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gps_groups • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '32.45', 'all_subgroups_known': False, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 3, 'aut_exponent': 8, 'aut_gen_orders': [4, 2, 2, 2, 2, 4, 4, 8], 'aut_gens': [[1, 2, 8, 16, 64, 512], [257, 2, 8, 152, 2112, 768], [257, 2, 8, 276, 320, 768], [1, 258, 8, 16, 64, 512], [257, 2, 8, 16, 64, 512], [1, 2186, 2312, 2160, 3136, 3584], [2441, 2, 8, 16, 2368, 640], [2145, 2, 8, 16, 3392, 1600], [1025, 386, 2312, 3536, 64, 512]], 'aut_group': None, 'aut_hash': 7723502086413711684, 'aut_nilpotency_class': 5, 'aut_nilpotent': True, 'aut_order': 32768, 'aut_permdeg': 144, 'aut_perms': [4856779561552612296866569563349078427483869000834606665097341666236615877765274361145288636663849146734007174586981347689506711019050537829460616580898489867653078662566144913291940809620073701457987783823688481673443564949725125461488205340640488343, 270077479228076560841174458477660024849878529455542266346342589706399145805967787047786977289998720675139962173518953339207616847155725700177487687217924411711158002981986352361602305191243387375136296618100025590172102041198625245307465618498823823, 23721121693857727312703897946236842591459405681152890763422139416051761747140977669696358228342950663567980678712457579977382714474826870161537780295758270940161360497845319551197520967821476314041284490301813788033759575510563966918727468190969104, 2197428676446943300470744579750085228090626608976400972572833054633226279860204281824006617208489681878041675794598152084052979340025460852458910021755392988875592530978246155434243287892762094522144471967010589348049257492086852280881552993916303490, 1789676738514711894261157985997999754326478146692780509553329908210982319620581765187902392584628067010520732835526278241311266327332202339216860642360304636032348159716877711455956913977811761730210692223831166376144998727306494737812900951845221933, 2374933555088035825978117913382201022848542607035453475546452788000947838151555372016695310284106594457636509515347484112407447802292179637208612076826751019339324403326051915495713820355605792705983747332580320082513687878502214260388762123048800916, 737490558657691268417916489472774953496200339844175254922776925612575222954668617998096543109342514311690297965760596673225659723423661466692550080938726908911473315053820389118605478621633608386936818616769444566046798715964675202356788729233389630, 3680205306047798063445735035515028355491855739336467052850325931902555286269819048669804357885643226137504840491003926937373638514060060116244985136201961761143553714139989760323323359337863482694506951791706393762552014478874400076370089499823450946], 'aut_phi_ratio': 16.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 2, 1, 3], [2, 4, 1, 2], [2, 16, 1, 6], [2, 16, 2, 1], [2, 32, 1, 3], [2, 32, 2, 1], [4, 4, 1, 4], [4, 8, 1, 4], [4, 16, 1, 6], [4, 32, 1, 12], [4, 32, 2, 1], [4, 64, 1, 2], [4, 64, 2, 1], [4, 64, 4, 4], [8, 16, 1, 4], [8, 32, 1, 6], [8, 64, 1, 14], [8, 64, 2, 2], [16, 128, 2, 2]], 'aut_supersolvable': True, 'aut_tex': 'C_2^2\\times C_2^4.C_2^6.C_2^3', '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': 8, 'autcentquo_group': None, 'autcentquo_hash': 2265801771948521539, 'autcentquo_nilpotent': True, 'autcentquo_order': 2048, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'C_2\\times C_2^4.C_2^4.C_2^2', 'cc_stats': [[1, 1, 1], [2, 1, 1], [2, 2, 3], [2, 4, 2], [2, 16, 8], [2, 32, 5], [4, 4, 4], [4, 8, 4], [4, 16, 6], [4, 32, 14], [4, 64, 20], [8, 16, 4], [8, 32, 6], [8, 64, 18], [16, 128, 4]], 'center_label': '2.1', 'center_order': 2, 'central_product': None, 'central_quotient': '2048.csd', 'commutator_count': 1, 'commutator_label': '128.1883', 'complements_known': False, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '2.1', '2.1', '2.1', '2.1', '2.1', '2.1', '2.1', '2.1'], 'composition_length': 12, 'conjugacy_classes_known': True, 'counter': 1204, 'cyclic': False, 'derived_length': 3, 'dihedral': False, 'direct_factorization': None, 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 2, 1, 3], [2, 4, 1, 2], [2, 16, 1, 8], [2, 32, 1, 5], [4, 4, 1, 4], [4, 8, 1, 4], [4, 16, 1, 2], [4, 16, 2, 2], [4, 32, 1, 10], [4, 32, 2, 2], [4, 64, 1, 8], [4, 64, 2, 6], [8, 16, 1, 4], [8, 32, 1, 4], [8, 32, 2, 1], [8, 64, 1, 6], [8, 64, 2, 6], [16, 128, 1, 2], [16, 128, 2, 1]], 'element_repr_type': 'PC', 'elementary': 2, 'eulerian_function': None, 'exponent': 16, 'exponents_of_order': [12], 'factors_of_aut_order': [2], 'factors_of_order': [2], 'faithful_reps': [[32, 1, 2]], 'familial': False, 'frattini_label': '256.53624', 'frattini_quotient': '16.14', 'hash': 10266607072772748, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 8, 'inner_gen_orders': [2, 4, 2, 4, 4, 8], 'inner_gens': [[1, 2186, 2312, 2160, 3136, 3584], [2441, 2, 8, 16, 2368, 640], [2305, 2, 8, 16, 320, 2816], [2145, 2, 8, 16, 3392, 1600], [1025, 2306, 264, 1296, 64, 512], [1025, 386, 2312, 3536, 64, 512]], 'inner_hash': 5421954460789486987, 'inner_nilpotent': True, 'inner_order': 2048, 'inner_split': None, 'inner_tex': '(C_2^2\\times D_4).D_4^2', 'inner_used': [1, 2, 3, 4, 6], 'irrC_degree': 32, 'irrQ_degree': 32, 'irrQ_dim': 32, 'irrR_degree': 32, 'irrep_stats': [[1, 32], [2, 40], [4, 12], [8, 10], [16, 4], [32, 2]], 'label': '4096.buh', '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': 'C4^2:C2^2.D4^2', 'ngens': 4, 'nilpotency_class': 6, 'nilpotent': True, 'normal_counts': [1, 1, 3, 7, 11, 19, 45, 63, 75, 67, 43, 15, 1], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 80, 'number_characteristic_subgroups': 263, 'number_conjugacy_classes': 100, 'number_divisions': 82, 'number_normal_subgroups': 351, 'number_subgroup_autclasses': None, 'number_subgroup_classes': None, 'number_subgroups': None, 'old_label': None, 'order': 4096, 'order_factorization_type': 7, 'order_stats': [[1, 1], [2, 303], [4, 1872], [8, 1408], [16, 512]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 2, 'outer_gen_orders': [2, 2, 2, 2], 'outer_gen_pows': [2304, 0, 0, 0], 'outer_gens': [[257, 2, 8, 152, 2112, 768], [257, 2, 8, 276, 320, 768], [1, 258, 8, 16, 64, 512], [257, 2, 8, 16, 64, 512]], 'outer_group': '16.14', 'outer_hash': 14, 'outer_nilpotent': True, 'outer_order': 16, 'outer_permdeg': 16, 'outer_perms': [1327912203241, 2966658509284, 4396538203938, 5800737348096], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^4', 'pc_rank': 6, 'perfect': False, 'permutation_degree': 64, 'pgroup': 2, 'primary_abelian_invariants': [2, 2, 2, 4], 'quasisimple': False, 'rank': 4, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 16], [2, 32], [4, 16], [8, 12], [16, 4], [32, 2]], 'representations': {'PC': {'code': '86709430660540582280311497452766676309293270556243913919329081163789999706287163063784691040575234972296579075350708476440215250', 'gens': [1, 2, 4, 5, 7, 10], 'pres': [12, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 52465, 61, 9362, 110979, 129604, 172, 76037, 263430, 99474, 3402, 17862, 1242, 246, 208903, 13111, 1219, 283, 430089, 38421, 23073, 42285, 12057, 13509, 357, 405514, 84502, 26458, 12742, 394, 20795]}, 'Perm': {'d': 64, 'gens': [1112424861405409730857493385545454790328794326915879359387458988688655724504920113, 8091918689082190539467971410850694915832320044636171050983756587929121558812854794061828, 10106514407168233782541740660660574328204309342372199720663037493163705075520556681528693, 1985695982293684077459513503086662992763993602373746416935144214319468975521178028825012, 3996737933847331015796940550744873651533970099636901732977167513672363920653319091511737, 5982848832525376121142008162954435140350332109648056783376295704881310569160578003007393, 3561826782484347301908674601538570625356398310534007350252707189412602009611397660735]}}, 'schur_multiplier': [2, 2, 2, 2, 2, 2, 4], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 2, 4], 'solvability_type': 5, 'solvable': True, 'subgroup_inclusions_known': False, 'subgroup_index_bound': 8, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_4^2:C_2^2.D_4^2', 'transitive_degree': None, 'wreath_data': None, 'wreath_product': False}
