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gps_subgroup_search • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'ambient': '11664.be', 'ambient_counter': 31, 'ambient_order': 11664, 'ambient_tex': '\\He_3^2:(C_2\\times D_4)', 'central': False, 'central_factor': False, 'centralizer_order': 1, 'characteristic': False, 'core_order': 1458, 'counter': 13, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '11664.be.4.g1', 'maximal': False, 'maximal_normal': False, 'metabelian': False, 'metacyclic': False, 'minimal': False, 'minimal_normal': False, 'nilpotent': False, 'normal': False, 'old_label': '4.g1', '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': 4, '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': '2916.ea', 'subgroup_hash': 3693867090096231779, 'subgroup_order': 2916, 'subgroup_tex': 'C_3^4:S_3^2', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '11664.be', 'aut_centralizer_order': None, 'aut_label': '4.g1', 'aut_quo_index': None, 'aut_stab_index': None, 'aut_weyl_group': None, 'aut_weyl_index': None, 'centralizer': '11664.a1', 'complements': None, 'conjugacy_class_count': 2, 'contained_in': ['2.d1'], 'contains': ['8.b1', '8.d1', '12.g1', '12.j1', '12.k1'], 'core': '8.b1', 'coset_action_label': None, 'count': 4, 'diagramx': [1948, -1, 8769, -1], 'generators': [20929020542665, 45603, 4536034458905765, 357386, 244, 1155045378234093, 711374899737600, 376784618112000], 'label': '11664.be.4.g1', 'mobius_quo': None, 'mobius_sub': 0, 'normal_closure': '2.d1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '2.d1', 'old_label': '4.g1', 'projective_image': '11664.be', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '4.g1', 'subgroup_fusion': None, 'weyl_group': '5832.ci'}
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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': 3, 'aut_exponent': 12, 'aut_gen_orders': [2, 6, 12, 3, 3, 3, 3, 3], 'aut_gens': [[3402106385166964, 355693738798535], [4557070732429469, 355693738860964], [4891835414102425, 1134209726581945], [5247690931883523, 2999898405589945], [3402106385247363, 355693738798535], [4113655597986484, 20929020721652], [4113574726549824, 355868051835476], [3778891003359363, 21010008280195], [3402106385166964, 732478233496195]], 'aut_group': '34992.ns', 'aut_hash': 5395324004515259423, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 34992, 'aut_permdeg': 66, 'aut_perms': [521858103572185645447050180455616673563732069150414642098195715988820082788877203883719873574, 252893273483599920512736537502011762813601131386804065519710048079796999482028331140981625666, 339741019892978225485262972555278559868953308913752257885788132370529163232895968912461919481, 511526758187999293166803681543972911173765034631327466405398968680077922805058341903881231595, 387397890476933702914495819672759095480527224819386767865365965952073438328001652815462233916, 512098958896568963577190920609372738180805453999699390290320731738753467130324367288049349190, 49655496793879140035657359570583540425955454862178352468397224110764560101237321609003034528, 27925373836691748990250507851931783081465872884898484823812000057615235197572578986329], 'aut_phi_ratio': 36.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 27, 2, 1], [2, 81, 1, 1], [3, 2, 2, 1], [3, 4, 1, 1], [3, 6, 2, 1], [3, 9, 2, 1], [3, 12, 2, 1], [3, 12, 3, 1], [3, 12, 6, 1], [3, 18, 1, 1], [3, 18, 2, 1], [3, 18, 4, 1], [3, 36, 2, 4], [3, 36, 4, 1], [6, 54, 2, 1], [6, 81, 2, 1], [6, 81, 4, 1], [6, 162, 1, 1], [6, 162, 2, 2], [6, 162, 4, 1]], 'aut_supersolvable': False, 'aut_tex': '\\He_3^2:(S_3\\times D_4)', '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': 12, 'autcentquo_group': '34992.ns', 'autcentquo_hash': 5395324004515259423, 'autcentquo_nilpotent': False, 'autcentquo_order': 34992, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': '\\He_3^2:(S_3\\times D_4)', 'cc_stats': [[1, 1, 1], [2, 27, 2], [2, 81, 1], [3, 2, 2], [3, 4, 1], [3, 6, 2], [3, 9, 2], [3, 12, 11], [3, 18, 7], [3, 36, 12], [6, 54, 2], [6, 81, 6], [6, 162, 9]], 'center_label': '1.1', 'center_order': 1, 'central_product': False, 'central_quotient': '2916.ea', 'commutator_count': 1, 'commutator_label': '243.37', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '3.1', '3.1', '3.1', '3.1', '3.1', '3.1'], 'composition_length': 8, 'conjugacy_classes_known': True, 'counter': 105, 'cyclic': False, 'derived_length': 3, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 27, 1, 2], [2, 81, 1, 1], [3, 2, 1, 2], [3, 4, 1, 1], [3, 6, 2, 1], [3, 9, 2, 1], [3, 12, 1, 3], [3, 12, 2, 4], [3, 18, 1, 3], [3, 18, 2, 2], [3, 36, 1, 2], [3, 36, 2, 5], [6, 54, 1, 2], [6, 81, 2, 3], [6, 162, 1, 3], [6, 162, 2, 3]], 'element_repr_type': 'Perm', 'elementary': 1, 'eulerian_function': 24, 'exponent': 6, 'exponents_of_order': [6, 2], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [[12, 0, 6], [12, 1, 3]], 'familial': False, 'frattini_label': '9.2', 'frattini_quotient': '324.167', 'hash': 3693867090096231779, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 6, 'inner_gen_orders': [6, 6], 'inner_gens': [[3402106385166964, 1510645587937949], [4557070732394186, 355693738798535]], 'inner_hash': 3693867090096231779, 'inner_nilpotent': False, 'inner_order': 2916, 'inner_split': True, 'inner_tex': 'C_3^4:S_3^2', 'inner_used': [1, 2], 'irrC_degree': 12, 'irrQ_degree': 12, 'irrQ_dim': 12, 'irrR_degree': 12, 'irrep_stats': [[1, 12], [2, 18], [4, 15], [12, 9], [18, 4]], 'label': '2916.ea', 'linC_count': 9, 'linC_degree': 12, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 12, 'linQ_degree_count': 3, 'linQ_dim': 12, 'linQ_dim_count': 3, 'linR_count': 3, 'linR_degree': 12, 'maximal_subgroups_known': True, 'metabelian': False, 'metacyclic': False, 'monomial': True, 'name': 'C3^4:S3^2', 'ngens': 2, '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], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 25, 'number_characteristic_subgroups': 15, 'number_conjugacy_classes': 58, 'number_divisions': 39, 'number_normal_subgroups': 33, 'number_subgroup_autclasses': 328, 'number_subgroup_classes': 641, 'number_subgroups': 14367, 'old_label': None, 'order': 2916, 'order_factorization_type': 32, 'order_stats': [[1, 1], [2, 135], [3, 728], [6, 2052]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 6, 'outer_gen_orders': [2, 6], 'outer_gen_pows': [2623194826070403, 355693738665145], 'outer_gens': [[4912751973381123, 1845410269656362], [3423209842125482, 2999898405589945]], 'outer_group': '12.4', 'outer_hash': 4, 'outer_nilpotent': False, 'outer_order': 12, 'outer_permdeg': 5, 'outer_perms': [55, 49], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'D_6', 'pc_rank': 6, 'perfect': False, 'permutation_degree': 18, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 3], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 10], [4, 9], [8, 6], [12, 3], [18, 4], [24, 3]], 'representations': {'PC': {'code': '481245547502042790880504336723567231414345347817122110007200265995615', 'gens': [1, 3, 5, 6, 7, 8], 'pres': [8, -2, -3, -2, -3, -3, -3, -3, -3, 16, 46514, 31618, 66, 83715, 10571, 105124, 20172, 4820, 2308, 25925, 10381, 17301, 8237, 36294, 6070, 20759]}, 'Perm': {'d': 18, 'gens': [3402106385166964, 355693738798535]}}, 'schur_multiplier': [6], '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_3^4:S_3^2', 'transitive_degree': 18, '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': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 4, 'aut_exponent': 12, 'aut_gen_orders': [12, 12, 6], 'aut_gens': [[2979163906969437, 1133847166193275, 4536034458905765], [2644123839391173, 3000092920254145, 6046230224424813], [1866339282931919, 1510645588179120, 6046230224476922], [2267789007231930, 1510363935095189, 5669988841354294]], 'aut_group': '46656.jd', 'aut_hash': 1502020854965269241, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 46656, 'aut_permdeg': 126, 'aut_perms': [2581327856749975446076895697660288230705014838492689536561024440922223787760926678777274953459491588106886978957381472632301878893439378943838906225782260296976669210766688825261378534009427045049896900160458837, 8046446493998798212460292949504770853020397339755587412674298054955868230705632180385012550638036852550194479699972338476440516888088137116280728741352556955525200303251415522593426067954606397586023872834654174, 1530645687360915420669769148282045264688069167262288282661898303273991291802099866657832518305608999014167824485963682455817421418245723520458143537902513601748870242866109484289809897245168400872307375306442578], 'aut_phi_ratio': 12.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 18, 1, 1], [2, 54, 2, 1], [2, 81, 1, 1], [2, 81, 2, 1], [2, 162, 1, 1], [3, 2, 2, 1], [3, 4, 1, 1], [3, 12, 2, 1], [3, 24, 1, 1], [3, 24, 2, 2], [3, 36, 2, 1], [3, 72, 1, 1], [3, 72, 2, 1], [3, 144, 2, 1], [4, 486, 2, 1], [6, 36, 1, 2], [6, 36, 2, 1], [6, 108, 2, 2], [6, 162, 2, 1], [6, 216, 1, 1], [6, 216, 2, 2], [6, 324, 1, 1], [6, 324, 2, 3], [6, 324, 4, 1], [6, 648, 1, 1], [6, 648, 2, 1], [12, 972, 2, 1]], 'aut_supersolvable': False, 'aut_tex': '\\He_3^2:C_2\\wr 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': 12, 'autcentquo_group': '46656.jd', 'autcentquo_hash': 1502020854965269241, 'autcentquo_nilpotent': False, 'autcentquo_order': 46656, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': '\\He_3^2:C_2\\wr C_2^2', 'cc_stats': [[1, 1, 1], [2, 18, 1], [2, 54, 2], [2, 81, 3], [2, 162, 1], [3, 2, 2], [3, 4, 1], [3, 12, 2], [3, 24, 5], [3, 36, 2], [3, 72, 3], [3, 144, 2], [4, 486, 2], [6, 36, 4], [6, 108, 4], [6, 162, 2], [6, 216, 5], [6, 324, 11], [6, 648, 3], [12, 972, 2]], 'center_label': '1.1', 'center_order': 1, 'central_product': False, 'central_quotient': '11664.be', 'commutator_count': 1, 'commutator_label': '1458.1576', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '3.1', '3.1', '3.1', '3.1', '3.1', '3.1'], 'composition_length': 10, 'conjugacy_classes_known': True, 'counter': 31, 'cyclic': False, 'derived_length': 4, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 18, 1, 1], [2, 54, 1, 2], [2, 81, 1, 3], [2, 162, 1, 1], [3, 2, 1, 2], [3, 4, 1, 1], [3, 12, 1, 2], [3, 24, 1, 3], [3, 24, 2, 1], [3, 36, 1, 2], [3, 72, 1, 3], [3, 144, 1, 2], [4, 486, 1, 2], [6, 36, 1, 4], [6, 108, 1, 4], [6, 162, 1, 2], [6, 216, 1, 3], [6, 216, 2, 1], [6, 324, 1, 11], [6, 648, 1, 3], [12, 972, 1, 2]], 'element_repr_type': 'Perm', 'elementary': 1, 'eulerian_function': 10886400, 'exponent': 12, 'exponents_of_order': [6, 4], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [[12, 1, 4], [24, 0, 4], [24, 1, 4]], 'familial': False, 'frattini_label': '9.2', 'frattini_quotient': '1296.3531', 'hash': 7343738506780516409, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 12, 'inner_gen_orders': [6, 6, 6], 'inner_gens': [[2979163906969437, 355969124720882, 3423209842160906], [287883912505, 1133847166193275, 6025763444117612], [2267594572320719, 21378879463921, 4536034458905765]], 'inner_hash': 7343738506780516409, 'inner_nilpotent': False, 'inner_order': 11664, 'inner_split': False, 'inner_tex': '\\He_3^2:(C_2\\times D_4)', 'inner_used': [1, 2, 3], 'irrC_degree': 12, 'irrQ_degree': 12, 'irrQ_dim': 12, 'irrR_degree': 12, 'irrep_stats': [[1, 8], [2, 2], [4, 16], [8, 8], [12, 4], [16, 2], [18, 8], [24, 8], [36, 2]], 'label': '11664.be', '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': 'He3^2:(C2*D4)', '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], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 34, 'number_characteristic_subgroups': 11, 'number_conjugacy_classes': 58, 'number_divisions': 56, 'number_normal_subgroups': 27, 'number_subgroup_autclasses': 763, 'number_subgroup_classes': 1628, 'number_subgroups': 112427, 'old_label': None, 'order': 11664, 'order_factorization_type': 33, 'order_stats': [[1, 1], [2, 531], [3, 728], [4, 972], [6, 7488], [12, 1944]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': True, 'outer_exponent': 2, 'outer_gen_orders': [2, 2], 'outer_gen_pows': [45603, 2644573625246156], 'outer_gens': [[2644399305049437, 1133928073837292, 4891634668788748], [2267688021454530, 2623369139072164, 4912644720337893]], 'outer_group': '4.2', 'outer_hash': 2, 'outer_nilpotent': True, 'outer_order': 4, 'outer_permdeg': 4, 'outer_perms': [1, 6], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^2', 'pc_rank': 8, 'perfect': False, 'permutation_degree': 18, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 2], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 8], [2, 2], [4, 16], [8, 8], [12, 4], [16, 2], [18, 8], [24, 4], [36, 2], [48, 2]], 'representations': {'PC': {'code': '41196294641100265800881159880475059411503112181062100982911664781652930497772374293407404963127778680167921714025426131939536282340447', 'gens': [1, 2, 4, 5, 7, 8, 9, 10], 'pres': [10, 2, 2, 2, 3, 2, 3, 3, 3, 3, 3, 93680, 59801, 51, 154682, 363203, 209453, 90903, 258404, 35614, 1524, 56884, 144, 84485, 197775, 1465, 645126, 75616, 118466, 57996, 19786, 9716, 69127, 5777, 43227, 7247, 2937, 291618, 9768, 32449]}, 'Perm': {'d': 18, 'gens': [4536034458905765, 1133847166193275, 2979163906969437]}}, 'schur_multiplier': [2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 2], 'solvability_type': 12, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': '\\He_3^2:(C_2\\times D_4)', 'transitive_degree': 18, 'wreath_data': None, 'wreath_product': False}
