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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': False, 'ambient': '7776.ga', 'ambient_counter': 157, 'ambient_order': 7776, 'ambient_tex': 'C_6^3:S_3^2', 'central': False, 'central_factor': False, 'centralizer_order': 2, 'characteristic': False, 'core_order': 18, 'counter': 572, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '7776.ga.72.fh1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': False, 'minimal': False, 'minimal_normal': False, 'nilpotent': False, 'normal': False, 'old_label': '72.fh1', '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': 72, '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': '108.39', 'subgroup_hash': 39, 'subgroup_order': 108, 'subgroup_tex': 'C_3:S_3^2', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '7776.ga', 'aut_centralizer_order': None, 'aut_label': '72.fh1', 'aut_quo_index': None, 'aut_stab_index': None, 'aut_weyl_group': None, 'aut_weyl_index': None, 'centralizer': '3888.a1', 'complements': None, 'conjugacy_class_count': 4, 'contained_in': ['18.cf1', '24.bl1', '36.ce1'], 'contains': ['144.bq1', '144.cc1', '144.cu1', '216.hz1', '216.ic1', '216.ie1'], 'core': '432.c1', 'coset_action_label': None, 'count': 48, 'diagramx': None, 'generators': [20923272667543, 311087, 482630400, 12454200704, 41852285962216], 'label': '7776.ga.72.fh1', 'mobius_quo': None, 'mobius_sub': 0, 'normal_closure': '1.a1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '12.t1', 'old_label': '72.fh1', 'projective_image': '7776.ga', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '72.fh1', 'subgroup_fusion': None, 'weyl_group': '324.116'}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '4.2', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 5, 'aut_exponent': 24, 'aut_gen_orders': [6, 6, 8], 'aut_gens': [[1, 5064, 144, 45360, 3], [1, 40584, 45504, 80640, 4], [2, 85944, 80880, 144, 4], [1, 85944, 80880, 240, 4]], 'aut_group': '2592.fv', 'aut_hash': 9019849488891309561, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 2592, 'aut_permdeg': 12, 'aut_perms': [218863681, 335679362, 336032905], 'aut_phi_ratio': 72.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 3, 1, 1], [2, 9, 1, 1], [2, 27, 1, 1], [3, 2, 1, 1], [3, 2, 4, 1], [3, 4, 4, 1], [6, 6, 4, 1], [6, 18, 1, 1]], 'aut_supersolvable': False, 'aut_tex': 'S_3\\times C_3^2:\\GL(2,3)', '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': 24, 'autcentquo_group': '2592.fv', 'autcentquo_hash': 9019849488891309561, 'autcentquo_nilpotent': False, 'autcentquo_order': 2592, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'S_3\\times C_3^2:\\GL(2,3)', 'cc_stats': [[1, 1, 1], [2, 3, 1], [2, 9, 1], [2, 27, 1], [3, 2, 5], [3, 4, 4], [6, 6, 4], [6, 18, 1]], 'center_label': '1.1', 'center_order': 1, 'central_product': True, 'central_quotient': '108.39', 'commutator_count': 1, 'commutator_label': '27.5', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '3.1', '3.1', '3.1'], 'composition_length': 5, 'conjugacy_classes_known': True, 'counter': 39, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['18.4', 1], ['6.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 3, 1, 1], [2, 9, 1, 1], [2, 27, 1, 1], [3, 2, 1, 5], [3, 4, 1, 4], [6, 6, 1, 4], [6, 18, 1, 1]], 'element_repr_type': 'Perm', 'elementary': 1, 'eulerian_function': 168, 'exponent': 6, 'exponents_of_order': [3, 2], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [], 'familial': False, 'frattini_label': '1.1', 'frattini_quotient': '108.39', 'hash': 39, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 6, 'inner_gen_orders': [2, 2, 3, 3, 3], 'inner_gens': [[1, 5064, 144, 45360, 4], [1, 5064, 240, 80640, 3], [1, 5304, 144, 45360, 3], [1, 85704, 144, 45360, 3], [5, 5064, 144, 45360, 3]], 'inner_hash': 39, 'inner_nilpotent': False, 'inner_order': 108, 'inner_split': False, 'inner_tex': 'C_3:S_3^2', 'inner_used': [1, 2, 3, 4, 5], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 4], [2, 10], [4, 4]], 'label': '108.39', 'linC_count': 72, 'linC_degree': 6, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 6, 'linQ_degree_count': 72, 'linQ_dim': 6, 'linQ_dim_count': 72, 'linR_count': 72, 'linR_degree': 6, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'C3:S3^2', 'ngens': 5, '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': 9, 'number_characteristic_subgroups': 10, 'number_conjugacy_classes': 18, 'number_divisions': 18, 'number_normal_subgroups': 22, 'number_subgroup_autclasses': 34, 'number_subgroup_classes': 76, 'number_subgroups': 368, 'old_label': None, 'order': 108, 'order_factorization_type': 32, 'order_stats': [[1, 1], [2, 39], [3, 26], [6, 42]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 12, 'outer_gen_orders': [2, 3, 2, 2], 'outer_gen_pows': [0, 0, 5064, 5064], 'outer_gens': [[1, 5064, 144, 80784, 3], [1, 5064, 80640, 80784, 3], [1, 5064, 45600, 45504, 3], [1, 5064, 80640, 144, 3]], 'outer_group': '24.12', 'outer_hash': 12, 'outer_nilpotent': False, 'outer_order': 24, 'outer_permdeg': 4, 'outer_perms': [2, 4, 16, 7], 'outer_solvable': True, 'outer_supersolvable': False, 'outer_tex': 'S_4', 'pc_rank': 4, 'perfect': False, 'permutation_degree': 9, 'pgroup': 0, 'primary_abelian_invariants': [2, 2], 'quasisimple': False, 'rank': 3, 'rational': True, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 10], [4, 4]], 'representations': {'PC': {'code': 73347871936139, 'gens': [1, 2, 3, 5], 'pres': [5, -2, -3, -2, -3, -3, 41, 452, 42, 483, 314]}, 'GLZ': {'b': 3, 'd': 6, 'gens': [125101736062737953, 58301509072600689, 58415879068608930]}, 'GLFp': {'d': 4, 'p': 3, 'gens': [11495158, 22274384, 5504967, 25627796, 20098688]}, 'Perm': {'d': 9, 'gens': [1, 5064, 144, 45360, 3]}}, 'schur_multiplier': [6], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2], 'solvability_type': 7, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_3: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': 3, 'aut_exponent': 36, 'aut_gen_orders': [6, 6, 6, 6, 12, 6], 'aut_gens': [[712689240229075, 43159484847303, 397533486881388], [377930346465315, 355687907215457, 1067068514991303], [397545462219356, 355693655386442, 712695467535424], [754540568127307, 43153736828103, 1108914091352354], [1067074742162792, 20923269036533, 377924123076313], [712694984609388, 1308157116257, 397539235206322], [712689240348035, 732303873321095, 397533490520275]], 'aut_group': None, 'aut_hash': 4764413270895111565, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 93312, 'aut_permdeg': 114, 'aut_perms': [445998467213430455183371950673501990163526867035329777687108546599505610297755195025631276613995081861702719720921436860047327675464526537470764226061167902733283590300237259780491231273, 2252636996628713525212768301492362197947675963172841745747889094713450384712286140761808055470561574407584924036494687027067479954663836914136868258919610000538285172978661150662308135883, 1859063581631455878972518128096985207862930050331940286768584485968783112605831182878799237150090460995096287547369413817262573763933292402162640954961546407416876774811968056508755106100, 2524192866062423721479573602262645478313627521259806838205910883318904166870960206137445311366380926573835198478803123064589601577786847859046086903388955901395369684222025470210129433963, 2214235725718500773294501335837705359139333232985304803309881249592774743863029083038421772053222411318467557638220630920682685502639893964363674948259538051501079302011162981202208269730, 1244827525649229859888805190936857756283050236461528196346232626574904176395743993702091841143879642202756948347052669588352513016630695482541867782200135892576931995468547033913782714920], 'aut_phi_ratio': 36.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 3, 1, 2], [2, 3, 2, 1], [2, 9, 2, 1], [2, 54, 2, 1], [2, 162, 2, 1], [3, 2, 1, 2], [3, 3, 2, 1], [3, 4, 1, 1], [3, 6, 2, 1], [3, 6, 3, 1], [3, 12, 3, 1], [3, 72, 1, 1], [3, 144, 1, 1], [4, 54, 2, 1], [4, 162, 2, 1], [6, 2, 1, 2], [6, 3, 2, 3], [6, 4, 1, 1], [6, 6, 1, 4], [6, 6, 2, 6], [6, 6, 3, 3], [6, 6, 6, 2], [6, 9, 4, 2], [6, 12, 1, 2], [6, 12, 2, 2], [6, 12, 3, 3], [6, 12, 6, 2], [6, 18, 2, 1], [6, 18, 4, 1], [6, 18, 6, 2], [6, 18, 12, 1], [6, 54, 4, 1], [6, 72, 1, 1], [6, 108, 2, 1], [6, 108, 4, 1], [6, 144, 1, 1], [6, 162, 4, 1], [6, 216, 2, 1], [9, 72, 2, 1], [9, 144, 2, 1], [12, 54, 4, 1], [12, 108, 2, 1], [12, 108, 4, 1], [12, 162, 4, 1], [18, 72, 2, 1], [18, 144, 2, 1], [18, 216, 4, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_2\\times C_6^2.C_3^4.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': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^2', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 36, 'autcentquo_group': None, 'autcentquo_hash': 2074683354935437783, 'autcentquo_nilpotent': False, 'autcentquo_order': 23328, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'C_6^2.C_3^4.C_2^3', 'cc_stats': [[1, 1, 1], [2, 1, 1], [2, 3, 4], [2, 9, 2], [2, 54, 2], [2, 162, 2], [3, 2, 2], [3, 3, 2], [3, 4, 1], [3, 6, 5], [3, 12, 3], [3, 72, 1], [3, 144, 1], [4, 54, 2], [4, 162, 2], [6, 2, 2], [6, 3, 6], [6, 4, 1], [6, 6, 37], [6, 9, 8], [6, 12, 27], [6, 18, 30], [6, 54, 4], [6, 72, 1], [6, 108, 6], [6, 144, 1], [6, 162, 4], [6, 216, 2], [9, 72, 2], [9, 144, 2], [12, 54, 4], [12, 108, 6], [12, 162, 4], [18, 72, 2], [18, 144, 2], [18, 216, 4]], 'center_label': '2.1', 'center_order': 2, 'central_product': True, 'central_quotient': '3888.gu', 'commutator_count': 2, 'commutator_label': '972.590', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '2.1', '3.1', '3.1', '3.1', '3.1', '3.1'], 'composition_length': 10, 'conjugacy_classes_known': True, 'counter': 157, 'cyclic': False, 'derived_length': 3, 'dihedral': False, 'direct_factorization': [['2.1', 1], ['6.1', 1], ['648.281', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 3, 1, 4], [2, 9, 1, 2], [2, 54, 1, 2], [2, 162, 1, 2], [3, 2, 1, 2], [3, 3, 2, 1], [3, 4, 1, 1], [3, 6, 1, 3], [3, 6, 2, 1], [3, 12, 1, 3], [3, 72, 1, 1], [3, 144, 1, 1], [4, 54, 1, 2], [4, 162, 1, 2], [6, 2, 1, 2], [6, 3, 2, 3], [6, 4, 1, 1], [6, 6, 1, 15], [6, 6, 2, 11], [6, 9, 2, 4], [6, 12, 1, 11], [6, 12, 2, 8], [6, 18, 1, 14], [6, 18, 2, 8], [6, 54, 2, 2], [6, 72, 1, 1], [6, 108, 1, 2], [6, 108, 2, 2], [6, 144, 1, 1], [6, 162, 2, 2], [6, 216, 1, 2], [9, 72, 1, 2], [9, 144, 1, 2], [12, 54, 2, 2], [12, 108, 1, 2], [12, 108, 2, 2], [12, 162, 2, 2], [18, 72, 1, 2], [18, 144, 1, 2], [18, 216, 1, 4]], 'element_repr_type': 'Perm', 'elementary': 1, 'eulerian_function': 816480, 'exponent': 36, 'exponents_of_order': [5, 5], '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': '864.4692', 'hash': 1987921750283067703, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 36, 'inner_gen_orders': [12, 6, 12], 'inner_gens': [[712689240229075, 355687910908032, 397545940930865], [754540568255062, 43159484847303, 1108920797193982], [712694988240368, 1320611144934, 397533486881388]], 'inner_hash': 1441392995906330158, 'inner_nilpotent': False, 'inner_order': 3888, 'inner_split': True, 'inner_tex': 'S_3\\times C_3^3:S_4', 'inner_used': [1, 2, 3], 'irrC_degree': 12, 'irrQ_degree': 12, 'irrQ_dim': 12, 'irrR_degree': 12, 'irrep_stats': [[1, 8], [2, 20], [3, 40], [4, 8], [6, 80], [12, 30]], 'label': '7776.ga', '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': 'C6^3:S3^2', '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, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 72, 'number_characteristic_subgroups': 54, 'number_conjugacy_classes': 186, 'number_divisions': 138, 'number_normal_subgroups': 98, 'number_subgroup_autclasses': 1612, 'number_subgroup_classes': 3972, 'number_subgroups': 82438, 'old_label': None, 'order': 7776, 'order_factorization_type': 33, 'order_stats': [[1, 1], [2, 463], [3, 296], [4, 432], [6, 3344], [9, 432], [12, 1512], [18, 1296]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': True, 'outer_exponent': 6, 'outer_gen_orders': [2, 2, 6], 'outer_gen_pows': [0, 0, 0], 'outer_gens': [[712689240226865, 43159485013151, 397533486899356], [712689240229075, 43159481218503, 397533490510188], [397545462051949, 1320611206411, 712683009569388]], 'outer_group': '24.15', 'outer_hash': 15, 'outer_nilpotent': True, 'outer_order': 24, 'outer_permdeg': 9, 'outer_perms': [24, 40320, 723], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^2\\times C_6', 'pc_rank': None, '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, 20], [3, 8], [4, 8], [6, 48], [12, 38], [24, 8]], 'representations': {'PC': {'code': '10023141228343465678174942243459781929434705403130831641419086306831607147172956141189781009756844335507721408568485246300543654751418', 'gens': [1, 2, 4, 6, 8, 10], 'pres': [10, 2, 2, 3, 2, 3, 3, 3, 2, 3, 2, 47040, 151401, 51, 84482, 201123, 110653, 32063, 113, 196804, 150614, 3624, 494, 252005, 188655, 9385, 2555, 765, 235, 30246, 552967, 48997, 21647, 7257, 237, 136088, 22718, 194409, 16239, 24349, 8159]}, 'Perm': {'d': 18, 'gens': [712689240229075, 43159484847303, 397533486881388]}}, 'schur_multiplier': [2, 2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 2], 'solvability_type': 11, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': 'C_6^3:S_3^2', 'transitive_degree': 36, 'wreath_data': None, 'wreath_product': False}
