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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': False, 'ambient': '15552.il', 'ambient_counter': 220, 'ambient_order': 15552, 'ambient_tex': '(C_3\\times S_3^3):S_4', 'central': False, 'central_factor': False, 'centralizer_order': 6, 'characteristic': False, 'core_order': 81, 'counter': 108, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '15552.il.24.k1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': False, 'minimal': False, 'minimal_normal': False, 'nilpotent': False, 'normal': False, 'old_label': '24.k1', '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': 24, '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': '648.651', 'subgroup_hash': 651, 'subgroup_order': 648, 'subgroup_tex': 'S_3\\times C_3^2:C_{12}', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '15552.il', 'aut_centralizer_order': None, 'aut_label': '24.k1', 'aut_quo_index': None, 'aut_stab_index': None, 'aut_weyl_group': None, 'aut_weyl_index': None, 'centralizer': '2592.g1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['12.j1', '12.l1', '12.x1'], 'contains': ['48.d1', '48.m1', '48.n1', '72.i1', '72.bo1', '72.bp1', '72.bq1', '72.br1'], 'core': '192.a1', 'coset_action_label': None, 'count': 6, 'diagramx': None, 'generators': [106900861682, 10555323166080, 3, 40320, 268280812800, 7707571200, 275505755280], 'label': '15552.il.24.k1', 'mobius_quo': None, 'mobius_sub': 0, 'normal_closure': '1.a1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '6.g1', 'old_label': '24.k1', 'projective_image': '15552.il', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '24.k1', 'subgroup_fusion': None, 'weyl_group': '432.759'}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '24.9', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 5, 'aut_exponent': 24, 'aut_gen_orders': [24, 6, 6, 8, 6, 2], 'aut_gens': [[1, 12, 36, 216], [79, 144, 480, 216], [89, 156, 36, 432], [151, 12, 408, 432], [149, 156, 492, 432], [17, 96, 558, 432], [1, 24, 396, 432]], 'aut_group': None, 'aut_hash': 2482002321052132324, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 20736, 'aut_permdeg': 26, 'aut_perms': [401497229440125960137659859, 113740121222837855672365890, 245771230994635823347270589, 78351475242382844578036425, 302375733768749281035012634, 385973377915696257583100458], 'aut_phi_ratio': 96.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 3, 2, 1], [3, 1, 2, 1], [3, 2, 1, 1], [3, 2, 2, 1], [3, 2, 4, 1], [3, 2, 8, 1], [3, 4, 4, 1], [3, 4, 8, 1], [4, 9, 2, 1], [4, 27, 2, 1], [6, 1, 2, 1], [6, 2, 1, 1], [6, 2, 2, 1], [6, 2, 4, 1], [6, 2, 8, 1], [6, 3, 4, 1], [6, 4, 4, 1], [6, 4, 8, 1], [6, 6, 8, 1], [6, 6, 16, 1], [12, 9, 4, 1], [12, 18, 2, 1], [12, 18, 4, 1], [12, 27, 4, 1]], 'aut_supersolvable': False, 'aut_tex': '\\PSU(3,2).D_6^2.C_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': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^3', '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, 1, 1], [2, 3, 2], [3, 1, 2], [3, 2, 15], [3, 4, 12], [4, 9, 2], [4, 27, 2], [6, 1, 2], [6, 2, 15], [6, 3, 4], [6, 4, 12], [6, 6, 24], [12, 9, 4], [12, 18, 6], [12, 27, 4]], 'center_label': '6.2', 'center_order': 6, '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', '2.1', '3.1', '3.1', '3.1', '3.1'], 'composition_length': 7, 'conjugacy_classes_known': True, 'counter': 651, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['3.1', 1], ['36.7', 1], ['6.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 3, 1, 2], [3, 1, 2, 1], [3, 2, 1, 5], [3, 2, 2, 5], [3, 4, 1, 4], [3, 4, 2, 4], [4, 9, 2, 1], [4, 27, 2, 1], [6, 1, 2, 1], [6, 2, 1, 5], [6, 2, 2, 5], [6, 3, 2, 2], [6, 4, 1, 4], [6, 4, 2, 4], [6, 6, 1, 8], [6, 6, 2, 8], [12, 9, 4, 1], [12, 18, 2, 1], [12, 18, 4, 1], [12, 27, 4, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 4368, 'exponent': 12, 'exponents_of_order': [4, 3], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [], 'familial': False, 'frattini_label': '2.1', 'frattini_quotient': '324.166', 'hash': 651, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 6, 'inner_gen_orders': [2, 3, 6, 3], 'inner_gens': [[1, 24, 180, 216], [25, 12, 36, 216], [73, 12, 36, 432], [1, 12, 468, 216]], '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], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 24], [2, 60], [4, 24]], 'label': '648.651', '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': True, 'metacyclic': False, 'monomial': True, 'name': 'S3*C3^2:C12', 'ngens': 7, '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], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 26, 'number_characteristic_subgroups': 28, 'number_conjugacy_classes': 108, 'number_divisions': 66, 'number_normal_subgroups': 92, 'number_subgroup_autclasses': 131, 'number_subgroup_classes': 394, 'number_subgroups': 1436, 'old_label': None, 'order': 648, 'order_factorization_type': 33, 'order_stats': [[1, 1], [2, 7], [3, 80], [4, 72], [6, 236], [12, 252]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': True, 'outer_exponent': 12, 'outer_gen_orders': [2, 2, 2, 2, 3, 2, 2], 'outer_gen_pows': [0, 0, 0, 0, 0, 1, 1], 'outer_gens': [[11, 168, 36, 432], [11, 24, 186, 216], [5, 24, 180, 432], [5, 12, 42, 216], [1, 156, 36, 216], [1, 96, 60, 216], [1, 72, 120, 216]], 'outer_group': '192.1537', 'outer_hash': 1537, 'outer_nilpotent': False, 'outer_order': 192, 'outer_permdeg': 10, 'outer_perms': [40447, 1, 289, 127, 45360, 367920, 806400], 'outer_solvable': True, 'outer_supersolvable': False, 'outer_tex': 'C_2^3\\times S_4', 'pc_rank': 4, 'perfect': False, 'permutation_degree': 16, 'pgroup': 0, 'primary_abelian_invariants': [2, 4, 3], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 24], [4, 29], [8, 9]], 'representations': {'PC': {'code': 1277856469629965371042924955, 'gens': [1, 4, 5, 7], 'pres': [7, -2, -2, -3, -3, -2, -3, -3, 14, 36, 675, 6304, 102, 6053, 622]}, 'GLZN': {'d': 2, 'p': 18, 'gens': [99161, 1746, 5839, 40831, 80956, 96397, 5995]}, 'Perm': {'d': 16, 'gens': [136, 87700843440, 325, 1037836800, 1394852704560, 2615348736000, 435]}}, 'schur_multiplier': [6], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 12], 'solvability_type': 7, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'S_3\\times C_3^2:C_{12}', 'transitive_degree': 72, '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': '4.2', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 4, 'aut_exponent': 36, 'aut_gen_orders': [6, 4, 6, 6, 12], 'aut_gens': [[100673799962, 88216128865, 1315469030521], [2704487345309, 2704966349281, 1332713088122], [187812173069, 456053069665, 8321745640345], [7935228365281, 7973112310829, 5873049907322], [7935707324882, 2704966349285, 9335864966666], [7966885288346, 7973112311525, 5873049909509]], 'aut_group': None, 'aut_hash': 4593880105280533620, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 186624, 'aut_permdeg': 126, 'aut_perms': [4179268539655598952197241908224876267026742279083108283636949922207882007255167957277022923082325276214641245586879913607434656296424683223625782021126395018772333093125777575141982372378982086976133457362774699, 8626751507956610982365969881076132231690499904589603562566112942060745965800132377547632358677756705513501889710240601031087053849777631894335562081720270242510892374597815748833904988591234883086793077916952384, 5038611013011433609460007553669033060750510052849396523153197017526693534480403442578451598120876721613088608982075245943212157458353338812564108944469315118229193889232547208161104043772954506635826320961345168, 11835095308255973980794503930665482737473293040867198773526046375116655462321836952899590642853654820658360617666940042198332023397035331554199762772161694983332790952400587411396484659525539140571566830292421679, 20118941268782373820268156871287690164355528880136368112661944993185765883533403303476489909163438875183490725585017411962432730614254469835782958438077523323815331143931557218306990194625445634992138012704040603], 'aut_phi_ratio': 36.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 3, 1, 1], [2, 9, 2, 1], [2, 18, 1, 1], [2, 27, 1, 1], [2, 27, 2, 1], [2, 54, 1, 1], [2, 81, 1, 1], [2, 108, 1, 1], [2, 324, 1, 1], [3, 2, 1, 1], [3, 6, 1, 1], [3, 8, 1, 1], [3, 12, 1, 2], [3, 16, 1, 1], [3, 24, 1, 1], [3, 288, 3, 1], [4, 108, 1, 1], [4, 324, 1, 1], [4, 324, 2, 1], [4, 972, 2, 1], [6, 6, 1, 2], [6, 12, 1, 3], [6, 12, 2, 1], [6, 18, 2, 2], [6, 24, 1, 3], [6, 24, 2, 1], [6, 36, 2, 3], [6, 36, 4, 2], [6, 48, 1, 1], [6, 54, 1, 1], [6, 54, 2, 3], [6, 72, 1, 1], [6, 72, 2, 2], [6, 108, 1, 1], [6, 108, 2, 2], [6, 162, 1, 1], [6, 216, 1, 2], [6, 432, 1, 1], [6, 648, 1, 1], [6, 864, 3, 1], [9, 576, 3, 1], [12, 216, 1, 2], [12, 432, 1, 1], [12, 648, 1, 1], [12, 648, 2, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_3^4.C_2^4.D_6^2', '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': 36, 'autcentquo_group': None, 'autcentquo_hash': 4593880105280533620, 'autcentquo_nilpotent': False, 'autcentquo_order': 186624, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'C_3^4.C_2^4.D_6^2', 'cc_stats': [[1, 1, 1], [2, 3, 1], [2, 9, 2], [2, 18, 1], [2, 27, 3], [2, 54, 1], [2, 81, 1], [2, 108, 1], [2, 324, 1], [3, 2, 1], [3, 6, 1], [3, 8, 1], [3, 12, 2], [3, 16, 1], [3, 24, 1], [3, 288, 3], [4, 108, 1], [4, 324, 3], [4, 972, 2], [6, 6, 2], [6, 12, 5], [6, 18, 4], [6, 24, 5], [6, 36, 14], [6, 48, 1], [6, 54, 7], [6, 72, 5], [6, 108, 5], [6, 162, 1], [6, 216, 2], [6, 432, 1], [6, 648, 1], [6, 864, 3], [9, 576, 3], [12, 216, 2], [12, 432, 1], [12, 648, 3]], 'center_label': '1.1', 'center_order': 1, 'central_product': False, 'central_quotient': '15552.il', 'commutator_count': 2, 'commutator_label': '3888.ga', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '2.1', '2.1', '3.1', '3.1', '3.1', '3.1', '3.1'], 'composition_length': 11, 'conjugacy_classes_known': True, 'counter': 220, 'cyclic': False, 'derived_length': 4, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 3, 1, 1], [2, 9, 1, 2], [2, 18, 1, 1], [2, 27, 1, 3], [2, 54, 1, 1], [2, 81, 1, 1], [2, 108, 1, 1], [2, 324, 1, 1], [3, 2, 1, 1], [3, 6, 1, 1], [3, 8, 1, 1], [3, 12, 1, 2], [3, 16, 1, 1], [3, 24, 1, 1], [3, 288, 1, 3], [4, 108, 1, 1], [4, 324, 1, 3], [4, 972, 1, 2], [6, 6, 1, 2], [6, 12, 1, 3], [6, 12, 2, 1], [6, 18, 1, 2], [6, 18, 2, 1], [6, 24, 1, 3], [6, 24, 2, 1], [6, 36, 1, 6], [6, 36, 2, 4], [6, 48, 1, 1], [6, 54, 1, 5], [6, 54, 2, 1], [6, 72, 1, 3], [6, 72, 2, 1], [6, 108, 1, 3], [6, 108, 2, 1], [6, 162, 1, 1], [6, 216, 1, 2], [6, 432, 1, 1], [6, 648, 1, 1], [6, 864, 1, 3], [9, 576, 1, 3], [12, 216, 1, 2], [12, 432, 1, 1], [12, 648, 1, 3]], 'element_repr_type': 'Perm', 'elementary': 1, 'eulerian_function': 6420960, 'exponent': 36, 'exponents_of_order': [6, 5], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [[12, 0, 10], [12, 1, 2], [24, 0, 4], [24, 1, 2], [48, 1, 1]], 'familial': False, 'frattini_label': '1.1', 'frattini_quotient': '15552.il', 'hash': 3320700848296769751, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 36, 'inner_gen_orders': [4, 4, 6], 'inner_gens': [[100673799962, 456053070965, 2879359218005], [437328461069, 88216128865, 2966461303705], [4012161673685, 4041380772265, 1315469030521]], 'inner_hash': 3320700848296769751, 'inner_nilpotent': False, 'inner_order': 15552, 'inner_split': True, 'inner_tex': '(C_3\\times S_3^3):S_4', 'inner_used': [1, 2, 3], 'irrC_degree': 12, 'irrQ_degree': 12, 'irrQ_dim': 12, 'irrR_degree': 12, 'irrep_stats': [[1, 4], [2, 8], [3, 12], [6, 20], [8, 2], [12, 30], [16, 4], [24, 12], [48, 1]], 'label': '15552.il', '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': '(C3*S3^3):S4', '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, 0, 0, 0, 0, 0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 62, 'number_characteristic_subgroups': 21, 'number_conjugacy_classes': 93, 'number_divisions': 83, 'number_normal_subgroups': 35, 'number_subgroup_autclasses': 2386, 'number_subgroup_classes': 3451, 'number_subgroups': 210344, 'old_label': None, 'order': 15552, 'order_factorization_type': 33, 'order_stats': [[1, 1], [2, 687], [3, 944], [4, 3024], [6, 6360], [9, 1728], [12, 2808]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': True, 'outer_exponent': 6, 'outer_gen_orders': [2, 6], 'outer_gen_pows': [0, 0], 'outer_gens': [[100673799266, 88216130165, 1315469032705], [88139925362, 88216128869, 4222000702106]], 'outer_group': '12.4', 'outer_hash': 4, 'outer_nilpotent': False, 'outer_order': 12, 'outer_permdeg': 5, 'outer_perms': [7, 49], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'D_6', 'pc_rank': 6, 'perfect': False, 'permutation_degree': 16, 'pgroup': 0, 'primary_abelian_invariants': [2, 2], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 8], [3, 12], [6, 16], [8, 2], [12, 20], [16, 4], [24, 14], [48, 3]], 'representations': {'PC': {'code': '38981109283145851961651476249235157782143888774474234528586998810502906643116499503616721435911454261853465366338754383532034374478674324320104742204265281495045145712981', 'gens': [1, 2, 4, 6, 8, 10], 'pres': [11, 2, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 106524, 308309, 56, 140450, 40659, 138878, 118165, 124, 2644, 130685, 231676, 174069, 2018, 192, 66534, 33281, 99820, 1887, 811015, 114066, 202781, 15880, 2702, 260, 28520, 256627, 7158, 14297, 2439, 506900, 59431, 19864, 328, 17445, 52304, 17489]}, 'Perm': {'d': 16, 'gens': [100673799962, 88216128865, 1315469030521]}}, 'schur_multiplier': [2, 2, 2, 6], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2], 'solvability_type': 11, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': '(C_3\\times S_3^3):S_4', 'transitive_degree': 36, 'wreath_data': None, 'wreath_product': False}
