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gps_subgroup_search • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'ambient': '7776.jv', 'ambient_counter': 256, 'ambient_order': 7776, 'ambient_tex': 'C_6^3:S_3^2', 'central': False, 'central_factor': False, 'centralizer_order': 4, 'characteristic': False, 'core_order': 24, 'counter': 375, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '7776.jv.54.by1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': False, 'minimal': False, 'minimal_normal': False, 'nilpotent': False, 'normal': False, 'old_label': '54.by1', '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': 54, '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': '144.46', 'subgroup_hash': 46, 'subgroup_order': 144, 'subgroup_tex': 'C_{18}:D_4', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '7776.jv', 'aut_centralizer_order': None, 'aut_label': '54.by1', 'aut_quo_index': None, 'aut_stab_index': None, 'aut_weyl_group': None, 'aut_weyl_index': None, 'centralizer': '1944.b1', 'complements': None, 'conjugacy_class_count': 2, 'contained_in': ['18.bk1', '18.bq1', '27.e1'], 'contains': ['108.w1', '108.be1', '108.bm1', '108.de1', '108.fc1', '162.bc1'], 'core': '324.a1', 'coset_action_label': None, 'count': 18, 'diagramx': [8312, -1, 4610, -1], 'generators': [871, 2592, 4632, 216, 252, 3888], 'label': '7776.jv.54.by1', 'mobius_quo': None, 'mobius_sub': 0, 'normal_closure': '2.c1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '9.c1', 'old_label': '54.by1', 'projective_image': '3888.jh', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '54.by1', 'subgroup_fusion': None, 'weyl_group': '216.111'}
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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': 2, 'aut_exponent': 36, 'aut_gen_orders': [2, 2, 4, 6, 18], 'aut_gens': [[1, 2, 8], [1, 78, 8], [1, 2, 12], [73, 6, 12], [1, 2, 88], [1, 118, 8]], 'aut_group': '1728.18843', 'aut_hash': 18843, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 1728, 'aut_permdeg': 17, 'aut_perms': [82, 18498, 27978, 2890814941009, 23719803621916], 'aut_phi_ratio': 36.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 1, 2, 1], [2, 2, 2, 1], [2, 18, 2, 1], [3, 2, 1, 1], [4, 18, 2, 1], [6, 2, 1, 1], [6, 2, 2, 1], [6, 2, 4, 1], [9, 2, 3, 1], [18, 2, 3, 1], [18, 2, 6, 1], [18, 2, 12, 1]], 'aut_supersolvable': True, 'aut_tex': 'D_{18}:(C_6\\times D_4)', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 4, 'autcent_group': '32.27', 'autcent_hash': 27, 'autcent_nilpotent': True, 'autcent_order': 32, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^2\\wr C_2', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 18, 'autcentquo_group': '54.6', 'autcentquo_hash': 6, 'autcentquo_nilpotent': False, 'autcentquo_order': 54, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'C_9:C_6', 'cc_stats': [[1, 1, 1], [2, 1, 3], [2, 2, 2], [2, 18, 2], [3, 2, 1], [4, 18, 2], [6, 2, 7], [9, 2, 3], [18, 2, 21]], 'center_label': '4.2', 'center_order': 4, 'central_product': True, 'central_quotient': '36.4', 'commutator_count': 1, 'commutator_label': '18.2', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '3.1', '3.1'], 'composition_length': 6, 'conjugacy_classes_known': True, 'counter': 46, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['2.1', 1], ['72.8', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [2, 2, 1, 2], [2, 18, 1, 2], [3, 2, 1, 1], [4, 18, 1, 2], [6, 2, 1, 3], [6, 2, 2, 2], [9, 2, 3, 1], [18, 2, 3, 3], [18, 2, 6, 2]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 504, 'exponent': 36, 'exponents_of_order': [4, 2], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [], 'familial': False, 'frattini_label': '6.2', 'frattini_quotient': '24.14', 'hash': 46, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 18, 'inner_gen_orders': [2, 2, 9], 'inner_gens': [[1, 6, 8], [5, 2, 136], [1, 18, 8]], 'inner_hash': 4, 'inner_nilpotent': False, 'inner_order': 36, 'inner_split': False, 'inner_tex': 'D_{18}', 'inner_used': [1, 2, 3], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 8], [2, 34]], 'label': '144.46', 'linC_count': 48, 'linC_degree': 3, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 8, 'linQ_degree_count': 4, 'linQ_dim': 8, 'linQ_dim_count': 4, 'linR_count': 12, 'linR_degree': 4, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'C18:D4', 'ngens': 6, 'nilpotency_class': -1, 'nilpotent': False, 'normal_counts': [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': 14, 'number_characteristic_subgroups': 15, 'number_conjugacy_classes': 42, 'number_divisions': 22, 'number_normal_subgroups': 35, 'number_subgroup_autclasses': 48, 'number_subgroup_classes': 81, 'number_subgroups': 295, 'old_label': None, 'order': 144, 'order_factorization_type': 32, 'order_stats': [[1, 1], [2, 43], [3, 2], [4, 36], [6, 14], [9, 6], [18, 42]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 12, 'outer_gen_orders': [2, 2, 12], 'outer_gen_pows': [0, 0, 0], 'outer_gens': [[5, 74, 8], [73, 74, 8], [73, 74, 108]], 'outer_group': '48.45', 'outer_hash': 45, 'outer_nilpotent': True, 'outer_order': 48, 'outer_permdeg': 9, 'outer_perms': [24, 41064, 46108], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_6\\times D_4', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 15, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 2], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 8], [2, 6], [4, 2], [6, 4], [12, 2]], 'representations': {'PC': {'code': 619359845302024408707, 'gens': [1, 2, 4], 'pres': [6, -2, -2, -2, -2, -3, -3, 73, 31, 1641, 69, 1930, 118, 1739]}, 'GLZN': {'d': 2, 'p': 36, 'gens': [47089, 1342214, 47305, 1656323, 46801, 70633]}, 'Perm': {'d': 15, 'gens': [14377673647, 289, 54, 288, 107304059520, 201348690480]}}, 'schur_multiplier': [2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 2], 'solvability_type': 7, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_{18}:D_4', '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': '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, 12, 6, 6, 6, 6], 'aut_gens': [[1, 2, 12, 72, 432], [6261, 6082, 12, 360, 5124], [4081, 3166, 5532, 6840, 5928], [6233, 3182, 204, 3960, 1260], [309, 7082, 5532, 6660, 3156], [337, 1114, 204, 4248, 3192], [297, 4630, 5340, 72, 540]], 'aut_group': None, 'aut_hash': 7651468306769606931, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 93312, 'aut_permdeg': 66, 'aut_perms': [282804845978339080447998836825540887372657346871730082974886972680973889057140642325878049121, 121007795160202893854916703756924213043111554533086731800184928167121612400955120387519332501, 482673924725391725892073720652976040729861930707244985926982058857702993190847837304357246138, 287733258868832890448205721610029257170397525207526725260783315324962278005891463301356307587, 5826757683221887958303106948850469750647768838114094101892796506040493455277639575507162434, 8370540617792243106604347529685743369225487857597954444505455729057124583241602701341988181], 'aut_phi_ratio': 36.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 3, 1, 2], [2, 6, 2, 1], [2, 27, 2, 1], [2, 81, 2, 1], [2, 162, 2, 1], [3, 2, 1, 1], [3, 3, 2, 1], [3, 6, 3, 1], [3, 8, 1, 1], [3, 16, 1, 1], [3, 18, 1, 1], [3, 24, 2, 1], [3, 48, 3, 1], [3, 144, 1, 1], [4, 6, 2, 1], [4, 162, 2, 1], [6, 2, 1, 1], [6, 3, 2, 1], [6, 6, 1, 2], [6, 6, 3, 1], [6, 8, 1, 1], [6, 9, 2, 2], [6, 12, 2, 1], [6, 16, 1, 1], [6, 18, 1, 1], [6, 18, 3, 2], [6, 18, 4, 1], [6, 24, 2, 1], [6, 27, 4, 1], [6, 36, 6, 1], [6, 48, 3, 1], [6, 54, 1, 2], [6, 81, 4, 1], [6, 108, 2, 1], [6, 144, 1, 1], [6, 162, 4, 1], [6, 216, 2, 1], [6, 216, 4, 1], [9, 18, 2, 1], [9, 144, 2, 1], [12, 12, 2, 1], [12, 18, 4, 1], [12, 36, 6, 1], [12, 108, 2, 1], [12, 162, 4, 1], [18, 18, 2, 1], [18, 54, 2, 2], [18, 108, 4, 1], [18, 144, 2, 1], [36, 108, 4, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_6^2.C_3^4.C_2^5', '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': 6340318746942157276, '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, 2], [2, 6, 2], [2, 27, 2], [2, 81, 2], [2, 162, 2], [3, 2, 1], [3, 3, 2], [3, 6, 3], [3, 8, 1], [3, 16, 1], [3, 18, 1], [3, 24, 2], [3, 48, 3], [3, 144, 1], [4, 6, 2], [4, 162, 2], [6, 2, 1], [6, 3, 2], [6, 6, 5], [6, 8, 1], [6, 9, 4], [6, 12, 2], [6, 16, 1], [6, 18, 11], [6, 24, 2], [6, 27, 4], [6, 36, 6], [6, 48, 3], [6, 54, 2], [6, 81, 4], [6, 108, 2], [6, 144, 1], [6, 162, 4], [6, 216, 6], [9, 18, 2], [9, 144, 2], [12, 12, 2], [12, 18, 4], [12, 36, 6], [12, 108, 2], [12, 162, 4], [18, 18, 2], [18, 54, 4], [18, 108, 4], [18, 144, 2], [36, 108, 4]], 'center_label': '2.1', 'center_order': 2, 'central_product': True, 'central_quotient': '3888.jh', 'commutator_count': 1, 'commutator_label': '972.525', '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': 256, 'cyclic': False, 'derived_length': 3, 'dihedral': False, 'direct_factorization': [['162.19', 1], ['2.1', 1], ['24.12', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 3, 1, 2], [2, 6, 1, 2], [2, 27, 1, 2], [2, 81, 1, 2], [2, 162, 1, 2], [3, 2, 1, 1], [3, 3, 2, 1], [3, 6, 1, 3], [3, 8, 1, 1], [3, 16, 1, 1], [3, 18, 1, 1], [3, 24, 2, 1], [3, 48, 1, 3], [3, 144, 1, 1], [4, 6, 1, 2], [4, 162, 1, 2], [6, 2, 1, 1], [6, 3, 2, 1], [6, 6, 1, 5], [6, 8, 1, 1], [6, 9, 2, 2], [6, 12, 1, 2], [6, 16, 1, 1], [6, 18, 1, 7], [6, 18, 2, 2], [6, 24, 2, 1], [6, 27, 2, 2], [6, 36, 1, 6], [6, 48, 1, 3], [6, 54, 1, 2], [6, 81, 2, 2], [6, 108, 1, 2], [6, 144, 1, 1], [6, 162, 2, 2], [6, 216, 1, 2], [6, 216, 2, 2], [9, 18, 1, 2], [9, 144, 1, 2], [12, 12, 1, 2], [12, 18, 2, 2], [12, 36, 1, 6], [12, 108, 1, 2], [12, 162, 2, 2], [18, 18, 1, 2], [18, 54, 1, 4], [18, 108, 1, 4], [18, 144, 1, 2], [36, 108, 1, 4]], 'element_repr_type': 'PC', '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': [[18, 1, 6]], 'familial': False, 'frattini_label': '9.2', 'frattini_quotient': '864.4691', 'hash': 271183728978066164, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 36, 'inner_gen_orders': [18, 6, 3, 6, 18], 'inner_gens': [[1, 7066, 2748, 6552, 432], [1157, 2, 5532, 3960, 1116], [5473, 5354, 12, 2664, 3168], [1297, 3890, 5196, 72, 3024], [1, 7598, 2892, 5256, 432]], 'inner_hash': 1865359641169939977, 'inner_nilpotent': False, 'inner_order': 3888, 'inner_split': True, 'inner_tex': 'C_3^3:S_3\\times S_4', 'inner_used': [1, 2, 3, 4], 'irrC_degree': 18, 'irrQ_degree': 18, 'irrQ_dim': 18, 'irrR_degree': 18, 'irrep_stats': [[1, 8], [2, 20], [3, 24], [4, 8], [6, 36], [9, 16], [12, 6], [18, 12]], 'label': '7776.jv', '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': 10, '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': 58, 'number_characteristic_subgroups': 47, 'number_conjugacy_classes': 130, 'number_divisions': 110, 'number_normal_subgroups': 89, 'number_subgroup_autclasses': 1266, 'number_subgroup_classes': 3002, 'number_subgroups': 80746, 'old_label': None, 'order': 7776, 'order_factorization_type': 33, 'order_stats': [[1, 1], [2, 559], [3, 404], [4, 336], [6, 3572], [9, 324], [12, 1176], [18, 972], [36, 432]], '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': [[5377, 290, 2652, 360, 744], [5377, 326, 2652, 360, 744], [3493, 190, 2940, 3960, 2160]], 'outer_group': '24.15', 'outer_hash': 15, 'outer_nilpotent': True, 'outer_order': 24, 'outer_permdeg': 9, 'outer_perms': [40320, 720, 27], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^2\\times C_6', 'pc_rank': 5, 'perfect': False, 'permutation_degree': 15, '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, 36], [12, 10], [18, 20]], 'representations': {'PC': {'code': '35624107197229684526166280597460147191393645246111738832437255190306504829994333559374208255098257646673493016784306764697113866966022409215048551410859', 'gens': [1, 2, 4, 6, 8], 'pres': [10, 2, 2, 3, 2, 3, 2, 3, 2, 3, 3, 17280, 141321, 51, 164402, 109923, 110653, 26063, 113, 145204, 4814, 65124, 393125, 118815, 63205, 13355, 13185, 175, 372966, 31116, 8026, 44657, 117387, 21157, 11087, 3417, 3187, 237, 77778, 77788, 28118, 3828, 7618, 2228, 358, 259219]}, 'Perm': {'d': 15, 'gens': [147, 274468100972, 3634290, 178685, 87657654961, 316805, 6706344954, 6227337842, 80784, 240]}}, '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': 54, 'wreath_data': None, 'wreath_product': False}
