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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': True, 'ambient': '972.357', 'ambient_counter': 357, 'ambient_order': 972, 'ambient_tex': '(C_3^2\\times C_9):C_{12}', 'central': False, 'central_factor': False, 'centralizer_order': 54, 'characteristic': False, 'core_order': 6, 'counter': 100, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '972.357.54.f1.c1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': False, 'old_label': '54.f1.c1', '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': 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': '18.5', 'subgroup_hash': 5, 'subgroup_order': 18, 'subgroup_tex': 'C_3\\times C_6', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '972.357', 'aut_centralizer_order': 162, 'aut_label': '54.f1', 'aut_quo_index': None, 'aut_stab_index': 54, 'aut_weyl_group': '12.4', 'aut_weyl_index': 8748, 'centralizer': '18.e1.a1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['18.e1.a1'], 'contains': ['108.f1.c1', '162.a1.c1', '162.f1.a1', '162.f1.b1', '162.f1.c1'], 'core': '162.a1.c1', 'coset_action_label': None, 'count': 18, 'diagramx': [1296, -1, 1880, -1, 1086, -1, 703, -1], 'generators': [6, 760, 660], 'label': '972.357.54.f1.c1', 'mobius_quo': None, 'mobius_sub': 0, 'normal_closure': '2.a1.a1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '18.e1.a1', 'old_label': '54.f1.c1', 'projective_image': '486.173', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '54.f1.c1', 'subgroup_fusion': None, 'weyl_group': '1.1'}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': True, 'abelian_quotient': '18.5', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 4, 'aut_exponent': 24, 'aut_gen_orders': [2, 3], 'aut_gens': [[1, 3], [1, 17], [7, 3]], 'aut_group': '48.29', 'aut_hash': 29, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 48, 'aut_permdeg': 8, 'aut_perms': [475, 23888], 'aut_phi_ratio': 8.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [3, 1, 8, 1], [6, 1, 8, 1]], 'aut_supersolvable': False, 'aut_tex': '\\GL(2,3)', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 24, 'autcent_group': '48.29', 'autcent_hash': 29, 'autcent_nilpotent': False, 'autcent_order': 48, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': False, 'autcent_tex': '\\GL(2,3)', 'autcentquo_abelian': True, 'autcentquo_cyclic': True, 'autcentquo_exponent': 1, 'autcentquo_group': '1.1', 'autcentquo_hash': 1, 'autcentquo_nilpotent': True, 'autcentquo_order': 1, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'C_1', 'cc_stats': [[1, 1, 1], [2, 1, 1], [3, 1, 8], [6, 1, 8]], 'center_label': '18.5', 'center_order': 18, 'central_product': True, 'central_quotient': '1.1', 'commutator_count': 0, 'commutator_label': '1.1', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '3.1', '3.1'], 'composition_length': 3, 'conjugacy_classes_known': True, 'counter': 5, 'cyclic': False, 'derived_length': 1, 'dihedral': False, 'direct_factorization': [['2.1', 1], ['3.1', 2]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [3, 1, 2, 4], [6, 1, 2, 4]], 'element_repr_type': 'PC', 'elementary': 3, 'eulerian_function': 3, 'exponent': 6, 'exponents_of_order': [2, 1], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [], 'familial': False, 'frattini_label': '1.1', 'frattini_quotient': '18.5', 'hash': 5, 'hyperelementary': 3, 'inner_abelian': True, 'inner_cyclic': True, 'inner_exponent': 1, 'inner_gen_orders': [1, 1], 'inner_gens': [[1, 3], [1, 3]], 'inner_hash': 1, 'inner_nilpotent': True, 'inner_order': 1, 'inner_split': True, 'inner_tex': 'C_1', 'inner_used': [], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 18]], 'label': '18.5', 'linC_count': 72, 'linC_degree': 2, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 4, 'linQ_degree_count': 18, 'linQ_dim': 4, 'linQ_dim_count': 18, 'linR_count': 18, 'linR_degree': 4, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C3*C6', 'ngens': 3, 'nilpotency_class': 1, 'nilpotent': True, 'normal_counts': [0, 0, 0, 0, 0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 4, 'number_characteristic_subgroups': 4, 'number_conjugacy_classes': 18, 'number_divisions': 10, 'number_normal_subgroups': 12, 'number_subgroup_autclasses': 6, 'number_subgroup_classes': 12, 'number_subgroups': 12, 'old_label': None, 'order': 18, 'order_factorization_type': 22, 'order_stats': [[1, 1], [2, 1], [3, 8], [6, 8]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 24, 'outer_gen_orders': [2, 3], 'outer_gen_pows': [0, 0], 'outer_gens': [[1, 17], [7, 3]], 'outer_group': '48.29', 'outer_hash': 29, 'outer_nilpotent': False, 'outer_order': 48, 'outer_permdeg': 8, 'outer_perms': [475, 23888], 'outer_solvable': True, 'outer_supersolvable': False, 'outer_tex': '\\GL(2,3)', 'pc_rank': 2, 'perfect': False, 'permutation_degree': 8, 'pgroup': 0, 'primary_abelian_invariants': [2, 3, 3], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 2], [2, 8]], 'representations': {'PC': {'code': 277, 'gens': [1, 2], 'pres': [3, -3, -2, -3, 16]}, 'GLZ': {'b': 3, 'd': 4, 'gens': [35931151, 16858245]}, 'GLFp': {'d': 2, 'p': 7, 'gens': [1030, 1374]}, 'Perm': {'d': 8, 'gens': [5040, 240, 4]}}, 'schur_multiplier': [3], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [3, 6], 'solvability_type': 1, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_3\\times C_6', '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': '12.2', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 3, 'aut_exponent': 18, 'aut_gen_orders': [9, 6, 6, 6, 6], 'aut_gens': [[1, 12, 36, 108], [457, 660, 36, 432], [385, 24, 360, 852], [655, 660, 36, 780], [13, 660, 720, 900], [407, 660, 396, 804]], 'aut_group': None, 'aut_hash': 3048360342115462777, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 104976, 'aut_permdeg': 180, 'aut_perms': [193500830048752219480935448860758409847578856428418454189417610290799890860149868904748501610875895800245407263992743507421632749924549826516005550499021244884187526166378628250119568421230843209748950422289133085457922978471712431630210036568170030580967537824324723161222401664143975127866740767656173077100249165094726157043181, 168581476071801977713958686135745574542755946950870110397358279043213260730052051790668735298204466365743604416409672289761813964007171937390814416836289649829221451871461342835756639967194746942963620263722743258861418318223195227896575196707764683462576229206381248596922983320747467747789953696754123888520172011163267538758027, 157375352766790715119024784800347077468541494421893440945167622782554274848132029163895182498120042774635417748565773017562902235453242678480468315617212032964399996448916376414894936027038393709475732567974193799442455581196247755661602317637461246606659281589398792204904960573141078525401633813296653982484627095947690418123013, 121693383789126622283274086217949590934153299190904492745438827173985394943676775487086995823957879954903050206729656778096583120203046836994635486407877580643305245715349342307070702985961605445676563484256117804795250931687918723368749121324077694466303580394302215212117721940223139125213913078321151126128123785578804493698135, 25783159682655908228336304556296726586524541174750037280152029162856275286532387320906766354110338511333264630266532229572978717299689947193495853864131057821095796311240517439911983963032635082015515833634265446944984279006443596266682733494045119787385916658992963203510893450109628936495423438001386819745806729564802952076250], 'aut_phi_ratio': 324.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [3, 2, 1, 1], [3, 2, 3, 1], [3, 6, 1, 1], [3, 6, 2, 1], [3, 9, 2, 1], [3, 18, 2, 1], [3, 18, 6, 1], [4, 81, 2, 1], [6, 2, 1, 1], [6, 2, 3, 1], [6, 6, 1, 1], [6, 6, 2, 1], [6, 9, 2, 1], [6, 18, 2, 1], [6, 18, 6, 1], [9, 6, 9, 1], [12, 81, 4, 1], [18, 6, 9, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_3.C_3^5.D_6^2', 'autcent_abelian': True, 'autcent_cyclic': True, 'autcent_exponent': 2, 'autcent_group': '2.1', 'autcent_hash': 1, 'autcent_nilpotent': True, 'autcent_order': 2, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 18, 'autcentquo_group': '52488.sz', 'autcentquo_hash': 4618096886160246644, 'autcentquo_nilpotent': False, 'autcentquo_order': 52488, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'C_3^5.S_3^3', 'cc_stats': [[1, 1, 1], [2, 1, 1], [3, 2, 4], [3, 6, 3], [3, 9, 2], [3, 18, 8], [4, 81, 2], [6, 2, 4], [6, 6, 3], [6, 9, 2], [6, 18, 8], [9, 6, 9], [12, 81, 4], [18, 6, 9]], 'center_label': '2.1', 'center_order': 2, 'central_product': False, 'central_quotient': '486.173', 'commutator_count': 1, 'commutator_label': '81.11', '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'], 'composition_length': 7, 'conjugacy_classes_known': True, 'counter': 357, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [3, 2, 1, 4], [3, 6, 1, 3], [3, 9, 2, 1], [3, 18, 2, 4], [4, 81, 2, 1], [6, 2, 1, 4], [6, 6, 1, 3], [6, 9, 2, 1], [6, 18, 2, 4], [9, 6, 3, 3], [12, 81, 4, 1], [18, 6, 3, 3]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 4368, 'exponent': 36, 'exponents_of_order': [5, 2], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [], 'familial': False, 'frattini_label': '18.5', 'frattini_quotient': '54.13', 'hash': 357, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 18, 'inner_gen_orders': [6, 3, 3, 9], 'inner_gens': [[1, 24, 720, 576], [25, 12, 36, 108], [397, 12, 36, 108], [613, 12, 36, 108]], 'inner_hash': 173, 'inner_nilpotent': False, 'inner_order': 486, 'inner_split': True, 'inner_tex': '(C_3^2\\times C_9):C_6', 'inner_used': [1, 2, 3, 4], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 12], [2, 24], [6, 24]], 'label': '972.357', 'linC_count': 243, 'linC_degree': 8, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 20, 'linQ_degree_count': 27, 'linQ_dim': 22, 'linQ_dim_count': 27, 'linR_count': 135, 'linR_degree': 10, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': '(C3^2*C9):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], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 20, 'number_characteristic_subgroups': 20, 'number_conjugacy_classes': 60, 'number_divisions': 34, 'number_normal_subgroups': 42, 'number_subgroup_autclasses': 94, 'number_subgroup_classes': 164, 'number_subgroups': 1484, 'old_label': None, 'order': 972, 'order_factorization_type': 32, 'order_stats': [[1, 1], [2, 1], [3, 188], [4, 162], [6, 188], [9, 54], [12, 324], [18, 54]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 6, 'outer_gen_orders': [2, 3, 6, 6], 'outer_gen_pows': [0, 0, 0, 0], 'outer_gens': [[5, 12, 360, 588], [1, 12, 36, 132], [5, 24, 36, 540], [11, 336, 72, 132]], 'outer_group': '216.113', 'outer_hash': 113, 'outer_nilpotent': False, 'outer_order': 216, 'outer_permdeg': 13, 'outer_perms': [3999720376, 3991736280, 2410047376, 642822503], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_6^2:S_3', 'pc_rank': 4, 'perfect': False, 'permutation_degree': 34, 'pgroup': 0, 'primary_abelian_invariants': [4, 3], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 2], [2, 11], [4, 9], [6, 6], [18, 6]], 'representations': {'PC': {'code': 245730950302216648677651029677626309972383, 'gens': [1, 4, 5, 6], 'pres': [7, -2, -2, -3, -3, -3, 3, -3, 14, 36, 675, 25204, 11981, 3168, 24197, 9840, 5311, 166, 31758]}, 'Perm': {'d': 34, 'gens': [840853835740061812372432980916183809, 1110687348321604532385217834573115520, 16, 10032927234699328770515468081781697920, 840, 18470435429862645724585328196109455360, 27696690635106531822232825510982265600]}}, 'schur_multiplier': [3, 3], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [12], 'solvability_type': 7, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': '(C_3^2\\times C_9):C_{12}', 'transitive_degree': 324, 'wreath_data': None, 'wreath_product': False}
