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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': True, 'ambient': '1056.916', 'ambient_counter': 916, 'ambient_order': 1056, 'ambient_tex': 'D_{12}\\times D_{22}', 'central': False, 'central_factor': False, 'centralizer_order': 96, 'characteristic': False, 'core_order': 2, 'counter': 123, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '1056.916.264.d1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': False, 'old_label': '264.d1', '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': 264, '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': '4.2', 'subgroup_hash': 2, 'subgroup_order': 4, 'subgroup_tex': 'C_2^2', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '1056.916', 'aut_centralizer_order': 3840, 'aut_label': '264.d1', 'aut_quo_index': None, 'aut_stab_index': 22, 'aut_weyl_group': '2.1', 'aut_weyl_index': 84480, 'centralizer': '11.a1', 'complements': None, 'conjugacy_class_count': 2, 'contained_in': ['24.b1', '88.d1', '132.c1', '132.f1', '132.i1'], 'contains': ['528.a1', '528.d1'], 'core': '528.a1', 'coset_action_label': None, 'count': 22, 'diagramx': [4951, -1, 4442, -1], 'generators': [4, 528], 'label': '1056.916.264.d1', 'mobius_quo': None, 'mobius_sub': -24, 'normal_closure': '24.b1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '11.a1', 'old_label': '264.d1', 'projective_image': '528.164', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '264.d1', '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': '4.2', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 6, 'aut_gen_orders': [2, 3], 'aut_gens': [[1, 2], [3, 2], [2, 3]], 'aut_group': '6.1', 'aut_hash': 1, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 6, 'aut_permdeg': 3, 'aut_perms': [1, 4], 'aut_phi_ratio': 3.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 3, 1]], 'aut_supersolvable': True, 'aut_tex': 'S_3', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 6, 'autcent_group': '6.1', 'autcent_hash': 1, 'autcent_nilpotent': False, 'autcent_order': 6, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'S_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, 3]], 'center_label': '4.2', 'center_order': 4, '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', '2.1'], 'composition_length': 2, 'conjugacy_classes_known': True, 'counter': 2, 'cyclic': False, 'derived_length': 1, 'dihedral': True, 'direct_factorization': [['2.1', 2]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3]], 'element_repr_type': 'PC', 'elementary': 2, 'eulerian_function': 1, 'exponent': 2, 'exponents_of_order': [2], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2], 'faithful_reps': [], 'familial': True, 'frattini_label': '1.1', 'frattini_quotient': '4.2', 'hash': 2, 'hyperelementary': 2, 'inner_abelian': True, 'inner_cyclic': True, 'inner_exponent': 1, 'inner_gen_orders': [1, 1], 'inner_gens': [[1, 2], [1, 2]], '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, 4]], 'label': '4.2', 'linC_count': 3, 'linC_degree': 2, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 2, 'linQ_degree_count': 3, 'linQ_dim': 2, 'linQ_dim_count': 3, 'linR_count': 3, 'linR_degree': 2, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C2^2', 'ngens': 2, 'nilpotency_class': 1, 'nilpotent': True, 'normal_counts': [0, 0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 2, 'number_characteristic_subgroups': 2, 'number_conjugacy_classes': 4, 'number_divisions': 4, 'number_normal_subgroups': 5, 'number_subgroup_autclasses': 3, 'number_subgroup_classes': 5, 'number_subgroups': 5, 'old_label': None, 'order': 4, 'order_factorization_type': 2, 'order_stats': [[1, 1], [2, 3]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 6, 'outer_gen_orders': [2, 3], 'outer_gen_pows': [0, 0], 'outer_gens': [[3, 2], [2, 3]], 'outer_group': '6.1', 'outer_hash': 1, 'outer_nilpotent': False, 'outer_order': 6, 'outer_permdeg': 3, 'outer_perms': [1, 4], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'S_3', 'pc_rank': 2, 'perfect': False, 'permutation_degree': 4, 'pgroup': 2, 'primary_abelian_invariants': [2, 2], 'quasisimple': False, 'rank': 2, 'rational': True, 'rational_characters_known': True, 'ratrep_stats': [[1, 4]], 'representations': {'PC': {'code': 0, 'gens': [1, 2], 'pres': [2, -2, 2]}, 'GLZ': {'b': 3, 'd': 2, 'gens': [14, 12]}, 'GLFp': {'d': 2, 'p': 3, 'gens': [55, 56]}, 'Perm': {'d': 4, 'gens': [6, 1]}}, 'schur_multiplier': [2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2], 'solvability_type': 1, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_2^2', 'transitive_degree': 4, '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': '16.14', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 660, 'aut_gen_orders': [10, 20, 20, 12, 20, 60, 12], 'aut_gens': [[1, 2, 4, 8], [1, 3, 340, 825], [529, 267, 196, 105], [529, 794, 1013, 857], [529, 442, 964, 873], [529, 882, 869, 425], [1, 90, 485, 680], [529, 707, 485, 344]], 'aut_group': None, 'aut_hash': 4743563984708094474, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 168960, 'aut_permdeg': 68, 'aut_perms': [222117737880002627578960371226641255950008540172014669797833035720005553237155361507911876698809, 2187958811470004570521922012176519487471546574598853955169558720043997705272421980430827996230762, 365841284824882739151257510186286423978976352515530523082794568772844115119079246902482405392187, 1132823232234650890854789621718672453505583574886394877491085447681335158097295372816068552864528, 2113213540142045633935095638201218174166247916236242386769282371228639794164130980601285699112764, 74483136742922477408527921207911828358224881090179789704789860503721999395114333994595111659896, 2113690530149121218608517680739718747276967969032508867769004830718489249229750641474806942319117], 'aut_phi_ratio': 528.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 1, 2, 1], [2, 6, 4, 1], [2, 11, 4, 1], [2, 66, 4, 1], [3, 2, 1, 1], [4, 2, 2, 1], [4, 22, 2, 1], [6, 2, 1, 1], [6, 2, 2, 1], [6, 22, 4, 1], [11, 2, 5, 1], [12, 2, 4, 1], [12, 22, 4, 1], [22, 2, 5, 1], [22, 2, 10, 1], [22, 12, 20, 1], [33, 4, 5, 1], [44, 4, 10, 1], [66, 4, 5, 1], [66, 4, 10, 1], [132, 4, 20, 1]], 'aut_supersolvable': True, 'aut_tex': '(C_2^3\\times C_{66}).C_5.C_2^6', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 4, 'autcent_group': '128.1578', 'autcent_hash': 1578, 'autcent_nilpotent': True, 'autcent_order': 128, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^3\\wr C_2', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 330, 'autcentquo_group': '1320.142', 'autcentquo_hash': 142, 'autcentquo_nilpotent': False, 'autcentquo_order': 1320, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'D_6\\times F_{11}', 'cc_stats': [[1, 1, 1], [2, 1, 3], [2, 6, 4], [2, 11, 4], [2, 66, 4], [3, 2, 1], [4, 2, 2], [4, 22, 2], [6, 2, 3], [6, 22, 4], [11, 2, 5], [12, 2, 4], [12, 22, 4], [22, 2, 15], [22, 12, 20], [33, 4, 5], [44, 4, 10], [66, 4, 15], [132, 4, 20]], 'center_label': '4.2', 'center_order': 4, 'central_product': True, 'central_quotient': '264.34', 'commutator_count': 1, 'commutator_label': '66.4', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '2.1', '3.1', '11.1'], 'composition_length': 7, 'conjugacy_classes_known': True, 'counter': 916, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['2.1', 1], ['22.1', 1], ['24.6', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [2, 6, 1, 4], [2, 11, 1, 4], [2, 66, 1, 4], [3, 2, 1, 1], [4, 2, 1, 2], [4, 22, 1, 2], [6, 2, 1, 3], [6, 22, 1, 4], [11, 2, 5, 1], [12, 2, 2, 2], [12, 22, 2, 2], [22, 2, 5, 3], [22, 12, 5, 4], [33, 4, 5, 1], [44, 4, 5, 2], [66, 4, 5, 3], [132, 4, 10, 2]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 2178540, 'exponent': 132, 'exponents_of_order': [5, 1, 1], 'factors_of_aut_order': [2, 3, 5, 11], 'factors_of_order': [2, 3, 11], 'faithful_reps': [], 'familial': False, 'frattini_label': '2.1', 'frattini_quotient': '528.164', 'hash': 916, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 66, 'inner_gen_orders': [1, 2, 2, 66], 'inner_gens': [[1, 2, 4, 8], [1, 2, 4, 184], [1, 2, 4, 872], [1, 882, 196, 8]], 'inner_hash': 34, 'inner_nilpotent': False, 'inner_order': 264, 'inner_split': False, 'inner_tex': 'S_3\\times D_{22}', 'inner_used': [2, 3, 4], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 16], [2, 60], [4, 50]], 'label': '1056.916', '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': 'D12*D22', '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, 0, 0, 0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 23, 'number_characteristic_subgroups': 26, 'number_conjugacy_classes': 126, 'number_divisions': 48, 'number_normal_subgroups': 132, 'number_subgroup_autclasses': 136, 'number_subgroup_classes': 472, 'number_subgroups': 5696, 'old_label': None, 'order': 1056, 'order_factorization_type': 311, 'order_stats': [[1, 1], [2, 335], [3, 2], [4, 48], [6, 94], [11, 10], [12, 96], [22, 270], [33, 20], [44, 40], [66, 60], [132, 80]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': True, 'outer_exponent': 20, 'outer_gen_orders': [10, 4, 10, 20], 'outer_gen_pows': [352, 0, 0, 704], 'outer_gens': [[529, 971, 916, 680], [529, 883, 436, 521], [1, 971, 5, 280], [529, 443, 1013, 248]], 'outer_group': '640.18080', 'outer_hash': 18080, 'outer_nilpotent': True, 'outer_order': 640, 'outer_permdeg': 160, 'outer_perms': [324212236389553427248085251650430540858530393699295537640074167347442293942154525669310311271632011927955424175375213480057931324561675589196458040523887663190255310779438935730205553779365345297809427415105808282770581067471634063648685527039706539703047796370729205855932905975316696, 196297918025229288377842098948833838135287222015929775226023722668497761146339501717282290471996234274539802501285950282292098981592367430089442284506282399495553831378345568791905390507830922454833211871650200903367975993945901006151304092329904774978025868912875557659185110689121216, 85576893386805164748434295949610699840738390935674514744718116333157124835793299092893654582799897401673978668284274226065328729805198323641441107114761177039658169316659011397113688194984365391947485277423064041519016820990788528089669005526788376243054569966598571921505787511795734, 317928661642235206225913745210353859378361334833614605578841323156332062505336421125041565192551035442851855286549312744758064745897770218424557604371558406245888339845979632196293245013419380756050729421300655938411099303125218545324329615246017708361864176263764730624973945576184264], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_5:(C_2.D_4^2)', 'pc_rank': 4, 'perfect': False, 'permutation_degree': 20, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 2, 2], 'quasisimple': False, 'rank': 4, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 16], [2, 12], [4, 4], [10, 8], [20, 6], [40, 2]], 'representations': {'PC': {'code': 19731723705547429756645888921204736560000019, 'gens': [1, 2, 3, 4], 'pres': [7, -2, -2, -2, -2, -2, -3, -11, 2586, 6121, 80, 6451, 6038, 102, 15468, 3379, 166, 11780]}, 'GLZN': {'d': 2, 'p': 66, 'gens': [291149, 6612431, 433456, 6806977, 6613265, 287893, 12362371]}, 'Perm': {'d': 20, 'gens': [6423384156578664, 42001, 40320, 42720, 744, 3, 134491868718278400]}}, 'schur_multiplier': [2, 2, 2, 2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 2, 2], 'solvability_type': 7, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'D_{12}\\times D_{22}', 'transitive_degree': 264, 'wreath_data': None, 'wreath_product': False}
