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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': True, 'ambient': '4400.q', 'ambient_counter': 17, 'ambient_order': 4400, 'ambient_tex': 'C_{44}:C_{10}^2', 'central': False, 'central_factor': False, 'centralizer_order': 200, 'characteristic': False, 'core_order': 1, 'counter': 95, 'cyclic': True, 'direct': None, 'hall': 0, 'label': '4400.q.2200.c1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': True, 'minimal_normal': False, 'nilpotent': True, 'normal': False, 'old_label': '2200.c1', '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': 2200, 'quotient_simple': None, 'quotient_solvable': None, 'quotient_supersolvable': None, 'quotient_tex': None, 'simple': True, 'solvable': True, 'special_labels': [], 'split': None, 'standard_generators': False, 'stem': False, 'subgroup': '2.1', 'subgroup_hash': 1, 'subgroup_order': 2, 'subgroup_tex': 'C_2', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '4400.q', 'aut_centralizer_order': 1600, 'aut_label': '2200.c1', 'aut_quo_index': None, 'aut_stab_index': 88, 'aut_weyl_group': '1.1', 'aut_weyl_index': 140800, 'centralizer': '22.a1', 'complements': None, 'conjugacy_class_count': 4, 'contained_in': ['200.c1', '440.e1', '440.f1', '1100.c1', '1100.d1'], 'contains': ['4400.a1'], 'core': '4400.a1', 'coset_action_label': None, 'count': 88, 'diagramx': [8146, -1, 2570, -1], 'generators': [50], 'label': '4400.q.2200.c1', 'mobius_quo': None, 'mobius_sub': 0, 'normal_closure': '100.b1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '22.a1', 'old_label': '2200.c1', 'projective_image': '4400.q', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '2200.c1', 'subgroup_fusion': None, 'weyl_group': '1.1'}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': True, 'abelian_quotient': '2.1', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': True, 'aut_cyclic': True, 'aut_derived_length': 0, 'aut_exponent': 1, 'aut_gen_orders': [], 'aut_gens': [[1]], 'aut_group': '1.1', 'aut_hash': 1, 'aut_nilpotency_class': 0, 'aut_nilpotent': True, 'aut_order': 1, 'aut_permdeg': 1, 'aut_perms': [], 'aut_phi_ratio': 1.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_1', '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': 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]], 'center_label': '2.1', 'center_order': 2, 'central_product': False, 'central_quotient': '1.1', 'commutator_count': 0, 'commutator_label': '1.1', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1'], 'composition_length': 1, 'conjugacy_classes_known': True, 'counter': 1, 'cyclic': True, 'derived_length': 1, 'dihedral': True, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1]], 'element_repr_type': 'PC', 'elementary': 2, 'eulerian_function': 1, 'exponent': 2, 'exponents_of_order': [1], 'factors_of_aut_order': [], 'factors_of_order': [2], 'faithful_reps': [[1, 1, 1]], 'familial': True, 'frattini_label': '1.1', 'frattini_quotient': '2.1', 'hash': 1, 'hyperelementary': 2, 'inner_abelian': True, 'inner_cyclic': True, 'inner_exponent': 1, 'inner_gen_orders': [1], 'inner_gens': [[1]], '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, 2]], 'label': '2.1', 'linC_count': 1, 'linC_degree': 1, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 1, 'linQ_degree_count': 1, 'linQ_dim': 1, 'linQ_dim_count': 1, 'linR_count': 1, 'linR_degree': 1, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C2', 'ngens': 1, 'nilpotency_class': 1, 'nilpotent': True, 'normal_counts': [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': 2, 'number_divisions': 2, 'number_normal_subgroups': 2, 'number_subgroup_autclasses': 2, 'number_subgroup_classes': 2, 'number_subgroups': 2, 'old_label': None, 'order': 2, 'order_factorization_type': 1, 'order_stats': [[1, 1], [2, 1]], 'outer_abelian': True, 'outer_cyclic': True, 'outer_equivalence': False, 'outer_exponent': 1, 'outer_gen_orders': [], 'outer_gen_pows': [], 'outer_gens': [], 'outer_group': '1.1', 'outer_hash': 1, 'outer_nilpotent': True, 'outer_order': 1, 'outer_permdeg': 1, 'outer_perms': [], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_1', 'pc_rank': 1, 'perfect': False, 'permutation_degree': 2, 'pgroup': 2, 'primary_abelian_invariants': [2], 'quasisimple': False, 'rank': 1, 'rational': True, 'rational_characters_known': True, 'ratrep_stats': [[1, 2]], 'representations': {'PC': {'code': 0, 'gens': [1], 'pres': [1, -2]}, 'GLZ': {'b': 3, 'd': 2, 'gens': [12]}, 'Lie': [{'d': 2, 'q': 2, 'gens': [6], 'family': 'CSOPlus'}, {'d': 2, 'q': 2, 'gens': [6], 'family': 'COPlus'}, {'d': 1, 'q': 2, 'gens': [1], 'family': 'AGL'}, {'d': 1, 'q': 2, 'gens': [1], 'family': 'AGammaL'}, {'d': 1, 'q': 2, 'gens': [1], 'family': 'ASigmaL'}], 'GLFp': {'d': 2, 'p': 2, 'gens': [11]}, 'Perm': {'d': 2, 'gens': [1]}}, 'schur_multiplier': [], 'semidirect_product': False, 'simple': True, 'smith_abelian_invariants': [2], 'solvability_type': 0, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_2', 'transitive_degree': 2, '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': '200.52', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 220, 'aut_gen_orders': [20, 20, 4, 2, 20, 20], 'aut_gens': [[1, 10, 100], [2201, 2312, 500], [2207, 1410, 3900], [2203, 1518, 2105], [2209, 1816, 2100], [2203, 2117, 3100], [9, 4117, 500]], 'aut_group': None, 'aut_hash': 4556555602398024433, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 140800, 'aut_permdeg': 143, 'aut_perms': [12015751494156987555228925170674841645285555612291729579892723173022219690491084970898005697873641988273022318634035353356017295878330386042649153558633678410669636204803784611604498640942485272343137965691663038661016137892324917817275130460238874, 34234043115761097408650625404044300178817648649207541662666950452908523371633518098799372841736058945870514981898616269139981295658105720232603382576928698168432743540456440976401080062257928776488254532278547035457155983355571391510524372873659051, 8512535885989870061115561989373365960047924591236996131176497854544087537268841500176830451950734942487995813059023539814467429235124168236925254922019548728703867408994996692964425350056424602562257562173282335468787494473488638386995413742061495, 27003507023037487978963349309051487164952841595684586651043782463415420275280056766747720748511757779992548599713104739714142487338656283515960073099165561817357779911116206414867017882325554321112175980025794930699768631352349028844115897948467415, 4728971208827476279992173687378745848140006803372570465648510965062635224894433003646380690921220144432035142857955551726579750956266031997924729084411365106769252299588791419709217367671939992206422661422833437325762752782244758122029929805441301, 4650049406057583608219906061847526750652541303361457662243508896951314410411785574691464475614829329265579972784429455127314798544820879422547901716007147150417922486906200586564595936362497308608083681437393409827404637775978732892149652157406858], 'aut_phi_ratio': 88.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 1, 2, 1], [2, 22, 4, 1], [4, 2, 2, 1], [5, 1, 4, 1], [5, 11, 5, 4], [10, 1, 4, 1], [10, 1, 8, 1], [10, 11, 5, 4], [10, 11, 10, 4], [10, 22, 16, 1], [10, 22, 20, 4], [11, 10, 1, 1], [20, 2, 8, 1], [20, 22, 10, 4], [22, 10, 1, 1], [22, 10, 2, 1], [44, 10, 4, 1], [55, 10, 4, 1], [110, 10, 4, 1], [110, 10, 8, 1], [220, 10, 16, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_{55}.(C_2^4\\times C_{20}).C_2^3', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 20, 'autcent_group': '640.19303', 'autcent_hash': 19303, 'autcent_nilpotent': False, 'autcent_order': 640, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^2\\wr C_2\\times F_5', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 110, 'autcentquo_group': '220.7', 'autcentquo_hash': 7, 'autcentquo_nilpotent': False, 'autcentquo_order': 220, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'C_2\\times F_{11}', 'cc_stats': [[1, 1, 1], [2, 1, 3], [2, 22, 4], [4, 2, 2], [5, 1, 4], [5, 11, 20], [10, 1, 12], [10, 11, 60], [10, 22, 96], [11, 10, 1], [20, 2, 8], [20, 22, 40], [22, 10, 3], [44, 10, 4], [55, 10, 4], [110, 10, 12], [220, 10, 16]], 'center_label': '20.5', 'center_order': 20, 'central_product': True, 'central_quotient': '220.7', 'commutator_count': 1, 'commutator_label': '22.2', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '5.1', '5.1', '11.1'], 'composition_length': 7, 'conjugacy_classes_known': True, 'counter': 17, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['2.1', 1], ['440.9', 1], ['5.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [2, 22, 1, 4], [4, 2, 1, 2], [5, 1, 4, 1], [5, 11, 4, 5], [10, 1, 4, 3], [10, 11, 4, 15], [10, 22, 4, 24], [11, 10, 1, 1], [20, 2, 4, 2], [20, 22, 4, 10], [22, 10, 1, 3], [44, 10, 2, 2], [55, 10, 4, 1], [110, 10, 4, 3], [220, 10, 8, 2]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 187488, 'exponent': 220, 'exponents_of_order': [4, 2, 1], 'factors_of_aut_order': [2, 5, 11], 'factors_of_order': [2, 5, 11], 'faithful_reps': [], 'familial': False, 'frattini_label': '2.1', 'frattini_quotient': '2200.a', 'hash': 7612526101672382494, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 110, 'inner_gen_orders': [1, 10, 22], 'inner_gens': [[1, 10, 100], [1, 10, 3900], [1, 610, 100]], 'inner_hash': 7, 'inner_nilpotent': False, 'inner_order': 220, 'inner_split': False, 'inner_tex': 'C_2\\times F_{11}', 'inner_used': [2, 3], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 200], [2, 50], [10, 40]], 'label': '4400.q', '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': 'C44:C10^2', '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, 0, 0, 0, 0, 0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 38, 'number_characteristic_subgroups': 23, 'number_conjugacy_classes': 290, 'number_divisions': 82, 'number_normal_subgroups': 168, 'number_subgroup_autclasses': 96, 'number_subgroup_classes': 432, 'number_subgroups': 3200, 'old_label': None, 'order': 4400, 'order_factorization_type': 321, 'order_stats': [[1, 1], [2, 91], [4, 4], [5, 224], [10, 2784], [11, 10], [20, 896], [22, 30], [44, 40], [55, 40], [110, 120], [220, 160]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': True, 'outer_exponent': 20, 'outer_gen_orders': [2, 2, 4, 4, 10], 'outer_gen_pows': [3300, 0, 0, 0, 0], 'outer_gens': [[1, 3310, 100], [2201, 10, 2100], [3, 18, 2100], [2201, 10, 105], [1, 2211, 100]], 'outer_group': '640.19303', 'outer_hash': 19303, 'outer_nilpotent': False, 'outer_order': 640, 'outer_permdeg': 13, 'outer_perms': [11652480, 3669120, 79, 1041596640, 3612239396], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^2\\wr C_2\\times F_5', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 22, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 2, 5, 5], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 8], [2, 2], [4, 48], [8, 12], [10, 4], [20, 2], [40, 4], [80, 2]], 'representations': {'PC': {'code': '1659446987753833766528716812695289313362450303009', 'gens': [1, 3, 5], 'pres': [7, -2, -5, -2, -5, -2, -2, -11, 14, 58, 13668, 4400, 102, 14299, 1286, 124, 11780, 2967]}, 'GLFp': {'d': 4, 'p': 11, 'gens': [12531822321003912, 37668179739389305, 42497879883498899, 33418192856010432, 26794159036998355, 36687300164533435, 17156201816241749]}, 'Perm': {'d': 22, 'gens': [56213599674603571241, 68, 58999653623681971268, 3669908, 10360195344788890361, 110236175353650085241, 12428870892684744960]}}, 'schur_multiplier': [2, 2, 10], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 10, 10], 'solvability_type': 7, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_{44}:C_{10}^2', 'transitive_degree': 440, 'wreath_data': None, 'wreath_product': False}
