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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': True, 'ambient': '13134.f', 'ambient_counter': 6, 'ambient_order': 13134, 'ambient_tex': 'C_{199}:C_{66}', 'central': False, 'central_factor': False, 'centralizer_order': 66, 'characteristic': False, 'core_order': 3, 'counter': 9, 'cyclic': True, 'direct': None, 'hall': 66, 'label': '13134.f.199.a1.a1', 'maximal': True, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': False, 'old_label': '199.a1.a1', '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': 199, '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': '66.4', 'subgroup_hash': 4, 'subgroup_order': 66, 'subgroup_tex': 'C_{66}', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '13134.f', 'aut_centralizer_order': None, 'aut_label': '199.a1', 'aut_quo_index': None, 'aut_stab_index': None, 'aut_weyl_group': None, 'aut_weyl_index': None, 'centralizer': '199.a1.a1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['1.a1.a1'], 'contains': ['398.a1.a1', '597.a1.a1', '2189.a1.a1'], 'core': '4378.a1.a1', 'coset_action_label': None, 'count': 199, 'diagramx': [2426, -1, 8831, -1, 4216, -1, 9444, -1], 'generators': [11, 2, 4378], 'label': '13134.f.199.a1.a1', 'mobius_quo': None, 'mobius_sub': -1, 'normal_closure': '1.a1.a1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '199.a1.a1', 'old_label': '199.a1.a1', 'projective_image': '4378.a', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '199.a1.a1', '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': '66.4', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': True, 'aut_cyclic': False, 'aut_derived_length': 1, 'aut_exponent': 10, 'aut_gen_orders': [2, 10], 'aut_gens': [[1], [43], [53]], 'aut_group': '20.5', 'aut_hash': 5, 'aut_nilpotency_class': 1, 'aut_nilpotent': True, 'aut_order': 20, 'aut_permdeg': 9, 'aut_perms': [720, 40353], 'aut_phi_ratio': 1.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [3, 1, 2, 1], [6, 1, 2, 1], [11, 1, 10, 1], [22, 1, 10, 1], [33, 1, 20, 1], [66, 1, 20, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_2\\times C_{10}', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 10, 'autcent_group': '20.5', 'autcent_hash': 5, 'autcent_nilpotent': True, 'autcent_order': 20, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2\\times C_{10}', '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, 2], [6, 1, 2], [11, 1, 10], [22, 1, 10], [33, 1, 20], [66, 1, 20]], 'center_label': '66.4', 'center_order': 66, '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', '11.1'], 'composition_length': 3, 'conjugacy_classes_known': True, 'counter': 4, 'cyclic': True, 'derived_length': 1, 'dihedral': False, 'direct_factorization': [['11.1', 1], ['2.1', 1], ['3.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [3, 1, 2, 1], [6, 1, 2, 1], [11, 1, 10, 1], [22, 1, 10, 1], [33, 1, 20, 1], [66, 1, 20, 1]], 'element_repr_type': 'PC', 'elementary': 66, 'eulerian_function': 1, 'exponent': 66, 'exponents_of_order': [1, 1, 1], 'factors_of_aut_order': [2, 5], 'factors_of_order': [2, 3, 11], 'faithful_reps': [[1, 0, 20]], 'familial': True, 'frattini_label': '1.1', 'frattini_quotient': '66.4', 'hash': 4, 'hyperelementary': 66, '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': 20, 'irrQ_dim': 20, 'irrR_degree': 2, 'irrep_stats': [[1, 66]], 'label': '66.4', 'linC_count': 20, 'linC_degree': 1, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 12, 'linQ_degree_count': 3, 'linQ_dim': 12, 'linQ_dim_count': 3, 'linR_count': 10, 'linR_degree': 2, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C66', 'ngens': 3, 'nilpotency_class': 1, 'nilpotent': True, 'normal_counts': [0, 0, 0, 0, 0, 0, 0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 8, 'number_characteristic_subgroups': 8, 'number_conjugacy_classes': 66, 'number_divisions': 8, 'number_normal_subgroups': 8, 'number_subgroup_autclasses': 8, 'number_subgroup_classes': 8, 'number_subgroups': 8, 'old_label': None, 'order': 66, 'order_factorization_type': 11, 'order_stats': [[1, 1], [2, 1], [3, 2], [6, 2], [11, 10], [22, 10], [33, 20], [66, 20]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 10, 'outer_gen_orders': [2, 10], 'outer_gen_pows': [0, 0], 'outer_gens': [[43], [53]], 'outer_group': '20.5', 'outer_hash': 5, 'outer_nilpotent': True, 'outer_order': 20, 'outer_permdeg': 9, 'outer_perms': [720, 40353], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2\\times C_{10}', 'pc_rank': 1, 'perfect': False, 'permutation_degree': 16, 'pgroup': 0, 'primary_abelian_invariants': [2, 3, 11], 'quasisimple': False, 'rank': 1, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 2], [2, 2], [10, 2], [20, 2]], 'representations': {'PC': {'code': 2467019, 'gens': [1], 'pres': [3, -2, -3, -11, 6, 22]}, 'GLFp': {'d': 2, 'p': 23, 'gens': [160116]}, 'Perm': {'d': 16, 'gens': [1307674368000, 12454041600, 36288000]}}, 'schur_multiplier': [], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [66], 'solvability_type': 0, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_{66}', 'transitive_degree': 66, 'wreath_data': None, 'wreath_product': False}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': False, 'abelian_quotient': '66.4', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 39402, 'aut_gen_orders': [198, 398], 'aut_gens': [[1, 22], [1, 12496], [8977, 4400]], 'aut_group': '78804.a', 'aut_hash': 1732360149213810607, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 78804, 'aut_permdeg': 201, 'aut_perms': [7844765794801755003323891217213786104500286325232778825737630311203202404850187064785395461078108897743163910941402173937572581043336717191733551683006435433818603880831873985800142390645654421690299127723513071931862347673132765079843083405686019779497320988713910073232164854718126143953689530991100795370986031405739191593521697268170391131707501096317508861440940136140609, 127610229096837571379784503460062648927393768117326363189564907280347054450802612278184810282119171816925360914140868304740257817637875082353981431799087995729482947593308312562139980966403722021025855999026905227035138573993184533834909038697328358623728369855844896624629934853661417086512521719895007042556437492798849398164909667418837403746966116579782032254878727783477421], 'aut_phi_ratio': 19.9, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 199, 1, 1], [3, 1, 2, 1], [6, 199, 2, 1], [11, 199, 1, 10], [22, 199, 1, 10], [33, 199, 2, 10], [66, 199, 2, 10], [199, 22, 9, 1], [597, 22, 18, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_2\\times F_{199}', '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': 39402, 'autcentquo_group': '39402.d', 'autcentquo_hash': 6821401462869295210, 'autcentquo_nilpotent': False, 'autcentquo_order': 39402, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'F_{199}', 'cc_stats': [[1, 1, 1], [2, 199, 1], [3, 1, 2], [6, 199, 2], [11, 199, 10], [22, 199, 10], [33, 199, 20], [66, 199, 20], [199, 22, 9], [597, 22, 18]], 'center_label': '3.1', 'center_order': 3, 'central_product': True, 'central_quotient': '4378.a', 'commutator_count': 1, 'commutator_label': '199.1', 'complements_known': True, 'complete': False, 'complex_characters_known': False, 'composition_factors': ['2.1', '3.1', '11.1', '199.1'], 'composition_length': 4, 'conjugacy_classes_known': True, 'counter': 6, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['3.1', 1], ['4378.a', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 199, 1, 1], [3, 1, 2, 1], [6, 199, 2, 1], [11, 199, 10, 1], [22, 199, 10, 1], [33, 199, 20, 1], [66, 199, 20, 1], [199, 22, 9, 1], [597, 22, 18, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 1440, 'exponent': 13134, 'exponents_of_order': [1, 1, 1, 1], 'factors_of_aut_order': [2, 3, 11, 199], 'factors_of_order': [2, 3, 11, 199], 'faithful_reps': [[22, 0, 18]], 'familial': False, 'frattini_label': '1.1', 'frattini_quotient': '13134.f', 'hash': 12, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 4378, 'inner_gen_orders': [22, 199], 'inner_gens': [[1, 10384], [2773, 22]], 'inner_hash': 1, 'inner_nilpotent': False, 'inner_order': 4378, 'inner_split': True, 'inner_tex': 'C_{199}:C_{22}', 'inner_used': [1, 2], 'irrC_degree': 22, 'irrQ_degree': 396, 'irrQ_dim': None, 'irrR_degree': None, 'irrep_stats': [[1, 66], [22, 27]], 'label': '13134.f', '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': True, 'monomial': True, 'name': 'C199:C66', 'ngens': 4, 'nilpotency_class': -1, 'nilpotent': False, 'normal_counts': [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': 46, 'number_characteristic_subgroups': 10, 'number_conjugacy_classes': 93, 'number_divisions': 10, 'number_normal_subgroups': 10, 'number_subgroup_autclasses': 16, 'number_subgroup_classes': 16, 'number_subgroups': 1204, 'old_label': None, 'order': 13134, 'order_factorization_type': 11, 'order_stats': [[1, 1], [2, 199], [3, 2], [6, 398], [11, 1990], [22, 1990], [33, 3980], [66, 3980], [199, 198], [597, 396]], 'outer_abelian': True, 'outer_cyclic': True, 'outer_equivalence': False, 'outer_exponent': 18, 'outer_gen_orders': [18], 'outer_gen_pows': [18], 'outer_gens': [[1, 9086]], 'outer_group': '18.2', 'outer_hash': 2, 'outer_nilpotent': True, 'outer_order': 18, 'outer_permdeg': 11, 'outer_perms': [3859800], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_{18}', 'pc_rank': 2, 'perfect': False, 'permutation_degree': 202, 'pgroup': 0, 'primary_abelian_invariants': [2, 3, 11], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': False, 'ratrep_stats': [[1, 2], [2, 2], [10, 2], [20, 2], [198, 1], [396, 1]], 'representations': {'PC': {'code': '31370899927240120358130079340545055', 'gens': [1, 3], 'pres': [4, -2, -11, -3, -199, 8, 124610, 13602, 46, 78147, 54391]}, 'GLFp': {'d': 2, 'p': 199, 'gens': [7880799, 1040239127]}, 'Perm': {'d': 202, 'gens': [788717715775379712147044883354113803238779741712176774503593236359568043071535222678707832421615822549229750026246094776642794817601281250418805260077961068512740531321929441349809910248430158712002445828331447620399662889469470639504688903718292949569357677734702193391481020240952428847333079808817352115455206255835073607377287982653800782681422860939486823694278550669400, 3, 1593208585881059268929057010943344089257063492719445276108336696961159274386117195065000839039811462843456833185122103754109056016938679434804905286969248996260524755514005102279076618054585461070075195745437159037499059518252028715592436547845006513606523826229209705152575722736980854632055061530519335153281496082300365554908606411574913297154440143419164768100521951480680, 160886344556907055407607556799828226265389254239163704741381840835073325819198754993630057919500682366731012489348152972582074037157462508083173024454589007206100945501648888505849401735727064108564272273717902269077863438899795538499179779467359841236424207725331370859648977133918159472474224866790854724954885160396513494173529641548264809950635855884474746935175522182136560]}}, 'schur_multiplier': [], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [66], 'solvability_type': 6, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_{199}:C_{66}', 'transitive_degree': 597, 'wreath_data': None, 'wreath_product': False}
