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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': False, 'ambient': '9828.a', 'ambient_counter': 1, 'ambient_order': 9828, 'ambient_tex': '\\PSL(2,27)', 'central': False, 'central_factor': False, 'centralizer_order': 2, 'characteristic': False, 'core_order': 1, 'counter': 3, 'cyclic': False, 'direct': None, 'hall': 14, 'label': '9828.a.351.a1.a1', 'maximal': True, 'maximal_normal': False, 'metabelian': True, 'metacyclic': None, 'minimal': False, 'minimal_normal': False, 'nilpotent': False, 'normal': False, 'old_label': '351.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': 351, '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': '28.3', 'subgroup_hash': 3, 'subgroup_order': 28, 'subgroup_tex': 'D_{14}', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '9828.a', 'aut_centralizer_order': None, 'aut_label': '351.a1', 'aut_quo_index': None, 'aut_stab_index': None, 'aut_weyl_group': None, 'aut_weyl_index': None, 'centralizer': '4914.a1.a1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['1.a1.a1'], 'contains': ['702.a1.a1', '702.b1.a1', '702.b1.a2', '2457.a1.a1'], 'core': '9828.a1.a1', 'coset_action_label': None, 'count': 351, 'diagramx': [6969, -1, 3436, -1, 7681, -1, 6587, -1], 'generators': [35102, 244158, 416238], 'label': '9828.a.351.a1.a1', 'mobius_quo': None, 'mobius_sub': -1, 'normal_closure': '1.a1.a1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '351.a1.a1', 'old_label': '351.a1.a1', 'projective_image': '9828.a', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '351.a1.a1', 'subgroup_fusion': None, 'weyl_group': '14.1'}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '4.2', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 42, 'aut_gen_orders': [6, 14], 'aut_gens': [[1, 2], [1, 6], [11, 2]], 'aut_group': '84.7', 'aut_hash': 7, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 84, 'aut_permdeg': 9, 'aut_perms': [6055, 203047], 'aut_phi_ratio': 7.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 7, 2, 1], [7, 2, 3, 1], [14, 2, 3, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_2\\times F_7', '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': 42, 'autcentquo_group': '42.1', 'autcentquo_hash': 1, 'autcentquo_nilpotent': False, 'autcentquo_order': 42, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'F_7', 'cc_stats': [[1, 1, 1], [2, 1, 1], [2, 7, 2], [7, 2, 3], [14, 2, 3]], 'center_label': '2.1', 'center_order': 2, 'central_product': True, 'central_quotient': '14.1', 'commutator_count': 1, 'commutator_label': '7.1', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '7.1'], 'composition_length': 3, 'conjugacy_classes_known': True, 'counter': 3, 'cyclic': False, 'derived_length': 2, 'dihedral': True, 'direct_factorization': [['14.1', 1], ['2.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 7, 1, 2], [7, 2, 3, 1], [14, 2, 3, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 3, 'exponent': 14, 'exponents_of_order': [2, 1], 'factors_of_aut_order': [2, 3, 7], 'factors_of_order': [2, 7], 'faithful_reps': [[2, 1, 3]], 'familial': True, 'frattini_label': '1.1', 'frattini_quotient': '28.3', 'hash': 3, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 14, 'inner_gen_orders': [2, 7], 'inner_gens': [[1, 26], [5, 2]], 'inner_hash': 1, 'inner_nilpotent': False, 'inner_order': 14, 'inner_split': True, 'inner_tex': 'D_7', 'inner_used': [1, 2], 'irrC_degree': 2, 'irrQ_degree': 6, 'irrQ_dim': 6, 'irrR_degree': 2, 'irrep_stats': [[1, 4], [2, 6]], 'label': '28.3', 'linC_count': 3, 'linC_degree': 2, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 6, 'linQ_degree_count': 1, 'linQ_dim': 6, 'linQ_dim_count': 1, 'linR_count': 3, 'linR_degree': 2, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'D14', 'ngens': 3, 'nilpotency_class': -1, 'nilpotent': False, 'normal_counts': [0, 0, 0, 0, 0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 5, 'number_characteristic_subgroups': 5, 'number_conjugacy_classes': 10, 'number_divisions': 6, 'number_normal_subgroups': 7, 'number_subgroup_autclasses': 8, 'number_subgroup_classes': 10, 'number_subgroups': 28, 'old_label': None, 'order': 28, 'order_factorization_type': 22, 'order_stats': [[1, 1], [2, 15], [7, 6], [14, 6]], 'outer_abelian': True, 'outer_cyclic': True, 'outer_equivalence': False, 'outer_exponent': 6, 'outer_gen_orders': [6], 'outer_gen_pows': [0], 'outer_gens': [[15, 10]], 'outer_group': '6.2', 'outer_hash': 2, 'outer_nilpotent': True, 'outer_order': 6, 'outer_permdeg': 5, 'outer_perms': [28], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_6', 'pc_rank': 2, 'perfect': False, 'permutation_degree': 9, 'pgroup': 0, 'primary_abelian_invariants': [2, 2], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [6, 2]], 'representations': {'PC': {'code': 28266197, 'gens': [1, 2], 'pres': [3, -2, -2, -7, 157, 16, 218]}, 'GLZ': {'b': 3, 'd': 6, 'gens': [26761470676238614, 122882691976452762]}, 'GLFp': {'d': 2, 'p': 7, 'gens': [351, 2059, 2064]}, 'Perm': {'d': 9, 'gens': [5166, 1, 50646]}}, 'schur_multiplier': [2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2], 'solvability_type': 6, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'D_{14}', 'transitive_degree': 14, 'wreath_data': None, 'wreath_product': False}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '1.1', 'all_subgroups_known': True, 'almost_simple': True, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 1, 'aut_exponent': 3276, 'aut_gen_orders': [26, 3, 3], 'aut_gens': [[373980, 1487], [374034, 225981], [528435, 17687], [236716, 363133]], 'aut_group': '58968.a', 'aut_hash': 6214424669380498256, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 58968, 'aut_permdeg': 351, 'aut_perms': [38162656969030292324487522539883223984863336485114218066598191855240381267883882585663307009529751995394782321874708040458483880347764459826502353754325302489218356661183649775459089956024901429971208908321357096752039762254815338343652597422693873770601657703034146533365678060644236003720330615169413668459288115812124901952003697859152434510170096481795963142535507860117484884679197978080802229317929051242629515661261643053336863511813716526666032162721910348275637025504765636556014233750545685869232668092178454090874995412072029405668175289275456515968238680135191313475713259380207458227402727583939443581414269684598641507001363037956030096102665843815136979815221977624624474794019162499737841792195321194169870573208964006458573707, 13567063985316294450048928563696256042809899608130186656390075015744980281561015324211986019427272229361021179704619898751024192611678016571141840135400767136766120357529036509393188638958266934630338229030322237144045849110884812333241725030115244797252957465830145365810670212537197808060063506799896941310546493452735014531043570151723754338028966645079297370719565472446359854171352965100517192406846733059012320817425565899766605890003838366930082267220454265188155761161343690293004523567799287741847289331223752042442898495393651771564277981911447317063986355680290076418548955676813731275293042696315668033361008802591082753286254643299447177235968756342772213517897568770880716309431639592467266655863971125646038336486162685385472211, 20679638651903668469010102960260812141278521002420268207518520447359377146037033068356362890417031970965835083827530048024440213297734187822536280492220874670639801238342325553522595171060511239261753223332165803591402548228738145105449894482586310503039595327186283398969484174775522329307991851367933237886148929905562252955677720449008433574286373841126530616975163963376857508524995086172447953320384742113240105721411205418870471129596344610103315888031242731245952547277502067760602726582265941881790223387944863273042590730833887015761263029213848827627123635097172858454011014931291614746416987185494539483627154228261946804176857990046266030364976875670932400239286764670555677856435127003784003148293002524031960826976752102290618023], 'aut_phi_ratio': 22.75, 'aut_solvable': False, 'aut_stats': [[1, 1, 1, 1], [2, 351, 1, 1], [3, 364, 2, 1], [7, 702, 3, 1], [13, 756, 3, 2], [14, 702, 3, 1]], 'aut_supersolvable': False, 'aut_tex': '\\PSL(2,27).C_6', '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': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 3276, 'autcentquo_group': '58968.a', 'autcentquo_hash': 6214424669380498256, 'autcentquo_nilpotent': False, 'autcentquo_order': 58968, 'autcentquo_solvable': False, 'autcentquo_supersolvable': False, 'autcentquo_tex': '\\PSL(2,27).C_6', 'cc_stats': [[1, 1, 1], [2, 351, 1], [3, 364, 2], [7, 702, 3], [13, 756, 6], [14, 702, 3]], 'center_label': '1.1', 'center_order': 1, 'central_product': False, 'central_quotient': '9828.a', 'commutator_count': 1, 'commutator_label': '9828.a', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['9828.a'], 'composition_length': 1, 'conjugacy_classes_known': True, 'counter': 1, 'cyclic': False, 'derived_length': 0, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 351, 1, 1], [3, 364, 2, 1], [7, 702, 3, 1], [13, 756, 6, 1], [14, 702, 3, 1]], 'element_repr_type': 'Lie', 'elementary': 1, 'eulerian_function': 1572, 'exponent': 546, 'exponents_of_order': [3, 2, 1, 1], 'factors_of_aut_order': [2, 3, 7, 13], 'factors_of_order': [2, 3, 7, 13], 'faithful_reps': [[13, 0, 2], [26, 1, 6], [27, 1, 1], [28, 1, 6]], 'familial': True, 'frattini_label': '1.1', 'frattini_quotient': '9828.a', 'hash': 7765825457534789434, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 546, 'inner_gen_orders': [13, 3], 'inner_gens': [[373980, 13799], [77293, 1487]], 'inner_hash': 7765825457534789434, 'inner_nilpotent': False, 'inner_order': 9828, 'inner_split': True, 'inner_tex': '\\PSL(2,27)', 'inner_used': [1, 2], 'irrC_degree': 13, 'irrQ_degree': 26, 'irrQ_dim': 26, 'irrR_degree': 26, 'irrep_stats': [[1, 1], [13, 2], [26, 6], [27, 1], [28, 6]], 'label': '9828.a', 'linC_count': 2, 'linC_degree': 13, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 26, 'linQ_degree_count': 1, 'linQ_dim': 26, 'linQ_dim_count': 1, 'linR_count': 7, 'linR_degree': 26, 'maximal_subgroups_known': True, 'metabelian': False, 'metacyclic': False, 'monomial': False, 'name': 'PSL(2,27)', 'ngens': 2, '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, 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': 7, 'number_characteristic_subgroups': 2, 'number_conjugacy_classes': 16, 'number_divisions': 6, 'number_normal_subgroups': 2, 'number_subgroup_autclasses': 15, 'number_subgroup_classes': 16, 'number_subgroups': 5286, 'old_label': None, 'order': 9828, 'order_factorization_type': 321, 'order_stats': [[1, 1], [2, 351], [3, 728], [7, 2106], [13, 4536], [14, 2106]], 'outer_abelian': True, 'outer_cyclic': True, 'outer_equivalence': False, 'outer_exponent': 6, 'outer_gen_orders': [6], 'outer_gen_pows': [19684], 'outer_gens': [[72295, 447567]], 'outer_group': '6.2', 'outer_hash': 2, 'outer_nilpotent': True, 'outer_order': 6, 'outer_permdeg': 5, 'outer_perms': [28], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_6', 'pc_rank': None, 'perfect': True, 'permutation_degree': 28, 'pgroup': 0, 'primary_abelian_invariants': [], 'quasisimple': True, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 1], [26, 1], [27, 1], [78, 2], [168, 1]], 'representations': {'Lie': [{'d': 2, 'q': 27, 'gens': [1487, 373980], 'family': 'PSL'}, {'d': 2, 'q': 27, 'family': 'PSU'}, {'d': 3, 'q': 27, 'gens': [775944739, 2541866359795], 'family': 'Omega'}, {'d': 3, 'q': 27, 'family': 'POmega'}], 'Perm': {'d': 28, 'gens': [380309071898006557168262400, 11711058550567061734868211621]}}, 'schur_multiplier': [2], 'semidirect_product': False, 'simple': True, 'smith_abelian_invariants': [], 'solvability_type': 13, 'solvable': False, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': '\\PSL(2,27)', 'transitive_degree': 28, 'wreath_data': None, 'wreath_product': False}
