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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': True, 'ambient': '56882.2', 'ambient_counter': 2, 'ambient_order': 56882, 'ambient_tex': 'C_{1673}:C_{34}', 'central': False, 'central_factor': False, 'centralizer_order': 3346, 'characteristic': True, 'core_order': 239, 'counter': 8, 'cyclic': True, 'direct': False, 'hall': 239, 'label': '56882.2.238.a1.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': True, 'minimal_normal': True, 'nilpotent': True, 'normal': True, 'old_label': '238.a1.a1', 'outer_equivalence': False, 'perfect': False, 'proper': True, 'quotient': '238.4', 'quotient_Agroup': True, 'quotient_abelian': True, 'quotient_cyclic': True, 'quotient_hash': 4, 'quotient_metabelian': True, 'quotient_nilpotent': True, 'quotient_order': 238, 'quotient_simple': False, 'quotient_solvable': True, 'quotient_supersolvable': True, 'quotient_tex': 'C_{238}', 'simple': True, 'solvable': True, 'special_labels': ['D', 'L1', 'D1', 'C3'], 'split': True, 'standard_generators': False, 'stem': False, 'subgroup': '239.1', 'subgroup_hash': 1, 'subgroup_order': 239, 'subgroup_tex': 'C_{239}', 'supersolvable': True, 'sylow': 239}
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gps_subgroup_data • Show schema
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{'ambient': '56882.2', 'aut_centralizer_order': None, 'aut_label': '238.a1', 'aut_quo_index': None, 'aut_stab_index': None, 'aut_weyl_group': None, 'aut_weyl_index': None, 'centralizer': '17.a1.a1', 'complements': ['239.a1.a1'], 'conjugacy_class_count': 1, 'contained_in': ['14.a1.a1', '34.a1.a1', '119.a1.a1'], 'contains': ['56882.a1.a1'], 'core': '238.a1.a1', 'coset_action_label': None, 'count': 1, 'diagramx': [8158, 7303, 3802, 7144, 6571, 5657, 5320, 5657], 'generators': [238], 'label': '56882.2.238.a1.a1', 'mobius_quo': -1, 'mobius_sub': -1, 'normal_closure': '238.a1.a1', 'normal_contained_in': ['14.a1.a1', '34.a1.a1', '119.a1.a1'], 'normal_contains': ['56882.a1.a1'], 'normalizer': '1.a1.a1', 'old_label': '238.a1.a1', 'projective_image': '56882.2', 'quotient_action_image': '17.1', 'quotient_action_kernel': '14.2', 'quotient_action_kernel_order': 14, 'quotient_fusion': None, 'short_label': '238.a1.a1', 'subgroup_fusion': None, 'weyl_group': '17.1'}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': True, 'abelian_quotient': '239.1', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': True, 'aut_cyclic': True, 'aut_derived_length': 1, 'aut_exponent': 238, 'aut_gen_orders': [238], 'aut_gens': [[1], [185]], 'aut_group': '238.4', 'aut_hash': 4, 'aut_nilpotency_class': 1, 'aut_nilpotent': True, 'aut_order': 238, 'aut_permdeg': 26, 'aut_perms': [15538239713024067207168000], 'aut_phi_ratio': 1.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [239, 1, 238, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_{238}', 'autcent_abelian': True, 'autcent_cyclic': True, 'autcent_exponent': 238, 'autcent_group': '238.4', 'autcent_hash': 4, 'autcent_nilpotent': True, 'autcent_order': 238, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_{238}', '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], [239, 1, 238]], 'center_label': '239.1', 'center_order': 239, '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': ['239.1'], 'composition_length': 1, 'conjugacy_classes_known': True, 'counter': 1, 'cyclic': True, 'derived_length': 1, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [239, 1, 238, 1]], 'element_repr_type': 'PC', 'elementary': 239, 'eulerian_function': 1, 'exponent': 239, 'exponents_of_order': [1], 'factors_of_aut_order': [2, 7, 17], 'factors_of_order': [239], 'faithful_reps': [[1, 0, 238]], 'familial': True, 'frattini_label': '1.1', 'frattini_quotient': '239.1', 'hash': 1, 'hyperelementary': 239, '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': 238, 'irrQ_dim': 238, 'irrR_degree': 2, 'irrep_stats': [[1, 239]], 'label': '239.1', 'linC_count': 238, 'linC_degree': 1, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 238, 'linQ_degree_count': 1, 'linQ_dim': 238, 'linQ_dim_count': 1, 'linR_count': 119, 'linR_degree': 2, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C239', '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': 239, 'number_divisions': 2, 'number_normal_subgroups': 2, 'number_subgroup_autclasses': 2, 'number_subgroup_classes': 2, 'number_subgroups': 2, 'old_label': None, 'order': 239, 'order_factorization_type': 1, 'order_stats': [[1, 1], [239, 238]], 'outer_abelian': True, 'outer_cyclic': True, 'outer_equivalence': False, 'outer_exponent': 238, 'outer_gen_orders': [238], 'outer_gen_pows': [0], 'outer_gens': [[185]], 'outer_group': '238.4', 'outer_hash': 4, 'outer_nilpotent': True, 'outer_order': 238, 'outer_permdeg': 26, 'outer_perms': [15538239713024067207168000], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_{238}', 'pc_rank': 1, 'perfect': False, 'permutation_degree': 239, 'pgroup': 239, 'primary_abelian_invariants': [239], 'quasisimple': False, 'rank': 1, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 1], [238, 1]], 'representations': {'PC': {'code': 0, 'gens': [1], 'pres': [1, -239]}, 'Lie': [{'d': 1, 'q': 239, 'family': 'ASigmaL'}], 'GLFp': {'d': 2, 'p': 239, 'gens': [13652159]}, 'Perm': {'d': 239, 'gens': [16878603844979076580848519339502766803015135977780907614445640178545918526922755487345989778161557678798280541815151385649878367160975668584886276509508948749147248964256825974171298841215929844050778544406026688455121165697202950750082801934697357666436908527383462589455613080150433448708089759627350126270791596181208644402648548221913567394350548088433443916256849345161133023282323036596594746951538311168000000000000000000000000000000000000000000000000000000000]}}, 'schur_multiplier': [], 'semidirect_product': False, 'simple': True, 'smith_abelian_invariants': [239], 'solvability_type': 0, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_{239}', 'transitive_degree': 239, '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': '238.4', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 170646, 'aut_gen_orders': [6, 238, 239], 'aut_gens': [[1, 17], [1, 24361], [1, 51187], [239, 17]], 'aut_group': None, 'aut_hash': 4076511750313232849, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 341292, 'aut_permdeg': 245, 'aut_perms': [1709036715395560100279223980329775736135092541154435210090789769897145023800523488337365227023242139326450222056541548476624123270165337664611030943435688224225464484093100553019739242662640581753167722323329877970708175636617668920487275406555204379151961458585148047262769537910014680455264208971798012687809939381119831825867263941200575702732992093832153875875318039552974423230633939319229616181171394555999912931372681828141351412627044573033094717849178496438244474879999999, 2700559611319054884306555429814111174516480162323486226314380447639492683605265524289022227089702105662585245236797302510608331605239258003406979973048662377519058162410877210278280792052463184027218936416359547116603489856741304347161346842950172956028423323946396204553046250077926166556861223570851340799357030838040351851341053598010283863534519624483867686853787385083718833868580730311288522634539678826453323492721242884636130387691709717472640641698825617401206909669843513, 2090462989058802237193758113671438550270844022437582280731177221907416641328611665230974618416485216632445458900902326988885253259914971345543863352415386377323653103849900483321634042181079240160582027665927861209727702999119507468967654486938026919002780750677809791102214928571510549657813380712668898660125460222160669086850433928207230116903479632213844028136185145854650157286049384839049789944079593025290882863223462814117341477640216736214968371105546547616151583619762680], 'aut_phi_ratio': 14.9375, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [7, 1, 6, 1], [14, 1, 6, 1], [17, 239, 1, 16], [34, 239, 1, 16], [119, 239, 6, 16], [238, 239, 6, 16], [239, 17, 14, 1], [478, 17, 14, 1], [1673, 17, 84, 1], [3346, 17, 84, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_{239}:(C_2\\times C_{714})', 'autcent_abelian': True, 'autcent_cyclic': True, 'autcent_exponent': 6, 'autcent_group': '6.2', 'autcent_hash': 2, 'autcent_nilpotent': True, 'autcent_order': 6, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_6', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 56882, 'autcentquo_group': '56882.18', 'autcentquo_hash': 18, 'autcentquo_nilpotent': False, 'autcentquo_order': 56882, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'F_{239}', 'cc_stats': [[1, 1, 1], [2, 1, 1], [7, 1, 6], [14, 1, 6], [17, 239, 16], [34, 239, 16], [119, 239, 96], [238, 239, 96], [239, 17, 14], [478, 17, 14], [1673, 17, 84], [3346, 17, 84]], 'center_label': '14.2', 'center_order': 14, 'central_product': True, 'central_quotient': '4063.a', 'commutator_count': 1, 'commutator_label': '239.1', 'complements_known': True, 'complete': False, 'complex_characters_known': False, 'composition_factors': ['2.1', '7.1', '17.1', '239.1'], 'composition_length': 4, 'conjugacy_classes_known': True, 'counter': 2, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['2.1', 1], ['4063.a', 1], ['7.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [7, 1, 6, 1], [14, 1, 6, 1], [17, 239, 16, 1], [34, 239, 16, 1], [119, 239, 96, 1], [238, 239, 96, 1], [239, 17, 14, 1], [478, 17, 14, 1], [1673, 17, 84, 1], [3346, 17, 84, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 6912, 'exponent': 56882, 'exponents_of_order': [1, 1, 1, 1], 'factors_of_aut_order': [2, 3, 7, 17, 239], 'factors_of_order': [2, 7, 17, 239], 'faithful_reps': [[17, 0, 84]], 'familial': False, 'frattini_label': '1.1', 'frattini_quotient': '56882.2', 'hash': 2, 'hyperelementary': 17, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 4063, 'inner_gen_orders': [17, 239], 'inner_gens': [[1, 3587], [53313, 17]], 'inner_hash': 1, 'inner_nilpotent': False, 'inner_order': 4063, 'inner_split': True, 'inner_tex': 'C_{239}:C_{17}', 'inner_used': [1, 2], 'irrC_degree': 17, 'irrQ_degree': 1428, 'irrQ_dim': None, 'irrR_degree': None, 'irrep_stats': [[1, 238], [17, 196]], 'label': '56882.2', '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': 'C1673:C34', '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': 72, 'number_characteristic_subgroups': 12, 'number_conjugacy_classes': 434, 'number_divisions': 12, 'number_normal_subgroups': 12, 'number_subgroup_autclasses': 16, 'number_subgroup_classes': 16, 'number_subgroups': 968, 'old_label': None, 'order': 56882, 'order_factorization_type': 11, 'order_stats': [[1, 1], [2, 1], [7, 6], [14, 6], [17, 3824], [34, 3824], [119, 22944], [238, 22944], [239, 238], [478, 238], [1673, 1428], [3346, 1428]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 42, 'outer_gen_orders': [2, 42], 'outer_gen_pows': [0, 5], 'outer_gens': [[1, 56865], [1, 26809]], 'outer_group': '84.15', 'outer_hash': 15, 'outer_nilpotent': True, 'outer_order': 84, 'outer_permdeg': 14, 'outer_perms': [39916800, 6267341673], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2\\times C_{42}', 'pc_rank': 2, 'perfect': False, 'permutation_degree': 248, 'pgroup': 0, 'primary_abelian_invariants': [2, 7, 17], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': False, 'ratrep_stats': [[1, 2], [6, 2], [16, 2], [96, 2], [238, 2], [1428, 2]], 'representations': {'PC': {'code': '3611583054715498946916545426150331', 'gens': [1, 2], 'pres': [4, -17, -2, -7, -239, 28697, 21, 86090, 94, 803491]}, 'GLFp': {'d': 2, 'p': 239, 'gens': [13652159, 1706490036, 3249156960]}, 'Perm': {'d': 248, 'gens': [1, 46230, 1709489882309756244932096124400939731599171770922927971291039766434998183252245690246580322624258772197207306259275221634031292887683273788758968070268693293802317166423710268160053057829190590713445950332792027851113465085396078116279361022143562369959865793818497485790405651519943122698508887582314038726259635937215160250864848414185228400676872263572266245630823271781387264027135423154981916862327046428954049534154566955820536263527767859212310585857410305082220089124521971200, 211952535934989380816330611877660538736466441283651362789029394409899716177795664642117827652615847087056722981565992911666335691553552493602890581017267560223980510805051017999025800019366753163752576631902418643816213722355379317441856677740752566895847490881195459472839535935909646939964310771372759808962160160715741760006731243256216318593974753728024765884287712213996003205266399921553187302056961486907169814030544091570005105987335000710607836689808254015094119854485615289600]}}, 'schur_multiplier': [], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [238], 'solvability_type': 6, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_{1673}:C_{34}', 'transitive_degree': 3346, 'wreath_data': None, 'wreath_product': False}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': True, 'abelian_quotient': '238.4', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': True, 'aut_cyclic': False, 'aut_derived_length': 1, 'aut_exponent': 48, 'aut_gen_orders': [2, 48], 'aut_gens': [[1], [69], [3]], 'aut_group': '96.59', 'aut_hash': 59, 'aut_nilpotency_class': 1, 'aut_nilpotent': True, 'aut_order': 96, 'aut_permdeg': 21, 'aut_perms': [2439325392333218664, 19984155209275203], 'aut_phi_ratio': 1.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [7, 1, 6, 1], [14, 1, 6, 1], [17, 1, 16, 1], [34, 1, 16, 1], [119, 1, 96, 1], [238, 1, 96, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_2\\times C_{48}', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 48, 'autcent_group': '96.59', 'autcent_hash': 59, 'autcent_nilpotent': True, 'autcent_order': 96, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2\\times C_{48}', '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], [7, 1, 6], [14, 1, 6], [17, 1, 16], [34, 1, 16], [119, 1, 96], [238, 1, 96]], 'center_label': '238.4', 'center_order': 238, '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', '7.1', '17.1'], 'composition_length': 3, 'conjugacy_classes_known': True, 'counter': 4, 'cyclic': True, 'derived_length': 1, 'dihedral': False, 'direct_factorization': [['17.1', 1], ['2.1', 1], ['7.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [7, 1, 6, 1], [14, 1, 6, 1], [17, 1, 16, 1], [34, 1, 16, 1], [119, 1, 96, 1], [238, 1, 96, 1]], 'element_repr_type': 'PC', 'elementary': 238, 'eulerian_function': 1, 'exponent': 238, 'exponents_of_order': [1, 1, 1], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 7, 17], 'faithful_reps': [[1, 0, 96]], 'familial': True, 'frattini_label': '1.1', 'frattini_quotient': '238.4', 'hash': 4, 'hyperelementary': 238, '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': 96, 'irrQ_dim': 96, 'irrR_degree': 2, 'irrep_stats': [[1, 238]], 'label': '238.4', 'linC_count': 96, 'linC_degree': 1, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 22, 'linQ_degree_count': 3, 'linQ_dim': 22, 'linQ_dim_count': 3, 'linR_count': 48, 'linR_degree': 2, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C238', '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': 238, 'number_divisions': 8, 'number_normal_subgroups': 8, 'number_subgroup_autclasses': 8, 'number_subgroup_classes': 8, 'number_subgroups': 8, 'old_label': None, 'order': 238, 'order_factorization_type': 11, 'order_stats': [[1, 1], [2, 1], [7, 6], [14, 6], [17, 16], [34, 16], [119, 96], [238, 96]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 48, 'outer_gen_orders': [2, 48], 'outer_gen_pows': [0, 0], 'outer_gens': [[69], [3]], 'outer_group': '96.59', 'outer_hash': 59, 'outer_nilpotent': True, 'outer_order': 96, 'outer_permdeg': 21, 'outer_perms': [2439325392333218664, 19984155209275203], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2\\times C_{48}', 'pc_rank': 1, 'perfect': False, 'permutation_degree': 26, 'pgroup': 0, 'primary_abelian_invariants': [2, 7, 17], 'quasisimple': False, 'rank': 1, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 2], [6, 2], [16, 2], [96, 2]], 'representations': {'PC': {'code': 33435743, 'gens': [1], 'pres': [3, -2, -7, -17, 6, 46]}, 'GLFp': {'d': 2, 'p': 239, 'gens': [1064849869]}, 'Perm': {'d': 26, 'gens': [15511210043330985984000000, 155112100433309859840000, 334764638208000]}}, 'schur_multiplier': [], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [238], 'solvability_type': 0, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_{238}', 'transitive_degree': 238, 'wreath_data': None, 'wreath_product': False}
